id	sid	tid	token	lemma	pos
ejpam-2553	1	1	european	european	PROPN
ejpam-2553	1	2	journal	journal	PROPN
ejpam-2553	1	3	of	of	ADP
ejpam-2553	1	4	pure	pure	ADJ
ejpam-2553	1	5	and	and	CCONJ
ejpam-2553	1	6	applied	apply	VERB
ejpam-2553	1	7	mathematics	mathematic	NOUN
ejpam-2553	1	8	vol	vol	NOUN
ejpam-2553	1	9	.	.	PROPN
ejpam-2553	2	1	9	9	NUM
ejpam-2553	2	2	,	,	PUNCT
ejpam-2553	2	3	no	no	INTJ
ejpam-2553	2	4	.	.	NOUN
ejpam-2553	2	5	4	4	NUM
ejpam-2553	2	6	,	,	PUNCT
ejpam-2553	2	7	2016	2016	NUM
ejpam-2553	2	8	,	,	PUNCT
ejpam-2553	2	9	402	402	NUM
ejpam-2553	2	10	-	-	SYM
ejpam-2553	2	11	418	418	NUM
ejpam-2553	2	12	issn	issn	PROPN
ejpam-2553	2	13	1307	1307	NUM
ejpam-2553	2	14	-	-	SYM
ejpam-2553	2	15	5543	5543	NUM
ejpam-2553	2	16	–	–	PUNCT
ejpam-2553	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2553	2	18	on	on	ADP
ejpam-2553	2	19	regular	regular	ADJ
ejpam-2553	2	20	multiplicative	multiplicative	ADJ
ejpam-2553	2	21	hyperrings	hyperring	NOUN
ejpam-2553	2	22	reza	reza	PROPN
ejpam-2553	2	23	ameri∗	ameri∗	PROPN
ejpam-2553	2	24	,	,	PUNCT
ejpam-2553	2	25	ali	ali	PROPN
ejpam-2553	2	26	kordi	kordi	PROPN
ejpam-2553	2	27	school	school	PROPN
ejpam-2553	2	28	of	of	ADP
ejpam-2553	2	29	mathematics	mathematic	NOUN
ejpam-2553	2	30	,	,	PUNCT
ejpam-2553	2	31	statistics	statistic	NOUN
ejpam-2553	2	32	and	and	CCONJ
ejpam-2553	2	33	computer	computer	NOUN
ejpam-2553	2	34	science	science	NOUN
ejpam-2553	2	35	,	,	PUNCT
ejpam-2553	2	36	college	college	NOUN
ejpam-2553	2	37	of	of	ADP
ejpam-2553	2	38	sciences	science	NOUN
ejpam-2553	2	39	,	,	PUNCT
ejpam-2553	2	40	university	university	NOUN
ejpam-2553	2	41	of	of	ADP
ejpam-2553	2	42	tehran	tehran	PROPN
ejpam-2553	2	43	,	,	PUNCT
ejpam-2553	2	44	p.o	p.o	PROPN
ejpam-2553	2	45	.	.	PROPN
ejpam-2553	2	46	box	box	PROPN
ejpam-2553	2	47	:	:	PUNCT
ejpam-2553	2	48	14155	14155	NUM
ejpam-2553	2	49	-	-	SYM
ejpam-2553	2	50	6455	6455	NUM
ejpam-2553	2	51	,	,	PUNCT
ejpam-2553	2	52	tehran	tehran	PROPN
ejpam-2553	2	53	,	,	PUNCT
ejpam-2553	3	1	iran	iran	PROPN
ejpam-2553	3	2	abstract	abstract	ADJ
ejpam-2553	3	3	.	.	PUNCT
ejpam-2553	4	1	we	we	PRON
ejpam-2553	4	2	introduce	introduce	VERB
ejpam-2553	4	3	and	and	CCONJ
ejpam-2553	4	4	study	study	VERB
ejpam-2553	4	5	regular	regular	ADJ
ejpam-2553	4	6	multiplicative	multiplicative	ADJ
ejpam-2553	4	7	hyperrings	hyperring	NOUN
ejpam-2553	4	8	,	,	PUNCT
ejpam-2553	4	9	as	as	ADP
ejpam-2553	4	10	a	a	DET
ejpam-2553	4	11	generalization	generalization	NOUN
ejpam-2553	4	12	of	of	ADP
ejpam-2553	4	13	classical	classical	ADJ
ejpam-2553	4	14	rings	ring	NOUN
ejpam-2553	4	15	.	.	PUNCT
ejpam-2553	5	1	also	also	ADV
ejpam-2553	5	2	,	,	PUNCT
ejpam-2553	5	3	we	we	PRON
ejpam-2553	5	4	use	use	VERB
ejpam-2553	5	5	the	the	DET
ejpam-2553	5	6	fundamental	fundamental	ADJ
ejpam-2553	5	7	relation	relation	NOUN
ejpam-2553	5	8	γ∗	γ∗	NOUN
ejpam-2553	5	9	on	on	ADP
ejpam-2553	5	10	a	a	DET
ejpam-2553	5	11	given	give	VERB
ejpam-2553	5	12	regular	regular	ADJ
ejpam-2553	5	13	multiplicative	multiplicative	ADJ
ejpam-2553	5	14	hyperring	hyperring	NOUN
ejpam-2553	5	15	r	r	NOUN
ejpam-2553	5	16	and	and	CCONJ
ejpam-2553	5	17	prove	prove	VERB
ejpam-2553	5	18	that	that	SCONJ
ejpam-2553	5	19	the	the	DET
ejpam-2553	5	20	fundamental	fundamental	ADJ
ejpam-2553	5	21	ring	ring	NOUN
ejpam-2553	5	22	r	r	NOUN
ejpam-2553	5	23	/	/	SYM
ejpam-2553	5	24	γ∗	γ∗	NOUN
ejpam-2553	5	25	of	of	ADP
ejpam-2553	5	26	r	r	NOUN
ejpam-2553	5	27	is	be	AUX
ejpam-2553	5	28	a	a	DET
ejpam-2553	5	29	regular	regular	ADJ
ejpam-2553	5	30	ring	ring	NOUN
ejpam-2553	5	31	.	.	PUNCT
ejpam-2553	6	1	finally	finally	ADV
ejpam-2553	6	2	,	,	PUNCT
ejpam-2553	6	3	we	we	PRON
ejpam-2553	6	4	investigate	investigate	VERB
ejpam-2553	6	5	the	the	DET
ejpam-2553	6	6	algebraic	algebraic	ADJ
ejpam-2553	6	7	properties	property	NOUN
ejpam-2553	6	8	of	of	ADP
ejpam-2553	6	9	m(r	m(r	PROPN
ejpam-2553	6	10	)	)	PUNCT
ejpam-2553	6	11	,	,	PUNCT
ejpam-2553	6	12	the	the	DET
ejpam-2553	6	13	regular	regular	ADJ
ejpam-2553	6	14	hyperideal	hyperideal	NOUN
ejpam-2553	6	15	of	of	ADP
ejpam-2553	6	16	r	r	NOUN
ejpam-2553	6	17	,	,	PUNCT
ejpam-2553	6	18	generated	generate	VERB
ejpam-2553	6	19	by	by	ADP
ejpam-2553	6	20	all	all	DET
ejpam-2553	6	21	elements	element	NOUN
ejpam-2553	6	22	of	of	ADP
ejpam-2553	6	23	r	r	NOUN
ejpam-2553	6	24	such	such	ADJ
ejpam-2553	6	25	that	that	SCONJ
ejpam-2553	6	26	its	its	PRON
ejpam-2553	6	27	generated	generate	VERB
ejpam-2553	6	28	hyperideal	hyperideal	NOUN
ejpam-2553	6	29	is	be	AUX
ejpam-2553	6	30	regular	regular	ADJ
ejpam-2553	6	31	.	.	PUNCT
ejpam-2553	7	1	2010	2010	NUM
ejpam-2553	7	2	mathematics	mathematic	NOUN
ejpam-2553	7	3	subject	subject	NOUN
ejpam-2553	7	4	classifications	classification	NOUN
ejpam-2553	7	5	:	:	PUNCT
ejpam-2553	7	6	20n20	20n20	NUM
ejpam-2553	7	7	key	key	ADJ
ejpam-2553	7	8	words	word	NOUN
ejpam-2553	7	9	and	and	CCONJ
ejpam-2553	7	10	phrases	phrase	NOUN
ejpam-2553	7	11	:	:	PUNCT
ejpam-2553	7	12	multiplicative	multiplicative	ADJ
ejpam-2553	7	13	hyperring	hyperring	NOUN
ejpam-2553	7	14	,	,	PUNCT
ejpam-2553	7	15	regular	regular	ADJ
ejpam-2553	7	16	hyperring	hyperring	NOUN
ejpam-2553	7	17	,	,	PUNCT
ejpam-2553	7	18	fundamental	fundamental	ADJ
ejpam-2553	7	19	relation	relation	NOUN
ejpam-2553	7	20	,	,	PUNCT
ejpam-2553	7	21	hyperideal	hyperideal	NOUN
ejpam-2553	7	22	,	,	PUNCT
ejpam-2553	7	23	regular	regular	ADJ
ejpam-2553	7	24	element	element	NOUN
ejpam-2553	7	25	,	,	PUNCT
ejpam-2553	7	26	idempotent	idempotent	ADJ
ejpam-2553	7	27	1	1	NUM
ejpam-2553	7	28	.	.	PUNCT
ejpam-2553	8	1	introduction	introduction	NOUN
ejpam-2553	8	2	and	and	CCONJ
ejpam-2553	8	3	primary	primary	ADJ
ejpam-2553	8	4	the	the	DET
ejpam-2553	8	5	theory	theory	NOUN
ejpam-2553	8	6	of	of	ADP
ejpam-2553	8	7	hyperstructures	hyperstructure	NOUN
ejpam-2553	8	8	has	have	AUX
ejpam-2553	8	9	been	be	AUX
ejpam-2553	8	10	introduced	introduce	VERB
ejpam-2553	8	11	by	by	ADP
ejpam-2553	8	12	marty	marty	PROPN
ejpam-2553	8	13	in	in	ADP
ejpam-2553	8	14	1934	1934	NUM
ejpam-2553	8	15	during	during	ADP
ejpam-2553	8	16	the	the	DET
ejpam-2553	8	17	8th	8th	ADJ
ejpam-2553	8	18	congress	congress	NOUN
ejpam-2553	8	19	of	of	ADP
ejpam-2553	8	20	the	the	DET
ejpam-2553	8	21	scandinavian	scandinavian	ADJ
ejpam-2553	8	22	mathematicians	mathematician	NOUN
ejpam-2553	8	23	[	[	X
ejpam-2553	8	24	19	19	NUM
ejpam-2553	8	25	]	]	PUNCT
ejpam-2553	8	26	.	.	PUNCT
ejpam-2553	9	1	marty	marty	PROPN
ejpam-2553	9	2	introduced	introduce	VERB
ejpam-2553	9	3	hypergroups	hypergroup	NOUN
ejpam-2553	9	4	as	as	ADP
ejpam-2553	9	5	a	a	DET
ejpam-2553	9	6	generalization	generalization	NOUN
ejpam-2553	9	7	of	of	ADP
ejpam-2553	9	8	groups	group	NOUN
ejpam-2553	9	9	.	.	PUNCT
ejpam-2553	10	1	he	he	PRON
ejpam-2553	10	2	published	publish	VERB
ejpam-2553	10	3	some	some	DET
ejpam-2553	10	4	notes	note	NOUN
ejpam-2553	10	5	on	on	ADP
ejpam-2553	10	6	hypergroups	hypergroup	NOUN
ejpam-2553	10	7	,	,	PUNCT
ejpam-2553	10	8	using	use	VERB
ejpam-2553	10	9	them	they	PRON
ejpam-2553	10	10	in	in	ADP
ejpam-2553	10	11	different	different	ADJ
ejpam-2553	10	12	contexts	context	NOUN
ejpam-2553	10	13	as	as	ADP
ejpam-2553	10	14	algebraic	algebraic	ADJ
ejpam-2553	10	15	functions	function	NOUN
ejpam-2553	10	16	,	,	PUNCT
ejpam-2553	10	17	rational	rational	ADJ
ejpam-2553	10	18	fractions	fraction	NOUN
ejpam-2553	10	19	,	,	PUNCT
ejpam-2553	10	20	non	non	ADJ
ejpam-2553	10	21	-	-	ADJ
ejpam-2553	10	22	commutative	commutative	ADJ
ejpam-2553	10	23	groups	group	NOUN
ejpam-2553	10	24	and	and	CCONJ
ejpam-2553	10	25	then	then	ADV
ejpam-2553	10	26	many	many	ADJ
ejpam-2553	10	27	researchers	researcher	NOUN
ejpam-2553	10	28	have	have	AUX
ejpam-2553	10	29	been	be	AUX
ejpam-2553	10	30	worked	work	VERB
ejpam-2553	10	31	on	on	ADP
ejpam-2553	10	32	this	this	DET
ejpam-2553	10	33	new	new	ADJ
ejpam-2553	10	34	field	field	NOUN
ejpam-2553	10	35	of	of	ADP
ejpam-2553	10	36	modern	modern	ADJ
ejpam-2553	10	37	algebra	algebra	NOUN
ejpam-2553	10	38	and	and	CCONJ
ejpam-2553	10	39	developed	develop	VERB
ejpam-2553	10	40	it	it	PRON
ejpam-2553	10	41	.	.	PUNCT
ejpam-2553	11	1	it	it	PRON
ejpam-2553	11	2	was	be	AUX
ejpam-2553	11	3	later	later	ADV
ejpam-2553	11	4	observed	observe	VERB
ejpam-2553	11	5	that	that	SCONJ
ejpam-2553	11	6	the	the	DET
ejpam-2553	11	7	theory	theory	NOUN
ejpam-2553	11	8	of	of	ADP
ejpam-2553	11	9	hyperstructures	hyperstructure	NOUN
ejpam-2553	11	10	has	have	VERB
ejpam-2553	11	11	many	many	ADJ
ejpam-2553	11	12	applications	application	NOUN
ejpam-2553	11	13	in	in	ADP
ejpam-2553	11	14	both	both	CCONJ
ejpam-2553	11	15	pure	pure	ADJ
ejpam-2553	11	16	and	and	CCONJ
ejpam-2553	11	17	applied	applied	ADJ
ejpam-2553	11	18	sciences	science	NOUN
ejpam-2553	11	19	;	;	PUNCT
ejpam-2553	11	20	for	for	ADP
ejpam-2553	11	21	example	example	NOUN
ejpam-2553	11	22	,	,	PUNCT
ejpam-2553	11	23	semi	semi	NOUN
ejpam-2553	11	24	-	-	NOUN
ejpam-2553	11	25	hypergroups	hypergroup	NOUN
ejpam-2553	11	26	are	be	AUX
ejpam-2553	11	27	the	the	DET
ejpam-2553	11	28	simplest	simple	ADJ
ejpam-2553	11	29	algebraic	algebraic	ADJ
ejpam-2553	11	30	hyperstructures	hyperstructure	VERB
ejpam-2553	11	31	that	that	PRON
ejpam-2553	11	32	possess	possess	VERB
ejpam-2553	11	33	the	the	DET
ejpam-2553	11	34	properties	property	NOUN
ejpam-2553	11	35	of	of	ADP
ejpam-2553	11	36	closure	closure	NOUN
ejpam-2553	11	37	and	and	CCONJ
ejpam-2553	11	38	associativity	associativity	NOUN
ejpam-2553	11	39	.	.	PUNCT
ejpam-2553	12	1	the	the	DET
ejpam-2553	12	2	theory	theory	NOUN
ejpam-2553	12	3	of	of	ADP
ejpam-2553	12	4	hyperstructures	hyperstructure	NOUN
ejpam-2553	12	5	has	have	AUX
ejpam-2553	12	6	been	be	AUX
ejpam-2553	12	7	widely	widely	ADV
ejpam-2553	12	8	reviewed	review	VERB
ejpam-2553	12	9	[	[	X
ejpam-2553	12	10	8	8	NUM
ejpam-2553	12	11	,	,	PUNCT
ejpam-2553	12	12	9	9	NUM
ejpam-2553	12	13	,	,	PUNCT
ejpam-2553	12	14	12	12	NUM
ejpam-2553	12	15	,	,	PUNCT
ejpam-2553	12	16	19	19	NUM
ejpam-2553	12	17	,	,	PUNCT
ejpam-2553	12	18	31	31	NUM
ejpam-2553	12	19	]	]	PUNCT
ejpam-2553	12	20	.	.	PUNCT
ejpam-2553	13	1	in	in	ADP
ejpam-2553	13	2	[	[	X
ejpam-2553	13	3	9	9	NUM
ejpam-2553	13	4	]	]	SYM
ejpam-2553	13	5	corsini	corsini	ADJ
ejpam-2553	13	6	and	and	CCONJ
ejpam-2553	13	7	leoreanu	leoreanu	PROPN
ejpam-2553	13	8	-	-	PUNCT
ejpam-2553	13	9	fotea	fotea	NOUN
ejpam-2553	13	10	have	have	AUX
ejpam-2553	13	11	collected	collect	VERB
ejpam-2553	13	12	numerous	numerous	ADJ
ejpam-2553	13	13	applications	application	NOUN
ejpam-2553	13	14	of	of	ADP
ejpam-2553	13	15	algebraic	algebraic	ADJ
ejpam-2553	13	16	hyperstructures	hyperstructure	NOUN
ejpam-2553	13	17	,	,	PUNCT
ejpam-2553	13	18	especially	especially	ADV
ejpam-2553	13	19	those	those	PRON
ejpam-2553	13	20	from	from	ADP
ejpam-2553	13	21	the	the	DET
ejpam-2553	13	22	last	last	ADJ
ejpam-2553	13	23	fifteen	fifteen	NUM
ejpam-2553	13	24	years	year	NOUN
ejpam-2553	13	25	to	to	ADP
ejpam-2553	13	26	the	the	DET
ejpam-2553	13	27	following	follow	VERB
ejpam-2553	13	28	subjects	subject	NOUN
ejpam-2553	13	29	:	:	PUNCT
ejpam-2553	13	30	geometry	geometry	NOUN
ejpam-2553	13	31	,	,	PUNCT
ejpam-2553	13	32	hypergraphs	hypergraph	NOUN
ejpam-2553	13	33	,	,	PUNCT
ejpam-2553	13	34	binary	binary	ADJ
ejpam-2553	13	35	relations	relation	NOUN
ejpam-2553	13	36	,	,	PUNCT
ejpam-2553	13	37	lattices	lattice	NOUN
ejpam-2553	13	38	,	,	PUNCT
ejpam-2553	13	39	fuzzy	fuzzy	ADJ
ejpam-2553	13	40	sets	set	NOUN
ejpam-2553	13	41	and	and	CCONJ
ejpam-2553	13	42	rough	rough	ADJ
ejpam-2553	13	43	sets	set	NOUN
ejpam-2553	13	44	,	,	PUNCT
ejpam-2553	13	45	automata	automata	NOUN
ejpam-2553	13	46	,	,	PUNCT
ejpam-2553	13	47	cryptography	cryptography	NOUN
ejpam-2553	13	48	,	,	PUNCT
ejpam-2553	13	49	codes	code	NOUN
ejpam-2553	13	50	,	,	PUNCT
ejpam-2553	13	51	median	median	NOUN
ejpam-2553	13	52	algebras	algebra	NOUN
ejpam-2553	13	53	,	,	PUNCT
ejpam-2553	13	54	relation	relation	NOUN
ejpam-2553	13	55	algebras	algebra	NOUN
ejpam-2553	13	56	,	,	PUNCT
ejpam-2553	13	57	artificial	artificial	ADJ
ejpam-2553	13	58	intelligence	intelligence	NOUN
ejpam-2553	13	59	,	,	PUNCT
ejpam-2553	13	60	and	and	CCONJ
ejpam-2553	13	61	probabilities	probability	NOUN
ejpam-2553	13	62	.	.	PUNCT
ejpam-2553	14	1	the	the	DET
ejpam-2553	14	2	hyperrings	hyperring	NOUN
ejpam-2553	14	3	were	be	AUX
ejpam-2553	14	4	introduced	introduce	VERB
ejpam-2553	14	5	and	and	CCONJ
ejpam-2553	14	6	studied	study	VERB
ejpam-2553	14	7	by	by	ADP
ejpam-2553	14	8	krasner	krasner	NOUN
ejpam-2553	15	1	[	[	X
ejpam-2553	15	2	18	18	NUM
ejpam-2553	15	3	]	]	PUNCT
ejpam-2553	15	4	,	,	PUNCT
ejpam-2553	15	5	nakasis	nakasis	NOUN
ejpam-2553	16	1	[	[	X
ejpam-2553	16	2	21	21	NUM
ejpam-2553	16	3	]	]	PUNCT
ejpam-2553	16	4	,	,	PUNCT
ejpam-2553	16	5	massouros	massouro	NOUN
ejpam-2553	17	1	[	[	X
ejpam-2553	17	2	19	19	NUM
ejpam-2553	17	3	]	]	PUNCT
ejpam-2553	17	4	and	and	CCONJ
ejpam-2553	17	5	especially	especially	ADV
ejpam-2553	17	6	studied	study	VERB
ejpam-2553	17	7	by	by	ADP
ejpam-2553	17	8	davvaz	davvaz	NOUN
ejpam-2553	17	9	and	and	CCONJ
ejpam-2553	17	10	leoreanu	leoreanu	NOUN
ejpam-2553	17	11	-	-	PUNCT
ejpam-2553	17	12	fotea	fotea	NOUN
ejpam-2553	18	1	[	[	X
ejpam-2553	18	2	13	13	NUM
ejpam-2553	18	3	]	]	PUNCT
ejpam-2553	18	4	,	,	PUNCT
ejpam-2553	18	5	zahedi	zahedi	PROPN
ejpam-2553	18	6	and	and	CCONJ
ejpam-2553	18	7	ameri	ameri	PROPN
ejpam-2553	18	8	[	[	X
ejpam-2553	18	9	33	33	NUM
ejpam-2553	18	10	]	]	PUNCT
ejpam-2553	18	11	,	,	PUNCT
ejpam-2553	18	12	ameri	ameri	PROPN
ejpam-2553	18	13	and	and	CCONJ
ejpam-2553	18	14	∗corresponding	∗corresponde	VERB
ejpam-2553	18	15	author	author	NOUN
ejpam-2553	18	16	.	.	PUNCT
ejpam-2553	19	1	email	email	NOUN
ejpam-2553	19	2	addresses	address	NOUN
ejpam-2553	19	3	:	:	PUNCT
ejpam-2553	19	4	rameri@ut.ac.ir	rameri@ut.ac.ir	PROPN
ejpam-2553	19	5	(	(	PUNCT
ejpam-2553	19	6	r.	r.	PROPN
ejpam-2553	19	7	ameri	ameri	PROPN
ejpam-2553	19	8	)	)	PUNCT
ejpam-2553	19	9	,	,	PUNCT
ejpam-2553	19	10	ali.kordi@gmail.com	ali.kordi@gmail.com	X
ejpam-2553	19	11	(	(	PUNCT
ejpam-2553	19	12	a.	a.	NOUN
ejpam-2553	19	13	kordi	kordi	PROPN
ejpam-2553	19	14	)	)	PUNCT
ejpam-2553	19	15	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2553	20	1	402	402	NUM
ejpam-2553	20	2	c	c	X
ejpam-2553	20	3	©	©	PROPN
ejpam-2553	20	4	2016	2016	NUM
ejpam-2553	20	5	ejpam	ejpam	VERB
ejpam-2553	20	6	all	all	DET
ejpam-2553	20	7	rights	right	NOUN
ejpam-2553	20	8	reserved	reserve	VERB
ejpam-2553	20	9	.	.	PUNCT
ejpam-2553	21	1	r.	r.	PROPN
ejpam-2553	21	2	ameri	ameri	PROPN
ejpam-2553	21	3	,	,	PUNCT
ejpam-2553	21	4	a.	a.	PROPN
ejpam-2553	21	5	kordi	kordi	PROPN
ejpam-2553	21	6	/	/	PUNCT
ejpam-2553	21	7	eur	eur	PROPN
ejpam-2553	21	8	.	.	PUNCT
ejpam-2553	22	1	j.	j.	PROPN
ejpam-2553	22	2	pure	pure	PROPN
ejpam-2553	22	3	appl	appl	PROPN
ejpam-2553	22	4	.	.	PROPN
ejpam-2553	22	5	math	math	PROPN
ejpam-2553	22	6	,	,	PUNCT
ejpam-2553	22	7	9	9	NUM
ejpam-2553	22	8	(	(	PUNCT
ejpam-2553	22	9	2016	2016	NUM
ejpam-2553	22	10	)	)	PUNCT
ejpam-2553	22	11	,	,	PUNCT
ejpam-2553	22	12	402	402	NUM
ejpam-2553	22	13	-	-	SYM
ejpam-2553	22	14	418	418	NUM
ejpam-2553	22	15	403	403	NUM
ejpam-2553	22	16	norouzi	norouzi	NOUN
ejpam-2553	23	1	[	[	X
ejpam-2553	23	2	1	1	NUM
ejpam-2553	23	3	,	,	PUNCT
ejpam-2553	23	4	2	2	NUM
ejpam-2553	23	5	]	]	PUNCT
ejpam-2553	23	6	.	.	PUNCT
ejpam-2553	24	1	the	the	DET
ejpam-2553	24	2	study	study	NOUN
ejpam-2553	24	3	on	on	ADP
ejpam-2553	24	4	hyperrings	hyperring	NOUN
ejpam-2553	24	5	in	in	ADP
ejpam-2553	24	6	[	[	X
ejpam-2553	24	7	31	31	NUM
ejpam-2553	24	8	]	]	PUNCT
ejpam-2553	24	9	ends	end	VERB
ejpam-2553	24	10	with	with	ADP
ejpam-2553	24	11	an	an	DET
ejpam-2553	24	12	outline	outline	NOUN
ejpam-2553	24	13	of	of	ADP
ejpam-2553	24	14	applications	application	NOUN
ejpam-2553	24	15	in	in	ADP
ejpam-2553	24	16	chemistry	chemistry	NOUN
ejpam-2553	24	17	and	and	CCONJ
ejpam-2553	24	18	physics	physics	NOUN
ejpam-2553	24	19	,	,	PUNCT
ejpam-2553	24	20	analyzing	analyze	VERB
ejpam-2553	24	21	several	several	ADJ
ejpam-2553	24	22	special	special	ADJ
ejpam-2553	24	23	kinds	kind	NOUN
ejpam-2553	24	24	of	of	ADP
ejpam-2553	24	25	hyperstructures	hyperstructure	NOUN
ejpam-2553	24	26	:	:	PUNCT
ejpam-2553	24	27	e	e	NOUN
ejpam-2553	24	28	-	-	NOUN
ejpam-2553	24	29	hyperstructures	hyperstructure	NOUN
ejpam-2553	24	30	and	and	CCONJ
ejpam-2553	24	31	transposition	transposition	NOUN
ejpam-2553	24	32	hypergroups	hypergroup	NOUN
ejpam-2553	24	33	.	.	PUNCT
ejpam-2553	25	1	the	the	DET
ejpam-2553	25	2	theory	theory	NOUN
ejpam-2553	25	3	of	of	ADP
ejpam-2553	25	4	suitable	suitable	ADJ
ejpam-2553	25	5	modified	modified	ADJ
ejpam-2553	25	6	hyperstructures	hyperstructure	NOUN
ejpam-2553	25	7	can	can	AUX
ejpam-2553	25	8	serve	serve	VERB
ejpam-2553	25	9	as	as	ADP
ejpam-2553	25	10	a	a	DET
ejpam-2553	25	11	mathematical	mathematical	ADJ
ejpam-2553	25	12	background	background	NOUN
ejpam-2553	25	13	in	in	ADP
ejpam-2553	25	14	the	the	DET
ejpam-2553	25	15	field	field	NOUN
ejpam-2553	25	16	of	of	ADP
ejpam-2553	25	17	quantum	quantum	ADJ
ejpam-2553	25	18	communication	communication	NOUN
ejpam-2553	25	19	systems	system	NOUN
ejpam-2553	25	20	.	.	PUNCT
ejpam-2553	26	1	a	a	DET
ejpam-2553	26	2	well	well	ADV
ejpam-2553	26	3	-	-	PUNCT
ejpam-2553	26	4	known	know	VERB
ejpam-2553	26	5	type	type	NOUN
ejpam-2553	26	6	of	of	ADP
ejpam-2553	26	7	a	a	DET
ejpam-2553	26	8	hyperring	hyperring	NOUN
ejpam-2553	26	9	,	,	PUNCT
ejpam-2553	26	10	called	call	VERB
ejpam-2553	26	11	the	the	DET
ejpam-2553	26	12	krasner	krasner	NOUN
ejpam-2553	26	13	hyperring	hyperre	VERB
ejpam-2553	26	14	[	[	X
ejpam-2553	26	15	18	18	NUM
ejpam-2553	26	16	]	]	PUNCT
ejpam-2553	26	17	.	.	PUNCT
ejpam-2553	27	1	krasner	krasner	PROPN
ejpam-2553	27	2	hyperrings	hyperring	NOUN
ejpam-2553	27	3	are	be	AUX
ejpam-2553	27	4	essentially	essentially	ADV
ejpam-2553	27	5	hyperrings	hyperring	NOUN
ejpam-2553	27	6	,	,	PUNCT
ejpam-2553	27	7	with	with	ADP
ejpam-2553	27	8	approximately	approximately	ADV
ejpam-2553	27	9	modified	modify	VERB
ejpam-2553	27	10	axioms	axiom	NOUN
ejpam-2553	27	11	in	in	SCONJ
ejpam-2553	27	12	which	which	DET
ejpam-2553	27	13	addition	addition	NOUN
ejpam-2553	27	14	is	be	AUX
ejpam-2553	27	15	a	a	DET
ejpam-2553	27	16	hyperoperation	hyperoperation	NOUN
ejpam-2553	27	17	,	,	PUNCT
ejpam-2553	27	18	while	while	SCONJ
ejpam-2553	27	19	the	the	DET
ejpam-2553	27	20	multiplication	multiplication	NOUN
ejpam-2553	27	21	is	be	AUX
ejpam-2553	27	22	an	an	DET
ejpam-2553	27	23	operation	operation	NOUN
ejpam-2553	27	24	.	.	PUNCT
ejpam-2553	28	1	then	then	ADV
ejpam-2553	28	2	,	,	PUNCT
ejpam-2553	28	3	this	this	DET
ejpam-2553	28	4	concept	concept	NOUN
ejpam-2553	28	5	has	have	AUX
ejpam-2553	28	6	been	be	AUX
ejpam-2553	28	7	studied	study	VERB
ejpam-2553	28	8	by	by	ADP
ejpam-2553	28	9	a	a	DET
ejpam-2553	28	10	variety	variety	NOUN
ejpam-2553	28	11	of	of	ADP
ejpam-2553	28	12	authors	author	NOUN
ejpam-2553	28	13	.	.	PUNCT
ejpam-2553	29	1	some	some	DET
ejpam-2553	29	2	principal	principal	ADJ
ejpam-2553	29	3	notions	notion	NOUN
ejpam-2553	29	4	of	of	ADP
ejpam-2553	29	5	hyperring	hyperring	NOUN
ejpam-2553	29	6	theory	theory	NOUN
ejpam-2553	29	7	can	can	AUX
ejpam-2553	29	8	be	be	AUX
ejpam-2553	29	9	found	find	VERB
ejpam-2553	29	10	in	in	ADP
ejpam-2553	29	11	[	[	X
ejpam-2553	29	12	12	12	NUM
ejpam-2553	29	13	,	,	PUNCT
ejpam-2553	29	14	13	13	NUM
ejpam-2553	29	15	,	,	PUNCT
ejpam-2553	29	16	20	20	NUM
ejpam-2553	29	17	,	,	PUNCT
ejpam-2553	29	18	30	30	NUM
ejpam-2553	29	19	,	,	PUNCT
ejpam-2553	29	20	32	32	NUM
ejpam-2553	29	21	]	]	PUNCT
ejpam-2553	29	22	.	.	PUNCT
ejpam-2553	30	1	the	the	DET
ejpam-2553	30	2	another	another	DET
ejpam-2553	30	3	type	type	NOUN
ejpam-2553	30	4	of	of	ADP
ejpam-2553	30	5	hyperrings	hyperring	NOUN
ejpam-2553	30	6	was	be	AUX
ejpam-2553	30	7	introduced	introduce	VERB
ejpam-2553	30	8	by	by	ADP
ejpam-2553	30	9	rota	rota	NOUN
ejpam-2553	30	10	in	in	ADP
ejpam-2553	30	11	1982	1982	NUM
ejpam-2553	30	12	which	which	PRON
ejpam-2553	30	13	the	the	DET
ejpam-2553	30	14	multiplication	multiplication	NOUN
ejpam-2553	30	15	is	be	AUX
ejpam-2553	30	16	a	a	DET
ejpam-2553	30	17	hyperoperation	hyperoperation	NOUN
ejpam-2553	30	18	,	,	PUNCT
ejpam-2553	30	19	while	while	SCONJ
ejpam-2553	30	20	the	the	DET
ejpam-2553	30	21	addition	addition	NOUN
ejpam-2553	30	22	is	be	AUX
ejpam-2553	30	23	an	an	DET
ejpam-2553	30	24	operation	operation	NOUN
ejpam-2553	30	25	,	,	PUNCT
ejpam-2553	30	26	and	and	CCONJ
ejpam-2553	30	27	it	it	PRON
ejpam-2553	30	28	is	be	AUX
ejpam-2553	30	29	called	call	VERB
ejpam-2553	30	30	it	it	PRON
ejpam-2553	30	31	a	a	DET
ejpam-2553	30	32	multiplicative	multiplicative	ADJ
ejpam-2553	30	33	hyperring	hyperring	NOUN
ejpam-2553	30	34	(	(	PUNCT
ejpam-2553	30	35	for	for	ADP
ejpam-2553	30	36	more	more	ADJ
ejpam-2553	30	37	details	detail	NOUN
ejpam-2553	30	38	see	see	VERB
ejpam-2553	30	39	[	[	X
ejpam-2553	30	40	26–29	26–29	NOUN
ejpam-2553	30	41	]	]	X
ejpam-2553	30	42	)	)	PUNCT
ejpam-2553	30	43	which	which	PRON
ejpam-2553	30	44	was	be	AUX
ejpam-2553	30	45	subsequently	subsequently	ADV
ejpam-2553	30	46	investigated	investigate	VERB
ejpam-2553	30	47	by	by	ADP
ejpam-2553	30	48	olson	olson	NOUN
ejpam-2553	30	49	and	and	CCONJ
ejpam-2553	30	50	ward	ward	VERB
ejpam-2553	31	1	[	[	X
ejpam-2553	31	2	22	22	NUM
ejpam-2553	31	3	]	]	PUNCT
ejpam-2553	31	4	and	and	CCONJ
ejpam-2553	31	5	many	many	ADJ
ejpam-2553	31	6	others	other	NOUN
ejpam-2553	31	7	.	.	PUNCT
ejpam-2553	32	1	de	de	X
ejpam-2553	32	2	salvo	salvo	NOUN
ejpam-2553	33	1	[	[	X
ejpam-2553	33	2	14	14	NUM
ejpam-2553	33	3	]	]	PUNCT
ejpam-2553	33	4	introduced	introduce	VERB
ejpam-2553	33	5	hyperrings	hyperring	NOUN
ejpam-2553	33	6	in	in	ADP
ejpam-2553	33	7	which	which	PRON
ejpam-2553	33	8	the	the	DET
ejpam-2553	33	9	additions	addition	NOUN
ejpam-2553	33	10	and	and	CCONJ
ejpam-2553	33	11	the	the	DET
ejpam-2553	33	12	multiplications	multiplication	NOUN
ejpam-2553	33	13	are	be	AUX
ejpam-2553	33	14	hyperoperations	hyperoperation	NOUN
ejpam-2553	33	15	.	.	PUNCT
ejpam-2553	34	1	moreover	moreover	ADV
ejpam-2553	34	2	,	,	PUNCT
ejpam-2553	34	3	there	there	PRON
ejpam-2553	34	4	exists	exist	VERB
ejpam-2553	34	5	other	other	ADJ
ejpam-2553	34	6	types	type	NOUN
ejpam-2553	34	7	of	of	ADP
ejpam-2553	34	8	hyperrings	hyperring	NOUN
ejpam-2553	34	9	that	that	PRON
ejpam-2553	34	10	both	both	CCONJ
ejpam-2553	34	11	the	the	DET
ejpam-2553	34	12	addition	addition	NOUN
ejpam-2553	34	13	and	and	CCONJ
ejpam-2553	34	14	multiplication	multiplication	NOUN
ejpam-2553	34	15	are	be	AUX
ejpam-2553	34	16	hyperoperations	hyperoperation	NOUN
ejpam-2553	34	17	and	and	CCONJ
ejpam-2553	34	18	instead	instead	ADV
ejpam-2553	34	19	associativity	associativity	NOUN
ejpam-2553	34	20	,	,	PUNCT
ejpam-2553	34	21	commutativity	commutativity	NOUN
ejpam-2553	34	22	and	and	CCONJ
ejpam-2553	34	23	distributivity	distributivity	NOUN
ejpam-2553	34	24	satisfy	satisfy	NOUN
ejpam-2553	34	25	in	in	ADP
ejpam-2553	34	26	weak	weak	ADJ
ejpam-2553	34	27	associativity	associativity	NOUN
ejpam-2553	34	28	,	,	PUNCT
ejpam-2553	34	29	weak	weak	ADJ
ejpam-2553	34	30	commutativity	commutativity	NOUN
ejpam-2553	34	31	and	and	CCONJ
ejpam-2553	34	32	weak	weak	ADJ
ejpam-2553	34	33	distributivity	distributivity	NOUN
ejpam-2553	34	34	,	,	PUNCT
ejpam-2553	34	35	which	which	PRON
ejpam-2553	34	36	is	be	AUX
ejpam-2553	34	37	called	call	VERB
ejpam-2553	34	38	hv	hv	NOUN
ejpam-2553	34	39	-	-	PUNCT
ejpam-2553	34	40	hyperrings	hyperring	NOUN
ejpam-2553	34	41	,	,	PUNCT
ejpam-2553	34	42	this	this	DET
ejpam-2553	34	43	type	type	NOUN
ejpam-2553	34	44	of	of	ADP
ejpam-2553	34	45	hyperrings	hyperring	NOUN
ejpam-2553	34	46	can	can	AUX
ejpam-2553	34	47	be	be	AUX
ejpam-2553	34	48	seen	see	VERB
ejpam-2553	34	49	in	in	ADP
ejpam-2553	34	50	[	[	X
ejpam-2553	34	51	31	31	NUM
ejpam-2553	34	52	,	,	PUNCT
ejpam-2553	34	53	32	32	NUM
ejpam-2553	34	54	]	]	PUNCT
ejpam-2553	34	55	.	.	PUNCT
ejpam-2553	35	1	also	also	ADV
ejpam-2553	35	2	,	,	PUNCT
ejpam-2553	35	3	there	there	PRON
ejpam-2553	35	4	are	be	VERB
ejpam-2553	35	5	other	other	ADJ
ejpam-2553	35	6	types	type	NOUN
ejpam-2553	35	7	of	of	ADP
ejpam-2553	35	8	hyperrings	hyperring	NOUN
ejpam-2553	35	9	which	which	PRON
ejpam-2553	35	10	were	be	AUX
ejpam-2553	35	11	completely	completely	ADV
ejpam-2553	35	12	studied	study	VERB
ejpam-2553	35	13	in	in	ADP
ejpam-2553	35	14	[	[	X
ejpam-2553	35	15	12	12	NUM
ejpam-2553	35	16	]	]	PUNCT
ejpam-2553	35	17	.	.	PUNCT
ejpam-2553	36	1	these	these	DET
ejpam-2553	36	2	hyperrings	hyperring	NOUN
ejpam-2553	36	3	are	be	AUX
ejpam-2553	36	4	studied	study	VERB
ejpam-2553	36	5	by	by	ADP
ejpam-2553	36	6	rahnamai	rahnamai	PROPN
ejpam-2553	36	7	barghi	barghi	PROPN
ejpam-2553	37	1	[	[	X
ejpam-2553	37	2	25	25	NUM
ejpam-2553	37	3	]	]	PUNCT
ejpam-2553	37	4	.	.	PUNCT
ejpam-2553	37	5	procesi	procesi	PROPN
ejpam-2553	37	6	and	and	CCONJ
ejpam-2553	37	7	rota	rota	NOUN
ejpam-2553	37	8	in	in	ADP
ejpam-2553	37	9	[	[	X
ejpam-2553	37	10	23	23	NUM
ejpam-2553	37	11	]	]	PUNCT
ejpam-2553	37	12	have	have	AUX
ejpam-2553	37	13	studied	study	VERB
ejpam-2553	37	14	ring	ring	NOUN
ejpam-2553	37	15	of	of	ADP
ejpam-2553	37	16	fractions	fraction	NOUN
ejpam-2553	37	17	in	in	ADP
ejpam-2553	37	18	krasner	krasner	NOUN
ejpam-2553	37	19	hyperrings	hyperring	NOUN
ejpam-2553	37	20	and	and	CCONJ
ejpam-2553	37	21	also	also	ADV
ejpam-2553	37	22	they	they	PRON
ejpam-2553	37	23	conceptualized	conceptualize	VERB
ejpam-2553	37	24	in	in	ADP
ejpam-2553	37	25	[	[	X
ejpam-2553	37	26	24	24	NUM
ejpam-2553	37	27	]	]	X
ejpam-2553	37	28	the	the	DET
ejpam-2553	37	29	notion	notion	NOUN
ejpam-2553	37	30	of	of	ADP
ejpam-2553	37	31	primeness	primeness	NOUN
ejpam-2553	37	32	of	of	ADP
ejpam-2553	37	33	hyperideal	hyperideal	NOUN
ejpam-2553	37	34	in	in	ADP
ejpam-2553	37	35	a	a	DET
ejpam-2553	37	36	multiplicative	multiplicative	ADJ
ejpam-2553	37	37	hyperring	hyperring	NOUN
ejpam-2553	37	38	,	,	PUNCT
ejpam-2553	37	39	and	and	CCONJ
ejpam-2553	37	40	in	in	ADP
ejpam-2553	37	41	[	[	X
ejpam-2553	37	42	10	10	NUM
ejpam-2553	37	43	]	]	PUNCT
ejpam-2553	37	44	,	,	PUNCT
ejpam-2553	37	45	dasgupta	dasgupta	PROPN
ejpam-2553	37	46	extended	extend	VERB
ejpam-2553	37	47	the	the	DET
ejpam-2553	37	48	prime	prime	ADJ
ejpam-2553	37	49	and	and	CCONJ
ejpam-2553	37	50	primary	primary	ADJ
ejpam-2553	37	51	hyperideals	hyperideal	NOUN
ejpam-2553	37	52	in	in	ADP
ejpam-2553	37	53	multiplicative	multiplicative	ADJ
ejpam-2553	37	54	hyperrings	hyperring	NOUN
ejpam-2553	37	55	.	.	PUNCT
ejpam-2553	38	1	asokkumar	asokkumar	PROPN
ejpam-2553	38	2	and	and	CCONJ
ejpam-2553	38	3	velrajan	velrajan	NOUN
ejpam-2553	38	4	[	[	X
ejpam-2553	38	5	4	4	NUM
ejpam-2553	38	6	,	,	PUNCT
ejpam-2553	38	7	5	5	NUM
ejpam-2553	38	8	]	]	PUNCT
ejpam-2553	38	9	have	have	AUX
ejpam-2553	38	10	studied	study	VERB
ejpam-2553	38	11	von	von	PROPN
ejpam-2553	38	12	neumann	neumann	PROPN
ejpam-2553	38	13	regularity	regularity	PROPN
ejpam-2553	38	14	in	in	ADP
ejpam-2553	38	15	krasner	krasner	NOUN
ejpam-2553	38	16	hyperrings	hyperring	NOUN
ejpam-2553	38	17	.	.	PUNCT
ejpam-2553	39	1	a	a	DET
ejpam-2553	39	2	special	special	ADJ
ejpam-2553	39	3	equivalence	equivalence	NOUN
ejpam-2553	39	4	relations	relation	NOUN
ejpam-2553	39	5	which	which	PRON
ejpam-2553	39	6	is	be	AUX
ejpam-2553	39	7	called	call	VERB
ejpam-2553	39	8	fundamental	fundamental	ADJ
ejpam-2553	39	9	relations	relation	NOUN
ejpam-2553	39	10	play	play	VERB
ejpam-2553	39	11	important	important	ADJ
ejpam-2553	39	12	roles	role	NOUN
ejpam-2553	39	13	in	in	ADP
ejpam-2553	39	14	the	the	DET
ejpam-2553	39	15	the	the	DET
ejpam-2553	39	16	theory	theory	NOUN
ejpam-2553	39	17	of	of	ADP
ejpam-2553	39	18	algebraic	algebraic	PROPN
ejpam-2553	39	19	hyperstructures	hyperstructure	NOUN
ejpam-2553	39	20	.	.	PUNCT
ejpam-2553	40	1	the	the	DET
ejpam-2553	40	2	fundamental	fundamental	ADJ
ejpam-2553	40	3	relations	relation	NOUN
ejpam-2553	40	4	are	be	AUX
ejpam-2553	40	5	one	one	NUM
ejpam-2553	40	6	of	of	ADP
ejpam-2553	40	7	the	the	DET
ejpam-2553	40	8	most	most	ADV
ejpam-2553	40	9	important	important	ADJ
ejpam-2553	40	10	and	and	CCONJ
ejpam-2553	40	11	interesting	interesting	ADJ
ejpam-2553	40	12	concepts	concept	NOUN
ejpam-2553	40	13	in	in	ADP
ejpam-2553	40	14	algebraic	algebraic	PROPN
ejpam-2553	40	15	hyperstructures	hyperstructure	NOUN
ejpam-2553	40	16	that	that	SCONJ
ejpam-2553	40	17	ordinary	ordinary	ADJ
ejpam-2553	40	18	algebraic	algebraic	ADJ
ejpam-2553	40	19	structures	structure	NOUN
ejpam-2553	40	20	are	be	AUX
ejpam-2553	40	21	derived	derive	VERB
ejpam-2553	40	22	from	from	ADP
ejpam-2553	40	23	algebraic	algebraic	ADJ
ejpam-2553	40	24	hyperstructures	hyperstructure	NOUN
ejpam-2553	40	25	by	by	ADP
ejpam-2553	40	26	them	they	PRON
ejpam-2553	40	27	.	.	PUNCT
ejpam-2553	41	1	the	the	DET
ejpam-2553	41	2	fundamental	fundamental	ADJ
ejpam-2553	41	3	relation	relation	NOUN
ejpam-2553	41	4	β∗	β∗	NOUN
ejpam-2553	41	5	on	on	ADP
ejpam-2553	41	6	hypergroups	hypergroup	NOUN
ejpam-2553	41	7	was	be	AUX
ejpam-2553	41	8	defined	define	VERB
ejpam-2553	41	9	by	by	ADP
ejpam-2553	41	10	koskas	koskas	NOUN
ejpam-2553	42	1	[	[	X
ejpam-2553	42	2	17	17	NUM
ejpam-2553	42	3	]	]	PUNCT
ejpam-2553	42	4	,	,	PUNCT
ejpam-2553	42	5	mainly	mainly	ADV
ejpam-2553	42	6	studied	study	VERB
ejpam-2553	42	7	by	by	ADP
ejpam-2553	42	8	corsini	corsini	PROPN
ejpam-2553	42	9	[	[	X
ejpam-2553	42	10	19	19	NUM
ejpam-2553	42	11	]	]	PUNCT
ejpam-2553	42	12	,	,	PUNCT
ejpam-2553	42	13	freni	freni	PROPN
ejpam-2553	42	14	[	[	X
ejpam-2553	42	15	15	15	NUM
ejpam-2553	42	16	,	,	PUNCT
ejpam-2553	42	17	16	16	NUM
ejpam-2553	42	18	]	]	PUNCT
ejpam-2553	42	19	,	,	PUNCT
ejpam-2553	42	20	vougiouklis	vougioukli	VERB
ejpam-2553	42	21	[	[	X
ejpam-2553	42	22	32	32	NUM
ejpam-2553	42	23	]	]	PUNCT
ejpam-2553	42	24	(	(	PUNCT
ejpam-2553	42	25	for	for	ADP
ejpam-2553	42	26	more	more	ADJ
ejpam-2553	42	27	details	detail	NOUN
ejpam-2553	42	28	about	about	ADP
ejpam-2553	42	29	hyperrings	hyperring	NOUN
ejpam-2553	42	30	and	and	CCONJ
ejpam-2553	42	31	fundamental	fundamental	ADJ
ejpam-2553	42	32	relations	relation	NOUN
ejpam-2553	42	33	on	on	ADP
ejpam-2553	42	34	hyperrings	hyperring	NOUN
ejpam-2553	42	35	see	see	VERB
ejpam-2553	42	36	[	[	X
ejpam-2553	42	37	1	1	NUM
ejpam-2553	42	38	,	,	PUNCT
ejpam-2553	42	39	2	2	NUM
ejpam-2553	42	40	,	,	PUNCT
ejpam-2553	42	41	11	11	NUM
ejpam-2553	42	42	,	,	PUNCT
ejpam-2553	42	43	12	12	NUM
ejpam-2553	42	44	,	,	PUNCT
ejpam-2553	42	45	30	30	NUM
ejpam-2553	42	46	,	,	PUNCT
ejpam-2553	42	47	32	32	NUM
ejpam-2553	42	48	]	]	PUNCT
ejpam-2553	42	49	)	)	PUNCT
ejpam-2553	42	50	.	.	PUNCT
ejpam-2553	43	1	in	in	ADP
ejpam-2553	43	2	this	this	DET
ejpam-2553	43	3	paper	paper	NOUN
ejpam-2553	43	4	we	we	PRON
ejpam-2553	43	5	consider	consider	VERB
ejpam-2553	43	6	the	the	DET
ejpam-2553	43	7	classes	class	NOUN
ejpam-2553	43	8	of	of	ADP
ejpam-2553	43	9	multiplicative	multiplicative	ADJ
ejpam-2553	43	10	hyperring	hyperring	NOUN
ejpam-2553	43	11	as	as	ADP
ejpam-2553	43	12	a	a	DET
ejpam-2553	43	13	hyperstructures	hyperstructure	NOUN
ejpam-2553	43	14	(	(	PUNCT
ejpam-2553	43	15	r,+	r,+	NUM
ejpam-2553	43	16	,	,	PUNCT
ejpam-2553	43	17	.	.	PUNCT
ejpam-2553	43	18	)	)	PUNCT
ejpam-2553	43	19	,	,	PUNCT
ejpam-2553	43	20	where	where	SCONJ
ejpam-2553	43	21	(	(	PUNCT
ejpam-2553	43	22	r,+	r,+	NUM
ejpam-2553	43	23	)	)	PUNCT
ejpam-2553	43	24	is	be	AUX
ejpam-2553	43	25	an	an	DET
ejpam-2553	43	26	abelian	abelian	ADJ
ejpam-2553	43	27	group	group	NOUN
ejpam-2553	43	28	,	,	PUNCT
ejpam-2553	43	29	(	(	PUNCT
ejpam-2553	43	30	r,+	r,+	PRON
ejpam-2553	43	31	)	)	PUNCT
ejpam-2553	43	32	is	be	AUX
ejpam-2553	43	33	a	a	DET
ejpam-2553	43	34	semihypergroup	semihypergroup	NOUN
ejpam-2553	43	35	and	and	CCONJ
ejpam-2553	43	36	the	the	DET
ejpam-2553	43	37	hyperoperation	hyperoperation	NOUN
ejpam-2553	43	38	”	"	PUNCT
ejpam-2553	43	39	.	.	PUNCT
ejpam-2553	43	40	”	"	PUNCT
ejpam-2553	43	41	is	be	AUX
ejpam-2553	43	42	distributive	distributive	ADJ
ejpam-2553	43	43	with	with	ADP
ejpam-2553	43	44	respect	respect	NOUN
ejpam-2553	43	45	to	to	ADP
ejpam-2553	43	46	the	the	DET
ejpam-2553	43	47	operation	operation	NOUN
ejpam-2553	43	48	”	"	PUNCT
ejpam-2553	44	1	+	+	CCONJ
ejpam-2553	44	2	”	"	PUNCT
ejpam-2553	44	3	,	,	PUNCT
ejpam-2553	44	4	i.e.	i.e.	X
ejpam-2553	44	5	a.(b+	a.(b+	X
ejpam-2553	44	6	c	c	X
ejpam-2553	44	7	)	)	PUNCT
ejpam-2553	44	8	⊆	⊆	NUM
ejpam-2553	44	9	a.b+	a.b+	NUM
ejpam-2553	44	10	a.c	a.c	PROPN
ejpam-2553	44	11	.	.	PUNCT
ejpam-2553	45	1	the	the	DET
ejpam-2553	45	2	purpose	purpose	NOUN
ejpam-2553	45	3	of	of	ADP
ejpam-2553	45	4	this	this	DET
ejpam-2553	45	5	paper	paper	NOUN
ejpam-2553	45	6	is	be	AUX
ejpam-2553	45	7	the	the	DET
ejpam-2553	45	8	study	study	NOUN
ejpam-2553	45	9	study	study	NOUN
ejpam-2553	45	10	regular	regular	ADJ
ejpam-2553	45	11	multiplicative	multiplicative	ADJ
ejpam-2553	45	12	hyperrings	hyperring	NOUN
ejpam-2553	45	13	.	.	PUNCT
ejpam-2553	46	1	in	in	ADP
ejpam-2553	46	2	this	this	DET
ejpam-2553	46	3	regards	regard	NOUN
ejpam-2553	46	4	we	we	PRON
ejpam-2553	46	5	study	study	VERB
ejpam-2553	46	6	the	the	DET
ejpam-2553	46	7	properties	property	NOUN
ejpam-2553	46	8	of	of	ADP
ejpam-2553	46	9	regular	regular	ADJ
ejpam-2553	46	10	multiplicative	multiplicative	ADJ
ejpam-2553	46	11	hyperring	hyperring	NOUN
ejpam-2553	46	12	r	r	NOUN
ejpam-2553	46	13	and	and	CCONJ
ejpam-2553	46	14	obtain	obtain	VERB
ejpam-2553	46	15	some	some	DET
ejpam-2553	46	16	results	result	NOUN
ejpam-2553	46	17	.	.	PUNCT
ejpam-2553	47	1	we	we	PRON
ejpam-2553	47	2	will	will	AUX
ejpam-2553	47	3	proceed	proceed	VERB
ejpam-2553	47	4	to	to	PART
ejpam-2553	47	5	use	use	VERB
ejpam-2553	47	6	the	the	DET
ejpam-2553	47	7	fundamental	fundamental	ADJ
ejpam-2553	47	8	relation	relation	NOUN
ejpam-2553	47	9	γ∗	γ∗	NOUN
ejpam-2553	47	10	on	on	ADP
ejpam-2553	47	11	r	r	NOUN
ejpam-2553	47	12	and	and	CCONJ
ejpam-2553	47	13	prove	prove	VERB
ejpam-2553	47	14	that	that	SCONJ
ejpam-2553	47	15	the	the	DET
ejpam-2553	47	16	fundamental	fundamental	ADJ
ejpam-2553	47	17	ring	ring	NOUN
ejpam-2553	47	18	r	r	NOUN
ejpam-2553	47	19	/	/	SYM
ejpam-2553	47	20	γ∗	γ∗	NOUN
ejpam-2553	47	21	of	of	ADP
ejpam-2553	47	22	r	r	NOUN
ejpam-2553	47	23	is	be	AUX
ejpam-2553	47	24	regular	regular	ADJ
ejpam-2553	47	25	whenever	whenever	SCONJ
ejpam-2553	47	26	r	r	NOUN
ejpam-2553	47	27	is	be	AUX
ejpam-2553	47	28	regular	regular	ADJ
ejpam-2553	47	29	.	.	PUNCT
ejpam-2553	48	1	also	also	ADV
ejpam-2553	48	2	,	,	PUNCT
ejpam-2553	48	3	we	we	PRON
ejpam-2553	48	4	show	show	VERB
ejpam-2553	48	5	that	that	SCONJ
ejpam-2553	48	6	this	this	DET
ejpam-2553	48	7	process	process	NOUN
ejpam-2553	48	8	make	make	VERB
ejpam-2553	48	9	a	a	DET
ejpam-2553	48	10	functor	functor	NOUN
ejpam-2553	48	11	from	from	ADP
ejpam-2553	48	12	the	the	DET
ejpam-2553	48	13	category	category	NOUN
ejpam-2553	48	14	of	of	ADP
ejpam-2553	48	15	regular	regular	ADJ
ejpam-2553	48	16	multiplicative	multiplicative	ADJ
ejpam-2553	48	17	hyperrings	hyperring	NOUN
ejpam-2553	48	18	to	to	ADP
ejpam-2553	48	19	the	the	DET
ejpam-2553	48	20	category	category	NOUN
ejpam-2553	48	21	of	of	ADP
ejpam-2553	48	22	regular	regular	ADJ
ejpam-2553	48	23	rings	ring	NOUN
ejpam-2553	48	24	.	.	PUNCT
ejpam-2553	49	1	finally	finally	ADV
ejpam-2553	49	2	,	,	PUNCT
ejpam-2553	49	3	the	the	DET
ejpam-2553	49	4	notion	notion	NOUN
ejpam-2553	49	5	of	of	ADP
ejpam-2553	49	6	regular	regular	ADJ
ejpam-2553	49	7	hyperideal	hyperideal	ADJ
ejpam-2553	49	8	m(r	m(r	PROPN
ejpam-2553	49	9	)	)	PUNCT
ejpam-2553	49	10	,	,	PUNCT
ejpam-2553	49	11	consisting	consist	VERB
ejpam-2553	49	12	of	of	ADP
ejpam-2553	49	13	the	the	DET
ejpam-2553	49	14	elements	element	NOUN
ejpam-2553	49	15	of	of	ADP
ejpam-2553	49	16	r	r	NOUN
ejpam-2553	49	17	such	such	ADJ
ejpam-2553	49	18	that	that	SCONJ
ejpam-2553	49	19	the	the	DET
ejpam-2553	49	20	generated	generate	VERB
ejpam-2553	49	21	hyperideal	hyperideal	NOUN
ejpam-2553	49	22	by	by	ADP
ejpam-2553	49	23	these	these	DET
ejpam-2553	49	24	elements	element	NOUN
ejpam-2553	49	25	are	be	AUX
ejpam-2553	49	26	regular	regular	ADJ
ejpam-2553	49	27	hyperideals	hyperideal	NOUN
ejpam-2553	49	28	,	,	PUNCT
ejpam-2553	49	29	are	be	AUX
ejpam-2553	49	30	introduced	introduce	VERB
ejpam-2553	49	31	and	and	CCONJ
ejpam-2553	49	32	its	its	PRON
ejpam-2553	49	33	basic	basic	ADJ
ejpam-2553	49	34	properties	property	NOUN
ejpam-2553	49	35	are	be	AUX
ejpam-2553	49	36	investigated	investigate	VERB
ejpam-2553	49	37	.	.	PUNCT
ejpam-2553	50	1	r.	r.	PROPN
ejpam-2553	50	2	ameri	ameri	PROPN
ejpam-2553	50	3	,	,	PUNCT
ejpam-2553	50	4	a.	a.	PROPN
ejpam-2553	50	5	kordi	kordi	PROPN
ejpam-2553	50	6	/	/	PUNCT
ejpam-2553	50	7	eur	eur	PROPN
ejpam-2553	50	8	.	.	PUNCT
ejpam-2553	51	1	j.	j.	PROPN
ejpam-2553	51	2	pure	pure	PROPN
ejpam-2553	51	3	appl	appl	PROPN
ejpam-2553	51	4	.	.	PROPN
ejpam-2553	51	5	math	math	PROPN
ejpam-2553	51	6	,	,	PUNCT
ejpam-2553	51	7	9	9	NUM
ejpam-2553	51	8	(	(	PUNCT
ejpam-2553	51	9	2016	2016	NUM
ejpam-2553	51	10	)	)	PUNCT
ejpam-2553	51	11	,	,	PUNCT
ejpam-2553	51	12	402	402	NUM
ejpam-2553	51	13	-	-	SYM
ejpam-2553	51	14	418	418	NUM
ejpam-2553	51	15	404	404	NUM
ejpam-2553	51	16	2	2	NUM
ejpam-2553	51	17	.	.	PUNCT
ejpam-2553	52	1	regular	regular	ADJ
ejpam-2553	52	2	multiplicative	multiplicative	ADJ
ejpam-2553	52	3	hyperring	hyperring	NOUN
ejpam-2553	52	4	recall	recall	NOUN
ejpam-2553	52	5	that	that	SCONJ
ejpam-2553	52	6	a	a	DET
ejpam-2553	52	7	hyperoperation	hyperoperation	NOUN
ejpam-2553	52	8	”	"	PUNCT
ejpam-2553	52	9	.	.	PUNCT
ejpam-2553	52	10	”	"	PUNCT
ejpam-2553	53	1	on	on	ADP
ejpam-2553	53	2	nonempty	nonempty	ADV
ejpam-2553	53	3	set	set	VERB
ejpam-2553	53	4	h	h	NOUN
ejpam-2553	53	5	is	be	AUX
ejpam-2553	53	6	a	a	DET
ejpam-2553	53	7	mapping	mapping	NOUN
ejpam-2553	53	8	of	of	ADP
ejpam-2553	53	9	h	h	NOUN
ejpam-2553	53	10	×	×	PROPN
ejpam-2553	53	11	h	h	NOUN
ejpam-2553	53	12	into	into	ADP
ejpam-2553	53	13	the	the	DET
ejpam-2553	53	14	family	family	NOUN
ejpam-2553	53	15	of	of	ADP
ejpam-2553	53	16	all	all	DET
ejpam-2553	53	17	nonempty	nonempty	ADJ
ejpam-2553	53	18	subsets	subset	NOUN
ejpam-2553	53	19	of	of	ADP
ejpam-2553	53	20	h.	h.	PROPN
ejpam-2553	53	21	let	let	VERB
ejpam-2553	53	22	”	"	PUNCT
ejpam-2553	53	23	.	.	PUNCT
ejpam-2553	53	24	”	"	PUNCT
ejpam-2553	54	1	be	be	AUX
ejpam-2553	54	2	a	a	DET
ejpam-2553	54	3	hyperoperation	hyperoperation	NOUN
ejpam-2553	54	4	on	on	ADP
ejpam-2553	54	5	h.	h.	PROPN
ejpam-2553	54	6	then	then	ADV
ejpam-2553	54	7	,	,	PUNCT
ejpam-2553	54	8	(	(	PUNCT
ejpam-2553	54	9	h	h	NOUN
ejpam-2553	54	10	,	,	PUNCT
ejpam-2553	54	11	.	.	PUNCT
ejpam-2553	54	12	)	)	PUNCT
ejpam-2553	55	1	is	be	AUX
ejpam-2553	55	2	called	call	VERB
ejpam-2553	55	3	a	a	DET
ejpam-2553	55	4	hypergroupoid	hypergroupoid	NOUN
ejpam-2553	55	5	.	.	PUNCT
ejpam-2553	56	1	we	we	PRON
ejpam-2553	56	2	can	can	AUX
ejpam-2553	56	3	extend	extend	VERB
ejpam-2553	56	4	the	the	DET
ejpam-2553	56	5	hyperoperation	hyperoperation	NOUN
ejpam-2553	56	6	on	on	ADP
ejpam-2553	56	7	h	h	NOUN
ejpam-2553	56	8	to	to	ADP
ejpam-2553	56	9	subsets	subset	NOUN
ejpam-2553	56	10	of	of	ADP
ejpam-2553	56	11	h	h	NOUN
ejpam-2553	56	12	as	as	SCONJ
ejpam-2553	56	13	follows	follow	VERB
ejpam-2553	56	14	.	.	PUNCT
ejpam-2553	57	1	for	for	ADP
ejpam-2553	57	2	a	a	DET
ejpam-2553	57	3	,	,	PUNCT
ejpam-2553	57	4	b	b	PROPN
ejpam-2553	57	5	⊆	⊆	NUM
ejpam-2553	57	6	h	h	NOUN
ejpam-2553	57	7	and	and	CCONJ
ejpam-2553	57	8	h	h	NOUN
ejpam-2553	57	9	∈	∈	PROPN
ejpam-2553	57	10	h	h	NOUN
ejpam-2553	57	11	,	,	PUNCT
ejpam-2553	57	12	then	then	ADV
ejpam-2553	57	13	ab	ab	PROPN
ejpam-2553	57	14	=	=	PUNCT
ejpam-2553	57	15	∪a∈a	∪a∈a	PROPN
ejpam-2553	57	16	,	,	PUNCT
ejpam-2553	57	17	b∈bab	b∈bab	PROPN
ejpam-2553	57	18	,	,	PUNCT
ejpam-2553	57	19	ah=	ah=	PROPN
ejpam-2553	57	20	a{h	a{h	PROPN
ejpam-2553	57	21	}	}	PUNCT
ejpam-2553	57	22	,	,	PUNCT
ejpam-2553	57	23	hb	hb	X
ejpam-2553	57	24	=	=	PUNCT
ejpam-2553	57	25	{	{	PUNCT
ejpam-2553	57	26	h}b	h}b	PROPN
ejpam-2553	57	27	.	.	PUNCT
ejpam-2553	58	1	a	a	DET
ejpam-2553	58	2	semihypergroup	semihypergroup	NOUN
ejpam-2553	58	3	is	be	AUX
ejpam-2553	58	4	a	a	DET
ejpam-2553	58	5	hypergroupoid	hypergroupoid	PROPN
ejpam-2553	58	6	(	(	PUNCT
ejpam-2553	58	7	h	h	NOUN
ejpam-2553	58	8	,	,	PUNCT
ejpam-2553	58	9	.	.	PUNCT
ejpam-2553	58	10	)	)	PUNCT
ejpam-2553	58	11	,	,	PUNCT
ejpam-2553	58	12	which	which	PRON
ejpam-2553	58	13	is	be	AUX
ejpam-2553	58	14	associative	associative	ADJ
ejpam-2553	58	15	,	,	PUNCT
ejpam-2553	58	16	that	that	ADV
ejpam-2553	58	17	is	is	ADV
ejpam-2553	58	18	(	(	PUNCT
ejpam-2553	58	19	a.b).c	a.b).c	NOUN
ejpam-2553	58	20	=	=	PUNCT
ejpam-2553	58	21	a.(b.c	a.(b.c	ADV
ejpam-2553	58	22	)	)	PUNCT
ejpam-2553	58	23	or	or	CCONJ
ejpam-2553	58	24	fall	fall	VERB
ejpam-2553	58	25	a	a	DET
ejpam-2553	58	26	,	,	PUNCT
ejpam-2553	58	27	b	b	NOUN
ejpam-2553	58	28	,	,	PUNCT
ejpam-2553	58	29	c	c	PROPN
ejpam-2553	58	30	∈	∈	PROPN
ejpam-2553	58	31	h.	h.	PROPN
ejpam-2553	59	1	a	a	DET
ejpam-2553	59	2	hypergroup	hypergroup	NOUN
ejpam-2553	59	3	is	be	AUX
ejpam-2553	59	4	a	a	DET
ejpam-2553	59	5	semihypergroup	semihypergroup	NOUN
ejpam-2553	59	6	(	(	PUNCT
ejpam-2553	59	7	h	h	NOUN
ejpam-2553	59	8	,	,	PUNCT
ejpam-2553	59	9	.	.	PUNCT
ejpam-2553	59	10	)	)	PUNCT
ejpam-2553	60	1	,	,	PUNCT
ejpam-2553	60	2	that	that	PRON
ejpam-2553	60	3	satisfies	satisfy	VERB
ejpam-2553	60	4	the	the	DET
ejpam-2553	60	5	reproduction	reproduction	NOUN
ejpam-2553	60	6	axioms	axiom	NOUN
ejpam-2553	60	7	,	,	PUNCT
ejpam-2553	60	8	that	that	PRON
ejpam-2553	60	9	is	be	AUX
ejpam-2553	60	10	a.h	a.h	PROPN
ejpam-2553	60	11	=	=	NOUN
ejpam-2553	60	12	h	h	NOUN
ejpam-2553	61	1	=	=	PUNCT
ejpam-2553	61	2	h.a	h.a	VERB
ejpam-2553	61	3	for	for	ADP
ejpam-2553	61	4	all	all	DET
ejpam-2553	61	5	a	a	DET
ejpam-2553	61	6	∈	∈	PROPN
ejpam-2553	61	7	h.	h.	NOUN
ejpam-2553	61	8	a	a	DET
ejpam-2553	61	9	non	non	ADJ
ejpam-2553	61	10	-	-	ADJ
ejpam-2553	61	11	empty	empty	ADJ
ejpam-2553	61	12	set	set	ADJ
ejpam-2553	61	13	r	r	NOUN
ejpam-2553	61	14	with	with	ADP
ejpam-2553	61	15	two	two	NUM
ejpam-2553	61	16	hyperoperations	hyperoperation	NOUN
ejpam-2553	61	17	+	+	CCONJ
ejpam-2553	61	18	and	and	CCONJ
ejpam-2553	61	19	.	.	PUNCT
ejpam-2553	62	1	is	be	AUX
ejpam-2553	62	2	said	say	VERB
ejpam-2553	62	3	to	to	PART
ejpam-2553	62	4	be	be	AUX
ejpam-2553	62	5	a	a	DET
ejpam-2553	62	6	hyperring	hyperring	NOUN
ejpam-2553	62	7	if	if	SCONJ
ejpam-2553	62	8	(	(	PUNCT
ejpam-2553	62	9	r,+	r,+	NUM
ejpam-2553	62	10	)	)	PUNCT
ejpam-2553	62	11	is	be	AUX
ejpam-2553	62	12	a	a	DET
ejpam-2553	62	13	canonical	canonical	ADJ
ejpam-2553	62	14	hypergroup	hypergroup	NOUN
ejpam-2553	62	15	,	,	PUNCT
ejpam-2553	62	16	(	(	PUNCT
ejpam-2553	62	17	r	r	NOUN
ejpam-2553	62	18	,	,	PUNCT
ejpam-2553	62	19	.	.	PUNCT
ejpam-2553	62	20	)	)	PUNCT
ejpam-2553	63	1	is	be	AUX
ejpam-2553	63	2	a	a	DET
ejpam-2553	63	3	semihypergroup	semihypergroup	NOUN
ejpam-2553	63	4	with	with	ADP
ejpam-2553	63	5	r.0	r.0	NOUN
ejpam-2553	63	6	=	=	SYM
ejpam-2553	63	7	0.r	0.r	NUM
ejpam-2553	63	8	=	=	SYM
ejpam-2553	63	9	0	0	NUM
ejpam-2553	63	10	for	for	ADP
ejpam-2553	63	11	all	all	DET
ejpam-2553	63	12	r	r	NOUN
ejpam-2553	63	13	∈	∈	NOUN
ejpam-2553	63	14	r	r	NOUN
ejpam-2553	63	15	(	(	PUNCT
ejpam-2553	63	16	0	0	NUM
ejpam-2553	63	17	as	as	ADP
ejpam-2553	63	18	a	a	DET
ejpam-2553	63	19	bilaterally	bilaterally	ADV
ejpam-2553	63	20	absorbing	absorbing	ADJ
ejpam-2553	63	21	element	element	NOUN
ejpam-2553	63	22	)	)	PUNCT
ejpam-2553	63	23	and	and	CCONJ
ejpam-2553	63	24	the	the	DET
ejpam-2553	63	25	hyperoperation	hyperoperation	NOUN
ejpam-2553	63	26	.	.	PUNCT
ejpam-2553	64	1	is	be	AUX
ejpam-2553	64	2	distributive	distributive	ADJ
ejpam-2553	64	3	with	with	ADP
ejpam-2553	64	4	respect	respect	NOUN
ejpam-2553	64	5	to	to	ADP
ejpam-2553	64	6	+	+	ADJ
ejpam-2553	64	7	,	,	PUNCT
ejpam-2553	64	8	i.e.	i.e.	X
ejpam-2553	64	9	,	,	PUNCT
ejpam-2553	64	10	for	for	ADP
ejpam-2553	64	11	every	every	DET
ejpam-2553	64	12	a	a	DET
ejpam-2553	64	13	,	,	PUNCT
ejpam-2553	64	14	b	b	NOUN
ejpam-2553	64	15	,	,	PUNCT
ejpam-2553	64	16	c	c	PROPN
ejpam-2553	64	17	∈	∈	PROPN
ejpam-2553	64	18	r	r	NOUN
ejpam-2553	64	19	;	;	PUNCT
ejpam-2553	65	1	a(b+	a(b+	DET
ejpam-2553	65	2	c	c	X
ejpam-2553	65	3	)	)	PUNCT
ejpam-2553	65	4	=	=	SYM
ejpam-2553	65	5	ab+	ab+	NOUN
ejpam-2553	65	6	ac	ac	PROPN
ejpam-2553	65	7	and	and	CCONJ
ejpam-2553	65	8	(	(	PUNCT
ejpam-2553	65	9	a+	a+	X
ejpam-2553	65	10	b)c	b)c	X
ejpam-2553	65	11	=	=	PUNCT
ejpam-2553	65	12	ac	ac	PROPN
ejpam-2553	65	13	+	+	CCONJ
ejpam-2553	65	14	bc	bc	PROPN
ejpam-2553	65	15	.	.	PUNCT
ejpam-2553	66	1	a	a	DET
ejpam-2553	66	2	multiplicative	multiplicative	ADJ
ejpam-2553	66	3	hyperring	hyperring	NOUN
ejpam-2553	66	4	is	be	AUX
ejpam-2553	66	5	an	an	DET
ejpam-2553	66	6	additive	additive	ADJ
ejpam-2553	66	7	commutative	commutative	ADJ
ejpam-2553	66	8	group	group	NOUN
ejpam-2553	66	9	(	(	PUNCT
ejpam-2553	66	10	r,+	r,+	NUM
ejpam-2553	66	11	)	)	PUNCT
ejpam-2553	66	12	endowed	endow	VERB
ejpam-2553	66	13	with	with	ADP
ejpam-2553	66	14	a	a	DET
ejpam-2553	66	15	hyperoperation	hyperoperation	NOUN
ejpam-2553	66	16	"	"	PUNCT
ejpam-2553	66	17	.	.	PUNCT
ejpam-2553	66	18	"	"	PUNCT
ejpam-2553	66	19	which	which	PRON
ejpam-2553	66	20	satisfies	satisfy	VERB
ejpam-2553	66	21	the	the	DET
ejpam-2553	66	22	following	follow	VERB
ejpam-2553	66	23	conditions	condition	NOUN
ejpam-2553	66	24	:	:	PUNCT
ejpam-2553	66	25	(	(	PUNCT
ejpam-2553	66	26	1	1	X
ejpam-2553	66	27	)	)	PUNCT
ejpam-2553	66	28	∀a	∀a	NOUN
ejpam-2553	66	29	,	,	PUNCT
ejpam-2553	66	30	b	b	X
ejpam-2553	66	31	,	,	PUNCT
ejpam-2553	66	32	c	c	PROPN
ejpam-2553	66	33	∈	∈	PROPN
ejpam-2553	66	34	r	r	NOUN
ejpam-2553	66	35	:	:	PUNCT
ejpam-2553	66	36	a(bc	a(bc	NUM
ejpam-2553	66	37	)	)	PUNCT
ejpam-2553	66	38	=	=	SYM
ejpam-2553	66	39	(	(	PUNCT
ejpam-2553	66	40	ab)c	ab)c	PROPN
ejpam-2553	66	41	;	;	PUNCT
ejpam-2553	66	42	(	(	PUNCT
ejpam-2553	66	43	2	2	X
ejpam-2553	66	44	)	)	PUNCT
ejpam-2553	66	45	∀a	∀a	NOUN
ejpam-2553	66	46	,	,	PUNCT
ejpam-2553	66	47	b	b	X
ejpam-2553	66	48	,	,	PUNCT
ejpam-2553	66	49	c	c	PROPN
ejpam-2553	66	50	∈	∈	PROPN
ejpam-2553	66	51	r	r	NOUN
ejpam-2553	66	52	:	:	PUNCT
ejpam-2553	66	53	(	(	PUNCT
ejpam-2553	66	54	a+	a+	X
ejpam-2553	66	55	b)c	b)c	X
ejpam-2553	66	56	⊆	⊆	NUM
ejpam-2553	66	57	ac	ac	PROPN
ejpam-2553	66	58	+	+	CCONJ
ejpam-2553	66	59	bc	bc	PROPN
ejpam-2553	66	60	,	,	PUNCT
ejpam-2553	66	61	a(b+	a(b+	NOUN
ejpam-2553	66	62	c	c	X
ejpam-2553	66	63	)	)	PUNCT
ejpam-2553	66	64	⊆	⊆	NUM
ejpam-2553	66	65	ab+	ab+	NOUN
ejpam-2553	66	66	ac	ac	PROPN
ejpam-2553	66	67	;	;	PUNCT
ejpam-2553	66	68	(	(	PUNCT
ejpam-2553	66	69	3	3	X
ejpam-2553	66	70	)	)	PUNCT
ejpam-2553	66	71	∀a	∀a	NOUN
ejpam-2553	66	72	,	,	PUNCT
ejpam-2553	66	73	b	b	X
ejpam-2553	66	74	∈	∈	PROPN
ejpam-2553	66	75	r	r	NOUN
ejpam-2553	66	76	:	:	PUNCT
ejpam-2553	66	77	(	(	PUNCT
ejpam-2553	66	78	−a)b	−a)b	ADJ
ejpam-2553	66	79	=	=	PRON
ejpam-2553	66	80	a(−b	a(−b	X
ejpam-2553	66	81	)	)	PUNCT
ejpam-2553	66	82	=	=	SYM
ejpam-2553	67	1	−(ab	−(ab	PROPN
ejpam-2553	67	2	)	)	PUNCT
ejpam-2553	67	3	.	.	PUNCT
ejpam-2553	68	1	if	if	SCONJ
ejpam-2553	68	2	in	in	ADP
ejpam-2553	68	3	(	(	PUNCT
ejpam-2553	68	4	2	2	X
ejpam-2553	68	5	)	)	PUNCT
ejpam-2553	68	6	we	we	PRON
ejpam-2553	68	7	have	have	VERB
ejpam-2553	68	8	equalities	equality	NOUN
ejpam-2553	68	9	instead	instead	ADV
ejpam-2553	68	10	of	of	ADP
ejpam-2553	68	11	inclusions	inclusion	NOUN
ejpam-2553	68	12	,	,	PUNCT
ejpam-2553	68	13	then	then	ADV
ejpam-2553	68	14	we	we	PRON
ejpam-2553	68	15	say	say	VERB
ejpam-2553	68	16	that	that	SCONJ
ejpam-2553	68	17	the	the	DET
ejpam-2553	68	18	multiplicative	multiplicative	ADJ
ejpam-2553	68	19	hyperring	hyperring	NOUN
ejpam-2553	68	20	is	be	AUX
ejpam-2553	68	21	strongly	strongly	ADV
ejpam-2553	68	22	distributive	distributive	ADJ
ejpam-2553	68	23	.	.	PUNCT
ejpam-2553	69	1	definition	definition	NOUN
ejpam-2553	69	2	1	1	NUM
ejpam-2553	69	3	.	.	PUNCT
ejpam-2553	70	1	let	let	VERB
ejpam-2553	70	2	r	r	PRON
ejpam-2553	70	3	be	be	AUX
ejpam-2553	70	4	a	a	DET
ejpam-2553	70	5	multiplicative	multiplicative	ADJ
ejpam-2553	70	6	hyperring	hyperring	NOUN
ejpam-2553	70	7	.	.	PUNCT
ejpam-2553	71	1	then	then	ADV
ejpam-2553	71	2	(	(	PUNCT
ejpam-2553	71	3	i	i	NOUN
ejpam-2553	71	4	)	)	PUNCT
ejpam-2553	71	5	an	an	DET
ejpam-2553	71	6	element	element	NOUN
ejpam-2553	71	7	e	e	X
ejpam-2553	71	8	∈	∈	NOUN
ejpam-2553	71	9	r	r	NOUN
ejpam-2553	71	10	is	be	AUX
ejpam-2553	71	11	said	say	VERB
ejpam-2553	71	12	to	to	PART
ejpam-2553	71	13	be	be	AUX
ejpam-2553	71	14	a	a	DET
ejpam-2553	71	15	left(resp	left(resp	PROPN
ejpam-2553	71	16	.	.	PUNCT
ejpam-2553	72	1	right)identity	right)identity	NOUN
ejpam-2553	72	2	if	if	SCONJ
ejpam-2553	72	3	a	a	DET
ejpam-2553	72	4	∈	∈	PROPN
ejpam-2553	72	5	e.a	e.a	PROPN
ejpam-2553	72	6	(	(	PUNCT
ejpam-2553	72	7	resp	resp	PROPN
ejpam-2553	72	8	.	.	PUNCT
ejpam-2553	73	1	a	a	DET
ejpam-2553	73	2	∈	∈	PROPN
ejpam-2553	73	3	a.e	a.e	PROPN
ejpam-2553	73	4	)	)	PUNCT
ejpam-2553	73	5	for	for	ADP
ejpam-2553	73	6	a	a	DET
ejpam-2553	73	7	∈	∈	PROPN
ejpam-2553	73	8	r.	r.	NOUN
ejpam-2553	73	9	an	an	DET
ejpam-2553	73	10	element	element	NOUN
ejpam-2553	73	11	e	e	NOUN
ejpam-2553	73	12	is	be	AUX
ejpam-2553	73	13	called	call	VERB
ejpam-2553	73	14	an	an	DET
ejpam-2553	73	15	identity	identity	NOUN
ejpam-2553	73	16	element	element	NOUN
ejpam-2553	73	17	if	if	SCONJ
ejpam-2553	73	18	it	it	PRON
ejpam-2553	73	19	is	be	AUX
ejpam-2553	73	20	both	both	PRON
ejpam-2553	73	21	left	left	ADJ
ejpam-2553	73	22	and	and	CCONJ
ejpam-2553	73	23	right	right	ADJ
ejpam-2553	73	24	identity	identity	NOUN
ejpam-2553	73	25	element	element	NOUN
ejpam-2553	73	26	.	.	PUNCT
ejpam-2553	74	1	(	(	PUNCT
ejpam-2553	74	2	ii	ii	NOUN
ejpam-2553	74	3	)	)	PUNCT
ejpam-2553	74	4	an	an	DET
ejpam-2553	74	5	element	element	NOUN
ejpam-2553	74	6	e	e	X
ejpam-2553	74	7	∈	∈	NOUN
ejpam-2553	74	8	r	r	NOUN
ejpam-2553	74	9	is	be	AUX
ejpam-2553	74	10	said	say	VERB
ejpam-2553	74	11	to	to	PART
ejpam-2553	74	12	be	be	AUX
ejpam-2553	74	13	a	a	DET
ejpam-2553	74	14	left(resp	left(resp	PROPN
ejpam-2553	74	15	.	.	PUNCT
ejpam-2553	75	1	right	right	ADJ
ejpam-2553	75	2	)	)	PUNCT
ejpam-2553	75	3	scalar	scalar	ADJ
ejpam-2553	75	4	identity	identity	NOUN
ejpam-2553	75	5	if	if	SCONJ
ejpam-2553	75	6	a	a	DET
ejpam-2553	75	7	=	=	NOUN
ejpam-2553	75	8	e.a(resp	e.a(resp	NOUN
ejpam-2553	75	9	.	.	PUNCT
ejpam-2553	75	10	,	,	PUNCT
ejpam-2553	75	11	a	a	DET
ejpam-2553	75	12	=	=	X
ejpam-2553	75	13	a.e	a.e	PROPN
ejpam-2553	75	14	)	)	PUNCT
ejpam-2553	75	15	for	for	ADP
ejpam-2553	75	16	a	a	DET
ejpam-2553	75	17	∈	∈	PROPN
ejpam-2553	75	18	r.	r.	NOUN
ejpam-2553	75	19	an	an	DET
ejpam-2553	75	20	element	element	NOUN
ejpam-2553	75	21	e	e	NOUN
ejpam-2553	75	22	is	be	AUX
ejpam-2553	75	23	called	call	VERB
ejpam-2553	75	24	an	an	DET
ejpam-2553	75	25	scalar	scalar	ADJ
ejpam-2553	75	26	identity	identity	NOUN
ejpam-2553	75	27	element	element	NOUN
ejpam-2553	75	28	if	if	SCONJ
ejpam-2553	75	29	it	it	PRON
ejpam-2553	75	30	is	be	AUX
ejpam-2553	75	31	both	both	PRON
ejpam-2553	75	32	left	leave	VERB
ejpam-2553	75	33	and	and	CCONJ
ejpam-2553	75	34	right	right	ADJ
ejpam-2553	75	35	scalar	scalar	ADJ
ejpam-2553	75	36	identity	identity	NOUN
ejpam-2553	75	37	element	element	NOUN
ejpam-2553	75	38	.	.	PUNCT
ejpam-2553	76	1	(	(	PUNCT
ejpam-2553	76	2	iii	iii	X
ejpam-2553	76	3	)	)	PUNCT
ejpam-2553	76	4	an	an	DET
ejpam-2553	76	5	element	element	NOUN
ejpam-2553	76	6	a	a	PRON
ejpam-2553	76	7	is	be	AUX
ejpam-2553	76	8	called	call	VERB
ejpam-2553	76	9	a	a	DET
ejpam-2553	76	10	left	left	ADJ
ejpam-2553	76	11	(	(	PUNCT
ejpam-2553	76	12	right	right	ADJ
ejpam-2553	76	13	)	)	PUNCT
ejpam-2553	76	14	invertible	invertible	ADJ
ejpam-2553	76	15	(	(	PUNCT
ejpam-2553	76	16	with	with	ADP
ejpam-2553	76	17	respect	respect	NOUN
ejpam-2553	76	18	to	to	ADP
ejpam-2553	76	19	e	e	NOUN
ejpam-2553	76	20	)	)	PUNCT
ejpam-2553	76	21	,	,	PUNCT
ejpam-2553	76	22	if	if	SCONJ
ejpam-2553	76	23	there	there	PRON
ejpam-2553	76	24	exists	exist	VERB
ejpam-2553	76	25	x	x	X
ejpam-2553	76	26	∈	∈	PROPN
ejpam-2553	76	27	r	r	NOUN
ejpam-2553	76	28	,	,	PUNCT
ejpam-2553	76	29	such	such	ADJ
ejpam-2553	76	30	that	that	SCONJ
ejpam-2553	76	31	e	e	PROPN
ejpam-2553	76	32	∈	∈	PROPN
ejpam-2553	76	33	xa(e	xa(e	X
ejpam-2553	76	34	∈	∈	PROPN
ejpam-2553	76	35	ax	ax	NOUN
ejpam-2553	76	36	)	)	PUNCT
ejpam-2553	76	37	and	and	CCONJ
ejpam-2553	76	38	a	a	PRON
ejpam-2553	76	39	is	be	AUX
ejpam-2553	76	40	called	call	VERB
ejpam-2553	76	41	invertible	invertible	ADJ
ejpam-2553	76	42	if	if	SCONJ
ejpam-2553	76	43	it	it	PRON
ejpam-2553	76	44	is	be	AUX
ejpam-2553	76	45	both	both	CCONJ
ejpam-2553	76	46	a	a	DET
ejpam-2553	76	47	left	left	NOUN
ejpam-2553	76	48	and	and	CCONJ
ejpam-2553	76	49	a	a	DET
ejpam-2553	76	50	right	right	NOUN
ejpam-2553	76	51	invertible	invertible	ADJ
ejpam-2553	76	52	.	.	PUNCT
ejpam-2553	77	1	a	a	DET
ejpam-2553	77	2	multiplicative	multiplicative	ADJ
ejpam-2553	77	3	hyperring	hyperring	NOUN
ejpam-2553	77	4	r	r	NOUN
ejpam-2553	77	5	is	be	AUX
ejpam-2553	77	6	called	call	VERB
ejpam-2553	77	7	a	a	DET
ejpam-2553	77	8	left	left	ADJ
ejpam-2553	77	9	(	(	PUNCT
ejpam-2553	77	10	right	right	ADJ
ejpam-2553	77	11	)	)	PUNCT
ejpam-2553	77	12	invertible	invertible	ADJ
ejpam-2553	77	13	if	if	SCONJ
ejpam-2553	77	14	every	every	DET
ejpam-2553	77	15	element	element	NOUN
ejpam-2553	77	16	of	of	ADP
ejpam-2553	77	17	r	r	NOUN
ejpam-2553	77	18	has	have	VERB
ejpam-2553	77	19	a	a	DET
ejpam-2553	77	20	left	left	ADJ
ejpam-2553	77	21	(	(	PUNCT
ejpam-2553	77	22	right	right	ADJ
ejpam-2553	77	23	)	)	PUNCT
ejpam-2553	77	24	invertible	invertible	ADJ
ejpam-2553	77	25	and	and	CCONJ
ejpam-2553	77	26	r	r	NOUN
ejpam-2553	77	27	is	be	AUX
ejpam-2553	77	28	called	call	VERB
ejpam-2553	77	29	invertible	invertible	ADJ
ejpam-2553	77	30	if	if	SCONJ
ejpam-2553	77	31	it	it	PRON
ejpam-2553	77	32	is	be	AUX
ejpam-2553	77	33	both	both	CCONJ
ejpam-2553	77	34	a	a	DET
ejpam-2553	77	35	left	left	NOUN
ejpam-2553	77	36	and	and	CCONJ
ejpam-2553	77	37	a	a	DET
ejpam-2553	77	38	right	right	ADJ
ejpam-2553	77	39	invertible	invertible	ADJ
ejpam-2553	77	40	.	.	PUNCT
ejpam-2553	78	1	denote	denote	VERB
ejpam-2553	78	2	the	the	DET
ejpam-2553	78	3	set	set	NOUN
ejpam-2553	78	4	of	of	ADP
ejpam-2553	78	5	all	all	DET
ejpam-2553	78	6	invertible	invertible	ADJ
ejpam-2553	78	7	elements	element	NOUN
ejpam-2553	78	8	in	in	ADP
ejpam-2553	78	9	r	r	NOUN
ejpam-2553	78	10	by	by	ADP
ejpam-2553	78	11	u(r	u(r	NOUN
ejpam-2553	78	12	)	)	PUNCT
ejpam-2553	79	1	(	(	PUNCT
ejpam-2553	79	2	with	with	ADP
ejpam-2553	79	3	respect	respect	NOUN
ejpam-2553	79	4	to	to	ADP
ejpam-2553	79	5	the	the	DET
ejpam-2553	79	6	identity	identity	NOUN
ejpam-2553	79	7	e	e	NOUN
ejpam-2553	79	8	by	by	ADP
ejpam-2553	79	9	ue(r	ue(r	NOUN
ejpam-2553	79	10	)	)	PUNCT
ejpam-2553	79	11	)	)	PUNCT
ejpam-2553	79	12	.	.	PUNCT
ejpam-2553	80	1	definition	definition	NOUN
ejpam-2553	80	2	2	2	NUM
ejpam-2553	80	3	.	.	PUNCT
ejpam-2553	81	1	let	let	VERB
ejpam-2553	81	2	r	r	PRON
ejpam-2553	81	3	be	be	AUX
ejpam-2553	81	4	a	a	DET
ejpam-2553	81	5	multiplicative	multiplicative	ADJ
ejpam-2553	81	6	hyperring	hyperring	NOUN
ejpam-2553	81	7	.	.	PUNCT
ejpam-2553	82	1	we	we	PRON
ejpam-2553	82	2	called	call	VERB
ejpam-2553	82	3	a	a	DET
ejpam-2553	82	4	∈	∈	PROPN
ejpam-2553	82	5	r	r	NOUN
ejpam-2553	82	6	is	be	AUX
ejpam-2553	82	7	regular	regular	ADJ
ejpam-2553	82	8	if	if	SCONJ
ejpam-2553	82	9	there	there	PRON
ejpam-2553	82	10	exists	exist	VERB
ejpam-2553	82	11	x	x	X
ejpam-2553	82	12	∈	∈	NOUN
ejpam-2553	82	13	r	r	NOUN
ejpam-2553	82	14	such	such	ADJ
ejpam-2553	82	15	that	that	SCONJ
ejpam-2553	82	16	a	a	DET
ejpam-2553	82	17	∈	∈	PROPN
ejpam-2553	82	18	axa	axa	NOUN
ejpam-2553	82	19	.	.	PUNCT
ejpam-2553	83	1	so	so	ADV
ejpam-2553	83	2	,	,	PUNCT
ejpam-2553	83	3	we	we	PRON
ejpam-2553	83	4	can	can	AUX
ejpam-2553	83	5	define	define	VERB
ejpam-2553	83	6	that	that	SCONJ
ejpam-2553	83	7	r	r	NOUN
ejpam-2553	83	8	is	be	AUX
ejpam-2553	83	9	regular	regular	ADJ
ejpam-2553	83	10	multiplicative	multiplicative	ADJ
ejpam-2553	83	11	hyperring	hyperring	NOUN
ejpam-2553	83	12	,	,	PUNCT
ejpam-2553	83	13	if	if	SCONJ
ejpam-2553	83	14	all	all	PRON
ejpam-2553	83	15	of	of	ADP
ejpam-2553	83	16	elements	element	NOUN
ejpam-2553	83	17	in	in	ADP
ejpam-2553	83	18	r	r	NOUN
ejpam-2553	83	19	are	be	AUX
ejpam-2553	83	20	regular	regular	ADJ
ejpam-2553	83	21	elements	element	NOUN
ejpam-2553	83	22	.	.	PUNCT
ejpam-2553	84	1	the	the	DET
ejpam-2553	84	2	set	set	NOUN
ejpam-2553	84	3	of	of	ADP
ejpam-2553	84	4	all	all	DET
ejpam-2553	84	5	regular	regular	ADJ
ejpam-2553	84	6	elements	element	NOUN
ejpam-2553	84	7	in	in	ADP
ejpam-2553	84	8	r	r	NOUN
ejpam-2553	84	9	is	be	AUX
ejpam-2553	84	10	denoted	denote	VERB
ejpam-2553	84	11	by	by	ADP
ejpam-2553	84	12	v	v	NOUN
ejpam-2553	84	13	(	(	PUNCT
ejpam-2553	84	14	r	r	NOUN
ejpam-2553	84	15	)	)	PUNCT
ejpam-2553	84	16	.	.	PUNCT
ejpam-2553	85	1	r.	r.	PROPN
ejpam-2553	85	2	ameri	ameri	PROPN
ejpam-2553	85	3	,	,	PUNCT
ejpam-2553	85	4	a.	a.	PROPN
ejpam-2553	85	5	kordi	kordi	PROPN
ejpam-2553	85	6	/	/	PUNCT
ejpam-2553	85	7	eur	eur	PROPN
ejpam-2553	85	8	.	.	PUNCT
ejpam-2553	86	1	j.	j.	PROPN
ejpam-2553	86	2	pure	pure	PROPN
ejpam-2553	86	3	appl	appl	PROPN
ejpam-2553	86	4	.	.	PROPN
ejpam-2553	86	5	math	math	PROPN
ejpam-2553	86	6	,	,	PUNCT
ejpam-2553	86	7	9	9	NUM
ejpam-2553	86	8	(	(	PUNCT
ejpam-2553	86	9	2016	2016	NUM
ejpam-2553	86	10	)	)	PUNCT
ejpam-2553	86	11	,	,	PUNCT
ejpam-2553	86	12	402	402	NUM
ejpam-2553	86	13	-	-	SYM
ejpam-2553	86	14	418	418	NUM
ejpam-2553	86	15	405	405	NUM
ejpam-2553	86	16	example	example	NOUN
ejpam-2553	86	17	1	1	NUM
ejpam-2553	86	18	.	.	PUNCT
ejpam-2553	87	1	let	let	VERB
ejpam-2553	87	2	(	(	PUNCT
ejpam-2553	87	3	r,+	r,+	NUM
ejpam-2553	87	4	,	,	PUNCT
ejpam-2553	87	5	.	.	PUNCT
ejpam-2553	87	6	)	)	PUNCT
ejpam-2553	88	1	be	be	AUX
ejpam-2553	88	2	the	the	DET
ejpam-2553	88	3	regular	regular	ADJ
ejpam-2553	88	4	commutative	commutative	ADJ
ejpam-2553	88	5	ring	ring	NOUN
ejpam-2553	88	6	with	with	ADP
ejpam-2553	88	7	an	an	DET
ejpam-2553	88	8	unitary	unitary	ADJ
ejpam-2553	88	9	element	element	NOUN
ejpam-2553	88	10	.	.	PUNCT
ejpam-2553	89	1	for	for	ADP
ejpam-2553	89	2	every	every	DET
ejpam-2553	89	3	subset	subset	NOUN
ejpam-2553	89	4	a∈	a∈	PROPN
ejpam-2553	89	5	p∗(r	p∗(r	PROPN
ejpam-2553	89	6	)	)	PUNCT
ejpam-2553	89	7	=	=	SYM
ejpam-2553	89	8	p(r)−	p(r)−	PROPN
ejpam-2553	89	9	{	{	PUNCT
ejpam-2553	89	10	;	;	PUNCT
ejpam-2553	89	11	}	}	PUNCT
ejpam-2553	89	12	,	,	PUNCT
ejpam-2553	89	13	|a|	|a|	NOUN
ejpam-2553	89	14	≥	≥	NOUN
ejpam-2553	89	15	2	2	NUM
ejpam-2553	89	16	,	,	PUNCT
ejpam-2553	89	17	and	and	CCONJ
ejpam-2553	89	18	1	1	NUM
ejpam-2553	89	19	∈	∈	NOUN
ejpam-2553	89	20	a	a	PRON
ejpam-2553	89	21	,	,	PUNCT
ejpam-2553	89	22	define	define	VERB
ejpam-2553	89	23	a	a	DET
ejpam-2553	89	24	multiplicative	multiplicative	ADJ
ejpam-2553	89	25	hyperring	hyperring	NOUN
ejpam-2553	89	26	(	(	PUNCT
ejpam-2553	89	27	ra,+	ra,+	ADJ
ejpam-2553	89	28	,	,	PUNCT
ejpam-2553	89	29	◦	◦	NOUN
ejpam-2553	89	30	)	)	PUNCT
ejpam-2553	89	31	,	,	PUNCT
ejpam-2553	89	32	where	where	SCONJ
ejpam-2553	89	33	ra	ra	PROPN
ejpam-2553	89	34	=	=	SYM
ejpam-2553	89	35	r	r	NOUN
ejpam-2553	89	36	and	and	CCONJ
ejpam-2553	89	37	for	for	ADP
ejpam-2553	89	38	all	all	DET
ejpam-2553	89	39	x	x	SYM
ejpam-2553	89	40	,	,	PUNCT
ejpam-2553	89	41	y	y	PROPN
ejpam-2553	89	42	∈	∈	PROPN
ejpam-2553	89	43	ra	ra	PROPN
ejpam-2553	89	44	,	,	PUNCT
ejpam-2553	89	45	x	x	VERB
ejpam-2553	89	46	◦	◦	NOUN
ejpam-2553	89	47	y	y	NOUN
ejpam-2553	90	1	=	=	SYM
ejpam-2553	90	2	{	{	PUNCT
ejpam-2553	90	3	xa	xa	PROPN
ejpam-2553	90	4	y|a	y|a	PROPN
ejpam-2553	90	5	∈	∈	PROPN
ejpam-2553	90	6	a	a	PRON
ejpam-2553	90	7	}	}	PUNCT
ejpam-2553	90	8	.	.	PUNCT
ejpam-2553	91	1	then	then	ADV
ejpam-2553	91	2	(	(	PUNCT
ejpam-2553	91	3	ra,+	ra,+	ADJ
ejpam-2553	91	4	,	,	PUNCT
ejpam-2553	91	5	◦	◦	NOUN
ejpam-2553	91	6	)	)	PUNCT
ejpam-2553	91	7	is	be	AUX
ejpam-2553	91	8	a	a	DET
ejpam-2553	91	9	regular	regular	ADJ
ejpam-2553	91	10	multiplicative	multiplicative	ADJ
ejpam-2553	91	11	hyperring	hyperring	NOUN
ejpam-2553	91	12	.	.	PUNCT
ejpam-2553	92	1	since	since	SCONJ
ejpam-2553	92	2	,	,	PUNCT
ejpam-2553	92	3	for	for	ADP
ejpam-2553	92	4	all	all	DET
ejpam-2553	92	5	a	a	DET
ejpam-2553	92	6	∈	∈	NOUN
ejpam-2553	92	7	r	r	NOUN
ejpam-2553	92	8	,	,	PUNCT
ejpam-2553	92	9	there	there	PRON
ejpam-2553	92	10	exists	exist	VERB
ejpam-2553	92	11	r	r	NOUN
ejpam-2553	92	12	∈	∈	PROPN
ejpam-2553	92	13	r	r	NOUN
ejpam-2553	92	14	such	such	DET
ejpam-2553	92	15	that	that	SCONJ
ejpam-2553	92	16	a	a	DET
ejpam-2553	92	17	=	=	X
ejpam-2553	92	18	ara	ara	NOUN
ejpam-2553	92	19	.	.	PUNCT
ejpam-2553	93	1	now	now	ADV
ejpam-2553	93	2	,	,	PUNCT
ejpam-2553	93	3	by	by	ADP
ejpam-2553	93	4	setting	set	VERB
ejpam-2553	93	5	x	x	PUNCT
ejpam-2553	93	6	=	=	PUNCT
ejpam-2553	93	7	r	r	NOUN
ejpam-2553	93	8	we	we	PRON
ejpam-2553	93	9	have	have	VERB
ejpam-2553	93	10	,	,	PUNCT
ejpam-2553	93	11	a	a	DET
ejpam-2553	93	12	◦	◦	NOUN
ejpam-2553	93	13	x	x	PUNCT
ejpam-2553	93	14	◦	◦	VERB
ejpam-2553	93	15	a	a	DET
ejpam-2553	93	16	=	=	X
ejpam-2553	93	17	{	{	PUNCT
ejpam-2553	93	18	asx	asx	NOUN
ejpam-2553	93	19	|s	|s	PROPN
ejpam-2553	93	20	∈	∈	PROPN
ejpam-2553	93	21	a	a	DET
ejpam-2553	93	22	}	}	PUNCT
ejpam-2553	93	23	◦	◦	NOUN
ejpam-2553	93	24	a	a	DET
ejpam-2553	93	25	=	=	PUNCT
ejpam-2553	93	26	{	{	PUNCT
ejpam-2553	93	27	asx	asx	PROPN
ejpam-2553	93	28	ta|s	ta|s	PROPN
ejpam-2553	93	29	,	,	PUNCT
ejpam-2553	93	30	t	t	PROPN
ejpam-2553	93	31	∈	∈	PROPN
ejpam-2553	93	32	a	a	DET
ejpam-2553	93	33	}	}	PUNCT
ejpam-2553	93	34	=	=	SYM
ejpam-2553	93	35	{	{	PUNCT
ejpam-2553	93	36	axast|s	axast|s	NOUN
ejpam-2553	93	37	,	,	PUNCT
ejpam-2553	93	38	t	t	PROPN
ejpam-2553	93	39	∈	∈	PROPN
ejpam-2553	93	40	a	a	DET
ejpam-2553	93	41	}	}	PUNCT
ejpam-2553	93	42	=	=	SYM
ejpam-2553	93	43	{	{	PUNCT
ejpam-2553	93	44	ast|s	ast|s	ADJ
ejpam-2553	93	45	,	,	PUNCT
ejpam-2553	93	46	t	t	PROPN
ejpam-2553	93	47	∈	∈	PROPN
ejpam-2553	93	48	a	a	PRON
ejpam-2553	93	49	}	}	PUNCT
ejpam-2553	93	50	,	,	PUNCT
ejpam-2553	93	51	since	since	SCONJ
ejpam-2553	93	52	1	1	NUM
ejpam-2553	93	53	∈	∈	PROPN
ejpam-2553	93	54	a	a	NOUN
ejpam-2553	93	55	,	,	PUNCT
ejpam-2553	93	56	we	we	PRON
ejpam-2553	93	57	have	have	VERB
ejpam-2553	93	58	a	a	DET
ejpam-2553	93	59	∈	∈	PROPN
ejpam-2553	93	60	a	a	DET
ejpam-2553	93	61	◦	◦	NOUN
ejpam-2553	93	62	x	x	PUNCT
ejpam-2553	93	63	◦	◦	NOUN
ejpam-2553	93	64	a.	a.	NOUN
ejpam-2553	93	65	hence	hence	ADV
ejpam-2553	93	66	(	(	PUNCT
ejpam-2553	93	67	ra,+	ra,+	ADJ
ejpam-2553	93	68	,	,	PUNCT
ejpam-2553	93	69	◦	◦	NOUN
ejpam-2553	93	70	)	)	PUNCT
ejpam-2553	93	71	is	be	AUX
ejpam-2553	93	72	regular	regular	ADJ
ejpam-2553	93	73	.	.	PUNCT
ejpam-2553	93	74	example	example	NOUN
ejpam-2553	94	1	2	2	NUM
ejpam-2553	94	2	.	.	X
ejpam-2553	95	1	let	let	VERB
ejpam-2553	95	2	(	(	PUNCT
ejpam-2553	95	3	r,+	r,+	NUM
ejpam-2553	95	4	,	,	PUNCT
ejpam-2553	95	5	.	.	PUNCT
ejpam-2553	95	6	)	)	PUNCT
ejpam-2553	96	1	be	be	AUX
ejpam-2553	96	2	a	a	DET
ejpam-2553	96	3	non	non	ADJ
ejpam-2553	96	4	-	-	ADJ
ejpam-2553	96	5	zero	zero	ADJ
ejpam-2553	96	6	regular	regular	ADJ
ejpam-2553	96	7	ring	ring	NOUN
ejpam-2553	96	8	and	and	CCONJ
ejpam-2553	96	9	for	for	ADP
ejpam-2553	96	10	all	all	DET
ejpam-2553	96	11	a	a	DET
ejpam-2553	96	12	,	,	PUNCT
ejpam-2553	96	13	b	b	X
ejpam-2553	96	14	∈	∈	PROPN
ejpam-2553	96	15	r	r	NOUN
ejpam-2553	96	16	,	,	PUNCT
ejpam-2553	96	17	define	define	VERB
ejpam-2553	96	18	a	a	DET
ejpam-2553	96	19	hyperoperation	hyperoperation	NOUN
ejpam-2553	96	20	a	a	DET
ejpam-2553	96	21	◦	◦	NOUN
ejpam-2553	96	22	b	b	NOUN
ejpam-2553	96	23	=	=	SYM
ejpam-2553	96	24	{	{	PUNCT
ejpam-2553	96	25	a.b	a.b	PROPN
ejpam-2553	96	26	,	,	PUNCT
ejpam-2553	96	27	2a.b	2a.b	NUM
ejpam-2553	96	28	,	,	PUNCT
ejpam-2553	96	29	3a.b	3a.b	NUM
ejpam-2553	96	30	,	,	PUNCT
ejpam-2553	96	31	.	.	PUNCT
ejpam-2553	96	32	.	.	PUNCT
ejpam-2553	97	1	.	.	PUNCT
ejpam-2553	97	2	}	}	PUNCT
ejpam-2553	97	3	.	.	PUNCT
ejpam-2553	98	1	then	then	ADV
ejpam-2553	98	2	(	(	PUNCT
ejpam-2553	98	3	r,+	r,+	NUM
ejpam-2553	98	4	,	,	PUNCT
ejpam-2553	98	5	◦	◦	NOUN
ejpam-2553	98	6	)	)	PUNCT
ejpam-2553	98	7	is	be	AUX
ejpam-2553	98	8	a	a	DET
ejpam-2553	98	9	regular	regular	ADJ
ejpam-2553	98	10	multiplicative	multiplicative	ADJ
ejpam-2553	98	11	hyperring	hyperring	NOUN
ejpam-2553	98	12	,	,	PUNCT
ejpam-2553	98	13	which	which	PRON
ejpam-2553	98	14	is	be	AUX
ejpam-2553	98	15	not	not	PART
ejpam-2553	98	16	strongly	strongly	ADV
ejpam-2553	98	17	distributive	distributive	ADJ
ejpam-2553	98	18	.	.	PUNCT
ejpam-2553	99	1	also	also	ADV
ejpam-2553	99	2	,	,	PUNCT
ejpam-2553	99	3	for	for	ADP
ejpam-2553	99	4	all	all	DET
ejpam-2553	99	5	a	a	DET
ejpam-2553	99	6	∈	∈	NOUN
ejpam-2553	99	7	r	r	NOUN
ejpam-2553	99	8	,	,	PUNCT
ejpam-2553	99	9	there	there	PRON
ejpam-2553	99	10	exists	exist	VERB
ejpam-2553	99	11	r	r	NOUN
ejpam-2553	99	12	∈	∈	PROPN
ejpam-2553	99	13	r	r	NOUN
ejpam-2553	99	14	such	such	DET
ejpam-2553	99	15	that	that	SCONJ
ejpam-2553	99	16	a	a	DET
ejpam-2553	99	17	=	=	X
ejpam-2553	99	18	ara	ara	NOUN
ejpam-2553	99	19	.	.	PUNCT
ejpam-2553	100	1	now	now	ADV
ejpam-2553	100	2	by	by	ADP
ejpam-2553	100	3	setting	set	VERB
ejpam-2553	100	4	x	x	PUNCT
ejpam-2553	100	5	=	=	PUNCT
ejpam-2553	100	6	r	r	NOUN
ejpam-2553	100	7	we	we	PRON
ejpam-2553	100	8	have	have	VERB
ejpam-2553	100	9	a	a	DET
ejpam-2553	100	10	◦	◦	NOUN
ejpam-2553	100	11	x	x	PUNCT
ejpam-2553	100	12	◦	◦	VERB
ejpam-2553	100	13	a	a	DET
ejpam-2553	100	14	=	=	X
ejpam-2553	100	15	{	{	PUNCT
ejpam-2553	100	16	ar	ar	PROPN
ejpam-2553	100	17	,	,	PUNCT
ejpam-2553	100	18	2ar	2ar	NOUN
ejpam-2553	100	19	,	,	PUNCT
ejpam-2553	100	20	.	.	PUNCT
ejpam-2553	100	21	.	.	PUNCT
ejpam-2553	101	1	.	.	PUNCT
ejpam-2553	102	1	,	,	PUNCT
ejpam-2553	102	2	nar	nar	NOUN
ejpam-2553	102	3	,	,	PUNCT
ejpam-2553	102	4	.	.	PUNCT
ejpam-2553	102	5	.	.	PUNCT
ejpam-2553	103	1	.	.	PUNCT
ejpam-2553	103	2	}	}	PUNCT
ejpam-2553	104	1	◦	◦	VERB
ejpam-2553	104	2	a	a	DET
ejpam-2553	104	3	=	=	X
ejpam-2553	104	4	{	{	PUNCT
ejpam-2553	104	5	ara	ara	PROPN
ejpam-2553	104	6	,	,	PUNCT
ejpam-2553	104	7	2ara	2ara	PROPN
ejpam-2553	104	8	,	,	PUNCT
ejpam-2553	104	9	.	.	PUNCT
ejpam-2553	104	10	.	.	PUNCT
ejpam-2553	105	1	.	.	PUNCT
ejpam-2553	106	1	,	,	PUNCT
ejpam-2553	106	2	nara	nara	PROPN
ejpam-2553	106	3	,	,	PUNCT
ejpam-2553	106	4	.	.	PUNCT
ejpam-2553	106	5	.	.	PUNCT
ejpam-2553	106	6	.	.	PUNCT
ejpam-2553	106	7	}	}	PUNCT
ejpam-2553	106	8	.	.	PUNCT
ejpam-2553	107	1	then	then	ADV
ejpam-2553	107	2	a	a	DET
ejpam-2553	107	3	∈	∈	PROPN
ejpam-2553	107	4	a	a	DET
ejpam-2553	107	5	◦	◦	NOUN
ejpam-2553	107	6	x	x	PUNCT
ejpam-2553	107	7	◦	◦	NOUN
ejpam-2553	107	8	a.	a.	NOUN
ejpam-2553	107	9	2.1	2.1	NUM
ejpam-2553	107	10	.	.	PUNCT
ejpam-2553	108	1	applications	application	NOUN
ejpam-2553	108	2	of	of	ADP
ejpam-2553	108	3	the	the	DET
ejpam-2553	108	4	γ∗-relation	γ∗-relation	NOUN
ejpam-2553	108	5	in	in	ADP
ejpam-2553	108	6	regular	regular	ADJ
ejpam-2553	108	7	multiplicative	multiplicative	ADJ
ejpam-2553	108	8	hyperrings	hyperring	NOUN
ejpam-2553	108	9	let	let	VERB
ejpam-2553	108	10	(	(	PUNCT
ejpam-2553	108	11	r,+	r,+	NUM
ejpam-2553	108	12	,	,	PUNCT
ejpam-2553	108	13	.	.	PUNCT
ejpam-2553	108	14	)	)	PUNCT
ejpam-2553	109	1	be	be	AUX
ejpam-2553	109	2	a	a	DET
ejpam-2553	109	3	hyperring	hyperring	NOUN
ejpam-2553	109	4	.	.	PUNCT
ejpam-2553	110	1	we	we	PRON
ejpam-2553	110	2	define	define	VERB
ejpam-2553	110	3	the	the	DET
ejpam-2553	110	4	relation	relation	NOUN
ejpam-2553	110	5	γ	γ	NOUN
ejpam-2553	110	6	as	as	SCONJ
ejpam-2553	110	7	follows	follow	VERB
ejpam-2553	110	8	:	:	PUNCT
ejpam-2553	110	9	aγb	aγb	X
ejpam-2553	110	10	if	if	SCONJ
ejpam-2553	111	1	and	and	CCONJ
ejpam-2553	111	2	only	only	ADV
ejpam-2553	111	3	if	if	SCONJ
ejpam-2553	111	4	{	{	PUNCT
ejpam-2553	111	5	a	a	DET
ejpam-2553	111	6	,	,	PUNCT
ejpam-2553	111	7	b	b	NOUN
ejpam-2553	111	8	}	}	PUNCT
ejpam-2553	111	9	⊆	⊆	NUM
ejpam-2553	111	10	u	u	NOUN
ejpam-2553	111	11	where	where	SCONJ
ejpam-2553	111	12	u	u	NOUN
ejpam-2553	111	13	is	be	AUX
ejpam-2553	111	14	a	a	DET
ejpam-2553	111	15	finite	finite	ADJ
ejpam-2553	111	16	sum	sum	NOUN
ejpam-2553	111	17	of	of	ADP
ejpam-2553	111	18	finite	finite	ADJ
ejpam-2553	111	19	products	product	NOUN
ejpam-2553	111	20	of	of	ADP
ejpam-2553	111	21	elements	element	NOUN
ejpam-2553	111	22	of	of	ADP
ejpam-2553	111	23	r	r	NOUN
ejpam-2553	111	24	,	,	PUNCT
ejpam-2553	111	25	i.e.	i.e.	X
ejpam-2553	111	26	,	,	PUNCT
ejpam-2553	111	27	aγb⇔∃z1	aγb⇔∃z1	PROPN
ejpam-2553	111	28	,	,	PUNCT
ejpam-2553	111	29	.	.	PUNCT
ejpam-2553	111	30	.	.	PUNCT
ejpam-2553	112	1	.	.	PUNCT
ejpam-2553	113	1	,	,	PUNCT
ejpam-2553	113	2	zn	zn	PROPN
ejpam-2553	113	3	∈	∈	PROPN
ejpam-2553	113	4	r	r	NOUN
ejpam-2553	113	5	such	such	ADJ
ejpam-2553	113	6	that	that	SCONJ
ejpam-2553	113	7	{	{	PUNCT
ejpam-2553	113	8	a	a	PRON
ejpam-2553	113	9	,	,	PUNCT
ejpam-2553	113	10	b	b	NOUN
ejpam-2553	113	11	}	}	PUNCT
ejpam-2553	113	12	⊆	⊆	NUM
ejpam-2553	113	13	∑	∑	ADP
ejpam-2553	113	14	j∈j	j∈j	PROPN
ejpam-2553	113	15	∏	∏	PROPN
ejpam-2553	113	16	i∈i	i∈i	PROPN
ejpam-2553	113	17	j	j	PROPN
ejpam-2553	113	18	zi	zi	PROPN
ejpam-2553	113	19	;	;	PUNCT
ejpam-2553	113	20	i	i	PROPN
ejpam-2553	113	21	j	j	PROPN
ejpam-2553	113	22	,	,	PUNCT
ejpam-2553	113	23	j	j	PROPN
ejpam-2553	113	24	⊆	⊆	NUM
ejpam-2553	113	25	{	{	PUNCT
ejpam-2553	113	26	1	1	NUM
ejpam-2553	113	27	,	,	PUNCT
ejpam-2553	113	28	.	.	PUNCT
ejpam-2553	113	29	.	.	PUNCT
ejpam-2553	113	30	.	.	PUNCT
ejpam-2553	113	31	,	,	PUNCT
ejpam-2553	113	32	n	n	CCONJ
ejpam-2553	113	33	}	}	PUNCT
ejpam-2553	113	34	.	.	PUNCT
ejpam-2553	114	1	we	we	PRON
ejpam-2553	114	2	denote	denote	VERB
ejpam-2553	114	3	the	the	DET
ejpam-2553	114	4	transitive	transitive	ADJ
ejpam-2553	114	5	closure	closure	NOUN
ejpam-2553	114	6	of	of	ADP
ejpam-2553	114	7	γ	γ	X
ejpam-2553	114	8	by	by	ADP
ejpam-2553	114	9	γ∗.	γ∗.	ADJ
ejpam-2553	114	10	the	the	DET
ejpam-2553	114	11	relation	relation	NOUN
ejpam-2553	114	12	γ∗	γ∗	NOUN
ejpam-2553	114	13	as	as	ADP
ejpam-2553	114	14	the	the	DET
ejpam-2553	114	15	smallest	small	ADJ
ejpam-2553	114	16	equivalence	equivalence	NOUN
ejpam-2553	114	17	relation	relation	NOUN
ejpam-2553	114	18	on	on	ADP
ejpam-2553	114	19	a	a	DET
ejpam-2553	114	20	multiplicative	multiplicative	ADJ
ejpam-2553	114	21	hyperring	hyperring	NOUN
ejpam-2553	114	22	(	(	PUNCT
ejpam-2553	114	23	r,+	r,+	NUM
ejpam-2553	114	24	,	,	PUNCT
ejpam-2553	114	25	.	.	PUNCT
ejpam-2553	114	26	)	)	PUNCT
ejpam-2553	115	1	such	such	ADJ
ejpam-2553	115	2	that	that	SCONJ
ejpam-2553	115	3	the	the	DET
ejpam-2553	115	4	quotient	quotient	NOUN
ejpam-2553	115	5	r	r	NOUN
ejpam-2553	115	6	/	/	SYM
ejpam-2553	115	7	γ∗	γ∗	PROPN
ejpam-2553	115	8	,	,	PUNCT
ejpam-2553	115	9	the	the	DET
ejpam-2553	115	10	set	set	NOUN
ejpam-2553	115	11	of	of	ADP
ejpam-2553	115	12	all	all	DET
ejpam-2553	115	13	equivalence	equivalence	NOUN
ejpam-2553	115	14	classes	class	NOUN
ejpam-2553	115	15	,	,	PUNCT
ejpam-2553	115	16	is	be	AUX
ejpam-2553	115	17	a	a	DET
ejpam-2553	115	18	fundamental	fundamental	ADJ
ejpam-2553	115	19	ring	ring	NOUN
ejpam-2553	115	20	.	.	PUNCT
ejpam-2553	116	1	let	let	VERB
ejpam-2553	116	2	u	u	PRON
ejpam-2553	116	3	be	be	AUX
ejpam-2553	116	4	the	the	DET
ejpam-2553	116	5	set	set	NOUN
ejpam-2553	116	6	of	of	ADP
ejpam-2553	116	7	all	all	DET
ejpam-2553	116	8	finite	finite	ADJ
ejpam-2553	116	9	sums	sum	NOUN
ejpam-2553	116	10	of	of	ADP
ejpam-2553	116	11	products	product	NOUN
ejpam-2553	116	12	of	of	ADP
ejpam-2553	116	13	elements	element	NOUN
ejpam-2553	116	14	of	of	ADP
ejpam-2553	116	15	r	r	NOUN
ejpam-2553	116	16	we	we	PRON
ejpam-2553	116	17	can	can	AUX
ejpam-2553	116	18	rewrite	rewrite	VERB
ejpam-2553	116	19	the	the	DET
ejpam-2553	116	20	definition	definition	NOUN
ejpam-2553	116	21	of	of	ADP
ejpam-2553	116	22	γ∗	γ∗	NOUN
ejpam-2553	116	23	on	on	ADP
ejpam-2553	116	24	r	r	NOUN
ejpam-2553	116	25	as	as	SCONJ
ejpam-2553	116	26	follows	follow	VERB
ejpam-2553	116	27	:	:	PUNCT
ejpam-2553	116	28	aγ∗b⇔∃z1	aγ∗b⇔∃z1	NOUN
ejpam-2553	116	29	,	,	PUNCT
ejpam-2553	116	30	.	.	PUNCT
ejpam-2553	116	31	.	.	PUNCT
ejpam-2553	117	1	.	.	PUNCT
ejpam-2553	118	1	,	,	PUNCT
ejpam-2553	118	2	zn+1	zn+1	X
ejpam-2553	118	3	∈	∈	NOUN
ejpam-2553	118	4	r	r	NOUN
ejpam-2553	118	5	with	with	ADP
ejpam-2553	118	6	z1	z1	PROPN
ejpam-2553	118	7	=	=	SYM
ejpam-2553	118	8	a	a	PROPN
ejpam-2553	118	9	,	,	PUNCT
ejpam-2553	118	10	zn+1	zn+1	PROPN
ejpam-2553	118	11	=	=	SYM
ejpam-2553	118	12	b	b	PROPN
ejpam-2553	118	13	and	and	CCONJ
ejpam-2553	118	14	u1	u1	NOUN
ejpam-2553	118	15	,	,	PUNCT
ejpam-2553	118	16	.	.	PUNCT
ejpam-2553	118	17	.	.	PUNCT
ejpam-2553	118	18	.	.	PUNCT
ejpam-2553	119	1	,	,	PUNCT
ejpam-2553	119	2	un	un	PROPN
ejpam-2553	119	3	∈	∈	PROPN
ejpam-2553	119	4	u	u	PROPN
ejpam-2553	119	5	such	such	ADJ
ejpam-2553	119	6	that	that	SCONJ
ejpam-2553	119	7	{	{	PUNCT
ejpam-2553	119	8	zi	zi	NOUN
ejpam-2553	119	9	,	,	PUNCT
ejpam-2553	119	10	zi+1	zi+1	CCONJ
ejpam-2553	119	11	}	}	PUNCT
ejpam-2553	119	12	⊆	⊆	NUM
ejpam-2553	119	13	ui	ui	NOUN
ejpam-2553	119	14	for	for	ADP
ejpam-2553	119	15	i	i	PRON
ejpam-2553	119	16	∈	∈	PROPN
ejpam-2553	119	17	{	{	PUNCT
ejpam-2553	119	18	1	1	NUM
ejpam-2553	119	19	,	,	PUNCT
ejpam-2553	119	20	.	.	PUNCT
ejpam-2553	119	21	.	.	PUNCT
ejpam-2553	120	1	.	.	PUNCT
ejpam-2553	120	2	,	,	PUNCT
ejpam-2553	121	1	n	n	CCONJ
ejpam-2553	121	2	}	}	PUNCT
ejpam-2553	121	3	.	.	PUNCT
ejpam-2553	122	1	suppose	suppose	VERB
ejpam-2553	122	2	that	that	SCONJ
ejpam-2553	122	3	γ∗(a	γ∗(a	NOUN
ejpam-2553	122	4	)	)	PUNCT
ejpam-2553	122	5	is	be	AUX
ejpam-2553	122	6	the	the	DET
ejpam-2553	122	7	equivalence	equivalence	NOUN
ejpam-2553	122	8	class	class	NOUN
ejpam-2553	122	9	containing	contain	VERB
ejpam-2553	122	10	a	a	DET
ejpam-2553	122	11	∈	∈	PROPN
ejpam-2553	122	12	r.	r.	NOUN
ejpam-2553	122	13	then	then	ADV
ejpam-2553	122	14	,	,	PUNCT
ejpam-2553	122	15	both	both	CCONJ
ejpam-2553	122	16	the	the	DET
ejpam-2553	122	17	sum	sum	NOUN
ejpam-2553	122	18	⊕	⊕	PROPN
ejpam-2553	122	19	and	and	CCONJ
ejpam-2553	122	20	the	the	DET
ejpam-2553	122	21	product	product	NOUN
ejpam-2553	122	22	�	�	PROPN
ejpam-2553	122	23	in	in	ADP
ejpam-2553	122	24	r	r	PROPN
ejpam-2553	122	25	/	/	SYM
ejpam-2553	122	26	γ∗	γ∗	NOUN
ejpam-2553	122	27	are	be	AUX
ejpam-2553	122	28	defined	define	VERB
ejpam-2553	122	29	as	as	SCONJ
ejpam-2553	122	30	follows	follow	VERB
ejpam-2553	122	31	:	:	PUNCT
ejpam-2553	123	1	γ∗(a)⊕γ∗(b	γ∗(a)⊕γ∗(b	NUM
ejpam-2553	123	2	)	)	PUNCT
ejpam-2553	123	3	=	=	SYM
ejpam-2553	123	4	γ∗(c	γ∗(c	PROPN
ejpam-2553	123	5	)	)	PUNCT
ejpam-2553	123	6	for	for	ADP
ejpam-2553	123	7	all	all	DET
ejpam-2553	123	8	c	c	PROPN
ejpam-2553	123	9	∈	∈	PROPN
ejpam-2553	123	10	γ∗(a)+γ∗(b	γ∗(a)+γ∗(b	PROPN
ejpam-2553	123	11	)	)	PUNCT
ejpam-2553	123	12	and	and	CCONJ
ejpam-2553	123	13	γ∗(a)	γ∗(a)	NOUN
ejpam-2553	123	14	�	�	NOUN
ejpam-2553	123	15	γ∗(b	γ∗(b	NOUN
ejpam-2553	123	16	)	)	PUNCT
ejpam-2553	123	17	=	=	SYM
ejpam-2553	123	18	γ∗(d	γ∗(d	PROPN
ejpam-2553	123	19	)	)	PUNCT
ejpam-2553	123	20	for	for	ADP
ejpam-2553	123	21	all	all	DET
ejpam-2553	123	22	d	d	PROPN
ejpam-2553	123	23	∈	∈	PROPN
ejpam-2553	123	24	γ∗(a).γ∗(b	γ∗(a).γ∗(b	PROPN
ejpam-2553	123	25	)	)	PUNCT
ejpam-2553	123	26	.	.	PUNCT
ejpam-2553	124	1	then	then	ADV
ejpam-2553	124	2	r	r	X
ejpam-2553	124	3	/	/	SYM
ejpam-2553	124	4	γ∗	γ∗	NOUN
ejpam-2553	124	5	is	be	AUX
ejpam-2553	124	6	a	a	DET
ejpam-2553	124	7	ring	ring	NOUN
ejpam-2553	124	8	,	,	PUNCT
ejpam-2553	124	9	which	which	PRON
ejpam-2553	124	10	is	be	AUX
ejpam-2553	124	11	called	call	VERB
ejpam-2553	124	12	fundamental	fundamental	ADJ
ejpam-2553	124	13	ring	ring	NOUN
ejpam-2553	124	14	of	of	ADP
ejpam-2553	124	15	r	r	NOUN
ejpam-2553	124	16	(	(	PUNCT
ejpam-2553	124	17	see	see	VERB
ejpam-2553	124	18	also	also	ADV
ejpam-2553	124	19	[	[	X
ejpam-2553	124	20	31	31	NUM
ejpam-2553	124	21	]	]	NUM
ejpam-2553	124	22	)	)	PUNCT
ejpam-2553	124	23	.	.	PUNCT
ejpam-2553	125	1	theorem	theorem	NOUN
ejpam-2553	125	2	1	1	NUM
ejpam-2553	125	3	.	.	PUNCT
ejpam-2553	126	1	let	let	VERB
ejpam-2553	126	2	r	r	PRON
ejpam-2553	126	3	be	be	AUX
ejpam-2553	126	4	a	a	DET
ejpam-2553	126	5	regular	regular	ADJ
ejpam-2553	126	6	multiplicative	multiplicative	ADJ
ejpam-2553	126	7	hyperring	hyperring	NOUN
ejpam-2553	126	8	.	.	PUNCT
ejpam-2553	127	1	then	then	ADV
ejpam-2553	127	2	r	r	X
ejpam-2553	127	3	/	/	SYM
ejpam-2553	127	4	γ∗	γ∗	NOUN
ejpam-2553	127	5	is	be	AUX
ejpam-2553	127	6	regular	regular	ADJ
ejpam-2553	127	7	ring	ring	NOUN
ejpam-2553	127	8	.	.	PUNCT
ejpam-2553	128	1	proof	proof	NOUN
ejpam-2553	128	2	.	.	PUNCT
ejpam-2553	129	1	assume	assume	VERB
ejpam-2553	129	2	that	that	SCONJ
ejpam-2553	129	3	x	x	PUNCT
ejpam-2553	129	4	∈	∈	PROPN
ejpam-2553	129	5	r	r	NOUN
ejpam-2553	129	6	/	/	SYM
ejpam-2553	129	7	γ∗.	γ∗.	NOUN
ejpam-2553	129	8	thus	thus	ADV
ejpam-2553	129	9	there	there	PRON
ejpam-2553	129	10	is	be	VERB
ejpam-2553	129	11	a	a	DET
ejpam-2553	129	12	r	r	NOUN
ejpam-2553	129	13	∈	∈	NOUN
ejpam-2553	129	14	r	r	NOUN
ejpam-2553	129	15	such	such	ADJ
ejpam-2553	129	16	that	that	SCONJ
ejpam-2553	129	17	x	x	SYM
ejpam-2553	129	18	=	=	PUNCT
ejpam-2553	129	19	γ∗(r	γ∗(r	PROPN
ejpam-2553	129	20	)	)	PUNCT
ejpam-2553	129	21	.	.	PUNCT
ejpam-2553	130	1	since	since	SCONJ
ejpam-2553	130	2	r	r	NOUN
ejpam-2553	130	3	is	be	AUX
ejpam-2553	130	4	a	a	DET
ejpam-2553	130	5	regular	regular	ADJ
ejpam-2553	130	6	hyperring	hyperring	NOUN
ejpam-2553	130	7	,	,	PUNCT
ejpam-2553	130	8	then	then	ADV
ejpam-2553	130	9	there	there	PRON
ejpam-2553	130	10	exists	exist	VERB
ejpam-2553	130	11	r	r	NOUN
ejpam-2553	130	12	′	′	NOUN
ejpam-2553	130	13	∈	∈	NOUN
ejpam-2553	130	14	r	r	NOUN
ejpam-2553	131	1	such	such	ADJ
ejpam-2553	131	2	that	that	SCONJ
ejpam-2553	131	3	r	r	NOUN
ejpam-2553	131	4	∈	∈	NOUN
ejpam-2553	131	5	r	r	NOUN
ejpam-2553	131	6	r	r	NOUN
ejpam-2553	131	7	′r	′r	PROPN
ejpam-2553	131	8	.	.	PUNCT
ejpam-2553	132	1	so	so	ADV
ejpam-2553	132	2	γ∗(r	γ∗(r	X
ejpam-2553	132	3	)	)	PUNCT
ejpam-2553	132	4	=	=	SYM
ejpam-2553	133	1	γ∗(r	γ∗(r	ADJ
ejpam-2553	133	2	r	r	NOUN
ejpam-2553	133	3	′r	′r	PROPN
ejpam-2553	133	4	)	)	PUNCT
ejpam-2553	133	5	=	=	SYM
ejpam-2553	133	6	γ∗(r	γ∗(r	X
ejpam-2553	133	7	)	)	PUNCT
ejpam-2553	133	8	�	�	PROPN
ejpam-2553	133	9	γ∗(r	γ∗(r	PROPN
ejpam-2553	133	10	′	′	NOUN
ejpam-2553	133	11	)	)	PUNCT
ejpam-2553	133	12	�	�	PROPN
ejpam-2553	133	13	γ∗(r	γ∗(r	PROPN
ejpam-2553	133	14	)	)	PUNCT
ejpam-2553	133	15	.	.	PUNCT
ejpam-2553	134	1	therefore	therefore	ADV
ejpam-2553	134	2	r	r	NOUN
ejpam-2553	134	3	/	/	SYM
ejpam-2553	134	4	γ∗	γ∗	NOUN
ejpam-2553	134	5	is	be	AUX
ejpam-2553	134	6	a	a	DET
ejpam-2553	134	7	regular	regular	ADJ
ejpam-2553	134	8	ring	ring	NOUN
ejpam-2553	134	9	.	.	PUNCT
ejpam-2553	135	1	remark	remark	PROPN
ejpam-2553	135	2	1	1	NUM
ejpam-2553	135	3	.	.	PUNCT
ejpam-2553	136	1	the	the	DET
ejpam-2553	136	2	converse	converse	NOUN
ejpam-2553	136	3	of	of	ADP
ejpam-2553	136	4	theorem	theorem	NOUN
ejpam-2553	136	5	1	1	NUM
ejpam-2553	136	6	is	be	AUX
ejpam-2553	136	7	not	not	PART
ejpam-2553	136	8	valid	valid	ADJ
ejpam-2553	136	9	.	.	PUNCT
ejpam-2553	137	1	for	for	ADP
ejpam-2553	137	2	example	example	NOUN
ejpam-2553	137	3	let	let	VERB
ejpam-2553	137	4	(	(	PUNCT
ejpam-2553	137	5	r,+	r,+	NUM
ejpam-2553	137	6	,	,	PUNCT
ejpam-2553	137	7	.	.	PUNCT
ejpam-2553	137	8	)	)	PUNCT
ejpam-2553	138	1	be	be	AUX
ejpam-2553	138	2	a	a	DET
ejpam-2553	138	3	non	non	ADJ
ejpam-2553	138	4	regular	regular	ADJ
ejpam-2553	138	5	ring	ring	NOUN
ejpam-2553	138	6	.	.	PUNCT
ejpam-2553	139	1	consider	consider	VERB
ejpam-2553	139	2	(	(	PUNCT
ejpam-2553	139	3	r,+	r,+	NUM
ejpam-2553	139	4	,	,	PUNCT
ejpam-2553	139	5	.	.	PUNCT
ejpam-2553	139	6	)	)	PUNCT
ejpam-2553	140	1	as	as	ADP
ejpam-2553	140	2	a	a	DET
ejpam-2553	140	3	hyperring	hyperring	NOUN
ejpam-2553	140	4	under	under	ADP
ejpam-2553	140	5	operators	operator	NOUN
ejpam-2553	140	6	"	"	PUNCT
ejpam-2553	140	7	+	+	PROPN
ejpam-2553	140	8	"	"	PUNCT
ejpam-2553	140	9	and	and	CCONJ
ejpam-2553	140	10	"	"	PUNCT
ejpam-2553	140	11	.	.	PUNCT
ejpam-2553	140	12	"	"	PUNCT
ejpam-2553	140	13	.	.	PUNCT
ejpam-2553	141	1	clearly	clearly	ADV
ejpam-2553	141	2	r	r	X
ejpam-2553	141	3	/	/	SYM
ejpam-2553	141	4	γ∗	γ∗	NOUN
ejpam-2553	141	5	∼=	∼=	PROPN
ejpam-2553	141	6	r.	r.	PROPN
ejpam-2553	141	7	r.	r.	PROPN
ejpam-2553	141	8	ameri	ameri	PROPN
ejpam-2553	141	9	,	,	PUNCT
ejpam-2553	141	10	a.	a.	PROPN
ejpam-2553	141	11	kordi	kordi	PROPN
ejpam-2553	141	12	/	/	PUNCT
ejpam-2553	141	13	eur	eur	PROPN
ejpam-2553	141	14	.	.	PUNCT
ejpam-2553	142	1	j.	j.	PROPN
ejpam-2553	142	2	pure	pure	PROPN
ejpam-2553	142	3	appl	appl	PROPN
ejpam-2553	142	4	.	.	PROPN
ejpam-2553	142	5	math	math	PROPN
ejpam-2553	142	6	,	,	PUNCT
ejpam-2553	142	7	9	9	NUM
ejpam-2553	142	8	(	(	PUNCT
ejpam-2553	142	9	2016	2016	NUM
ejpam-2553	142	10	)	)	PUNCT
ejpam-2553	142	11	,	,	PUNCT
ejpam-2553	142	12	402	402	NUM
ejpam-2553	142	13	-	-	SYM
ejpam-2553	142	14	418	418	NUM
ejpam-2553	142	15	406	406	NUM
ejpam-2553	142	16	definition	definition	NOUN
ejpam-2553	142	17	3	3	NUM
ejpam-2553	142	18	.	.	PUNCT
ejpam-2553	143	1	a	a	DET
ejpam-2553	143	2	multiplicative	multiplicative	ADJ
ejpam-2553	143	3	hyperring	hyperring	NOUN
ejpam-2553	143	4	r	r	NOUN
ejpam-2553	143	5	is	be	AUX
ejpam-2553	143	6	said	say	VERB
ejpam-2553	143	7	multiplicatively	multiplicatively	ADV
ejpam-2553	143	8	n	n	CCONJ
ejpam-2553	143	9	-	-	PUNCT
ejpam-2553	143	10	complete	complete	ADJ
ejpam-2553	143	11	(	(	PUNCT
ejpam-2553	143	12	n	n	CCONJ
ejpam-2553	143	13	-	-	PUNCT
ejpam-2553	143	14	mc	mc	NOUN
ejpam-2553	143	15	)	)	PUNCT
ejpam-2553	143	16	if	if	SCONJ
ejpam-2553	143	17	for	for	ADP
ejpam-2553	143	18	all	all	DET
ejpam-2553	143	19	x1	x1	PROPN
ejpam-2553	143	20	,	,	PUNCT
ejpam-2553	143	21	.	.	PUNCT
ejpam-2553	143	22	.	.	PUNCT
ejpam-2553	143	23	.	.	PUNCT
ejpam-2553	144	1	,	,	PUNCT
ejpam-2553	144	2	xn	xn	PUNCT
ejpam-2553	145	1	∈	∈	PROPN
ejpam-2553	145	2	r	r	NOUN
ejpam-2553	145	3	we	we	PRON
ejpam-2553	145	4	have	have	VERB
ejpam-2553	145	5	γ	γ	X
ejpam-2553	145	6	(	(	PUNCT
ejpam-2553	145	7	n	n	ADV
ejpam-2553	145	8	∏	∏	PROPN
ejpam-2553	145	9	i=1	i=1	PROPN
ejpam-2553	145	10	x	x	PUNCT
ejpam-2553	145	11	i	i	NOUN
ejpam-2553	145	12	)	)	PUNCT
ejpam-2553	145	13	=	=	SYM
ejpam-2553	146	1	n	n	PRON
ejpam-2553	146	2	∏	∏	NUM
ejpam-2553	147	1	i=1	i=1	X
ejpam-2553	147	2	x	x	X
ejpam-2553	148	1	i	i	NOUN
ejpam-2553	148	2	.	.	PUNCT
ejpam-2553	149	1	also	also	ADV
ejpam-2553	149	2	[	[	X
ejpam-2553	149	3	7	7	NUM
ejpam-2553	149	4	]	]	PUNCT
ejpam-2553	149	5	,	,	PUNCT
ejpam-2553	149	6	we	we	PRON
ejpam-2553	149	7	called	call	VERB
ejpam-2553	149	8	that	that	SCONJ
ejpam-2553	149	9	a	a	DET
ejpam-2553	149	10	hyperring	hyperring	NOUN
ejpam-2553	149	11	r	r	NOUN
ejpam-2553	149	12	is	be	AUX
ejpam-2553	149	13	n	n	ADV
ejpam-2553	149	14	-	-	PUNCT
ejpam-2553	149	15	complete	complete	ADJ
ejpam-2553	149	16	if	if	SCONJ
ejpam-2553	149	17	for	for	ADP
ejpam-2553	149	18	all	all	PRON
ejpam-2553	149	19	(	(	PUNCT
ejpam-2553	149	20	k1	k1	NOUN
ejpam-2553	149	21	,	,	PUNCT
ejpam-2553	149	22	.	.	PUNCT
ejpam-2553	149	23	.	.	PUNCT
ejpam-2553	149	24	.	.	PUNCT
ejpam-2553	150	1	,	,	PUNCT
ejpam-2553	150	2	kn	kn	PROPN
ejpam-2553	150	3	)	)	PUNCT
ejpam-2553	150	4	∈	∈	PROPN
ejpam-2553	150	5	nn	nn	PROPN
ejpam-2553	150	6	and	and	CCONJ
ejpam-2553	150	7	for	for	ADP
ejpam-2553	150	8	all	all	PRON
ejpam-2553	150	9	(	(	PUNCT
ejpam-2553	150	10	x1	x1	PROPN
ejpam-2553	150	11	j	j	PROPN
ejpam-2553	150	12	,	,	PUNCT
ejpam-2553	150	13	.	.	PUNCT
ejpam-2553	150	14	.	.	PUNCT
ejpam-2553	151	1	.	.	PUNCT
ejpam-2553	152	1	,	,	PUNCT
ejpam-2553	152	2	x	x	X
ejpam-2553	152	3	iki	iki	PROPN
ejpam-2553	152	4	)	)	PUNCT
ejpam-2553	152	5	∈	∈	PROPN
ejpam-2553	152	6	rki	rki	NOUN
ejpam-2553	152	7	we	we	PRON
ejpam-2553	152	8	have	have	VERB
ejpam-2553	152	9	γ	γ	X
ejpam-2553	152	10	(	(	PUNCT
ejpam-2553	152	11	n	n	PROPN
ejpam-2553	152	12	∑	∑	PROPN
ejpam-2553	152	13	i=1	i=1	PROPN
ejpam-2553	153	1	(	(	PUNCT
ejpam-2553	153	2	ki	ki	PROPN
ejpam-2553	153	3	∏	∏	PROPN
ejpam-2553	153	4	j=1	j=1	NOUN
ejpam-2553	153	5	x	x	PUNCT
ejpam-2553	153	6	i	i	PRON
ejpam-2553	153	7	j	j	PROPN
ejpam-2553	153	8	)	)	PUNCT
ejpam-2553	153	9	)	)	PUNCT
ejpam-2553	154	1	=	=	SYM
ejpam-2553	155	1	n	n	PROPN
ejpam-2553	155	2	∑	∑	PROPN
ejpam-2553	155	3	i=1	i=1	PROPN
ejpam-2553	155	4	(	(	PUNCT
ejpam-2553	155	5	ki	ki	PROPN
ejpam-2553	155	6	∏	∏	PROPN
ejpam-2553	155	7	j=1	j=1	NOUN
ejpam-2553	155	8	x	x	PUNCT
ejpam-2553	155	9	i	i	PRON
ejpam-2553	155	10	j	j	PROPN
ejpam-2553	155	11	)	)	PUNCT
ejpam-2553	155	12	.	.	PUNCT
ejpam-2553	156	1	theorem	theorem	NOUN
ejpam-2553	156	2	2	2	NUM
ejpam-2553	156	3	.	.	PUNCT
ejpam-2553	157	1	let	let	VERB
ejpam-2553	157	2	r	r	PRON
ejpam-2553	157	3	be	be	AUX
ejpam-2553	157	4	a	a	DET
ejpam-2553	157	5	3	3	NUM
ejpam-2553	157	6	-	-	PUNCT
ejpam-2553	157	7	mc	mc	PROPN
ejpam-2553	157	8	multiplicative	multiplicative	ADJ
ejpam-2553	157	9	hyperring	hyperring	NOUN
ejpam-2553	157	10	.	.	PUNCT
ejpam-2553	158	1	if	if	SCONJ
ejpam-2553	158	2	r	r	NOUN
ejpam-2553	158	3	/	/	SYM
ejpam-2553	158	4	γ∗	γ∗	NOUN
ejpam-2553	158	5	is	be	AUX
ejpam-2553	158	6	a	a	DET
ejpam-2553	158	7	regular	regular	ADJ
ejpam-2553	158	8	ring	ring	NOUN
ejpam-2553	158	9	then	then	ADV
ejpam-2553	158	10	r	r	NOUN
ejpam-2553	158	11	is	be	AUX
ejpam-2553	158	12	a	a	DET
ejpam-2553	158	13	regular	regular	ADJ
ejpam-2553	158	14	multiplicative	multiplicative	ADJ
ejpam-2553	158	15	hyperring	hyperring	NOUN
ejpam-2553	158	16	.	.	PUNCT
ejpam-2553	159	1	proof	proof	NOUN
ejpam-2553	159	2	.	.	PUNCT
ejpam-2553	160	1	assume	assume	VERB
ejpam-2553	160	2	that	that	SCONJ
ejpam-2553	160	3	a	a	DET
ejpam-2553	160	4	∈	∈	PROPN
ejpam-2553	160	5	r.	r.	NOUN
ejpam-2553	160	6	then	then	ADV
ejpam-2553	160	7	there	there	PRON
ejpam-2553	160	8	exists	exist	VERB
ejpam-2553	160	9	r	r	NOUN
ejpam-2553	160	10	∈	∈	PROPN
ejpam-2553	160	11	r	r	NOUN
ejpam-2553	160	12	such	such	ADJ
ejpam-2553	160	13	that	that	SCONJ
ejpam-2553	160	14	γ∗(a	γ∗(a	NOUN
ejpam-2553	160	15	)	)	PUNCT
ejpam-2553	160	16	�	�	PROPN
ejpam-2553	160	17	γ∗(r	γ∗(r	PROPN
ejpam-2553	160	18	)	)	PUNCT
ejpam-2553	160	19	�	�	PROPN
ejpam-2553	160	20	γ∗(a	γ∗(a	PROPN
ejpam-2553	160	21	)	)	PUNCT
ejpam-2553	160	22	=	=	SYM
ejpam-2553	161	1	γ∗(a	γ∗(a	NOUN
ejpam-2553	161	2	)	)	PUNCT
ejpam-2553	161	3	.	.	PUNCT
ejpam-2553	162	1	then	then	ADV
ejpam-2553	162	2	γ∗(a	γ∗(a	NOUN
ejpam-2553	162	3	)	)	PUNCT
ejpam-2553	162	4	=	=	SYM
ejpam-2553	162	5	γ∗(ara	γ∗(ara	NOUN
ejpam-2553	162	6	)	)	PUNCT
ejpam-2553	162	7	.	.	PUNCT
ejpam-2553	163	1	thus	thus	ADV
ejpam-2553	163	2	a	a	DET
ejpam-2553	163	3	∈	∈	NOUN
ejpam-2553	163	4	γ∗(ara	γ∗(ara	NUM
ejpam-2553	163	5	)	)	PUNCT
ejpam-2553	163	6	=	=	SYM
ejpam-2553	163	7	ara	ara	NOUN
ejpam-2553	163	8	,	,	PUNCT
ejpam-2553	163	9	since	since	SCONJ
ejpam-2553	163	10	r	r	NOUN
ejpam-2553	163	11	is	be	AUX
ejpam-2553	163	12	3	3	NUM
ejpam-2553	163	13	-	-	SYM
ejpam-2553	163	14	mc	mc	NOUN
ejpam-2553	163	15	and	and	CCONJ
ejpam-2553	163	16	hence	hence	ADV
ejpam-2553	163	17	a	a	DET
ejpam-2553	163	18	∈	∈	PROPN
ejpam-2553	163	19	ara	ara	PROPN
ejpam-2553	163	20	.	.	PUNCT
ejpam-2553	163	21	proposition	proposition	NOUN
ejpam-2553	163	22	1	1	NUM
ejpam-2553	163	23	.	.	PUNCT
ejpam-2553	164	1	(	(	PUNCT
ejpam-2553	164	2	[	[	X
ejpam-2553	164	3	25	25	NUM
ejpam-2553	164	4	]	]	PUNCT
ejpam-2553	164	5	)	)	PUNCT
ejpam-2553	164	6	let	let	VERB
ejpam-2553	164	7	r	r	PRON
ejpam-2553	164	8	be	be	AUX
ejpam-2553	164	9	a	a	DET
ejpam-2553	164	10	commutative	commutative	ADJ
ejpam-2553	164	11	ring	ring	NOUN
ejpam-2553	164	12	and	and	CCONJ
ejpam-2553	164	13	a	a	DET
ejpam-2553	164	14	∈	∈	NOUN
ejpam-2553	164	15	v	v	NOUN
ejpam-2553	164	16	(	(	PUNCT
ejpam-2553	164	17	r	r	NOUN
ejpam-2553	164	18	)	)	PUNCT
ejpam-2553	164	19	.	.	PUNCT
ejpam-2553	165	1	then	then	ADV
ejpam-2553	165	2	there	there	PRON
ejpam-2553	165	3	is	be	VERB
ejpam-2553	165	4	a	a	DET
ejpam-2553	165	5	unique	unique	ADJ
ejpam-2553	165	6	x	x	SYM
ejpam-2553	165	7	∈	∈	NOUN
ejpam-2553	165	8	r	r	NOUN
ejpam-2553	165	9	with	with	ADP
ejpam-2553	165	10	axa	axa	NOUN
ejpam-2553	165	11	=	=	PUNCT
ejpam-2553	165	12	a	a	PROPN
ejpam-2553	165	13	and	and	CCONJ
ejpam-2553	165	14	xax	xax	PROPN
ejpam-2553	165	15	=	=	PROPN
ejpam-2553	165	16	x.	x.	NOUN
ejpam-2553	165	17	obviously	obviously	ADV
ejpam-2553	165	18	,	,	PUNCT
ejpam-2553	165	19	if	if	SCONJ
ejpam-2553	165	20	r	r	NOUN
ejpam-2553	165	21	is	be	AUX
ejpam-2553	165	22	a	a	DET
ejpam-2553	165	23	regular	regular	ADJ
ejpam-2553	165	24	commutative	commutative	ADJ
ejpam-2553	165	25	multiplicative	multiplicative	ADJ
ejpam-2553	165	26	hyperring	hyperring	NOUN
ejpam-2553	165	27	,	,	PUNCT
ejpam-2553	165	28	then	then	ADV
ejpam-2553	165	29	for	for	ADP
ejpam-2553	165	30	each	each	PRON
ejpam-2553	165	31	a	a	DET
ejpam-2553	165	32	∈	∈	NOUN
ejpam-2553	165	33	r	r	NOUN
ejpam-2553	165	34	there	there	PRON
ejpam-2553	165	35	exists	exist	VERB
ejpam-2553	165	36	a	a	DET
ejpam-2553	165	37	unique	unique	ADJ
ejpam-2553	165	38	element	element	NOUN
ejpam-2553	165	39	γ∗(x	γ∗(x	NOUN
ejpam-2553	165	40	)	)	PUNCT
ejpam-2553	165	41	in	in	ADP
ejpam-2553	165	42	r	r	NOUN
ejpam-2553	165	43	/	/	SYM
ejpam-2553	165	44	γ∗	γ∗	NOUN
ejpam-2553	165	45	such	such	ADJ
ejpam-2553	165	46	that	that	DET
ejpam-2553	165	47	γ∗(axa	γ∗(axa	NOUN
ejpam-2553	165	48	)	)	PUNCT
ejpam-2553	165	49	=	=	SYM
ejpam-2553	165	50	γ∗(a	γ∗(a	PROPN
ejpam-2553	165	51	)	)	PUNCT
ejpam-2553	165	52	and	and	CCONJ
ejpam-2553	165	53	γ∗(xax	γ∗(xax	NOUN
ejpam-2553	165	54	)	)	PUNCT
ejpam-2553	165	55	=	=	PUNCT
ejpam-2553	165	56	γ∗(x	γ∗(x	PROPN
ejpam-2553	165	57	)	)	PUNCT
ejpam-2553	165	58	.	.	PUNCT
ejpam-2553	166	1	theorem	theorem	NOUN
ejpam-2553	166	2	3	3	X
ejpam-2553	166	3	.	.	PUNCT
ejpam-2553	167	1	if	if	SCONJ
ejpam-2553	167	2	r	r	NOUN
ejpam-2553	167	3	is	be	AUX
ejpam-2553	167	4	a	a	DET
ejpam-2553	167	5	commutative	commutative	ADJ
ejpam-2553	167	6	3	3	NUM
ejpam-2553	167	7	-	-	PUNCT
ejpam-2553	167	8	mc	mc	PROPN
ejpam-2553	167	9	multiplicative	multiplicative	PROPN
ejpam-2553	167	10	hyperring	hyperring	NOUN
ejpam-2553	167	11	and	and	CCONJ
ejpam-2553	167	12	a	a	DET
ejpam-2553	167	13	∈	∈	NOUN
ejpam-2553	167	14	r	r	NOUN
ejpam-2553	167	15	be	be	VERB
ejpam-2553	167	16	a	a	DET
ejpam-2553	167	17	regular	regular	ADJ
ejpam-2553	167	18	element	element	NOUN
ejpam-2553	167	19	of	of	ADP
ejpam-2553	167	20	r.	r.	PROPN
ejpam-2553	167	21	then	then	ADV
ejpam-2553	167	22	there	there	PRON
ejpam-2553	167	23	exists	exist	VERB
ejpam-2553	167	24	x	x	X
ejpam-2553	167	25	∈	∈	NOUN
ejpam-2553	167	26	r	r	NOUN
ejpam-2553	167	27	such	such	DET
ejpam-2553	167	28	that	that	SCONJ
ejpam-2553	167	29	a	a	DET
ejpam-2553	167	30	∈	∈	PROPN
ejpam-2553	167	31	axa	axa	NOUN
ejpam-2553	167	32	and	and	CCONJ
ejpam-2553	167	33	x	x	SYM
ejpam-2553	167	34	∈	∈	PROPN
ejpam-2553	167	35	xax	xax	PROPN
ejpam-2553	167	36	.	.	PUNCT
ejpam-2553	167	37	proof	proof	NOUN
ejpam-2553	167	38	.	.	PUNCT
ejpam-2553	168	1	since	since	SCONJ
ejpam-2553	168	2	r	r	NOUN
ejpam-2553	168	3	is	be	AUX
ejpam-2553	168	4	a	a	DET
ejpam-2553	168	5	commutative	commutative	ADJ
ejpam-2553	168	6	multiplicative	multiplicative	ADJ
ejpam-2553	168	7	hyperring	hyperring	NOUN
ejpam-2553	168	8	then	then	ADV
ejpam-2553	168	9	it	it	PRON
ejpam-2553	168	10	is	be	AUX
ejpam-2553	168	11	easy	easy	ADJ
ejpam-2553	168	12	to	to	PART
ejpam-2553	168	13	check	check	VERB
ejpam-2553	168	14	that	that	PRON
ejpam-2553	168	15	r	r	NOUN
ejpam-2553	168	16	/	/	SYM
ejpam-2553	168	17	γ∗	γ∗	NOUN
ejpam-2553	168	18	is	be	AUX
ejpam-2553	168	19	commutative	commutative	ADJ
ejpam-2553	168	20	ring	ring	NOUN
ejpam-2553	168	21	.	.	PUNCT
ejpam-2553	169	1	as	as	ADP
ejpam-2553	169	2	a	a	DET
ejpam-2553	169	3	∈	∈	NOUN
ejpam-2553	169	4	r	r	NOUN
ejpam-2553	169	5	is	be	AUX
ejpam-2553	169	6	a	a	DET
ejpam-2553	169	7	regular	regular	ADJ
ejpam-2553	169	8	element	element	NOUN
ejpam-2553	169	9	then	then	ADV
ejpam-2553	169	10	γ∗(a	γ∗(a	NOUN
ejpam-2553	169	11	)	)	PUNCT
ejpam-2553	169	12	is	be	AUX
ejpam-2553	169	13	a	a	DET
ejpam-2553	169	14	regular	regular	ADJ
ejpam-2553	169	15	element	element	NOUN
ejpam-2553	169	16	of	of	ADP
ejpam-2553	169	17	r	r	NOUN
ejpam-2553	169	18	/	/	SYM
ejpam-2553	169	19	γ∗	γ∗	NOUN
ejpam-2553	169	20	,	,	PUNCT
ejpam-2553	169	21	by	by	ADP
ejpam-2553	169	22	theorem	theorem	NOUN
ejpam-2553	169	23	1	1	NUM
ejpam-2553	169	24	,	,	PUNCT
ejpam-2553	169	25	then	then	ADV
ejpam-2553	169	26	,	,	PUNCT
ejpam-2553	169	27	by	by	ADP
ejpam-2553	169	28	proposition	proposition	NOUN
ejpam-2553	169	29	1	1	NUM
ejpam-2553	169	30	,	,	PUNCT
ejpam-2553	169	31	there	there	PRON
ejpam-2553	169	32	is	be	VERB
ejpam-2553	169	33	a	a	DET
ejpam-2553	169	34	unique	unique	ADJ
ejpam-2553	169	35	γ∗(x	γ∗(x	NOUN
ejpam-2553	169	36	)	)	PUNCT
ejpam-2553	169	37	∈	∈	PROPN
ejpam-2553	169	38	r	r	NOUN
ejpam-2553	169	39	/	/	SYM
ejpam-2553	169	40	γ∗	γ∗	NOUN
ejpam-2553	169	41	for	for	ADP
ejpam-2553	169	42	x	x	PROPN
ejpam-2553	169	43	∈	∈	PROPN
ejpam-2553	169	44	r	r	NOUN
ejpam-2553	170	1	such	such	ADJ
ejpam-2553	170	2	that	that	SCONJ
ejpam-2553	170	3	γ∗(a)	γ∗(a)	NOUN
ejpam-2553	170	4	�	�	PROPN
ejpam-2553	170	5	γ∗(x	γ∗(x	PROPN
ejpam-2553	170	6	)	)	PUNCT
ejpam-2553	170	7	�	�	PROPN
ejpam-2553	170	8	γ∗(a	γ∗(a	PROPN
ejpam-2553	170	9	)	)	PUNCT
ejpam-2553	170	10	=	=	SYM
ejpam-2553	170	11	γ∗(a	γ∗(a	NOUN
ejpam-2553	170	12	)	)	PUNCT
ejpam-2553	170	13	γ∗(x)	γ∗(x)	PROPN
ejpam-2553	170	14	�	�	NOUN
ejpam-2553	170	15	γ∗(a	γ∗(a	NOUN
ejpam-2553	170	16	)	)	PUNCT
ejpam-2553	170	17	�	�	PROPN
ejpam-2553	170	18	γ∗(x	γ∗(x	PROPN
ejpam-2553	170	19	)	)	PUNCT
ejpam-2553	170	20	=	=	SYM
ejpam-2553	170	21	γ∗(x	γ∗(x	PROPN
ejpam-2553	170	22	)	)	PUNCT
ejpam-2553	170	23	.	.	PUNCT
ejpam-2553	171	1	then	then	ADV
ejpam-2553	171	2	γ∗(axa	γ∗(axa	NOUN
ejpam-2553	171	3	)	)	PUNCT
ejpam-2553	171	4	=	=	SYM
ejpam-2553	171	5	γ∗(a	γ∗(a	PROPN
ejpam-2553	171	6	)	)	PUNCT
ejpam-2553	171	7	and	and	CCONJ
ejpam-2553	171	8	γ∗(xax	γ∗(xax	NOUN
ejpam-2553	171	9	)	)	PUNCT
ejpam-2553	171	10	=	=	PUNCT
ejpam-2553	171	11	γ∗(x	γ∗(x	PROPN
ejpam-2553	171	12	)	)	PUNCT
ejpam-2553	171	13	.	.	PUNCT
ejpam-2553	172	1	therefore	therefore	ADV
ejpam-2553	172	2	,	,	PUNCT
ejpam-2553	172	3	a	a	DET
ejpam-2553	172	4	∈	∈	NOUN
ejpam-2553	172	5	axa	axa	NOUN
ejpam-2553	172	6	and	and	CCONJ
ejpam-2553	172	7	x	x	SYM
ejpam-2553	172	8	∈	∈	PROPN
ejpam-2553	172	9	xax	xax	PROPN
ejpam-2553	172	10	,	,	PUNCT
ejpam-2553	172	11	since	since	SCONJ
ejpam-2553	172	12	r	r	NOUN
ejpam-2553	172	13	is	be	AUX
ejpam-2553	172	14	3	3	NUM
ejpam-2553	172	15	-	-	SYM
ejpam-2553	172	16	mc	mc	PROPN
ejpam-2553	172	17	.	.	PROPN
ejpam-2553	172	18	definition	definition	NOUN
ejpam-2553	172	19	4	4	NUM
ejpam-2553	172	20	.	.	PUNCT
ejpam-2553	173	1	let	let	VERB
ejpam-2553	173	2	r	r	PRON
ejpam-2553	173	3	be	be	AUX
ejpam-2553	173	4	a	a	DET
ejpam-2553	173	5	multiplicative	multiplicative	ADJ
ejpam-2553	173	6	hyperring	hyperring	NOUN
ejpam-2553	173	7	.	.	PUNCT
ejpam-2553	174	1	then	then	ADV
ejpam-2553	174	2	we	we	PRON
ejpam-2553	174	3	called	call	VERB
ejpam-2553	174	4	that	that	DET
ejpam-2553	174	5	mn(r	mn(r	NOUN
ejpam-2553	174	6	)	)	PUNCT
ejpam-2553	174	7	,	,	PUNCT
ejpam-2553	174	8	as	as	ADP
ejpam-2553	174	9	the	the	DET
ejpam-2553	174	10	set	set	NOUN
ejpam-2553	174	11	of	of	ADP
ejpam-2553	174	12	all	all	DET
ejpam-2553	174	13	hypermatrices	hypermatrice	NOUN
ejpam-2553	174	14	of	of	ADP
ejpam-2553	174	15	r.	r.	PROPN
ejpam-2553	174	16	also	also	ADV
ejpam-2553	174	17	we	we	PRON
ejpam-2553	174	18	called	call	VERB
ejpam-2553	174	19	that	that	PRON
ejpam-2553	174	20	for	for	ADP
ejpam-2553	174	21	alla	alla	NOUN
ejpam-2553	174	22	=	=	PUNCT
ejpam-2553	174	23	(	(	PUNCT
ejpam-2553	174	24	ai	ai	INTJ
ejpam-2553	174	25	j)n×n	j)n×n	PROPN
ejpam-2553	174	26	,	,	PUNCT
ejpam-2553	174	27	b	b	NOUN
ejpam-2553	174	28	=	=	SYM
ejpam-2553	174	29	(	(	PUNCT
ejpam-2553	174	30	bi	bi	NOUN
ejpam-2553	174	31	j)n×n	j)n×n	PROPN
ejpam-2553	174	32	∈	∈	PROPN
ejpam-2553	174	33	p∗(mn(r)),a	p∗(mn(r)),a	PROPN
ejpam-2553	174	34	⊆b	⊆b	NOUN
ejpam-2553	174	35	if	if	SCONJ
ejpam-2553	174	36	and	and	CCONJ
ejpam-2553	174	37	only	only	ADV
ejpam-2553	174	38	if	if	SCONJ
ejpam-2553	174	39	ai	ai	VERB
ejpam-2553	174	40	j	j	PROPN
ejpam-2553	174	41	⊆	⊆	NUM
ejpam-2553	174	42	bi	bi	PROPN
ejpam-2553	174	43	j	j	PROPN
ejpam-2553	174	44	.	.	PUNCT
ejpam-2553	175	1	r.	r.	PROPN
ejpam-2553	175	2	ameri	ameri	PROPN
ejpam-2553	175	3	,	,	PUNCT
ejpam-2553	175	4	a.	a.	PROPN
ejpam-2553	175	5	kordi	kordi	PROPN
ejpam-2553	175	6	/	/	PUNCT
ejpam-2553	175	7	eur	eur	PROPN
ejpam-2553	175	8	.	.	PUNCT
ejpam-2553	176	1	j.	j.	PROPN
ejpam-2553	176	2	pure	pure	PROPN
ejpam-2553	176	3	appl	appl	PROPN
ejpam-2553	176	4	.	.	PROPN
ejpam-2553	176	5	math	math	PROPN
ejpam-2553	176	6	,	,	PUNCT
ejpam-2553	176	7	9	9	NUM
ejpam-2553	176	8	(	(	PUNCT
ejpam-2553	176	9	2016	2016	NUM
ejpam-2553	176	10	)	)	PUNCT
ejpam-2553	176	11	,	,	PUNCT
ejpam-2553	176	12	402	402	NUM
ejpam-2553	176	13	-	-	SYM
ejpam-2553	176	14	418	418	NUM
ejpam-2553	176	15	407	407	NUM
ejpam-2553	176	16	remark	remark	NOUN
ejpam-2553	176	17	2	2	NUM
ejpam-2553	176	18	.	.	PUNCT
ejpam-2553	177	1	let	let	AUX
ejpam-2553	177	2	r	r	PRON
ejpam-2553	177	3	be	be	AUX
ejpam-2553	177	4	a	a	DET
ejpam-2553	177	5	multiplicative	multiplicative	ADJ
ejpam-2553	177	6	hyperring	hyperring	NOUN
ejpam-2553	177	7	with	with	ADP
ejpam-2553	177	8	a	a	DET
ejpam-2553	177	9	scalar	scalar	ADJ
ejpam-2553	177	10	identity	identity	NOUN
ejpam-2553	177	11	1	1	NUM
ejpam-2553	177	12	and	and	CCONJ
ejpam-2553	177	13	mn(r	mn(r	NOUN
ejpam-2553	177	14	)	)	PUNCT
ejpam-2553	177	15	denotes	denote	VERB
ejpam-2553	177	16	the	the	DET
ejpam-2553	177	17	set	set	NOUN
ejpam-2553	177	18	of	of	ADP
ejpam-2553	177	19	all	all	DET
ejpam-2553	177	20	n×	n×	PRON
ejpam-2553	177	21	n	n	NOUN
ejpam-2553	177	22	matrices	matrix	NOUN
ejpam-2553	177	23	with	with	ADP
ejpam-2553	177	24	entries	entry	NOUN
ejpam-2553	177	25	in	in	ADP
ejpam-2553	177	26	r.	r.	PROPN
ejpam-2553	177	27	it	it	PRON
ejpam-2553	177	28	is	be	AUX
ejpam-2553	177	29	easy	easy	ADJ
ejpam-2553	177	30	to	to	PART
ejpam-2553	177	31	verify	verify	VERB
ejpam-2553	177	32	that	that	DET
ejpam-2553	177	33	mn(r	mn(r	NOUN
ejpam-2553	177	34	)	)	PUNCT
ejpam-2553	177	35	is	be	AUX
ejpam-2553	177	36	a	a	DET
ejpam-2553	177	37	non	non	ADJ
ejpam-2553	177	38	-	-	ADJ
ejpam-2553	177	39	commutative	commutative	ADJ
ejpam-2553	177	40	multiplicative	multiplicative	ADJ
ejpam-2553	177	41	hyperring	hyperring	NOUN
ejpam-2553	177	42	with	with	ADP
ejpam-2553	177	43	unitary	unitary	ADJ
ejpam-2553	177	44	element	element	NOUN
ejpam-2553	177	45	under	under	ADP
ejpam-2553	177	46	usual	usual	ADJ
ejpam-2553	177	47	matrix	matrix	NOUN
ejpam-2553	177	48	operations	operation	NOUN
ejpam-2553	177	49	.	.	PUNCT
ejpam-2553	178	1	let	let	VERB
ejpam-2553	178	2	a=	a=	VERB
ejpam-2553	178	3	(	(	PUNCT
ejpam-2553	178	4	ai	ai	VERB
ejpam-2553	178	5	j)n×n	j)n×n	PROPN
ejpam-2553	178	6	be	be	AUX
ejpam-2553	178	7	a	a	DET
ejpam-2553	178	8	matrix	matrix	NOUN
ejpam-2553	178	9	,	,	PUNCT
ejpam-2553	178	10	where	where	SCONJ
ejpam-2553	178	11	ars	ar	VERB
ejpam-2553	178	12	=	=	PUNCT
ejpam-2553	178	13	ab	ab	PROPN
ejpam-2553	178	14	for	for	ADP
ejpam-2553	178	15	some	some	DET
ejpam-2553	178	16	1	1	NUM
ejpam-2553	178	17	≤	≤	NOUN
ejpam-2553	178	18	r	r	NOUN
ejpam-2553	178	19	,	,	PUNCT
ejpam-2553	178	20	s	s	VERB
ejpam-2553	178	21	≤	≤	NUM
ejpam-2553	178	22	n	n	CCONJ
ejpam-2553	178	23	and	and	CCONJ
ejpam-2553	178	24	in	in	ADP
ejpam-2553	178	25	other	other	ADJ
ejpam-2553	178	26	positions	position	NOUN
ejpam-2553	178	27	ai	ai	VERB
ejpam-2553	178	28	j	j	PROPN
ejpam-2553	178	29	=	=	PROPN
ejpam-2553	178	30	0	0	PROPN
ejpam-2553	178	31	.	.	PUNCT
ejpam-2553	179	1	then	then	ADV
ejpam-2553	179	2	a	a	DET
ejpam-2553	179	3	=	=	SYM
ejpam-2553	179	4	bc	bc	PROPN
ejpam-2553	179	5	,	,	PUNCT
ejpam-2553	179	6	where	where	SCONJ
ejpam-2553	179	7	b	b	X
ejpam-2553	179	8	=	=	SYM
ejpam-2553	179	9	(	(	PUNCT
ejpam-2553	179	10	bi	bi	NOUN
ejpam-2553	179	11	j)n×n	j)n×n	PROPN
ejpam-2553	179	12	such	such	ADJ
ejpam-2553	179	13	that	that	DET
ejpam-2553	179	14	brs	brs	NOUN
ejpam-2553	179	15	=	=	PUNCT
ejpam-2553	179	16	a	a	PROPN
ejpam-2553	179	17	and	and	CCONJ
ejpam-2553	179	18	c	c	NOUN
ejpam-2553	179	19	=	=	SYM
ejpam-2553	179	20	(	(	PUNCT
ejpam-2553	179	21	ci	ci	PROPN
ejpam-2553	179	22	j	j	PROPN
ejpam-2553	179	23	)	)	PUNCT
ejpam-2553	179	24	such	such	ADJ
ejpam-2553	179	25	that	that	DET
ejpam-2553	179	26	css	css	PROPN
ejpam-2553	179	27	=	=	PROPN
ejpam-2553	179	28	b	b	PROPN
ejpam-2553	179	29	and	and	CCONJ
ejpam-2553	179	30	in	in	ADP
ejpam-2553	179	31	other	other	ADJ
ejpam-2553	179	32	entries	entry	NOUN
ejpam-2553	179	33	of	of	ADP
ejpam-2553	179	34	b	b	NOUN
ejpam-2553	179	35	,	,	PUNCT
ejpam-2553	179	36	c	c	VERB
ejpam-2553	179	37	we	we	PRON
ejpam-2553	179	38	have	have	VERB
ejpam-2553	179	39	bi	bi	PROPN
ejpam-2553	179	40	j	j	PROPN
ejpam-2553	179	41	and	and	CCONJ
ejpam-2553	179	42	ci	ci	PROPN
ejpam-2553	179	43	j	j	PROPN
ejpam-2553	180	1	=	=	PROPN
ejpam-2553	180	2	0	0	PROPN
ejpam-2553	180	3	.	.	PUNCT
ejpam-2553	180	4	recall	recall	VERB
ejpam-2553	180	5	that	that	SCONJ
ejpam-2553	180	6	r	r	NOUN
ejpam-2553	180	7	has	have	VERB
ejpam-2553	180	8	a	a	DET
ejpam-2553	180	9	zero	zero	NUM
ejpam-2553	180	10	absorbing	absorbing	NOUN
ejpam-2553	180	11	property	property	NOUN
ejpam-2553	180	12	if	if	SCONJ
ejpam-2553	180	13	for	for	ADP
ejpam-2553	180	14	all	all	DET
ejpam-2553	180	15	a	a	DET
ejpam-2553	180	16	∈	∈	NOUN
ejpam-2553	180	17	r	r	NOUN
ejpam-2553	180	18	,	,	PUNCT
ejpam-2553	180	19	0	0	NUM
ejpam-2553	180	20	◦	◦	NOUN
ejpam-2553	180	21	a	a	DET
ejpam-2553	180	22	=	=	NOUN
ejpam-2553	180	23	a	a	DET
ejpam-2553	180	24	◦	◦	NOUN
ejpam-2553	180	25	0=	0=	PUNCT
ejpam-2553	180	26	{	{	PUNCT
ejpam-2553	180	27	0	0	NUM
ejpam-2553	180	28	}	}	PUNCT
ejpam-2553	180	29	.	.	PUNCT
ejpam-2553	181	1	theorem	theorem	ADJ
ejpam-2553	181	2	4	4	NUM
ejpam-2553	181	3	.	.	PUNCT
ejpam-2553	182	1	let	let	VERB
ejpam-2553	182	2	(	(	PUNCT
ejpam-2553	182	3	r,+	r,+	NUM
ejpam-2553	182	4	,	,	PUNCT
ejpam-2553	182	5	.	.	PUNCT
ejpam-2553	182	6	)	)	PUNCT
ejpam-2553	183	1	be	be	AUX
ejpam-2553	183	2	a	a	DET
ejpam-2553	183	3	multiplicative	multiplicative	ADJ
ejpam-2553	183	4	hyperring	hyperring	NOUN
ejpam-2553	183	5	such	such	ADJ
ejpam-2553	183	6	that	that	SCONJ
ejpam-2553	183	7	it	it	PRON
ejpam-2553	183	8	has	have	VERB
ejpam-2553	183	9	zero	zero	NUM
ejpam-2553	183	10	absorbing	absorb	VERB
ejpam-2553	183	11	property	property	NOUN
ejpam-2553	183	12	.	.	PUNCT
ejpam-2553	184	1	then	then	ADV
ejpam-2553	184	2	mn(r)/γ	mn(r)/γ	VERB
ejpam-2553	184	3	∗	∗	NOUN
ejpam-2553	184	4	∼=	∼=	PART
ejpam-2553	184	5	mn(r	mn(r	NOUN
ejpam-2553	184	6	/	/	SYM
ejpam-2553	184	7	γ	γ	NOUN
ejpam-2553	184	8	∗	∗	NOUN
ejpam-2553	184	9	)	)	PUNCT
ejpam-2553	184	10	.	.	PUNCT
ejpam-2553	185	1	proof	proof	NOUN
ejpam-2553	185	2	.	.	PUNCT
ejpam-2553	186	1	consider	consider	VERB
ejpam-2553	186	2	the	the	DET
ejpam-2553	186	3	projection	projection	NOUN
ejpam-2553	186	4	homomorphism	homomorphism	PROPN
ejpam-2553	186	5	φ	φ	X
ejpam-2553	186	6	:	:	PUNCT
ejpam-2553	186	7	mn(r)→	mn(r)→	VERB
ejpam-2553	186	8	mn(r	mn(r	NOUN
ejpam-2553	186	9	/	/	SYM
ejpam-2553	186	10	γ∗	γ∗	NOUN
ejpam-2553	186	11	)	)	PUNCT
ejpam-2553	186	12	defined	define	VERB
ejpam-2553	186	13	by	by	ADP
ejpam-2553	186	14	φ((ai	φ((ai	PROPN
ejpam-2553	186	15	j)n×n	j)n×n	PROPN
ejpam-2553	186	16	)	)	PUNCT
ejpam-2553	186	17	=	=	PUNCT
ejpam-2553	187	1	(	(	PUNCT
ejpam-2553	187	2	γ∗(ai	γ∗(ai	PROPN
ejpam-2553	187	3	j))n×n	j))n×n	PROPN
ejpam-2553	187	4	for	for	ADP
ejpam-2553	187	5	all	all	PRON
ejpam-2553	187	6	ai	ai	VERB
ejpam-2553	187	7	j	j	PROPN
ejpam-2553	187	8	∈	∈	PROPN
ejpam-2553	187	9	r	r	NOUN
ejpam-2553	187	10	and	and	CCONJ
ejpam-2553	187	11	1≤	1≤	NUM
ejpam-2553	188	1	i	i	PRON
ejpam-2553	188	2	,	,	PUNCT
ejpam-2553	188	3	j	j	PROPN
ejpam-2553	188	4	≤	≤	PROPN
ejpam-2553	188	5	n.	n.	NOUN
ejpam-2553	188	6	we	we	PRON
ejpam-2553	188	7	denote	denote	VERB
ejpam-2553	188	8	the	the	DET
ejpam-2553	188	9	equivalence	equivalence	NOUN
ejpam-2553	188	10	relation	relation	NOUN
ejpam-2553	188	11	associated	associate	VERB
ejpam-2553	188	12	with	with	ADP
ejpam-2553	188	13	φ	φ	PROPN
ejpam-2553	188	14	by	by	ADP
ejpam-2553	188	15	ρ	ρ	PROPN
ejpam-2553	188	16	.	.	PUNCT
ejpam-2553	189	1	that	that	PRON
ejpam-2553	189	2	is	is	ADV
ejpam-2553	189	3	,	,	PUNCT
ejpam-2553	189	4	(	(	PUNCT
ejpam-2553	189	5	ai	ai	INTJ
ejpam-2553	189	6	j)n×nρ(bi	j)n×nρ(bi	PROPN
ejpam-2553	189	7	j)n×n	j)n×n	PROPN
ejpam-2553	189	8	⇐	⇐	ADJ
ejpam-2553	189	9	⇒(γ∗(ai	⇒(γ∗(ai	PROPN
ejpam-2553	189	10	j))n×n	j))n×n	PROPN
ejpam-2553	189	11	=(	=(	NOUN
ejpam-2553	189	12	γ∗(bi	γ∗(bi	PROPN
ejpam-2553	189	13	j))n×n	j))n×n	PROPN
ejpam-2553	189	14	,	,	PUNCT
ejpam-2553	189	15	∀ai	∀ai	PROPN
ejpam-2553	189	16	j	j	PROPN
ejpam-2553	189	17	,	,	PUNCT
ejpam-2553	189	18	bi	bi	NOUN
ejpam-2553	189	19	j	j	PROPN
ejpam-2553	189	20	∈	∈	PROPN
ejpam-2553	189	21	r	r	PROPN
ejpam-2553	189	22	,	,	PUNCT
ejpam-2553	189	23	1≤	1≤	NOUN
ejpam-2553	190	1	i	i	PROPN
ejpam-2553	190	2	,	,	PUNCT
ejpam-2553	190	3	j	j	PROPN
ejpam-2553	190	4	≤	≤	PROPN
ejpam-2553	190	5	n.	n.	NOUN
ejpam-2553	190	6	in	in	ADP
ejpam-2553	190	7	fact	fact	NOUN
ejpam-2553	190	8	,	,	PUNCT
ejpam-2553	190	9	ρ	ρ	PROPN
ejpam-2553	190	10	=	=	SYM
ejpam-2553	190	11	ker(φ	ker(φ	NOUN
ejpam-2553	190	12	)	)	PUNCT
ejpam-2553	190	13	.	.	PUNCT
ejpam-2553	191	1	since	since	SCONJ
ejpam-2553	191	2	φ	φ	PROPN
ejpam-2553	191	3	is	be	AUX
ejpam-2553	191	4	an	an	DET
ejpam-2553	191	5	epimorphism	epimorphism	NOUN
ejpam-2553	191	6	,	,	PUNCT
ejpam-2553	191	7	we	we	PRON
ejpam-2553	191	8	have	have	VERB
ejpam-2553	191	9	mn(r)/ρ	mn(r)/ρ	NOUN
ejpam-2553	191	10	=	=	SYM
ejpam-2553	191	11	mn(r)/ker(φ)∼=	mn(r)/ker(φ)∼=	NUM
ejpam-2553	191	12	mn(r	mn(r	NOUN
ejpam-2553	191	13	/	/	SYM
ejpam-2553	191	14	γ	γ	NOUN
ejpam-2553	191	15	∗	∗	NOUN
ejpam-2553	191	16	)	)	PUNCT
ejpam-2553	191	17	.	.	PUNCT
ejpam-2553	192	1	we	we	PRON
ejpam-2553	192	2	know	know	VERB
ejpam-2553	192	3	that	that	SCONJ
ejpam-2553	192	4	mn(r	mn(r	NOUN
ejpam-2553	192	5	/	/	SYM
ejpam-2553	192	6	γ∗	γ∗	PROPN
ejpam-2553	192	7	)	)	PUNCT
ejpam-2553	192	8	is	be	AUX
ejpam-2553	192	9	a	a	DET
ejpam-2553	192	10	ring	ring	NOUN
ejpam-2553	192	11	,	,	PUNCT
ejpam-2553	192	12	and	and	CCONJ
ejpam-2553	192	13	so	so	ADV
ejpam-2553	192	14	mn(r)/ρ	mn(r)/ρ	INTJ
ejpam-2553	192	15	is	be	AUX
ejpam-2553	192	16	a	a	DET
ejpam-2553	192	17	ring	ring	NOUN
ejpam-2553	192	18	.	.	PUNCT
ejpam-2553	193	1	thus	thus	ADV
ejpam-2553	193	2	γ∗	γ∗	VERB
ejpam-2553	193	3	⊆	⊆	NUM
ejpam-2553	193	4	ρ	ρ	NOUN
ejpam-2553	193	5	,	,	PUNCT
ejpam-2553	193	6	since	since	SCONJ
ejpam-2553	193	7	γ∗	γ∗	NOUN
ejpam-2553	193	8	is	be	AUX
ejpam-2553	193	9	the	the	DET
ejpam-2553	193	10	smallest	small	ADJ
ejpam-2553	193	11	equivalence	equivalence	NOUN
ejpam-2553	193	12	relation	relation	NOUN
ejpam-2553	193	13	on	on	ADP
ejpam-2553	193	14	mn(r	mn(r	NOUN
ejpam-2553	193	15	)	)	PUNCT
ejpam-2553	193	16	such	such	ADJ
ejpam-2553	193	17	that	that	SCONJ
ejpam-2553	193	18	mn(r)/γ∗	mn(r)/γ∗	PROPN
ejpam-2553	193	19	is	be	AUX
ejpam-2553	193	20	a	a	DET
ejpam-2553	193	21	ring	ring	NOUN
ejpam-2553	193	22	.	.	PUNCT
ejpam-2553	194	1	let	let	VERB
ejpam-2553	194	2	(	(	PUNCT
ejpam-2553	194	3	a′i	a′i	ADP
ejpam-2553	194	4	j)n×nρ(ai	j)n×nρ(ai	PROPN
ejpam-2553	194	5	j)n×n	j)n×n	PROPN
ejpam-2553	194	6	for	for	ADP
ejpam-2553	194	7	all	all	PRON
ejpam-2553	194	8	ai	ai	VERB
ejpam-2553	194	9	j	j	PROPN
ejpam-2553	194	10	∈	∈	PROPN
ejpam-2553	194	11	r	r	NOUN
ejpam-2553	194	12	and	and	CCONJ
ejpam-2553	194	13	1≤	1≤	NUM
ejpam-2553	195	1	i	i	PRON
ejpam-2553	195	2	,	,	PUNCT
ejpam-2553	195	3	j	j	PROPN
ejpam-2553	195	4	≤	≤	PROPN
ejpam-2553	195	5	n.	n.	NOUN
ejpam-2553	195	6	hence	hence	ADV
ejpam-2553	195	7	(	(	PUNCT
ejpam-2553	195	8	γ∗(a′i	γ∗(a′i	PROPN
ejpam-2553	195	9	j))n×n	j))n×n	PROPN
ejpam-2553	195	10	=	=	PUNCT
ejpam-2553	195	11	(	(	PUNCT
ejpam-2553	195	12	γ	γ	X
ejpam-2553	195	13	∗(ai	∗(ai	ADJ
ejpam-2553	195	14	j))n×n	j))n×n	ADJ
ejpam-2553	195	15	⇐	⇐	ADJ
ejpam-2553	195	16	⇒	⇒	PROPN
ejpam-2553	195	17	γ∗(a′i	γ∗(a′i	PROPN
ejpam-2553	195	18	j	j	PROPN
ejpam-2553	195	19	)	)	PUNCT
ejpam-2553	195	20	=	=	SYM
ejpam-2553	195	21	γ	γ	PROPN
ejpam-2553	195	22	∗(ai	∗(ai	PROPN
ejpam-2553	195	23	j	j	PROPN
ejpam-2553	195	24	)	)	PUNCT
ejpam-2553	195	25	,	,	PUNCT
ejpam-2553	195	26	∀ai	∀ai	PROPN
ejpam-2553	195	27	j	j	PROPN
ejpam-2553	195	28	∈	∈	PROPN
ejpam-2553	195	29	r.	r.	PROPN
ejpam-2553	195	30	then	then	ADV
ejpam-2553	195	31	we	we	PRON
ejpam-2553	195	32	conclude	conclude	VERB
ejpam-2553	195	33	that	that	SCONJ
ejpam-2553	195	34	ai	ai	VERB
ejpam-2553	195	35	j	j	PROPN
ejpam-2553	195	36	,	,	PUNCT
ejpam-2553	195	37	a′i	a′i	PROPN
ejpam-2553	196	1	j	j	PROPN
ejpam-2553	196	2	∈	∈	PROPN
ejpam-2553	196	3	m	m	VERB
ejpam-2553	196	4	∑	∑	PUNCT
ejpam-2553	196	5	s=1	s=1	X
ejpam-2553	196	6	ks	ks	PROPN
ejpam-2553	196	7	∏	∏	PROPN
ejpam-2553	196	8	t=1	t=1	PROPN
ejpam-2553	196	9	xst	xst	PROPN
ejpam-2553	196	10	for	for	ADP
ejpam-2553	196	11	some	some	PRON
ejpam-2553	196	12	(	(	PUNCT
ejpam-2553	196	13	xs1	xs1	PROPN
ejpam-2553	196	14	,	,	PUNCT
ejpam-2553	196	15	.	.	PUNCT
ejpam-2553	196	16	.	.	PUNCT
ejpam-2553	196	17	.	.	PUNCT
ejpam-2553	197	1	,	,	PUNCT
ejpam-2553	197	2	xsks	xsks	PROPN
ejpam-2553	197	3	)	)	PUNCT
ejpam-2553	198	1	∈	∈	PROPN
ejpam-2553	198	2	rks	rk	NOUN
ejpam-2553	198	3	and	and	CCONJ
ejpam-2553	198	4	1≤	1≤	NUM
ejpam-2553	199	1	i	i	PRON
ejpam-2553	199	2	,	,	PUNCT
ejpam-2553	199	3	j	j	PROPN
ejpam-2553	199	4	≤	≤	PROPN
ejpam-2553	199	5	n.	n.	NOUN
ejpam-2553	199	6	then	then	ADV
ejpam-2553	199	7	we	we	PRON
ejpam-2553	199	8	have	have	VERB
ejpam-2553	199	9	(	(	PUNCT
ejpam-2553	199	10	ai	ai	VERB
ejpam-2553	199	11	j)n×n	j)n×n	PROPN
ejpam-2553	199	12	,	,	PUNCT
ejpam-2553	199	13	(	(	PUNCT
ejpam-2553	199	14	a′i	a′i	PROPN
ejpam-2553	200	1	j)n×n	j)n×n	PROPN
ejpam-2553	200	2	∈	∈	PROPN
ejpam-2553	200	3	(	(	PUNCT
ejpam-2553	200	4	m	m	NOUN
ejpam-2553	200	5	∑	∑	PUNCT
ejpam-2553	200	6	s=1	s=1	X
ejpam-2553	200	7	ks	ks	X
ejpam-2553	200	8	∏	∏	PROPN
ejpam-2553	200	9	t=1	t=1	PROPN
ejpam-2553	200	10	x	x	PROPN
ejpam-2553	201	1	i	i	PRON
ejpam-2553	201	2	j	j	PROPN
ejpam-2553	201	3	st)n×n	st)n×n	PROPN
ejpam-2553	201	4	=	=	X
ejpam-2553	201	5	m	m	VERB
ejpam-2553	201	6	∑	∑	PUNCT
ejpam-2553	201	7	s=1	s=1	X
ejpam-2553	201	8	(	(	PUNCT
ejpam-2553	201	9	ks	ks	PROPN
ejpam-2553	201	10	∏	∏	PROPN
ejpam-2553	201	11	t=1	t=1	PROPN
ejpam-2553	201	12	x	x	PROPN
ejpam-2553	202	1	i	i	PRON
ejpam-2553	202	2	j	j	PROPN
ejpam-2553	202	3	st)n×n	st)n×n	PROPN
ejpam-2553	202	4	=	=	X
ejpam-2553	202	5	m	m	VERB
ejpam-2553	202	6	∑	∑	PUNCT
ejpam-2553	202	7	s=1	s=1	PROPN
ejpam-2553	202	8	n	n	CCONJ
ejpam-2553	202	9	∑	∑	ADV
ejpam-2553	202	10	i=1	i=1	PROPN
ejpam-2553	202	11	,	,	PUNCT
ejpam-2553	202	12	j=1	j=1	PROPN
ejpam-2553	202	13	ai	ai	VERB
ejpam-2553	202	14	j	j	PROPN
ejpam-2553	202	15	,	,	PUNCT
ejpam-2553	202	16	where	where	SCONJ
ejpam-2553	202	17	ai	ai	VERB
ejpam-2553	202	18	j	j	PROPN
ejpam-2553	202	19	=	=	PUNCT
ejpam-2553	202	20	(	(	PUNCT
ejpam-2553	202	21	bpq)n×n	bpq)n×n	PROPN
ejpam-2553	202	22	,	,	PUNCT
ejpam-2553	202	23	such	such	ADJ
ejpam-2553	202	24	that	that	DET
ejpam-2553	202	25	bpq	bpq	NOUN
ejpam-2553	202	26	=	=	SYM
ejpam-2553	202	27	¨	¨	X
ejpam-2553	202	28	∏ks	∏ks	NOUN
ejpam-2553	202	29	t=1	t=1	PROPN
ejpam-2553	202	30	x	x	SYM
ejpam-2553	203	1	i	i	PRON
ejpam-2553	203	2	j	j	PROPN
ejpam-2553	203	3	st	st	PROPN
ejpam-2553	204	1	if	if	SCONJ
ejpam-2553	204	2	p	p	X
ejpam-2553	204	3	=	=	VERB
ejpam-2553	204	4	i	i	PROPN
ejpam-2553	204	5	,	,	PUNCT
ejpam-2553	204	6	q	q	PROPN
ejpam-2553	204	7	=	=	SYM
ejpam-2553	204	8	j	j	PROPN
ejpam-2553	204	9	,	,	PUNCT
ejpam-2553	204	10	0	0	NUM
ejpam-2553	204	11	otherwise	otherwise	PROPN
ejpam-2553	204	12	r.	r.	PROPN
ejpam-2553	204	13	ameri	ameri	PROPN
ejpam-2553	204	14	,	,	PUNCT
ejpam-2553	204	15	a.	a.	PROPN
ejpam-2553	204	16	kordi	kordi	PROPN
ejpam-2553	204	17	/	/	PUNCT
ejpam-2553	204	18	eur	eur	PROPN
ejpam-2553	204	19	.	.	PUNCT
ejpam-2553	205	1	j.	j.	PROPN
ejpam-2553	205	2	pure	pure	PROPN
ejpam-2553	205	3	appl	appl	PROPN
ejpam-2553	205	4	.	.	PROPN
ejpam-2553	205	5	math	math	PROPN
ejpam-2553	205	6	,	,	PUNCT
ejpam-2553	205	7	9	9	NUM
ejpam-2553	205	8	(	(	PUNCT
ejpam-2553	205	9	2016	2016	NUM
ejpam-2553	205	10	)	)	PUNCT
ejpam-2553	205	11	,	,	PUNCT
ejpam-2553	205	12	402	402	NUM
ejpam-2553	205	13	-	-	SYM
ejpam-2553	205	14	418	418	NUM
ejpam-2553	205	15	408	408	NUM
ejpam-2553	205	16	now	now	ADV
ejpam-2553	205	17	by	by	ADP
ejpam-2553	205	18	the	the	DET
ejpam-2553	205	19	remark	remark	NOUN
ejpam-2553	205	20	2	2	NUM
ejpam-2553	205	21	,	,	PUNCT
ejpam-2553	205	22	we	we	PRON
ejpam-2553	205	23	have	have	AUX
ejpam-2553	205	24	ai	ai	PROPN
ejpam-2553	205	25	j	j	PROPN
ejpam-2553	206	1	=	=	PUNCT
ejpam-2553	207	1	bi	bi	PROPN
ejpam-2553	208	1	j(b	j(b	PROPN
ejpam-2553	208	2	j	j	PROPN
ejpam-2553	208	3	j)ks−1	j)ks−1	PROPN
ejpam-2553	208	4	for	for	ADP
ejpam-2553	208	5	s	s	NOUN
ejpam-2553	208	6	=	=	SYM
ejpam-2553	208	7	1	1	NUM
ejpam-2553	208	8	,	,	PUNCT
ejpam-2553	208	9	.	.	PUNCT
ejpam-2553	208	10	.	.	PUNCT
ejpam-2553	208	11	.	.	PUNCT
ejpam-2553	209	1	m	m	PROPN
ejpam-2553	209	2	,	,	PUNCT
ejpam-2553	209	3	where	where	SCONJ
ejpam-2553	209	4	bi	bi	PROPN
ejpam-2553	209	5	j	j	PROPN
ejpam-2553	209	6	=	=	PRON
ejpam-2553	209	7	(	(	PUNCT
ejpam-2553	209	8	cuv)n×n	cuv)n×n	PROPN
ejpam-2553	209	9	,	,	PUNCT
ejpam-2553	209	10	where	where	SCONJ
ejpam-2553	209	11	cuv	cuv	PROPN
ejpam-2553	209	12	=	=	PUNCT
ejpam-2553	209	13	¨	¨	NOUN
ejpam-2553	209	14	x	x	PUNCT
ejpam-2553	209	15	i	i	PRON
ejpam-2553	209	16	j	j	PROPN
ejpam-2553	209	17	st	st	PROPN
ejpam-2553	210	1	if	if	SCONJ
ejpam-2553	210	2	u=	u=	PROPN
ejpam-2553	210	3	s	s	NUM
ejpam-2553	210	4	,	,	PUNCT
ejpam-2553	210	5	v	v	NOUN
ejpam-2553	210	6	=	=	SYM
ejpam-2553	210	7	t	t	PROPN
ejpam-2553	210	8	,	,	PUNCT
ejpam-2553	210	9	0	0	NUM
ejpam-2553	211	1	otherwise	otherwise	ADV
ejpam-2553	211	2	so	so	ADV
ejpam-2553	211	3	we	we	PRON
ejpam-2553	211	4	have	have	VERB
ejpam-2553	211	5	{	{	PUNCT
ejpam-2553	211	6	(	(	PUNCT
ejpam-2553	211	7	ai	ai	INTJ
ejpam-2553	211	8	j)n×n	j)n×n	PROPN
ejpam-2553	211	9	,	,	PUNCT
ejpam-2553	211	10	(	(	PUNCT
ejpam-2553	211	11	a′i	a′i	ADP
ejpam-2553	211	12	j)n×n	j)n×n	PROPN
ejpam-2553	211	13	}	}	PUNCT
ejpam-2553	211	14	∈	∈	NOUN
ejpam-2553	211	15	m	m	VERB
ejpam-2553	211	16	∑	∑	PUNCT
ejpam-2553	211	17	s=1	s=1	PROPN
ejpam-2553	211	18	n	n	CCONJ
ejpam-2553	211	19	∑	∑	ADV
ejpam-2553	211	20	i=1	i=1	PROPN
ejpam-2553	211	21	,	,	PUNCT
ejpam-2553	211	22	j=1	j=1	PROPN
ejpam-2553	211	23	ai	ai	VERB
ejpam-2553	211	24	j	j	PROPN
ejpam-2553	211	25	=	=	PUNCT
ejpam-2553	211	26	m	m	VERB
ejpam-2553	211	27	∑	∑	PUNCT
ejpam-2553	211	28	s=1	s=1	PROPN
ejpam-2553	211	29	n	n	CCONJ
ejpam-2553	211	30	∑	∑	ADV
ejpam-2553	211	31	i=1	i=1	PROPN
ejpam-2553	211	32	,	,	PUNCT
ejpam-2553	211	33	j=1	j=1	ADJ
ejpam-2553	211	34	bi	bi	ADJ
ejpam-2553	211	35	j(b	j(b	PROPN
ejpam-2553	211	36	j	j	PROPN
ejpam-2553	211	37	j)ks−1	j)ks−1	PROPN
ejpam-2553	211	38	,	,	PUNCT
ejpam-2553	211	39	i.e.	i.e.	X
ejpam-2553	211	40	,	,	PUNCT
ejpam-2553	211	41	(	(	PUNCT
ejpam-2553	211	42	ai	ai	VERB
ejpam-2553	211	43	j)n×nγ(a′i	j)n×nγ(a′i	PROPN
ejpam-2553	211	44	j)n×n	j)n×n	PROPN
ejpam-2553	211	45	.	.	PUNCT
ejpam-2553	212	1	hence	hence	ADV
ejpam-2553	212	2	(	(	PUNCT
ejpam-2553	212	3	ai	ai	INTJ
ejpam-2553	212	4	j)n×nγ	j)n×nγ	PROPN
ejpam-2553	212	5	∗(a′i	∗(a′i	PROPN
ejpam-2553	213	1	j)n×n	j)n×n	PROPN
ejpam-2553	213	2	.	.	PUNCT
ejpam-2553	214	1	consequently	consequently	ADV
ejpam-2553	214	2	,	,	PUNCT
ejpam-2553	214	3	(	(	PUNCT
ejpam-2553	214	4	a′i	a′i	VERB
ejpam-2553	214	5	j)n×n	j)n×n	PROPN
ejpam-2553	214	6	∈	∈	PROPN
ejpam-2553	214	7	γ∗((ai	γ∗((ai	PUNCT
ejpam-2553	214	8	j)n×n	j)n×n	PROPN
ejpam-2553	214	9	)	)	PUNCT
ejpam-2553	214	10	and	and	CCONJ
ejpam-2553	214	11	therefore	therefore	ADV
ejpam-2553	214	12	ρ	ρ	PROPN
ejpam-2553	214	13	⊆	⊆	NUM
ejpam-2553	214	14	γ∗.	γ∗.	ADV
ejpam-2553	214	15	then	then	ADV
ejpam-2553	214	16	γ∗	γ∗	PROPN
ejpam-2553	214	17	=	=	PROPN
ejpam-2553	214	18	ρ	ρ	PROPN
ejpam-2553	214	19	and	and	CCONJ
ejpam-2553	214	20	so	so	ADV
ejpam-2553	214	21	mn(r)/γ∗	mn(r)/γ∗	VERB
ejpam-2553	214	22	∼=	∼=	PART
ejpam-2553	214	23	mn(r	mn(r	NOUN
ejpam-2553	214	24	/	/	SYM
ejpam-2553	214	25	γ∗	γ∗	NOUN
ejpam-2553	214	26	)	)	PUNCT
ejpam-2553	214	27	.	.	PUNCT
ejpam-2553	215	1	theorem	theorem	NOUN
ejpam-2553	215	2	5	5	NUM
ejpam-2553	215	3	.	.	PUNCT
ejpam-2553	216	1	let	let	VERB
ejpam-2553	216	2	r	r	PRON
ejpam-2553	216	3	be	be	AUX
ejpam-2553	216	4	a	a	DET
ejpam-2553	216	5	multiplicative	multiplicative	ADJ
ejpam-2553	216	6	hyperring	hyperring	NOUN
ejpam-2553	216	7	.	.	PUNCT
ejpam-2553	217	1	r	r	NOUN
ejpam-2553	217	2	is	be	AUX
ejpam-2553	217	3	n	n	ADV
ejpam-2553	217	4	-	-	PUNCT
ejpam-2553	217	5	complete	complete	ADJ
ejpam-2553	217	6	(	(	PUNCT
ejpam-2553	217	7	n	n	CCONJ
ejpam-2553	217	8	-	-	PUNCT
ejpam-2553	217	9	mc	mc	NOUN
ejpam-2553	217	10	)	)	PUNCT
ejpam-2553	218	1	if	if	SCONJ
ejpam-2553	218	2	and	and	CCONJ
ejpam-2553	218	3	only	only	ADV
ejpam-2553	218	4	if	if	SCONJ
ejpam-2553	218	5	mn(r	mn(r	NOUN
ejpam-2553	218	6	)	)	PUNCT
ejpam-2553	218	7	is	be	AUX
ejpam-2553	218	8	n	n	X
ejpam-2553	218	9	-	-	PUNCT
ejpam-2553	218	10	complete	complete	ADJ
ejpam-2553	218	11	(	(	PUNCT
ejpam-2553	218	12	n	n	CCONJ
ejpam-2553	218	13	-	-	PUNCT
ejpam-2553	218	14	mc	mc	NOUN
ejpam-2553	218	15	)	)	PUNCT
ejpam-2553	218	16	.	.	PUNCT
ejpam-2553	219	1	proof	proof	NOUN
ejpam-2553	219	2	.	.	PUNCT
ejpam-2553	220	1	(	(	PUNCT
ejpam-2553	220	2	⇒	⇒	NOUN
ejpam-2553	220	3	)	)	PUNCT
ejpam-2553	220	4	assume	assume	VERB
ejpam-2553	220	5	that	that	SCONJ
ejpam-2553	220	6	r	r	NOUN
ejpam-2553	220	7	is	be	AUX
ejpam-2553	220	8	n	n	ADV
ejpam-2553	220	9	-	-	PUNCT
ejpam-2553	220	10	complete	complete	ADJ
ejpam-2553	220	11	and	and	CCONJ
ejpam-2553	220	12	t	t	NOUN
ejpam-2553	220	13	=	=	PUNCT
ejpam-2553	220	14	(	(	PUNCT
ejpam-2553	220	15	ai	ai	VERB
ejpam-2553	220	16	j)n×n	j)n×n	PROPN
ejpam-2553	220	17	∈	∈	PROPN
ejpam-2553	220	18	γ	γ	X
ejpam-2553	220	19	(	(	PUNCT
ejpam-2553	220	20	∑n	∑n	PROPN
ejpam-2553	220	21	s=1	s=1	PROPN
ejpam-2553	220	22	(	(	PUNCT
ejpam-2553	220	23	∏ks	∏ks	NOUN
ejpam-2553	220	24	t=1(xst)n×n	t=1(xst)n×n	ADJ
ejpam-2553	220	25	)	)	PUNCT
ejpam-2553	220	26	)	)	PUNCT
ejpam-2553	220	27	,	,	PUNCT
ejpam-2553	220	28	then	then	ADV
ejpam-2553	220	29	{	{	PUNCT
ejpam-2553	220	30	(	(	PUNCT
ejpam-2553	220	31	ai	ai	VERB
ejpam-2553	220	32	j)n×n	j)n×n	PROPN
ejpam-2553	220	33	,	,	PUNCT
ejpam-2553	220	34	n	n	CCONJ
ejpam-2553	220	35	∑	∑	ADV
ejpam-2553	220	36	s=1	s=1	X
ejpam-2553	220	37	(	(	PUNCT
ejpam-2553	220	38	ks	ks	X
ejpam-2553	220	39	∏	∏	PROPN
ejpam-2553	220	40	t=1	t=1	PROPN
ejpam-2553	220	41	(	(	PUNCT
ejpam-2553	220	42	xst)n×n	xst)n×n	NOUN
ejpam-2553	220	43	)	)	PUNCT
ejpam-2553	220	44	}	}	PUNCT
ejpam-2553	220	45	⊆	⊆	NUM
ejpam-2553	220	46	n	n	PROPN
ejpam-2553	220	47	∑	∑	ADP
ejpam-2553	220	48	z=1	z=1	PROPN
ejpam-2553	220	49	(	(	PUNCT
ejpam-2553	220	50	wz	wz	ADP
ejpam-2553	220	51	∏	∏	PROPN
ejpam-2553	221	1	`	`	PUNCT
ejpam-2553	221	2	=	=	SYM
ejpam-2553	221	3	1	1	NUM
ejpam-2553	221	4	(	(	PUNCT
ejpam-2553	221	5	yuv	yuv	PROPN
ejpam-2553	221	6	)	)	PUNCT
ejpam-2553	221	7	z	z	NOUN
ejpam-2553	221	8	`	`	PUNCT
ejpam-2553	221	9	n×n	n×n	PROPN
ejpam-2553	221	10	)	)	PUNCT
ejpam-2553	221	11	.	.	PUNCT
ejpam-2553	222	1	now	now	ADV
ejpam-2553	222	2	,	,	PUNCT
ejpam-2553	222	3	for	for	ADP
ejpam-2553	222	4	convenience	convenience	NOUN
ejpam-2553	222	5	,	,	PUNCT
ejpam-2553	222	6	let	let	VERB
ejpam-2553	222	7	a=	a=	VERB
ejpam-2553	222	8	(	(	PUNCT
ejpam-2553	222	9	ai	ai	VERB
ejpam-2553	222	10	j)n×n	j)n×n	PROPN
ejpam-2553	222	11	=	=	PUNCT
ejpam-2553	222	12	n	n	NOUN
ejpam-2553	222	13	∑	∑	ADV
ejpam-2553	222	14	s=1	s=1	X
ejpam-2553	222	15	(	(	PUNCT
ejpam-2553	222	16	ks	ks	X
ejpam-2553	222	17	∏	∏	PROPN
ejpam-2553	222	18	t=1	t=1	PROPN
ejpam-2553	222	19	(	(	PUNCT
ejpam-2553	222	20	xst)n×n	xst)n×n	PROPN
ejpam-2553	222	21	)	)	PUNCT
ejpam-2553	222	22	and	and	CCONJ
ejpam-2553	222	23	b	b	X
ejpam-2553	223	1	=	=	SYM
ejpam-2553	223	2	(	(	PUNCT
ejpam-2553	223	3	bi	bi	NOUN
ejpam-2553	223	4	j)n×n	j)n×n	PROPN
ejpam-2553	223	5	=	=	PROPN
ejpam-2553	223	6	n	n	CCONJ
ejpam-2553	223	7	∑	∑	ADP
ejpam-2553	223	8	z=1	z=1	PROPN
ejpam-2553	223	9	(	(	PUNCT
ejpam-2553	223	10	wz	wz	ADP
ejpam-2553	223	11	∏	∏	PROPN
ejpam-2553	223	12	`	`	PUNCT
ejpam-2553	223	13	=	=	SYM
ejpam-2553	223	14	1	1	NUM
ejpam-2553	223	15	(	(	PUNCT
ejpam-2553	223	16	yuv	yuv	PROPN
ejpam-2553	223	17	)	)	PUNCT
ejpam-2553	223	18	z	z	NOUN
ejpam-2553	223	19	`	`	PUNCT
ejpam-2553	223	20	n×n	n×n	PROPN
ejpam-2553	223	21	)	)	PUNCT
ejpam-2553	223	22	,	,	PUNCT
ejpam-2553	223	23	then	then	ADV
ejpam-2553	223	24	ai	ai	VERB
ejpam-2553	223	25	j	j	PROPN
ejpam-2553	223	26	∈	∈	PROPN
ejpam-2553	223	27	bi	bi	PROPN
ejpam-2553	223	28	j	j	PROPN
ejpam-2553	223	29	,	,	PUNCT
ejpam-2553	223	30	ai	ai	VERB
ejpam-2553	223	31	j	j	PROPN
ejpam-2553	223	32	⊆	⊆	NUM
ejpam-2553	223	33	bi	bi	PROPN
ejpam-2553	223	34	j	j	PROPN
ejpam-2553	223	35	,	,	PUNCT
ejpam-2553	223	36	so	so	ADV
ejpam-2553	223	37	γ(ai	γ(ai	PROPN
ejpam-2553	223	38	j	j	PROPN
ejpam-2553	223	39	)	)	PUNCT
ejpam-2553	223	40	=	=	SYM
ejpam-2553	224	1	γ(bi	γ(bi	ADP
ejpam-2553	224	2	j),γ(ai	j),γ(ai	PROPN
ejpam-2553	224	3	j	j	PROPN
ejpam-2553	224	4	)	)	PUNCT
ejpam-2553	224	5	=	=	PROPN
ejpam-2553	225	1	γ(bi	γ(bi	DET
ejpam-2553	225	2	j	j	PROPN
ejpam-2553	225	3	)	)	PUNCT
ejpam-2553	225	4	.	.	PUNCT
ejpam-2553	226	1	since	since	SCONJ
ejpam-2553	226	2	r	r	NOUN
ejpam-2553	226	3	is	be	AUX
ejpam-2553	226	4	n	n	ADV
ejpam-2553	226	5	-	-	PUNCT
ejpam-2553	226	6	complete	complete	ADJ
ejpam-2553	226	7	,	,	PUNCT
ejpam-2553	226	8	then	then	ADV
ejpam-2553	226	9	ai	ai	VERB
ejpam-2553	226	10	j	j	PROPN
ejpam-2553	226	11	∈	∈	PROPN
ejpam-2553	226	12	γ(bi	γ(bi	PROPN
ejpam-2553	226	13	j	j	PROPN
ejpam-2553	226	14	)	)	PUNCT
ejpam-2553	226	15	=	=	PROPN
ejpam-2553	227	1	γ(ai	γ(ai	PROPN
ejpam-2553	227	2	j	j	PROPN
ejpam-2553	227	3	)	)	PUNCT
ejpam-2553	228	1	=	=	VERB
ejpam-2553	228	2	ai	ai	VERB
ejpam-2553	228	3	j	j	PROPN
ejpam-2553	228	4	,	,	PUNCT
ejpam-2553	228	5	i.e.	i.e.	X
ejpam-2553	228	6	,	,	PUNCT
ejpam-2553	228	7	ai	ai	VERB
ejpam-2553	228	8	j	j	PROPN
ejpam-2553	228	9	∈	∈	PROPN
ejpam-2553	228	10	ai	ai	VERB
ejpam-2553	228	11	j	j	PROPN
ejpam-2553	228	12	for	for	ADP
ejpam-2553	228	13	all	all	DET
ejpam-2553	228	14	1	1	NUM
ejpam-2553	228	15	≤	≤	NUM
ejpam-2553	229	1	i	i	PRON
ejpam-2553	229	2	,	,	PUNCT
ejpam-2553	229	3	j	j	PROPN
ejpam-2553	229	4	≤	≤	PROPN
ejpam-2553	229	5	n.	n.	NOUN
ejpam-2553	229	6	hence	hence	ADV
ejpam-2553	229	7	(	(	PUNCT
ejpam-2553	229	8	ai	ai	VERB
ejpam-2553	229	9	j)n×n	j)n×n	PROPN
ejpam-2553	229	10	∈	∈	PROPN
ejpam-2553	229	11	a	a	PRON
ejpam-2553	229	12	,	,	PUNCT
ejpam-2553	229	13	and	and	CCONJ
ejpam-2553	229	14	so	so	ADV
ejpam-2553	229	15	γ	γ	PROPN
ejpam-2553	229	16	(	(	PUNCT
ejpam-2553	229	17	∑n	∑n	PROPN
ejpam-2553	229	18	s=1	s=1	PROPN
ejpam-2553	229	19	(	(	PUNCT
ejpam-2553	229	20	∏ks	∏ks	NOUN
ejpam-2553	229	21	t=1(xst)n×n	t=1(xst)n×n	ADJ
ejpam-2553	229	22	)	)	PUNCT
ejpam-2553	229	23	)	)	PUNCT
ejpam-2553	230	1	⊆	⊆	NUM
ejpam-2553	230	2	a.	a.	NOUN
ejpam-2553	230	3	(	(	PUNCT
ejpam-2553	230	4	⇐	⇐	NOUN
ejpam-2553	230	5	)	)	PUNCT
ejpam-2553	230	6	suppose	suppose	VERB
ejpam-2553	230	7	that	that	SCONJ
ejpam-2553	230	8	mn(r	mn(r	NOUN
ejpam-2553	230	9	)	)	PUNCT
ejpam-2553	230	10	is	be	AUX
ejpam-2553	230	11	n	n	PRON
ejpam-2553	230	12	-	-	PUNCT
ejpam-2553	230	13	complete	complete	ADJ
ejpam-2553	230	14	.	.	PUNCT
ejpam-2553	231	1	let	let	VERB
ejpam-2553	231	2	x	x	SYM
ejpam-2553	231	3	∈	∈	PROPN
ejpam-2553	231	4	γ	γ	X
ejpam-2553	231	5	(	(	PUNCT
ejpam-2553	231	6	∑n	∑n	PROPN
ejpam-2553	231	7	i=1	i=1	PROPN
ejpam-2553	231	8	(	(	PUNCT
ejpam-2553	231	9	∏ki	∏ki	NOUN
ejpam-2553	231	10	j=1	j=1	NOUN
ejpam-2553	231	11	x	x	PUNCT
ejpam-2553	231	12	i	i	PRON
ejpam-2553	231	13	j	j	PROPN
ejpam-2553	231	14	)	)	PUNCT
ejpam-2553	231	15	)	)	PUNCT
ejpam-2553	231	16	.	.	PUNCT
ejpam-2553	232	1	thus	thus	ADV
ejpam-2553	232	2	{	{	PUNCT
ejpam-2553	232	3	x	x	INTJ
ejpam-2553	232	4	,	,	PUNCT
ejpam-2553	232	5	n	n	PROPN
ejpam-2553	232	6	∑	∑	PROPN
ejpam-2553	232	7	i=1	i=1	PROPN
ejpam-2553	232	8	(	(	PUNCT
ejpam-2553	232	9	ki	ki	PROPN
ejpam-2553	232	10	∏	∏	PROPN
ejpam-2553	232	11	j=1	j=1	NOUN
ejpam-2553	232	12	x	x	PUNCT
ejpam-2553	232	13	i	i	PRON
ejpam-2553	232	14	j	j	PROPN
ejpam-2553	232	15	)	)	PUNCT
ejpam-2553	232	16	}	}	PUNCT
ejpam-2553	232	17	⊆	⊆	NUM
ejpam-2553	232	18	n	n	PROPN
ejpam-2553	232	19	∑	∑	PROPN
ejpam-2553	232	20	s=1	s=1	X
ejpam-2553	232	21	(	(	PUNCT
ejpam-2553	232	22	ks	ks	PROPN
ejpam-2553	232	23	∏	∏	PROPN
ejpam-2553	232	24	t=1	t=1	PROPN
ejpam-2553	232	25	yst	yst	PROPN
ejpam-2553	232	26	)	)	PUNCT
ejpam-2553	232	27	.	.	PUNCT
ejpam-2553	233	1	since	since	SCONJ
ejpam-2553	233	2	mn(r	mn(r	NOUN
ejpam-2553	233	3	)	)	PUNCT
ejpam-2553	233	4	is	be	AUX
ejpam-2553	233	5	n	n	ADV
ejpam-2553	233	6	-	-	PUNCT
ejpam-2553	233	7	complete	complete	ADJ
ejpam-2553	233	8	then	then	ADV
ejpam-2553	233	9	x	x	SYM
ejpam-2553	233	10	∈	∈	PROPN
ejpam-2553	233	11	∑n	∑n	PROPN
ejpam-2553	233	12	i=1	i=1	PROPN
ejpam-2553	233	13	(	(	PUNCT
ejpam-2553	233	14	∏ki	∏ki	NOUN
ejpam-2553	233	15	j=1	j=1	NOUN
ejpam-2553	233	16	x	x	PUNCT
ejpam-2553	233	17	i	i	PRON
ejpam-2553	233	18	j	j	PROPN
ejpam-2553	233	19	)	)	PUNCT
ejpam-2553	233	20	,	,	PUNCT
ejpam-2553	233	21	i.e.	i.e.	X
ejpam-2553	233	22	,	,	PUNCT
ejpam-2553	233	23	r	r	NOUN
ejpam-2553	233	24	is	be	AUX
ejpam-2553	233	25	n	n	ADV
ejpam-2553	233	26	-	-	PUNCT
ejpam-2553	233	27	complete	complete	ADJ
ejpam-2553	233	28	.	.	PUNCT
ejpam-2553	234	1	proposition	proposition	NOUN
ejpam-2553	234	2	2	2	NUM
ejpam-2553	234	3	.	.	PUNCT
ejpam-2553	235	1	(	(	PUNCT
ejpam-2553	235	2	[	[	X
ejpam-2553	235	3	6	6	NUM
ejpam-2553	235	4	]	]	PUNCT
ejpam-2553	235	5	)	)	PUNCT
ejpam-2553	235	6	let	let	VERB
ejpam-2553	235	7	r	r	PRON
ejpam-2553	235	8	be	be	AUX
ejpam-2553	235	9	a	a	DET
ejpam-2553	235	10	strongly	strongly	ADV
ejpam-2553	235	11	distributive	distributive	ADJ
ejpam-2553	235	12	multiplicative	multiplicative	ADJ
ejpam-2553	235	13	hyperring	hyperring	NOUN
ejpam-2553	235	14	and	and	CCONJ
ejpam-2553	235	15	a	a	DET
ejpam-2553	235	16	∈	∈	PROPN
ejpam-2553	235	17	r.	r.	NOUN
ejpam-2553	235	18	if	if	SCONJ
ejpam-2553	235	19	there	there	PRON
ejpam-2553	235	20	exists	exist	VERB
ejpam-2553	235	21	x	x	X
ejpam-2553	235	22	∈	∈	NOUN
ejpam-2553	235	23	r	r	NOUN
ejpam-2553	235	24	and	and	CCONJ
ejpam-2553	235	25	c	c	NOUN
ejpam-2553	235	26	∈	∈	PROPN
ejpam-2553	236	1	axa−	axa−	PROPN
ejpam-2553	236	2	a	a	DET
ejpam-2553	236	3	such	such	ADJ
ejpam-2553	236	4	that	that	SCONJ
ejpam-2553	236	5	c	c	PROPN
ejpam-2553	236	6	is	be	AUX
ejpam-2553	236	7	regular	regular	ADJ
ejpam-2553	236	8	,	,	PUNCT
ejpam-2553	236	9	then	then	ADV
ejpam-2553	236	10	a	a	PRON
ejpam-2553	236	11	is	be	AUX
ejpam-2553	236	12	regular	regular	ADJ
ejpam-2553	236	13	.	.	PUNCT
ejpam-2553	237	1	r.	r.	PROPN
ejpam-2553	237	2	ameri	ameri	PROPN
ejpam-2553	237	3	,	,	PUNCT
ejpam-2553	237	4	a.	a.	PROPN
ejpam-2553	237	5	kordi	kordi	PROPN
ejpam-2553	237	6	/	/	PUNCT
ejpam-2553	237	7	eur	eur	PROPN
ejpam-2553	237	8	.	.	PUNCT
ejpam-2553	238	1	j.	j.	PROPN
ejpam-2553	238	2	pure	pure	PROPN
ejpam-2553	238	3	appl	appl	PROPN
ejpam-2553	238	4	.	.	PROPN
ejpam-2553	238	5	math	math	PROPN
ejpam-2553	238	6	,	,	PUNCT
ejpam-2553	238	7	9	9	NUM
ejpam-2553	238	8	(	(	PUNCT
ejpam-2553	238	9	2016	2016	NUM
ejpam-2553	238	10	)	)	PUNCT
ejpam-2553	238	11	,	,	PUNCT
ejpam-2553	238	12	402	402	NUM
ejpam-2553	238	13	-	-	SYM
ejpam-2553	238	14	418	418	NUM
ejpam-2553	238	15	409	409	NUM
ejpam-2553	238	16	in	in	ADP
ejpam-2553	238	17	1950	1950	NUM
ejpam-2553	238	18	,	,	PUNCT
ejpam-2553	238	19	brown	brown	NOUN
ejpam-2553	238	20	and	and	CCONJ
ejpam-2553	238	21	mccoy	mccoy	PROPN
ejpam-2553	239	1	[	[	X
ejpam-2553	239	2	3	3	NUM
ejpam-2553	239	3	]	]	PUNCT
ejpam-2553	239	4	proved	prove	VERB
ejpam-2553	239	5	that	that	SCONJ
ejpam-2553	239	6	,	,	PUNCT
ejpam-2553	239	7	r	r	NOUN
ejpam-2553	239	8	is	be	AUX
ejpam-2553	239	9	a	a	DET
ejpam-2553	239	10	regular	regular	ADJ
ejpam-2553	239	11	ring	ring	NOUN
ejpam-2553	239	12	if	if	SCONJ
ejpam-2553	239	13	and	and	CCONJ
ejpam-2553	239	14	only	only	ADV
ejpam-2553	239	15	if	if	SCONJ
ejpam-2553	239	16	mn(r	mn(r	NOUN
ejpam-2553	239	17	)	)	PUNCT
ejpam-2553	239	18	is	be	AUX
ejpam-2553	239	19	regular	regular	ADJ
ejpam-2553	239	20	.	.	PUNCT
ejpam-2553	240	1	now	now	ADV
ejpam-2553	240	2	we	we	PRON
ejpam-2553	240	3	can	can	AUX
ejpam-2553	240	4	extend	extend	VERB
ejpam-2553	240	5	it	it	PRON
ejpam-2553	240	6	by	by	ADP
ejpam-2553	240	7	the	the	DET
ejpam-2553	240	8	following	follow	VERB
ejpam-2553	240	9	theorem	theorem	NOUN
ejpam-2553	240	10	:	:	PUNCT
ejpam-2553	240	11	theorem	theorem	NOUN
ejpam-2553	240	12	6	6	NUM
ejpam-2553	240	13	.	.	PUNCT
ejpam-2553	241	1	let	let	VERB
ejpam-2553	241	2	r	r	PRON
ejpam-2553	241	3	be	be	AUX
ejpam-2553	241	4	a	a	DET
ejpam-2553	241	5	strongly	strongly	ADV
ejpam-2553	241	6	distributive	distributive	ADJ
ejpam-2553	241	7	multiplicative	multiplicative	ADJ
ejpam-2553	241	8	hyperring	hyperring	NOUN
ejpam-2553	241	9	.	.	PUNCT
ejpam-2553	242	1	r	r	NOUN
ejpam-2553	242	2	is	be	AUX
ejpam-2553	242	3	regular	regular	ADJ
ejpam-2553	242	4	if	if	SCONJ
ejpam-2553	242	5	and	and	CCONJ
ejpam-2553	242	6	only	only	ADV
ejpam-2553	242	7	if	if	SCONJ
ejpam-2553	242	8	mn(r	mn(r	NOUN
ejpam-2553	242	9	)	)	PUNCT
ejpam-2553	242	10	is	be	AUX
ejpam-2553	242	11	regular	regular	ADJ
ejpam-2553	242	12	multiplicative	multiplicative	ADJ
ejpam-2553	242	13	hyperring	hyperring	NOUN
ejpam-2553	242	14	.	.	PUNCT
ejpam-2553	243	1	proof	proof	NOUN
ejpam-2553	243	2	.	.	PUNCT
ejpam-2553	244	1	(	(	PUNCT
ejpam-2553	244	2	⇒	⇒	PROPN
ejpam-2553	244	3	)	)	PUNCT
ejpam-2553	244	4	first	first	ADV
ejpam-2553	244	5	of	of	ADP
ejpam-2553	244	6	all	all	PRON
ejpam-2553	244	7	we	we	PRON
ejpam-2553	244	8	’ll	’ll	AUX
ejpam-2553	244	9	prove	prove	VERB
ejpam-2553	244	10	it	it	PRON
ejpam-2553	244	11	for	for	ADP
ejpam-2553	244	12	n	n	NOUN
ejpam-2553	244	13	=	=	SYM
ejpam-2553	244	14	2	2	NUM
ejpam-2553	244	15	,	,	PUNCT
ejpam-2553	244	16	and	and	CCONJ
ejpam-2553	244	17	then	then	ADV
ejpam-2553	244	18	we	we	PRON
ejpam-2553	244	19	will	will	AUX
ejpam-2553	244	20	extend	extend	VERB
ejpam-2553	244	21	it	it	PRON
ejpam-2553	244	22	to	to	ADP
ejpam-2553	244	23	arbitrary	arbitrary	ADJ
ejpam-2553	244	24	n.	n.	NOUN
ejpam-2553	244	25	assume	assume	VERB
ejpam-2553	244	26	that	that	SCONJ
ejpam-2553	244	27	a=	a=	PROPN
ejpam-2553	244	28	�	�	PROPN
ejpam-2553	244	29	a	a	DET
ejpam-2553	244	30	b	b	PROPN
ejpam-2553	244	31	c	c	NOUN
ejpam-2553	244	32	d	d	X
ejpam-2553	244	33	�	�	PROPN
ejpam-2553	244	34	∈	∈	PROPN
ejpam-2553	244	35	m2(r	m2(r	PROPN
ejpam-2553	244	36	)	)	PUNCT
ejpam-2553	244	37	,	,	PUNCT
ejpam-2553	244	38	since	since	SCONJ
ejpam-2553	244	39	b	b	NOUN
ejpam-2553	244	40	is	be	AUX
ejpam-2553	244	41	regular	regular	ADJ
ejpam-2553	244	42	there	there	ADV
ejpam-2553	244	43	exists	exist	VERB
ejpam-2553	244	44	b′	b′	NUM
ejpam-2553	244	45	∈	∈	NOUN
ejpam-2553	244	46	r	r	NOUN
ejpam-2553	244	47	such	such	DET
ejpam-2553	244	48	that	that	DET
ejpam-2553	244	49	b	b	PROPN
ejpam-2553	244	50	∈	∈	PROPN
ejpam-2553	244	51	bb′b	bb′b	NOUN
ejpam-2553	244	52	.	.	PUNCT
ejpam-2553	245	1	if	if	SCONJ
ejpam-2553	245	2	we	we	PRON
ejpam-2553	245	3	set	set	VERB
ejpam-2553	245	4	,	,	PUNCT
ejpam-2553	245	5	x	x	SYM
ejpam-2553	245	6	=	=	PUNCT
ejpam-2553	245	7	�	�	PROPN
ejpam-2553	245	8	0	0	NUM
ejpam-2553	245	9	0	0	NUM
ejpam-2553	245	10	b′	b′	NUM
ejpam-2553	245	11	0	0	NUM
ejpam-2553	245	12	�	�	PROPN
ejpam-2553	245	13	∈	∈	PROPN
ejpam-2553	245	14	m2(r	m2(r	PROPN
ejpam-2553	245	15	)	)	PUNCT
ejpam-2553	245	16	and	and	CCONJ
ejpam-2553	245	17	denote	denote	VERB
ejpam-2553	245	18	b	b	PROPN
ejpam-2553	245	19	=	=	SYM
ejpam-2553	245	20	axa−	axa−	PROPN
ejpam-2553	245	21	a	a	PRON
ejpam-2553	245	22	then	then	ADV
ejpam-2553	245	23	by	by	ADP
ejpam-2553	245	24	an	an	DET
ejpam-2553	245	25	easy	easy	ADJ
ejpam-2553	245	26	calculation	calculation	NOUN
ejpam-2553	245	27	we	we	PRON
ejpam-2553	245	28	can	can	AUX
ejpam-2553	245	29	see	see	VERB
ejpam-2553	245	30	that	that	DET
ejpam-2553	245	31	b′	b′	NOUN
ejpam-2553	245	32	=	=	PUNCT
ejpam-2553	245	33	�	�	PROPN
ejpam-2553	245	34	bb′a−	bb′a−	PROPN
ejpam-2553	245	35	a	a	DET
ejpam-2553	245	36	bb′b−	bb′b−	PROPN
ejpam-2553	245	37	b	b	PROPN
ejpam-2553	245	38	d	d	NOUN
ejpam-2553	245	39	b′a−	b′a−	NOUN
ejpam-2553	245	40	c	c	PROPN
ejpam-2553	245	41	d	d	X
ejpam-2553	245	42	b′b−	b′b−	X
ejpam-2553	245	43	d	d	PROPN
ejpam-2553	245	44	�	�	PROPN
ejpam-2553	245	45	⊆	⊆	NUM
ejpam-2553	245	46	b.	b.	PROPN
ejpam-2553	245	47	now	now	ADV
ejpam-2553	245	48	,	,	PUNCT
ejpam-2553	245	49	we	we	PRON
ejpam-2553	245	50	can	can	AUX
ejpam-2553	245	51	consider	consider	VERB
ejpam-2553	245	52	b∗	b∗	ADJ
ejpam-2553	245	53	=	=	SYM
ejpam-2553	245	54	�	�	PROPN
ejpam-2553	245	55	s	s	PART
ejpam-2553	245	56	0	0	NUM
ejpam-2553	245	57	t	t	NOUN
ejpam-2553	245	58	u	u	PROPN
ejpam-2553	245	59	�	�	PROPN
ejpam-2553	245	60	∈	∈	PROPN
ejpam-2553	245	61	b′	b′	NOUN
ejpam-2553	245	62	,	,	PUNCT
ejpam-2553	245	63	where	where	SCONJ
ejpam-2553	245	64	s	s	VERB
ejpam-2553	245	65	∈	∈	PROPN
ejpam-2553	245	66	bb′a	bb′a	NOUN
ejpam-2553	245	67	−	−	PROPN
ejpam-2553	245	68	a	a	PROPN
ejpam-2553	245	69	,	,	PUNCT
ejpam-2553	245	70	t	t	PROPN
ejpam-2553	245	71	∈	∈	PROPN
ejpam-2553	245	72	d	d	X
ejpam-2553	245	73	b′a	b′a	NOUN
ejpam-2553	245	74	−	−	PROPN
ejpam-2553	245	75	c	c	NOUN
ejpam-2553	245	76	,	,	PUNCT
ejpam-2553	245	77	u	u	PROPN
ejpam-2553	245	78	∈	∈	PROPN
ejpam-2553	245	79	d	d	X
ejpam-2553	245	80	b′b	b′b	ADV
ejpam-2553	245	81	−	−	PROPN
ejpam-2553	245	82	d.	d.	NOUN
ejpam-2553	245	83	since	since	SCONJ
ejpam-2553	245	84	s	s	PROPN
ejpam-2553	245	85	,	,	PUNCT
ejpam-2553	245	86	u	u	PRON
ejpam-2553	245	87	are	be	AUX
ejpam-2553	245	88	regular	regular	ADJ
ejpam-2553	245	89	there	there	ADV
ejpam-2553	245	90	exist	exist	VERB
ejpam-2553	245	91	s′	s′	NOUN
ejpam-2553	245	92	,	,	PUNCT
ejpam-2553	245	93	u′	u′	PROPN
ejpam-2553	245	94	∈	∈	PROPN
ejpam-2553	245	95	r	r	NOUN
ejpam-2553	245	96	such	such	DET
ejpam-2553	245	97	that	that	DET
ejpam-2553	245	98	s	s	PROPN
ejpam-2553	245	99	∈	∈	PROPN
ejpam-2553	245	100	ss′s	ss′s	PROPN
ejpam-2553	245	101	,	,	PUNCT
ejpam-2553	245	102	u	u	PROPN
ejpam-2553	245	103	∈	∈	PROPN
ejpam-2553	245	104	uu′u	uu′u	NOUN
ejpam-2553	245	105	.	.	PUNCT
ejpam-2553	246	1	if	if	SCONJ
ejpam-2553	246	2	l	l	NOUN
ejpam-2553	246	3	=	=	SYM
ejpam-2553	246	4	�	�	PROPN
ejpam-2553	246	5	s′	s′	NUM
ejpam-2553	246	6	0	0	NUM
ejpam-2553	246	7	0	0	NUM
ejpam-2553	246	8	u′	u′	PROPN
ejpam-2553	246	9	�	�	PROPN
ejpam-2553	246	10	,	,	PUNCT
ejpam-2553	246	11	then	then	ADV
ejpam-2553	246	12	by	by	ADP
ejpam-2553	246	13	simple	simple	ADJ
ejpam-2553	246	14	calculation	calculation	NOUN
ejpam-2553	246	15	we	we	PRON
ejpam-2553	246	16	can	can	AUX
ejpam-2553	246	17	show	show	VERB
ejpam-2553	246	18	that	that	SCONJ
ejpam-2553	246	19	c	c	NOUN
ejpam-2553	246	20	=	=	PUNCT
ejpam-2553	246	21	b∗lb∗	b∗lb∗	NOUN
ejpam-2553	246	22	−	−	PROPN
ejpam-2553	246	23	b∗	b∗	ADJ
ejpam-2553	246	24	⊇	⊇	X
ejpam-2553	246	25	�	�	PROPN
ejpam-2553	246	26	ss′s−	ss′s−	PROPN
ejpam-2553	246	27	s	s	PART
ejpam-2553	246	28	0	0	NUM
ejpam-2553	246	29	−t	−t	NOUN
ejpam-2553	246	30	+	+	CCONJ
ejpam-2553	246	31	ts′s+	ts′s+	PROPN
ejpam-2553	247	1	uu′	uu′	PROPN
ejpam-2553	247	2	t	t	PROPN
ejpam-2553	247	3	uu′u−	uu′u−	PROPN
ejpam-2553	247	4	u	u	NOUN
ejpam-2553	247	5	�	�	PROPN
ejpam-2553	247	6	=	=	SYM
ejpam-2553	247	7	c	c	NOUN
ejpam-2553	247	8	′.	′.	NOUN
ejpam-2553	247	9	let	let	VERB
ejpam-2553	247	10	c∗	c∗	PROPN
ejpam-2553	247	11	=	=	SYM
ejpam-2553	247	12	�	�	PROPN
ejpam-2553	247	13	0	0	NUM
ejpam-2553	247	14	0	0	NUM
ejpam-2553	247	15	m	m	VERB
ejpam-2553	247	16	0	0	NUM
ejpam-2553	247	17	�	�	PROPN
ejpam-2553	247	18	∈	∈	PROPN
ejpam-2553	247	19	c	c	NOUN
ejpam-2553	247	20	′	′	NOUN
ejpam-2553	247	21	,	,	PUNCT
ejpam-2553	247	22	where	where	SCONJ
ejpam-2553	247	23	m	m	PROPN
ejpam-2553	247	24	∈	∈	PROPN
ejpam-2553	247	25	−t	−t	NOUN
ejpam-2553	247	26	+	+	CCONJ
ejpam-2553	247	27	ts′s+	ts′s+	NOUN
ejpam-2553	248	1	uu′	uu′	PROPN
ejpam-2553	248	2	t.	t.	PROPN
ejpam-2553	248	3	since	since	SCONJ
ejpam-2553	248	4	m	m	PROPN
ejpam-2553	248	5	is	be	AUX
ejpam-2553	248	6	regular	regular	ADJ
ejpam-2553	248	7	there	there	ADV
ejpam-2553	248	8	exists	exist	VERB
ejpam-2553	248	9	m′	m′	NOUN
ejpam-2553	248	10	∈	∈	PROPN
ejpam-2553	248	11	r	r	NOUN
ejpam-2553	248	12	,	,	PUNCT
ejpam-2553	248	13	such	such	ADJ
ejpam-2553	248	14	that	that	SCONJ
ejpam-2553	248	15	m	m	PROPN
ejpam-2553	248	16	∈	∈	PROPN
ejpam-2553	248	17	mm′m	mm′m	PROPN
ejpam-2553	248	18	.	.	PUNCT
ejpam-2553	249	1	finally	finally	ADV
ejpam-2553	249	2	,	,	PUNCT
ejpam-2553	249	3	if	if	SCONJ
ejpam-2553	249	4	k	k	PROPN
ejpam-2553	249	5	=	=	SYM
ejpam-2553	249	6	�	�	PROPN
ejpam-2553	249	7	0	0	NUM
ejpam-2553	249	8	m′	m′	NOUN
ejpam-2553	249	9	0	0	SYM
ejpam-2553	249	10	0	0	NUM
ejpam-2553	249	11	�	�	NOUN
ejpam-2553	249	12	we	we	PRON
ejpam-2553	249	13	can	can	AUX
ejpam-2553	249	14	see	see	VERB
ejpam-2553	249	15	that	that	SCONJ
ejpam-2553	249	16	0	0	NUM
ejpam-2553	249	17	∈	∈	NOUN
ejpam-2553	249	18	c∗kc∗	c∗kc∗	NOUN
ejpam-2553	250	1	−	−	NOUN
ejpam-2553	251	1	c∗.	c∗.	NOUN
ejpam-2553	251	2	so	so	ADV
ejpam-2553	251	3	,	,	PUNCT
ejpam-2553	251	4	c∗	c∗	PROPN
ejpam-2553	251	5	is	be	AUX
ejpam-2553	251	6	regular	regular	ADJ
ejpam-2553	251	7	,	,	PUNCT
ejpam-2553	251	8	then	then	ADV
ejpam-2553	251	9	by	by	ADP
ejpam-2553	251	10	the	the	DET
ejpam-2553	251	11	proposition	proposition	NOUN
ejpam-2553	251	12	2	2	NUM
ejpam-2553	251	13	,	,	PUNCT
ejpam-2553	251	14	b∗	b∗	ADV
ejpam-2553	251	15	is	be	AUX
ejpam-2553	251	16	regular	regular	ADJ
ejpam-2553	251	17	and	and	CCONJ
ejpam-2553	251	18	hence	hence	ADV
ejpam-2553	251	19	a	a	PRON
ejpam-2553	251	20	is	be	AUX
ejpam-2553	251	21	regular	regular	ADJ
ejpam-2553	251	22	.	.	PUNCT
ejpam-2553	252	1	therefore	therefore	ADV
ejpam-2553	252	2	,	,	PUNCT
ejpam-2553	252	3	for	for	ADP
ejpam-2553	252	4	n	n	NOUN
ejpam-2553	252	5	=	=	SYM
ejpam-2553	252	6	2	2	NUM
ejpam-2553	252	7	,	,	PUNCT
ejpam-2553	252	8	if	if	SCONJ
ejpam-2553	252	9	r	r	NOUN
ejpam-2553	252	10	is	be	AUX
ejpam-2553	252	11	regular	regular	ADJ
ejpam-2553	252	12	then	then	ADV
ejpam-2553	252	13	m2(r	m2(r	X
ejpam-2553	252	14	)	)	PUNCT
ejpam-2553	252	15	is	be	AUX
ejpam-2553	252	16	regular	regular	ADJ
ejpam-2553	252	17	.	.	PUNCT
ejpam-2553	253	1	since	since	SCONJ
ejpam-2553	253	2	m2(m2(r))∼=	m2(m2(r))∼=	NOUN
ejpam-2553	253	3	m4(r	m4(r	NOUN
ejpam-2553	253	4	)	)	PUNCT
ejpam-2553	253	5	,	,	PUNCT
ejpam-2553	253	6	then	then	ADV
ejpam-2553	253	7	m4(r	m4(r	NOUN
ejpam-2553	253	8	)	)	PUNCT
ejpam-2553	253	9	is	be	AUX
ejpam-2553	253	10	regular	regular	ADJ
ejpam-2553	253	11	.	.	PUNCT
ejpam-2553	254	1	thus	thus	ADV
ejpam-2553	254	2	by	by	ADP
ejpam-2553	254	3	continuing	continue	VERB
ejpam-2553	254	4	this	this	DET
ejpam-2553	254	5	process	process	NOUN
ejpam-2553	254	6	we	we	PRON
ejpam-2553	254	7	can	can	AUX
ejpam-2553	254	8	show	show	VERB
ejpam-2553	254	9	that	that	SCONJ
ejpam-2553	254	10	for	for	ADP
ejpam-2553	254	11	any	any	DET
ejpam-2553	254	12	positive	positive	ADJ
ejpam-2553	254	13	integer	integer	NOUN
ejpam-2553	254	14	k	k	PROPN
ejpam-2553	254	15	,	,	PUNCT
ejpam-2553	254	16	m2k(r	m2k(r	PROPN
ejpam-2553	254	17	)	)	PUNCT
ejpam-2553	254	18	is	be	AUX
ejpam-2553	254	19	regular	regular	ADJ
ejpam-2553	254	20	.	.	PUNCT
ejpam-2553	255	1	now	now	ADV
ejpam-2553	255	2	,	,	PUNCT
ejpam-2553	255	3	assume	assume	VERB
ejpam-2553	255	4	that	that	SCONJ
ejpam-2553	255	5	n	n	PRON
ejpam-2553	255	6	is	be	AUX
ejpam-2553	255	7	an	an	DET
ejpam-2553	255	8	arbitrary	arbitrary	ADJ
ejpam-2553	255	9	positive	positive	ADJ
ejpam-2553	255	10	integer	integer	NOUN
ejpam-2553	255	11	,	,	PUNCT
ejpam-2553	255	12	choose	choose	VERB
ejpam-2553	255	13	k	k	PRON
ejpam-2553	255	14	such	such	ADJ
ejpam-2553	255	15	that	that	SCONJ
ejpam-2553	255	16	2k	2k	PROPN
ejpam-2553	255	17	≥	≥	NOUN
ejpam-2553	255	18	n.	n.	NOUN
ejpam-2553	255	19	if	if	SCONJ
ejpam-2553	255	20	a∈	a∈	PROPN
ejpam-2553	255	21	mn(r	mn(r	NOUN
ejpam-2553	255	22	)	)	PUNCT
ejpam-2553	255	23	,	,	PUNCT
ejpam-2553	255	24	let	let	VERB
ejpam-2553	255	25	a1	a1	NOUN
ejpam-2553	255	26	be	be	AUX
ejpam-2553	255	27	the	the	DET
ejpam-2553	255	28	matrix	matrix	NOUN
ejpam-2553	255	29	of	of	ADP
ejpam-2553	255	30	m2k(r	m2k(r	PROPN
ejpam-2553	255	31	)	)	PUNCT
ejpam-2553	255	32	with	with	ADP
ejpam-2553	255	33	a	a	PRON
ejpam-2553	255	34	in	in	ADP
ejpam-2553	255	35	the	the	DET
ejpam-2553	255	36	upper	upper	ADJ
ejpam-2553	255	37	left	left	ADJ
ejpam-2553	255	38	-	-	PUNCT
ejpam-2553	255	39	hand	hand	NOUN
ejpam-2553	255	40	corner	corner	NOUN
ejpam-2553	255	41	and	and	CCONJ
ejpam-2553	255	42	zeros	zero	NOUN
ejpam-2553	255	43	elsewhere	elsewhere	ADV
ejpam-2553	255	44	.	.	PUNCT
ejpam-2553	256	1	assume	assume	VERB
ejpam-2553	256	2	that	that	SCONJ
ejpam-2553	256	3	a1	a1	PROPN
ejpam-2553	256	4	∈	∈	PROPN
ejpam-2553	256	5	m2k(r	m2k(r	PROPN
ejpam-2553	256	6	)	)	PUNCT
ejpam-2553	256	7	,	,	PUNCT
ejpam-2553	256	8	is	be	AUX
ejpam-2553	256	9	regular	regular	ADJ
ejpam-2553	256	10	,	,	PUNCT
ejpam-2553	256	11	then	then	ADV
ejpam-2553	256	12	there	there	PRON
ejpam-2553	256	13	exists	exist	VERB
ejpam-2553	256	14	an	an	DET
ejpam-2553	256	15	element	element	NOUN
ejpam-2553	256	16	t	t	PROPN
ejpam-2553	256	17	=	=	SYM
ejpam-2553	256	18	�	�	PROPN
ejpam-2553	256	19	b	b	PROPN
ejpam-2553	256	20	c	c	PROPN
ejpam-2553	256	21	d	d	X
ejpam-2553	256	22	e	e	X
ejpam-2553	256	23	�	�	PROPN
ejpam-2553	256	24	of	of	ADP
ejpam-2553	256	25	m2k(r	m2k(r	PROPN
ejpam-2553	256	26	)	)	PUNCT
ejpam-2553	256	27	such	such	ADJ
ejpam-2553	256	28	that	that	DET
ejpam-2553	256	29	a1	a1	NOUN
ejpam-2553	256	30	∈	∈	PROPN
ejpam-2553	256	31	a1ta1	a1ta1	NOUN
ejpam-2553	256	32	.	.	PUNCT
ejpam-2553	257	1	however	however	ADV
ejpam-2553	257	2	,	,	PUNCT
ejpam-2553	257	3	this	this	PRON
ejpam-2553	257	4	implies	imply	VERB
ejpam-2553	257	5	that	that	SCONJ
ejpam-2553	257	6	a∈	a∈	PROPN
ejpam-2553	257	7	aba	aba	PROPN
ejpam-2553	257	8	and	and	CCONJ
ejpam-2553	257	9	hence	hence	ADV
ejpam-2553	257	10	a	a	PRON
ejpam-2553	257	11	is	be	AUX
ejpam-2553	257	12	regular	regular	ADJ
ejpam-2553	257	13	.	.	PUNCT
ejpam-2553	258	1	(	(	PUNCT
ejpam-2553	258	2	⇐	⇐	ADJ
ejpam-2553	258	3	)	)	PUNCT
ejpam-2553	258	4	assume	assume	VERB
ejpam-2553	258	5	that	that	SCONJ
ejpam-2553	258	6	mn(r	mn(r	NOUN
ejpam-2553	258	7	)	)	PUNCT
ejpam-2553	258	8	is	be	AUX
ejpam-2553	258	9	regular	regular	ADJ
ejpam-2553	258	10	and	and	CCONJ
ejpam-2553	258	11	a	a	PRON
ejpam-2553	258	12	is	be	AUX
ejpam-2553	258	13	an	an	DET
ejpam-2553	258	14	arbitrary	arbitrary	ADJ
ejpam-2553	258	15	element	element	NOUN
ejpam-2553	258	16	of	of	ADP
ejpam-2553	258	17	r	r	NOUN
ejpam-2553	258	18	,	,	PUNCT
ejpam-2553	258	19	then	then	ADV
ejpam-2553	258	20	a=	a=	VERB
ejpam-2553	258	21			NOUN
ejpam-2553	258	22			NOUN
ejpam-2553	258	23			NOUN
ejpam-2553	258	24			NOUN
ejpam-2553	258	25			NOUN
ejpam-2553	258	26	a	a	DET
ejpam-2553	258	27	0	0	NUM
ejpam-2553	258	28	·	·	PUNCT
ejpam-2553	258	29	·	·	PUNCT
ejpam-2553	258	30	·	·	PUNCT
ejpam-2553	259	1	0	0	NUM
ejpam-2553	259	2	0	0	NUM
ejpam-2553	259	3	0	0	NUM
ejpam-2553	259	4	·	·	PUNCT
ejpam-2553	259	5	·	·	PUNCT
ejpam-2553	259	6	·	·	PUNCT
ejpam-2553	259	7	0	0	NUM
ejpam-2553	259	8	...	...	PUNCT
ejpam-2553	259	9	...	...	PUNCT
ejpam-2553	259	10	.	.	PUNCT
ejpam-2553	259	11	.	.	PUNCT
ejpam-2553	259	12	.	.	PUNCT
ejpam-2553	260	1	...	...	PUNCT
ejpam-2553	261	1	0	0	NUM
ejpam-2553	261	2	0	0	NUM
ejpam-2553	261	3	·	·	PUNCT
ejpam-2553	261	4	·	·	PUNCT
ejpam-2553	261	5	·	·	PUNCT
ejpam-2553	261	6	0	0	NUM
ejpam-2553	261	7			NOUN
ejpam-2553	261	8			NOUN
ejpam-2553	261	9			VERB
ejpam-2553	261	10			NOUN
ejpam-2553	261	11			PUNCT
ejpam-2553	261	12	r.	r.	PROPN
ejpam-2553	261	13	ameri	ameri	PROPN
ejpam-2553	261	14	,	,	PUNCT
ejpam-2553	261	15	a.	a.	PROPN
ejpam-2553	261	16	kordi	kordi	PROPN
ejpam-2553	261	17	/	/	PUNCT
ejpam-2553	261	18	eur	eur	PROPN
ejpam-2553	261	19	.	.	PUNCT
ejpam-2553	262	1	j.	j.	PROPN
ejpam-2553	262	2	pure	pure	PROPN
ejpam-2553	262	3	appl	appl	PROPN
ejpam-2553	262	4	.	.	PROPN
ejpam-2553	262	5	math	math	PROPN
ejpam-2553	262	6	,	,	PUNCT
ejpam-2553	262	7	9	9	NUM
ejpam-2553	262	8	(	(	PUNCT
ejpam-2553	262	9	2016	2016	NUM
ejpam-2553	262	10	)	)	PUNCT
ejpam-2553	262	11	,	,	PUNCT
ejpam-2553	262	12	402	402	NUM
ejpam-2553	262	13	-	-	SYM
ejpam-2553	262	14	418	418	NUM
ejpam-2553	262	15	410	410	NUM
ejpam-2553	262	16	is	be	AUX
ejpam-2553	262	17	in	in	ADP
ejpam-2553	262	18	mn(r	mn(r	NOUN
ejpam-2553	262	19	)	)	PUNCT
ejpam-2553	262	20	.	.	PUNCT
ejpam-2553	263	1	since	since	SCONJ
ejpam-2553	263	2	mn(r	mn(r	NOUN
ejpam-2553	263	3	)	)	PUNCT
ejpam-2553	263	4	is	be	AUX
ejpam-2553	263	5	regular	regular	ADJ
ejpam-2553	263	6	,	,	PUNCT
ejpam-2553	263	7	then	then	ADV
ejpam-2553	263	8	there	there	PRON
ejpam-2553	263	9	exists	exist	VERB
ejpam-2553	263	10	b	b	NOUN
ejpam-2553	263	11	=	=	SYM
ejpam-2553	263	12			PROPN
ejpam-2553	263	13			NOUN
ejpam-2553	263	14			PROPN
ejpam-2553	263	15	b11	b11	NOUN
ejpam-2553	263	16	·	·	PUNCT
ejpam-2553	263	17	·	·	PUNCT
ejpam-2553	263	18	·	·	PUNCT
ejpam-2553	264	1	b1n	b1n	X
ejpam-2553	264	2	...	...	PUNCT
ejpam-2553	264	3	.	.	PUNCT
ejpam-2553	264	4	.	.	PUNCT
ejpam-2553	264	5	.	.	PUNCT
ejpam-2553	265	1	...	...	PUNCT
ejpam-2553	266	1	bn1	bn1	PROPN
ejpam-2553	266	2	·	·	PUNCT
ejpam-2553	266	3	·	·	PUNCT
ejpam-2553	266	4	·	·	PUNCT
ejpam-2553	266	5	bnn	bnn	PROPN
ejpam-2553	266	6			PROPN
ejpam-2553	266	7			VERB
ejpam-2553	266	8			PUNCT
ejpam-2553	267	1	∈	∈	NOUN
ejpam-2553	267	2	mn(r	mn(r	NOUN
ejpam-2553	267	3	)	)	PUNCT
ejpam-2553	267	4	such	such	ADJ
ejpam-2553	267	5	that	that	SCONJ
ejpam-2553	267	6	a∈	a∈	PROPN
ejpam-2553	267	7	aba	aba	PROPN
ejpam-2553	267	8	,	,	PUNCT
ejpam-2553	267	9	this	this	PRON
ejpam-2553	267	10	means	mean	VERB
ejpam-2553	267	11	that	that	SCONJ
ejpam-2553	267	12	a	a	DET
ejpam-2553	267	13	∈	∈	PROPN
ejpam-2553	267	14	ab11a	ab11a	ADV
ejpam-2553	267	15	,	,	PUNCT
ejpam-2553	267	16	i.e.	i.e.	X
ejpam-2553	267	17	,	,	PUNCT
ejpam-2553	267	18	a	a	PRON
ejpam-2553	267	19	is	be	AUX
ejpam-2553	267	20	regular	regular	ADJ
ejpam-2553	267	21	.	.	PUNCT
ejpam-2553	268	1	theorem	theorem	ADJ
ejpam-2553	268	2	7	7	NUM
ejpam-2553	268	3	.	.	PUNCT
ejpam-2553	269	1	let	let	VERB
ejpam-2553	269	2	(	(	PUNCT
ejpam-2553	269	3	r,+	r,+	NUM
ejpam-2553	269	4	,	,	PUNCT
ejpam-2553	269	5	.	.	PUNCT
ejpam-2553	269	6	)	)	PUNCT
ejpam-2553	270	1	be	be	AUX
ejpam-2553	270	2	a	a	DET
ejpam-2553	270	3	commutative	commutative	ADJ
ejpam-2553	270	4	multiplicative	multiplicative	ADJ
ejpam-2553	270	5	hyperring	hyperring	NOUN
ejpam-2553	270	6	with	with	ADP
ejpam-2553	270	7	zero	zero	NUM
ejpam-2553	270	8	absorbing	absorbing	NOUN
ejpam-2553	270	9	property	property	NOUN
ejpam-2553	270	10	.	.	PUNCT
ejpam-2553	271	1	then	then	ADV
ejpam-2553	271	2	r[x]/γ∗	r[x]/γ∗	PROPN
ejpam-2553	271	3	∼=	∼=	PROPN
ejpam-2553	271	4	(	(	PUNCT
ejpam-2553	271	5	r	r	NOUN
ejpam-2553	271	6	/	/	SYM
ejpam-2553	271	7	γ∗)[x	γ∗)[x	PROPN
ejpam-2553	271	8	]	]	PUNCT
ejpam-2553	271	9	.	.	PUNCT
ejpam-2553	272	1	proof	proof	NOUN
ejpam-2553	272	2	.	.	PUNCT
ejpam-2553	273	1	consider	consider	VERB
ejpam-2553	273	2	the	the	DET
ejpam-2553	273	3	map	map	NOUN
ejpam-2553	273	4	φ	φ	X
ejpam-2553	273	5	:	:	PUNCT
ejpam-2553	273	6	r[x]→	r[x]→	NOUN
ejpam-2553	273	7	(	(	PUNCT
ejpam-2553	273	8	r	r	X
ejpam-2553	273	9	/	/	SYM
ejpam-2553	273	10	γ∗)[x	γ∗)[x	PROPN
ejpam-2553	273	11	]	]	PUNCT
ejpam-2553	273	12	defined	define	VERB
ejpam-2553	273	13	by	by	ADP
ejpam-2553	273	14	φ	φ	PROPN
ejpam-2553	273	15	(	(	PUNCT
ejpam-2553	273	16	∑n	∑n	PROPN
ejpam-2553	273	17	i=1	i=1	PROPN
ejpam-2553	273	18	ai	ai	VERB
ejpam-2553	273	19	x	x	PUNCT
ejpam-2553	273	20	i	i	NOUN
ejpam-2553	273	21	)	)	PUNCT
ejpam-2553	274	1	=	=	SYM
ejpam-2553	275	1	∑n	∑n	PROPN
ejpam-2553	275	2	i=1	i=1	PROPN
ejpam-2553	275	3	γ	γ	PROPN
ejpam-2553	275	4	∗(ai)x	∗(ai)x	NUM
ejpam-2553	276	1	i	i	PRON
ejpam-2553	276	2	.	.	PUNCT
ejpam-2553	277	1	by	by	ADP
ejpam-2553	277	2	[	[	X
ejpam-2553	277	3	13	13	NUM
ejpam-2553	277	4	,	,	PUNCT
ejpam-2553	277	5	theorem	theorem	VERB
ejpam-2553	277	6	5.6.5	5.6.5	PROPN
ejpam-2553	277	7	]	]	PUNCT
ejpam-2553	277	8	,	,	PUNCT
ejpam-2553	277	9	φ	φ	PROPN
ejpam-2553	277	10	is	be	AUX
ejpam-2553	277	11	a	a	DET
ejpam-2553	277	12	projection	projection	ADJ
ejpam-2553	277	13	homomorphism	homomorphism	NOUN
ejpam-2553	277	14	.	.	PUNCT
ejpam-2553	278	1	we	we	PRON
ejpam-2553	278	2	denote	denote	VERB
ejpam-2553	278	3	the	the	DET
ejpam-2553	278	4	equivalence	equivalence	NOUN
ejpam-2553	278	5	relation	relation	NOUN
ejpam-2553	278	6	associated	associate	VERB
ejpam-2553	278	7	with	with	ADP
ejpam-2553	278	8	φ	φ	PROPN
ejpam-2553	278	9	by	by	ADP
ejpam-2553	278	10	ρ	ρ	PROPN
ejpam-2553	278	11	.	.	PUNCT
ejpam-2553	279	1	that	that	PRON
ejpam-2553	279	2	is	is	ADV
ejpam-2553	279	3	,	,	PUNCT
ejpam-2553	279	4	(	(	PUNCT
ejpam-2553	279	5	n	n	CCONJ
ejpam-2553	279	6	∑	∑	ADV
ejpam-2553	279	7	i=1	i=1	PROPN
ejpam-2553	279	8	ai	ai	VERB
ejpam-2553	279	9	x	x	PUNCT
ejpam-2553	279	10	i)ρ	i)ρ	ADJ
ejpam-2553	279	11	(	(	PUNCT
ejpam-2553	279	12	n	n	CCONJ
ejpam-2553	279	13	∑	∑	ADP
ejpam-2553	279	14	i=1	i=1	PROPN
ejpam-2553	279	15	bi	bi	NOUN
ejpam-2553	279	16	x	x	PROPN
ejpam-2553	279	17	i)	i)	PROPN
ejpam-2553	279	18	⇐	⇐	PROPN
ejpam-2553	279	19	⇒	⇒	PROPN
ejpam-2553	279	20	n	n	CCONJ
ejpam-2553	279	21	∑	∑	NOUN
ejpam-2553	279	22	i=1	i=1	PROPN
ejpam-2553	279	23	γ∗(ai)x	γ∗(ai)x	PROPN
ejpam-2553	280	1	i	i	PRON
ejpam-2553	280	2	=	=	SYM
ejpam-2553	280	3	n	n	CCONJ
ejpam-2553	280	4	∑	∑	PROPN
ejpam-2553	280	5	i=1	i=1	PROPN
ejpam-2553	280	6	γ∗(bi)x	γ∗(bi)x	PROPN
ejpam-2553	281	1	i	i	PRON
ejpam-2553	281	2	,	,	PUNCT
ejpam-2553	281	3	∀ai	∀ai	PROPN
ejpam-2553	281	4	,	,	PUNCT
ejpam-2553	281	5	bi	bi	PROPN
ejpam-2553	281	6	∈	∈	PROPN
ejpam-2553	281	7	r.	r.	PROPN
ejpam-2553	281	8	since	since	SCONJ
ejpam-2553	281	9	φ	φ	PROPN
ejpam-2553	281	10	is	be	AUX
ejpam-2553	281	11	epimorphism	epimorphism	NOUN
ejpam-2553	281	12	we	we	PRON
ejpam-2553	281	13	have	have	VERB
ejpam-2553	281	14	r[x]/ρ	r[x]/ρ	NOUN
ejpam-2553	281	15	=	=	PUNCT
ejpam-2553	281	16	r[x]/ker(φ)∼=	r[x]/ker(φ)∼=	NOUN
ejpam-2553	281	17	(	(	PUNCT
ejpam-2553	281	18	r	r	NOUN
ejpam-2553	281	19	/	/	SYM
ejpam-2553	281	20	γ∗)[x	γ∗)[x	PROPN
ejpam-2553	281	21	]	]	PUNCT
ejpam-2553	281	22	.	.	PUNCT
ejpam-2553	282	1	we	we	PRON
ejpam-2553	282	2	know	know	VERB
ejpam-2553	282	3	that	that	SCONJ
ejpam-2553	282	4	(	(	PUNCT
ejpam-2553	282	5	r	r	X
ejpam-2553	282	6	/	/	SYM
ejpam-2553	282	7	γ∗)[x	γ∗)[x	PROPN
ejpam-2553	282	8	]	]	PUNCT
ejpam-2553	282	9	is	be	AUX
ejpam-2553	282	10	a	a	DET
ejpam-2553	282	11	ring	ring	NOUN
ejpam-2553	282	12	,	,	PUNCT
ejpam-2553	282	13	then	then	ADV
ejpam-2553	282	14	r[x]/ρ	r[x]/ρ	PROPN
ejpam-2553	282	15	is	be	AUX
ejpam-2553	282	16	a	a	DET
ejpam-2553	282	17	ring	ring	NOUN
ejpam-2553	282	18	.	.	PUNCT
ejpam-2553	283	1	since	since	SCONJ
ejpam-2553	283	2	γ∗	γ∗	PROPN
ejpam-2553	283	3	is	be	AUX
ejpam-2553	283	4	the	the	DET
ejpam-2553	283	5	smallest	small	ADJ
ejpam-2553	283	6	equivalence	equivalence	NOUN
ejpam-2553	283	7	relation	relation	NOUN
ejpam-2553	283	8	on	on	ADP
ejpam-2553	283	9	r[x	r[x	NOUN
ejpam-2553	283	10	]	]	PUNCT
ejpam-2553	283	11	such	such	ADJ
ejpam-2553	283	12	that	that	SCONJ
ejpam-2553	283	13	r[x]/γ∗	r[x]/γ∗	PROPN
ejpam-2553	283	14	is	be	AUX
ejpam-2553	283	15	a	a	DET
ejpam-2553	283	16	ring	ring	NOUN
ejpam-2553	283	17	,	,	PUNCT
ejpam-2553	283	18	then	then	ADV
ejpam-2553	283	19	γ∗	γ∗	VERB
ejpam-2553	283	20	⊆	⊆	NUM
ejpam-2553	283	21	ρ	ρ	NOUN
ejpam-2553	283	22	.	.	PUNCT
ejpam-2553	284	1	now	now	ADV
ejpam-2553	284	2	,	,	PUNCT
ejpam-2553	284	3	let	let	VERB
ejpam-2553	284	4	(	(	PUNCT
ejpam-2553	284	5	∑n	∑n	NOUN
ejpam-2553	284	6	i=1	i=1	PROPN
ejpam-2553	284	7	ai	ai	VERB
ejpam-2553	284	8	x	x	PUNCT
ejpam-2553	284	9	i)ρ	i)ρ	ADJ
ejpam-2553	284	10	(	(	PUNCT
ejpam-2553	284	11	∑n	∑n	PROPN
ejpam-2553	284	12	i=1	i=1	PROPN
ejpam-2553	284	13	bi	bi	NOUN
ejpam-2553	284	14	x	x	PROPN
ejpam-2553	284	15	i	i	PROPN
ejpam-2553	284	16	)	)	PUNCT
ejpam-2553	284	17	for	for	ADP
ejpam-2553	284	18	all	all	DET
ejpam-2553	284	19	bi	bi	PROPN
ejpam-2553	284	20	∈	∈	PROPN
ejpam-2553	284	21	r.	r.	PROPN
ejpam-2553	284	22	hence	hence	ADV
ejpam-2553	284	23	n	n	PROPN
ejpam-2553	284	24	∑	∑	PROPN
ejpam-2553	284	25	i=1	i=1	PROPN
ejpam-2553	284	26	γ∗(ai)x	γ∗(ai)x	PROPN
ejpam-2553	285	1	i	i	PRON
ejpam-2553	285	2	=	=	SYM
ejpam-2553	285	3	n	n	CCONJ
ejpam-2553	285	4	∑	∑	PROPN
ejpam-2553	285	5	i=1	i=1	PROPN
ejpam-2553	285	6	γ∗(bi)x	γ∗(bi)x	PROPN
ejpam-2553	285	7	i	i	PROPN
ejpam-2553	285	8	⇐	⇐	ADJ
ejpam-2553	285	9	⇒	⇒	PROPN
ejpam-2553	285	10	γ∗(ai	γ∗(ai	PROPN
ejpam-2553	285	11	)	)	PUNCT
ejpam-2553	286	1	=	=	PUNCT
ejpam-2553	286	2	γ	γ	PROPN
ejpam-2553	286	3	∗(bi	∗(bi	PROPN
ejpam-2553	286	4	)	)	PUNCT
ejpam-2553	286	5	,	,	PUNCT
ejpam-2553	286	6	∀ai	∀ai	PROPN
ejpam-2553	286	7	∈	∈	PROPN
ejpam-2553	286	8	r.	r.	PROPN
ejpam-2553	286	9	thus	thus	ADV
ejpam-2553	286	10	,	,	PUNCT
ejpam-2553	286	11	{	{	PUNCT
ejpam-2553	286	12	ai	ai	INTJ
ejpam-2553	286	13	,	,	PUNCT
ejpam-2553	286	14	bi	bi	NOUN
ejpam-2553	286	15	}	}	PUNCT
ejpam-2553	286	16	⊆	⊆	NUM
ejpam-2553	286	17	∏m	∏m	PROPN
ejpam-2553	286	18	s=1	s=1	SYM
ejpam-2553	286	19	ts	ts	NOUN
ejpam-2553	286	20	.	.	PUNCT
ejpam-2553	287	1	therefore	therefore	ADV
ejpam-2553	287	2	,	,	PUNCT
ejpam-2553	287	3	{	{	PUNCT
ejpam-2553	287	4	n	n	CCONJ
ejpam-2553	287	5	∑	∑	PROPN
ejpam-2553	287	6	i=1	i=1	PROPN
ejpam-2553	287	7	ai	ai	VERB
ejpam-2553	287	8	x	x	VERB
ejpam-2553	287	9	i	i	PRON
ejpam-2553	287	10	,	,	PUNCT
ejpam-2553	287	11	n	n	CCONJ
ejpam-2553	287	12	∑	∑	PROPN
ejpam-2553	287	13	i=1	i=1	PROPN
ejpam-2553	287	14	bi	bi	NOUN
ejpam-2553	287	15	x	x	PUNCT
ejpam-2553	287	16	i	i	PROPN
ejpam-2553	287	17	}	}	PUNCT
ejpam-2553	287	18	⊆	⊆	NUM
ejpam-2553	287	19	n	n	PROPN
ejpam-2553	287	20	∑	∑	PROPN
ejpam-2553	287	21	i=1	i=1	PROPN
ejpam-2553	287	22	(	(	PUNCT
ejpam-2553	287	23	m	m	VERB
ejpam-2553	287	24	∏	∏	X
ejpam-2553	287	25	s=1	s=1	X
ejpam-2553	288	1	ts)i	ts)i	NOUN
ejpam-2553	288	2	x	x	VERB
ejpam-2553	289	1	i	i	NOUN
ejpam-2553	289	2	.	.	PUNCT
ejpam-2553	290	1	this	this	PRON
ejpam-2553	290	2	means	mean	VERB
ejpam-2553	290	3	that	that	SCONJ
ejpam-2553	290	4	(	(	PUNCT
ejpam-2553	290	5	∑n	∑n	NOUN
ejpam-2553	290	6	i=1	i=1	PROPN
ejpam-2553	290	7	ai	ai	VERB
ejpam-2553	290	8	x	x	PUNCT
ejpam-2553	290	9	i)γ	i)γ	ADV
ejpam-2553	290	10	(	(	PUNCT
ejpam-2553	290	11	∑n	∑n	PROPN
ejpam-2553	290	12	i=1	i=1	PROPN
ejpam-2553	290	13	bi	bi	NOUN
ejpam-2553	290	14	x	x	PROPN
ejpam-2553	290	15	i	i	PROPN
ejpam-2553	290	16	)	)	PUNCT
ejpam-2553	290	17	.	.	PUNCT
ejpam-2553	291	1	therefore	therefore	ADV
ejpam-2553	291	2	(	(	PUNCT
ejpam-2553	291	3	∑n	∑n	PROPN
ejpam-2553	291	4	i=1	i=1	PROPN
ejpam-2553	291	5	ai	ai	VERB
ejpam-2553	291	6	x	x	PUNCT
ejpam-2553	291	7	i)γ∗	i)γ∗	PROPN
ejpam-2553	291	8	(	(	PUNCT
ejpam-2553	291	9	∑n	∑n	PROPN
ejpam-2553	291	10	i=1	i=1	PROPN
ejpam-2553	291	11	bi	bi	NOUN
ejpam-2553	291	12	x	x	PROPN
ejpam-2553	291	13	i	i	PROPN
ejpam-2553	291	14	)	)	PUNCT
ejpam-2553	291	15	.	.	PUNCT
ejpam-2553	292	1	henceρ	henceρ	PROPN
ejpam-2553	292	2	⊆	⊆	NUM
ejpam-2553	292	3	γ∗	γ∗	NOUN
ejpam-2553	292	4	,	,	PUNCT
ejpam-2553	292	5	i.e.	i.e.	X
ejpam-2553	292	6	,	,	PUNCT
ejpam-2553	292	7	ρ	ρ	PROPN
ejpam-2553	292	8	=	=	SYM
ejpam-2553	292	9	γ∗	γ∗	PROPN
ejpam-2553	292	10	,	,	PUNCT
ejpam-2553	292	11	and	and	CCONJ
ejpam-2553	292	12	so	so	ADV
ejpam-2553	292	13	r[x]/γ∗	r[x]/γ∗	PROPN
ejpam-2553	292	14	∼=	∼=	PROPN
ejpam-2553	292	15	(	(	PUNCT
ejpam-2553	292	16	r	r	NOUN
ejpam-2553	292	17	/	/	SYM
ejpam-2553	292	18	γ∗)[x	γ∗)[x	PROPN
ejpam-2553	292	19	]	]	PUNCT
ejpam-2553	292	20	.	.	PUNCT
ejpam-2553	293	1	remark	remark	PROPN
ejpam-2553	293	2	3	3	NUM
ejpam-2553	293	3	.	.	PUNCT
ejpam-2553	294	1	if	if	SCONJ
ejpam-2553	294	2	r	r	NOUN
ejpam-2553	294	3	is	be	AUX
ejpam-2553	294	4	a	a	DET
ejpam-2553	294	5	commutative	commutative	ADJ
ejpam-2553	294	6	multiplicative	multiplicative	ADJ
ejpam-2553	294	7	hyperring	hyperring	NOUN
ejpam-2553	294	8	with	with	ADP
ejpam-2553	294	9	zero	zero	NUM
ejpam-2553	294	10	absorbing	absorbing	NOUN
ejpam-2553	294	11	property	property	NOUN
ejpam-2553	294	12	and	and	CCONJ
ejpam-2553	294	13	r[x	r[x	NOUN
ejpam-2553	294	14	]	]	PUNCT
ejpam-2553	294	15	is	be	AUX
ejpam-2553	294	16	regular	regular	ADJ
ejpam-2553	294	17	,	,	PUNCT
ejpam-2553	294	18	then	then	ADV
ejpam-2553	294	19	r	r	NOUN
ejpam-2553	294	20	is	be	AUX
ejpam-2553	294	21	regular	regular	ADJ
ejpam-2553	294	22	.	.	PUNCT
ejpam-2553	295	1	but	but	CCONJ
ejpam-2553	295	2	the	the	DET
ejpam-2553	295	3	converse	converse	NOUN
ejpam-2553	295	4	is	be	AUX
ejpam-2553	295	5	not	not	PART
ejpam-2553	295	6	true	true	ADJ
ejpam-2553	295	7	.	.	PUNCT
ejpam-2553	296	1	r.	r.	PROPN
ejpam-2553	296	2	ameri	ameri	PROPN
ejpam-2553	296	3	,	,	PUNCT
ejpam-2553	296	4	a.	a.	PROPN
ejpam-2553	296	5	kordi	kordi	PROPN
ejpam-2553	296	6	/	/	PUNCT
ejpam-2553	296	7	eur	eur	PROPN
ejpam-2553	296	8	.	.	PUNCT
ejpam-2553	297	1	j.	j.	PROPN
ejpam-2553	297	2	pure	pure	PROPN
ejpam-2553	297	3	appl	appl	PROPN
ejpam-2553	297	4	.	.	PROPN
ejpam-2553	297	5	math	math	PROPN
ejpam-2553	297	6	,	,	PUNCT
ejpam-2553	297	7	9	9	NUM
ejpam-2553	297	8	(	(	PUNCT
ejpam-2553	297	9	2016	2016	NUM
ejpam-2553	297	10	)	)	PUNCT
ejpam-2553	297	11	,	,	PUNCT
ejpam-2553	297	12	402	402	NUM
ejpam-2553	297	13	-	-	SYM
ejpam-2553	297	14	418	418	NUM
ejpam-2553	297	15	411	411	NUM
ejpam-2553	297	16	2.2	2.2	NUM
ejpam-2553	297	17	.	.	PUNCT
ejpam-2553	298	1	some	some	DET
ejpam-2553	298	2	properties	property	NOUN
ejpam-2553	298	3	of	of	ADP
ejpam-2553	298	4	regular	regular	ADJ
ejpam-2553	298	5	multiplicative	multiplicative	ADJ
ejpam-2553	298	6	hyperrings	hyperring	NOUN
ejpam-2553	298	7	definition	definition	NOUN
ejpam-2553	298	8	5	5	NUM
ejpam-2553	298	9	.	.	PUNCT
ejpam-2553	299	1	let	let	VERB
ejpam-2553	299	2	r	r	PRON
ejpam-2553	299	3	be	be	AUX
ejpam-2553	299	4	a	a	DET
ejpam-2553	299	5	multiplicative	multiplicative	ADJ
ejpam-2553	299	6	hyperring	hyperring	NOUN
ejpam-2553	299	7	.	.	PUNCT
ejpam-2553	300	1	a	a	DET
ejpam-2553	300	2	subset	subset	NOUN
ejpam-2553	300	3	a	a	PRON
ejpam-2553	300	4	of	of	ADP
ejpam-2553	300	5	r	r	NOUN
ejpam-2553	300	6	is	be	AUX
ejpam-2553	300	7	idempotent	idempotent	ADJ
ejpam-2553	300	8	if	if	SCONJ
ejpam-2553	300	9	a	a	DET
ejpam-2553	300	10	⊆	⊆	NUM
ejpam-2553	300	11	a2	a2	NOUN
ejpam-2553	300	12	.	.	PUNCT
ejpam-2553	301	1	the	the	DET
ejpam-2553	301	2	set	set	NOUN
ejpam-2553	301	3	of	of	ADP
ejpam-2553	301	4	all	all	DET
ejpam-2553	301	5	idempotent	idempotent	ADJ
ejpam-2553	301	6	elements	element	NOUN
ejpam-2553	301	7	of	of	ADP
ejpam-2553	301	8	r	r	NOUN
ejpam-2553	301	9	is	be	AUX
ejpam-2553	301	10	denoted	denote	VERB
ejpam-2553	301	11	by	by	ADP
ejpam-2553	301	12	idem(r	idem(r	NOUN
ejpam-2553	301	13	)	)	PUNCT
ejpam-2553	301	14	.	.	PUNCT
ejpam-2553	302	1	definition	definition	NOUN
ejpam-2553	302	2	6	6	NUM
ejpam-2553	302	3	.	.	PUNCT
ejpam-2553	303	1	we	we	PRON
ejpam-2553	303	2	say	say	VERB
ejpam-2553	303	3	that	that	SCONJ
ejpam-2553	303	4	i	i	PRON
ejpam-2553	303	5	is	be	AUX
ejpam-2553	303	6	a	a	DET
ejpam-2553	303	7	hyperideal	hyperideal	NOUN
ejpam-2553	303	8	of	of	ADP
ejpam-2553	303	9	multiplicative	multiplicative	ADJ
ejpam-2553	303	10	hyperring	hyperring	NOUN
ejpam-2553	303	11	(	(	PUNCT
ejpam-2553	303	12	r,+	r,+	NUM
ejpam-2553	303	13	,	,	PUNCT
ejpam-2553	303	14	.	.	PUNCT
ejpam-2553	303	15	)	)	PUNCT
ejpam-2553	304	1	if	if	SCONJ
ejpam-2553	304	2	it	it	PRON
ejpam-2553	304	3	satisfies	satisfy	VERB
ejpam-2553	304	4	the	the	DET
ejpam-2553	304	5	following	follow	VERB
ejpam-2553	304	6	conditions	condition	NOUN
ejpam-2553	304	7	:	:	PUNCT
ejpam-2553	304	8	(	(	PUNCT
ejpam-2553	304	9	1	1	X
ejpam-2553	304	10	)	)	PUNCT
ejpam-2553	304	11	i	i	PRON
ejpam-2553	304	12	−	−	VERB
ejpam-2553	305	1	i	i	PRON
ejpam-2553	305	2	⊆	⊆	NUM
ejpam-2553	305	3	i	i	PRON
ejpam-2553	305	4	,	,	PUNCT
ejpam-2553	305	5	(	(	PUNCT
ejpam-2553	305	6	2	2	X
ejpam-2553	305	7	)	)	PUNCT
ejpam-2553	305	8	∀x	∀x	VERB
ejpam-2553	305	9	∈	∈	PROPN
ejpam-2553	306	1	i	i	PRON
ejpam-2553	306	2	,	,	PUNCT
ejpam-2553	306	3	r	r	NOUN
ejpam-2553	306	4	∈	∈	PROPN
ejpam-2553	306	5	r	r	NOUN
ejpam-2553	306	6	,	,	PUNCT
ejpam-2553	306	7	x	x	PUNCT
ejpam-2553	306	8	r	r	NOUN
ejpam-2553	306	9	∪	∪	NOUN
ejpam-2553	306	10	r	r	NOUN
ejpam-2553	306	11	x	x	SYM
ejpam-2553	306	12	⊆	⊆	NUM
ejpam-2553	306	13	i	i	PRON
ejpam-2553	306	14	.	.	PUNCT
ejpam-2553	307	1	definition	definition	NOUN
ejpam-2553	307	2	7	7	NUM
ejpam-2553	307	3	.	.	PUNCT
ejpam-2553	308	1	let	let	VERB
ejpam-2553	308	2	r	r	PRON
ejpam-2553	308	3	be	be	AUX
ejpam-2553	308	4	a	a	DET
ejpam-2553	308	5	multiplicative	multiplicative	ADJ
ejpam-2553	308	6	hyperring	hyperring	NOUN
ejpam-2553	308	7	.	.	PUNCT
ejpam-2553	309	1	the	the	DET
ejpam-2553	309	2	element	element	NOUN
ejpam-2553	309	3	a	a	DET
ejpam-2553	309	4	∈	∈	NOUN
ejpam-2553	309	5	r	r	NOUN
ejpam-2553	309	6	is	be	AUX
ejpam-2553	309	7	nilpotent	nilpotent	ADJ
ejpam-2553	309	8	,	,	PUNCT
ejpam-2553	309	9	if	if	SCONJ
ejpam-2553	309	10	there	there	PRON
ejpam-2553	309	11	exists	exist	VERB
ejpam-2553	309	12	an	an	DET
ejpam-2553	309	13	n	n	NOUN
ejpam-2553	309	14	such	such	ADJ
ejpam-2553	309	15	that	that	SCONJ
ejpam-2553	309	16	an	an	DET
ejpam-2553	309	17	=	=	X
ejpam-2553	309	18	{	{	PUNCT
ejpam-2553	309	19	0	0	NUM
ejpam-2553	309	20	}	}	PUNCT
ejpam-2553	309	21	.	.	PUNCT
ejpam-2553	310	1	denote	denote	VERB
ejpam-2553	310	2	the	the	DET
ejpam-2553	310	3	set	set	NOUN
ejpam-2553	310	4	of	of	ADP
ejpam-2553	310	5	all	all	DET
ejpam-2553	310	6	nilpotent	nilpotent	ADJ
ejpam-2553	310	7	elements	element	NOUN
ejpam-2553	310	8	of	of	ADP
ejpam-2553	310	9	r	r	NOUN
ejpam-2553	310	10	by	by	ADP
ejpam-2553	310	11	nil(r	nil(r	NOUN
ejpam-2553	310	12	)	)	PUNCT
ejpam-2553	310	13	.	.	PUNCT
ejpam-2553	311	1	definition	definition	NOUN
ejpam-2553	311	2	8	8	NUM
ejpam-2553	311	3	.	.	PUNCT
ejpam-2553	312	1	let	let	VERB
ejpam-2553	312	2	r	r	PRON
ejpam-2553	312	3	be	be	AUX
ejpam-2553	312	4	a	a	DET
ejpam-2553	312	5	multiplicative	multiplicative	ADJ
ejpam-2553	312	6	hyperring	hyperring	NOUN
ejpam-2553	312	7	and	and	CCONJ
ejpam-2553	312	8	x	x	SYM
ejpam-2553	312	9	∈	∈	PROPN
ejpam-2553	312	10	r.	r.	PROPN
ejpam-2553	312	11	then	then	ADV
ejpam-2553	312	12	a	a	DET
ejpam-2553	312	13	left(right	left(right	PROPN
ejpam-2553	312	14	)	)	PUNCT
ejpam-2553	312	15	annihilator	annihilator	NOUN
ejpam-2553	312	16	of	of	ADP
ejpam-2553	312	17	x	x	PUNCT
ejpam-2553	312	18	is	be	AUX
ejpam-2553	312	19	ann(x	ann(x	PROPN
ejpam-2553	312	20	)	)	PUNCT
ejpam-2553	312	21	=	=	SYM
ejpam-2553	312	22	{	{	PUNCT
ejpam-2553	312	23	r	r	NOUN
ejpam-2553	312	24	∈	∈	PROPN
ejpam-2553	312	25	r|r	r|r	NOUN
ejpam-2553	312	26	x	x	PUNCT
ejpam-2553	313	1	=	=	PUNCT
ejpam-2553	313	2	0	0	NUM
ejpam-2553	313	3	}	}	PUNCT
ejpam-2553	313	4	(	(	PUNCT
ejpam-2553	313	5	ann(x	ann(x	PROPN
ejpam-2553	313	6	)	)	PUNCT
ejpam-2553	313	7	=	=	SYM
ejpam-2553	313	8	{	{	PUNCT
ejpam-2553	313	9	r	r	NOUN
ejpam-2553	313	10	∈	∈	NOUN
ejpam-2553	313	11	r|x	r|x	X
ejpam-2553	314	1	r	r	NOUN
ejpam-2553	314	2	=	=	NOUN
ejpam-2553	314	3	0	0	NUM
ejpam-2553	314	4	}	}	PUNCT
ejpam-2553	314	5	)	)	PUNCT
ejpam-2553	314	6	.	.	PUNCT
ejpam-2553	315	1	for	for	ADP
ejpam-2553	315	2	a	a	DET
ejpam-2553	315	3	non	non	ADJ
ejpam-2553	315	4	-	-	ADJ
ejpam-2553	315	5	empty	empty	ADJ
ejpam-2553	315	6	subset	subset	NOUN
ejpam-2553	315	7	b	b	NOUN
ejpam-2553	315	8	of	of	ADP
ejpam-2553	315	9	a	a	DET
ejpam-2553	315	10	multiplicative	multiplicative	ADJ
ejpam-2553	315	11	hyperring	hyperring	NOUN
ejpam-2553	315	12	r	r	NOUN
ejpam-2553	315	13	,	,	PUNCT
ejpam-2553	315	14	the	the	DET
ejpam-2553	315	15	annihilator	annihilator	NOUN
ejpam-2553	315	16	of	of	ADP
ejpam-2553	315	17	b	b	PROPN
ejpam-2553	315	18	is	be	AUX
ejpam-2553	315	19	ann(b	ann(b	PROPN
ejpam-2553	315	20	)	)	PUNCT
ejpam-2553	315	21	=	=	PUNCT
ejpam-2553	316	1	∩{ann(x)|x	∩{ann(x)|x	NOUN
ejpam-2553	316	2	∈	∈	PROPN
ejpam-2553	316	3	b	b	NOUN
ejpam-2553	316	4	}	}	PUNCT
ejpam-2553	316	5	.	.	PUNCT
ejpam-2553	317	1	theorem	theorem	ADJ
ejpam-2553	317	2	8	8	NUM
ejpam-2553	317	3	.	.	PUNCT
ejpam-2553	318	1	let	let	VERB
ejpam-2553	318	2	r	r	PRON
ejpam-2553	318	3	be	be	AUX
ejpam-2553	318	4	a	a	DET
ejpam-2553	318	5	commutative	commutative	ADJ
ejpam-2553	318	6	multiplicative	multiplicative	ADJ
ejpam-2553	318	7	hyperring	hyperring	NOUN
ejpam-2553	318	8	with	with	ADP
ejpam-2553	318	9	a	a	DET
ejpam-2553	318	10	scalar	scalar	ADJ
ejpam-2553	318	11	identity	identity	NOUN
ejpam-2553	318	12	1	1	NUM
ejpam-2553	318	13	,	,	PUNCT
ejpam-2553	318	14	and	and	CCONJ
ejpam-2553	318	15	a	a	DET
ejpam-2553	318	16	∈	∈	PROPN
ejpam-2553	318	17	r.	r.	NOUN
ejpam-2553	318	18	then	then	ADV
ejpam-2553	318	19	we	we	PRON
ejpam-2553	318	20	have	have	VERB
ejpam-2553	318	21	the	the	DET
ejpam-2553	318	22	following	following	ADJ
ejpam-2553	318	23	statements	statement	NOUN
ejpam-2553	318	24	:	:	PUNCT
ejpam-2553	318	25	(	(	PUNCT
ejpam-2553	318	26	1	1	X
ejpam-2553	318	27	)	)	PUNCT
ejpam-2553	318	28	if	if	SCONJ
ejpam-2553	318	29	a	a	DET
ejpam-2553	318	30	∈	∈	PROPN
ejpam-2553	318	31	ara	ara	NOUN
ejpam-2553	318	32	for	for	ADP
ejpam-2553	318	33	r	r	PROPN
ejpam-2553	318	34	∈	∈	PROPN
ejpam-2553	318	35	r	r	NOUN
ejpam-2553	318	36	,	,	PUNCT
ejpam-2553	318	37	then	then	ADV
ejpam-2553	318	38	ar	ar	PROPN
ejpam-2553	318	39	∈	∈	PROPN
ejpam-2553	318	40	idem(r	idem(r	PROPN
ejpam-2553	318	41	)	)	PUNCT
ejpam-2553	318	42	.	.	PUNCT
ejpam-2553	319	1	(	(	PUNCT
ejpam-2553	319	2	2	2	X
ejpam-2553	319	3	)	)	PUNCT
ejpam-2553	319	4	v	v	NOUN
ejpam-2553	319	5	(	(	PUNCT
ejpam-2553	319	6	r)∩	r)∩	PROPN
ejpam-2553	319	7	nil(r	nil(r	PROPN
ejpam-2553	319	8	)	)	PUNCT
ejpam-2553	319	9	=	=	PRON
ejpam-2553	319	10	{	{	PUNCT
ejpam-2553	319	11	0	0	NUM
ejpam-2553	319	12	}	}	PUNCT
ejpam-2553	319	13	.	.	PUNCT
ejpam-2553	320	1	proof	proof	NOUN
ejpam-2553	320	2	.	.	PUNCT
ejpam-2553	321	1	(	(	PUNCT
ejpam-2553	321	2	1	1	X
ejpam-2553	321	3	)	)	PUNCT
ejpam-2553	321	4	this	this	PRON
ejpam-2553	321	5	is	be	AUX
ejpam-2553	321	6	clear	clear	ADJ
ejpam-2553	321	7	.	.	PUNCT
ejpam-2553	322	1	(	(	PUNCT
ejpam-2553	322	2	2	2	X
ejpam-2553	322	3	)	)	PUNCT
ejpam-2553	322	4	suppose	suppose	VERB
ejpam-2553	322	5	a	a	DET
ejpam-2553	322	6	∈	∈	PROPN
ejpam-2553	322	7	v	v	NOUN
ejpam-2553	322	8	(	(	PUNCT
ejpam-2553	322	9	r	r	NOUN
ejpam-2553	322	10	)	)	PUNCT
ejpam-2553	322	11	∩	∩	NOUN
ejpam-2553	322	12	nil(r	nil(r	NOUN
ejpam-2553	322	13	)	)	PUNCT
ejpam-2553	322	14	.	.	PUNCT
ejpam-2553	323	1	then	then	ADV
ejpam-2553	323	2	there	there	PRON
ejpam-2553	323	3	is	be	VERB
ejpam-2553	323	4	a	a	DET
ejpam-2553	323	5	r	r	NOUN
ejpam-2553	323	6	∈	∈	NOUN
ejpam-2553	323	7	r	r	NOUN
ejpam-2553	323	8	and	and	CCONJ
ejpam-2553	323	9	n	n	CCONJ
ejpam-2553	323	10	∈	∈	PROPN
ejpam-2553	323	11	n	n	PRON
ejpam-2553	323	12	such	such	ADJ
ejpam-2553	323	13	that	that	SCONJ
ejpam-2553	323	14	a	a	DET
ejpam-2553	323	15	∈	∈	PROPN
ejpam-2553	323	16	ara	ara	NOUN
ejpam-2553	323	17	and	and	CCONJ
ejpam-2553	323	18	an	an	DET
ejpam-2553	323	19	=	=	X
ejpam-2553	323	20	{	{	PUNCT
ejpam-2553	323	21	0	0	NUM
ejpam-2553	323	22	}	}	PUNCT
ejpam-2553	323	23	.	.	PUNCT
ejpam-2553	324	1	thus	thus	ADV
ejpam-2553	324	2	we	we	PRON
ejpam-2553	324	3	have	have	VERB
ejpam-2553	324	4	,	,	PUNCT
ejpam-2553	324	5	a	a	DET
ejpam-2553	324	6	∈	∈	PROPN
ejpam-2553	324	7	ara	ara	NOUN
ejpam-2553	324	8	⊆	⊆	NUM
ejpam-2553	324	9	(	(	PUNCT
ejpam-2553	324	10	ara)r(ara	ara)r(ara	PROPN
ejpam-2553	324	11	)	)	PUNCT
ejpam-2553	324	12	=	=	SYM
ejpam-2553	324	13	a4r3	a4r3	NOUN
ejpam-2553	325	1	=	=	SYM
ejpam-2553	325	2	(	(	PUNCT
ejpam-2553	325	3	ara)a2r2	ara)a2r2	NOUN
ejpam-2553	325	4	⊆	⊆	NUM
ejpam-2553	325	5	.	.	PUNCT
ejpam-2553	325	6	.	.	PUNCT
ejpam-2553	325	7	.	.	PUNCT
ejpam-2553	326	1	⊆	⊆	NUM
ejpam-2553	326	2	anr	anr	NOUN
ejpam-2553	326	3	′	′	NUM
ejpam-2553	326	4	=	=	PUNCT
ejpam-2553	326	5	{	{	PUNCT
ejpam-2553	326	6	0}r	0}r	NOUN
ejpam-2553	326	7	′	′	NUM
ejpam-2553	326	8	=	=	PUNCT
ejpam-2553	326	9	{	{	PUNCT
ejpam-2553	326	10	0	0	NUM
ejpam-2553	326	11	}	}	PUNCT
ejpam-2553	326	12	,	,	PUNCT
ejpam-2553	326	13	because	because	SCONJ
ejpam-2553	326	14	0.r	0.r	NUM
ejpam-2553	326	15	′	′	NUM
ejpam-2553	326	16	=	=	SYM
ejpam-2553	326	17	(	(	PUNCT
ejpam-2553	326	18	1−	1−	NUM
ejpam-2553	326	19	1).r	1).r	NUM
ejpam-2553	326	20	′	′	NUM
ejpam-2553	326	21	⊆	⊆	NUM
ejpam-2553	326	22	1.r	1.r	NUM
ejpam-2553	326	23	′	′	NUM
ejpam-2553	326	24	−	−	NOUN
ejpam-2553	326	25	1.r	1.r	NUM
ejpam-2553	326	26	′	′	NUM
ejpam-2553	327	1	=	=	PUNCT
ejpam-2553	328	1	{	{	PUNCT
ejpam-2553	328	2	r	r	NOUN
ejpam-2553	328	3	′	′	NOUN
ejpam-2553	328	4	}	}	PUNCT
ejpam-2553	328	5	−	−	PROPN
ejpam-2553	328	6	{	{	PUNCT
ejpam-2553	328	7	r	r	NOUN
ejpam-2553	328	8	′}=	′}=	PROPN
ejpam-2553	328	9	{	{	PUNCT
ejpam-2553	328	10	0	0	NUM
ejpam-2553	328	11	}	}	PUNCT
ejpam-2553	328	12	.	.	PUNCT
ejpam-2553	329	1	therefore	therefore	ADV
ejpam-2553	329	2	,	,	PUNCT
ejpam-2553	329	3	a	a	DET
ejpam-2553	329	4	=	=	ADJ
ejpam-2553	329	5	0	0	NUM
ejpam-2553	329	6	.	.	PUNCT
ejpam-2553	329	7	theorem	theorem	NOUN
ejpam-2553	329	8	9	9	NUM
ejpam-2553	329	9	.	.	PUNCT
ejpam-2553	329	10	suppose	suppose	VERB
ejpam-2553	329	11	that	that	SCONJ
ejpam-2553	329	12	r	r	NOUN
ejpam-2553	329	13	is	be	AUX
ejpam-2553	329	14	a	a	DET
ejpam-2553	329	15	commutative	commutative	ADJ
ejpam-2553	329	16	multiplicative	multiplicative	ADJ
ejpam-2553	329	17	hyperring	hyperring	NOUN
ejpam-2553	329	18	with	with	ADP
ejpam-2553	329	19	a	a	DET
ejpam-2553	329	20	scalar	scalar	ADJ
ejpam-2553	329	21	identity	identity	NOUN
ejpam-2553	329	22	1	1	NUM
ejpam-2553	329	23	.	.	PUNCT
ejpam-2553	330	1	then	then	ADV
ejpam-2553	330	2	we	we	PRON
ejpam-2553	330	3	have	have	VERB
ejpam-2553	330	4	the	the	DET
ejpam-2553	330	5	following	following	ADJ
ejpam-2553	330	6	statements	statement	NOUN
ejpam-2553	330	7	:	:	PUNCT
ejpam-2553	330	8	(	(	PUNCT
ejpam-2553	330	9	1	1	X
ejpam-2553	330	10	)	)	PUNCT
ejpam-2553	330	11	if	if	SCONJ
ejpam-2553	330	12	for	for	ADP
ejpam-2553	330	13	some	some	DET
ejpam-2553	330	14	u	u	PROPN
ejpam-2553	330	15	∈	∈	PROPN
ejpam-2553	330	16	u(r	u(r	PROPN
ejpam-2553	330	17	)	)	PUNCT
ejpam-2553	330	18	and	and	CCONJ
ejpam-2553	330	19	a	a	DET
ejpam-2553	330	20	∈	∈	PROPN
ejpam-2553	330	21	r	r	NOUN
ejpam-2553	330	22	,	,	PUNCT
ejpam-2553	330	23	a	a	DET
ejpam-2553	330	24	∈	∈	PROPN
ejpam-2553	330	25	aua	aua	PROPN
ejpam-2553	330	26	,	,	PUNCT
ejpam-2553	330	27	then	then	ADV
ejpam-2553	330	28	a	a	DET
ejpam-2553	330	29	∈	∈	PROPN
ejpam-2553	330	30	ve	ve	VERB
ejpam-2553	330	31	for	for	ADP
ejpam-2553	330	32	some	some	DET
ejpam-2553	330	33	v	v	NOUN
ejpam-2553	330	34	∈	∈	PROPN
ejpam-2553	330	35	u(r	u(r	NOUN
ejpam-2553	330	36	)	)	PUNCT
ejpam-2553	330	37	and	and	CCONJ
ejpam-2553	330	38	e	e	PROPN
ejpam-2553	330	39	∈	∈	PROPN
ejpam-2553	330	40	idem(r	idem(r	PROPN
ejpam-2553	330	41	)	)	PUNCT
ejpam-2553	330	42	.	.	PUNCT
ejpam-2553	331	1	(	(	PUNCT
ejpam-2553	331	2	2	2	X
ejpam-2553	331	3	)	)	PUNCT
ejpam-2553	331	4	if	if	SCONJ
ejpam-2553	331	5	ζ=	ζ=	ADJ
ejpam-2553	331	6	ue	ue	NOUN
ejpam-2553	331	7	for	for	ADP
ejpam-2553	331	8	some	some	DET
ejpam-2553	331	9	u	u	PROPN
ejpam-2553	331	10	∈	∈	PROPN
ejpam-2553	331	11	u(r	u(r	PROPN
ejpam-2553	331	12	)	)	PUNCT
ejpam-2553	331	13	and	and	CCONJ
ejpam-2553	331	14	e	e	PROPN
ejpam-2553	331	15	∈	∈	PROPN
ejpam-2553	331	16	idem(r	idem(r	PROPN
ejpam-2553	331	17	)	)	PUNCT
ejpam-2553	331	18	,	,	PUNCT
ejpam-2553	331	19	then	then	ADV
ejpam-2553	331	20	for	for	ADP
ejpam-2553	331	21	some	some	DET
ejpam-2553	331	22	v	v	NOUN
ejpam-2553	331	23	∈	∈	PROPN
ejpam-2553	331	24	u(r	u(r	NOUN
ejpam-2553	331	25	)	)	PUNCT
ejpam-2553	331	26	,	,	PUNCT
ejpam-2553	331	27	ζ	ζ	NOUN
ejpam-2553	331	28	⊆	⊆	NUM
ejpam-2553	331	29	ζvζ	ζvζ	NOUN
ejpam-2553	331	30	.	.	PUNCT
ejpam-2553	332	1	(	(	PUNCT
ejpam-2553	332	2	3	3	X
ejpam-2553	332	3	)	)	PUNCT
ejpam-2553	332	4	if	if	SCONJ
ejpam-2553	332	5	for	for	ADP
ejpam-2553	332	6	a	a	DET
ejpam-2553	332	7	∈	∈	PROPN
ejpam-2553	332	8	r	r	NOUN
ejpam-2553	332	9	,	,	PUNCT
ejpam-2553	332	10	there	there	PRON
ejpam-2553	332	11	exists	exist	VERB
ejpam-2553	332	12	b	b	PROPN
ejpam-2553	332	13	∈	∈	NOUN
ejpam-2553	332	14	r	r	NOUN
ejpam-2553	332	15	such	such	ADJ
ejpam-2553	332	16	that	that	SCONJ
ejpam-2553	332	17	ab	ab	PROPN
ejpam-2553	332	18	=	=	PUNCT
ejpam-2553	332	19	0	0	PROPN
ejpam-2553	332	20	,	,	PUNCT
ejpam-2553	332	21	with	with	ADP
ejpam-2553	332	22	a+	a+	DET
ejpam-2553	332	23	b	b	PROPN
ejpam-2553	332	24	∈	∈	PROPN
ejpam-2553	332	25	u(r	u(r	PROPN
ejpam-2553	332	26	)	)	PUNCT
ejpam-2553	332	27	,	,	PUNCT
ejpam-2553	332	28	then	then	ADV
ejpam-2553	332	29	a	a	PRON
ejpam-2553	332	30	is	be	AUX
ejpam-2553	332	31	regular	regular	ADJ
ejpam-2553	332	32	.	.	PUNCT
ejpam-2553	333	1	(	(	PUNCT
ejpam-2553	333	2	4	4	X
ejpam-2553	333	3	)	)	PUNCT
ejpam-2553	333	4	if	if	SCONJ
ejpam-2553	333	5	η	η	PROPN
ejpam-2553	333	6	=	=	PROPN
ejpam-2553	333	7	ue	ue	PROPN
ejpam-2553	333	8	for	for	ADP
ejpam-2553	333	9	some	some	DET
ejpam-2553	333	10	u	u	PROPN
ejpam-2553	333	11	∈	∈	PROPN
ejpam-2553	333	12	u(r	u(r	PROPN
ejpam-2553	333	13	)	)	PUNCT
ejpam-2553	333	14	and	and	CCONJ
ejpam-2553	333	15	e	e	X
ejpam-2553	333	16	∈	∈	PROPN
ejpam-2553	333	17	idem(r	idem(r	PROPN
ejpam-2553	333	18	)	)	PUNCT
ejpam-2553	333	19	and	and	CCONJ
ejpam-2553	333	20	|η2|	|η2|	NOUN
ejpam-2553	333	21	=	=	SYM
ejpam-2553	333	22	1	1	NUM
ejpam-2553	333	23	,	,	PUNCT
ejpam-2553	333	24	then	then	ADV
ejpam-2553	333	25	there	there	PRON
ejpam-2553	333	26	exists	exist	VERB
ejpam-2553	333	27	`	`	PUNCT
ejpam-2553	333	28	∈	∈	PROPN
ejpam-2553	333	29	η	η	PROPN
ejpam-2553	333	30	such	such	ADJ
ejpam-2553	333	31	that	that	SCONJ
ejpam-2553	333	32	`	`	PUNCT
ejpam-2553	333	33	b	b	X
ejpam-2553	333	34	=	=	SYM
ejpam-2553	333	35	0	0	NUM
ejpam-2553	333	36	for	for	ADP
ejpam-2553	333	37	b	b	PROPN
ejpam-2553	333	38	∈	∈	PROPN
ejpam-2553	333	39	r	r	NOUN
ejpam-2553	333	40	with	with	ADP
ejpam-2553	333	41	`	`	PUNCT
ejpam-2553	333	42	+	+	NUM
ejpam-2553	333	43	b	b	PROPN
ejpam-2553	333	44	∈	∈	PROPN
ejpam-2553	333	45	u(r	u(r	NOUN
ejpam-2553	333	46	)	)	PUNCT
ejpam-2553	333	47	.	.	PUNCT
ejpam-2553	334	1	r.	r.	PROPN
ejpam-2553	334	2	ameri	ameri	PROPN
ejpam-2553	334	3	,	,	PUNCT
ejpam-2553	334	4	a.	a.	PROPN
ejpam-2553	334	5	kordi	kordi	PROPN
ejpam-2553	334	6	/	/	PUNCT
ejpam-2553	334	7	eur	eur	PROPN
ejpam-2553	334	8	.	.	PUNCT
ejpam-2553	335	1	j.	j.	PROPN
ejpam-2553	335	2	pure	pure	PROPN
ejpam-2553	335	3	appl	appl	PROPN
ejpam-2553	335	4	.	.	PROPN
ejpam-2553	335	5	math	math	PROPN
ejpam-2553	335	6	,	,	PUNCT
ejpam-2553	335	7	9	9	NUM
ejpam-2553	335	8	(	(	PUNCT
ejpam-2553	335	9	2016	2016	NUM
ejpam-2553	335	10	)	)	PUNCT
ejpam-2553	335	11	,	,	PUNCT
ejpam-2553	335	12	402	402	NUM
ejpam-2553	335	13	-	-	SYM
ejpam-2553	335	14	418	418	NUM
ejpam-2553	335	15	412	412	NUM
ejpam-2553	335	16	proof	proof	NOUN
ejpam-2553	335	17	.	.	PUNCT
ejpam-2553	336	1	(	(	PUNCT
ejpam-2553	336	2	1	1	X
ejpam-2553	336	3	)	)	PUNCT
ejpam-2553	336	4	assume	assume	VERB
ejpam-2553	336	5	that	that	SCONJ
ejpam-2553	336	6	a	a	DET
ejpam-2553	336	7	∈	∈	PROPN
ejpam-2553	336	8	aua	aua	NOUN
ejpam-2553	336	9	for	for	ADP
ejpam-2553	336	10	some	some	DET
ejpam-2553	336	11	u	u	PROPN
ejpam-2553	336	12	∈	∈	PROPN
ejpam-2553	336	13	u(r	u(r	PROPN
ejpam-2553	336	14	)	)	PUNCT
ejpam-2553	336	15	.	.	PUNCT
ejpam-2553	337	1	since	since	SCONJ
ejpam-2553	337	2	u	u	NOUN
ejpam-2553	337	3	is	be	AUX
ejpam-2553	337	4	invertible	invertible	ADJ
ejpam-2553	337	5	then	then	ADV
ejpam-2553	337	6	there	there	PRON
ejpam-2553	337	7	is	be	VERB
ejpam-2553	337	8	a	a	DET
ejpam-2553	337	9	v	v	NOUN
ejpam-2553	337	10	∈	∈	PROPN
ejpam-2553	337	11	u(r	u(r	NOUN
ejpam-2553	337	12	)	)	PUNCT
ejpam-2553	337	13	such	such	ADJ
ejpam-2553	337	14	that	that	SCONJ
ejpam-2553	337	15	1	1	NUM
ejpam-2553	337	16	∈	∈	NOUN
ejpam-2553	337	17	uv	uv	NOUN
ejpam-2553	337	18	.	.	PUNCT
ejpam-2553	338	1	let	let	VERB
ejpam-2553	338	2	e	e	NOUN
ejpam-2553	338	3	=	=	SYM
ejpam-2553	338	4	au	au	PROPN
ejpam-2553	338	5	,	,	PUNCT
ejpam-2553	338	6	thus	thus	ADV
ejpam-2553	338	7	we	we	PRON
ejpam-2553	338	8	have	have	VERB
ejpam-2553	338	9	e2	e2	PROPN
ejpam-2553	338	10	=	=	SYM
ejpam-2553	338	11	(	(	PUNCT
ejpam-2553	338	12	au)2	au)2	X
ejpam-2553	338	13	=	=	SYM
ejpam-2553	338	14	(	(	PUNCT
ejpam-2553	338	15	au)(au	au)(au	NOUN
ejpam-2553	338	16	)	)	PUNCT
ejpam-2553	338	17	=	=	SYM
ejpam-2553	338	18	(	(	PUNCT
ejpam-2553	338	19	aua)u	aua)u	NOUN
ejpam-2553	338	20	⊇	⊇	NOUN
ejpam-2553	338	21	au=	au=	PROPN
ejpam-2553	338	22	e	e	PROPN
ejpam-2553	338	23	,	,	PUNCT
ejpam-2553	338	24	then	then	ADV
ejpam-2553	338	25	e	e	PROPN
ejpam-2553	338	26	∈	∈	PROPN
ejpam-2553	338	27	idem(r	idem(r	PROPN
ejpam-2553	338	28	)	)	PUNCT
ejpam-2553	338	29	.	.	PUNCT
ejpam-2553	339	1	hence	hence	ADV
ejpam-2553	339	2	ve	ve	NOUN
ejpam-2553	339	3	=	=	SYM
ejpam-2553	339	4	v(au	v(au	NOUN
ejpam-2553	339	5	)	)	PUNCT
ejpam-2553	339	6	=	=	SYM
ejpam-2553	339	7	a(vu	a(vu	NUM
ejpam-2553	339	8	)	)	PUNCT
ejpam-2553	339	9	⊇	⊇	NOUN
ejpam-2553	339	10	a.1	a.1	PUNCT
ejpam-2553	339	11	3	3	NUM
ejpam-2553	339	12	a.	a.	NOUN
ejpam-2553	339	13	(	(	PUNCT
ejpam-2553	339	14	2	2	NUM
ejpam-2553	339	15	)	)	PUNCT
ejpam-2553	339	16	since	since	SCONJ
ejpam-2553	339	17	u	u	PROPN
ejpam-2553	339	18	∈	∈	PROPN
ejpam-2553	339	19	u(r	u(r	PROPN
ejpam-2553	339	20	)	)	PUNCT
ejpam-2553	339	21	,	,	PUNCT
ejpam-2553	339	22	then	then	ADV
ejpam-2553	339	23	there	there	PRON
ejpam-2553	339	24	exists	exist	VERB
ejpam-2553	339	25	v	v	ADP
ejpam-2553	339	26	∈	∈	PROPN
ejpam-2553	339	27	u(r	u(r	NOUN
ejpam-2553	339	28	)	)	PUNCT
ejpam-2553	339	29	such	such	ADJ
ejpam-2553	339	30	that	that	SCONJ
ejpam-2553	339	31	1	1	NUM
ejpam-2553	339	32	∈	∈	PROPN
ejpam-2553	339	33	vu	vu	NOUN
ejpam-2553	339	34	.	.	PUNCT
ejpam-2553	340	1	thus	thus	ADV
ejpam-2553	340	2	we	we	PRON
ejpam-2553	340	3	have	have	VERB
ejpam-2553	340	4	:	:	PUNCT
ejpam-2553	340	5	e	e	NOUN
ejpam-2553	340	6	=	=	SYM
ejpam-2553	340	7	1.e	1.e	NUM
ejpam-2553	340	8	⊆	⊆	NUM
ejpam-2553	340	9	(	(	PUNCT
ejpam-2553	340	10	vu)e	vu)e	X
ejpam-2553	340	11	=	=	SYM
ejpam-2553	340	12	v(ue	v(ue	NOUN
ejpam-2553	340	13	)	)	PUNCT
ejpam-2553	340	14	=	=	SYM
ejpam-2553	341	1	vζ⇒	vζ⇒	NOUN
ejpam-2553	341	2	e	e	PROPN
ejpam-2553	341	3	⊆	⊆	NUM
ejpam-2553	341	4	e2	e2	PROPN
ejpam-2553	341	5	⊆	⊆	NUM
ejpam-2553	341	6	evζ⇒	evζ⇒	NOUN
ejpam-2553	341	7	ζ=	ζ=	VERB
ejpam-2553	341	8	ue	ue	PROPN
ejpam-2553	341	9	⊆	⊆	NUM
ejpam-2553	341	10	uevζ=	uevζ=	NUM
ejpam-2553	341	11	ζvζ	ζvζ	NOUN
ejpam-2553	341	12	,	,	PUNCT
ejpam-2553	341	13	i.e.	i.e.	X
ejpam-2553	341	14	,	,	PUNCT
ejpam-2553	341	15	ζ	ζ	NOUN
ejpam-2553	341	16	⊆	⊆	NUM
ejpam-2553	341	17	ζvζ	ζvζ	NOUN
ejpam-2553	341	18	for	for	ADP
ejpam-2553	341	19	some	some	DET
ejpam-2553	341	20	v	v	NOUN
ejpam-2553	341	21	∈	∈	PROPN
ejpam-2553	341	22	u(r	u(r	NOUN
ejpam-2553	341	23	)	)	PUNCT
ejpam-2553	341	24	.	.	PUNCT
ejpam-2553	342	1	(	(	PUNCT
ejpam-2553	342	2	3	3	X
ejpam-2553	342	3	)	)	PUNCT
ejpam-2553	342	4	let	let	VERB
ejpam-2553	342	5	u	u	PRON
ejpam-2553	342	6	=	=	PROPN
ejpam-2553	342	7	a	a	PROPN
ejpam-2553	342	8	+	+	X
ejpam-2553	342	9	b.	b.	NOUN
ejpam-2553	342	10	then	then	ADV
ejpam-2553	342	11	au	au	ADV
ejpam-2553	342	12	=	=	PUNCT
ejpam-2553	342	13	a(a	a(a	PROPN
ejpam-2553	343	1	+	+	SYM
ejpam-2553	343	2	b	b	X
ejpam-2553	343	3	)	)	PUNCT
ejpam-2553	343	4	⊆	⊆	NUM
ejpam-2553	343	5	a2	a2	NOUN
ejpam-2553	343	6	+	+	CCONJ
ejpam-2553	343	7	ab	ab	PROPN
ejpam-2553	343	8	=	=	PROPN
ejpam-2553	343	9	a2	a2	PROPN
ejpam-2553	343	10	.	.	PUNCT
ejpam-2553	344	1	since	since	SCONJ
ejpam-2553	344	2	u	u	NOUN
ejpam-2553	344	3	is	be	AUX
ejpam-2553	344	4	invertible	invertible	ADJ
ejpam-2553	344	5	,	,	PUNCT
ejpam-2553	344	6	so	so	SCONJ
ejpam-2553	344	7	there	there	PRON
ejpam-2553	344	8	is	be	VERB
ejpam-2553	344	9	a	a	DET
ejpam-2553	344	10	v	v	NOUN
ejpam-2553	344	11	∈	∈	PROPN
ejpam-2553	344	12	u(r	u(r	NOUN
ejpam-2553	344	13	)	)	PUNCT
ejpam-2553	344	14	such	such	ADJ
ejpam-2553	344	15	that	that	SCONJ
ejpam-2553	344	16	1	1	NUM
ejpam-2553	344	17	∈	∈	NOUN
ejpam-2553	344	18	uv	uv	NOUN
ejpam-2553	344	19	.	.	PUNCT
ejpam-2553	345	1	therefore	therefore	ADV
ejpam-2553	345	2	,	,	PUNCT
ejpam-2553	345	3	we	we	PRON
ejpam-2553	345	4	have	have	VERB
ejpam-2553	345	5	:	:	PUNCT
ejpam-2553	345	6	a	a	DET
ejpam-2553	345	7	∈	∈	PROPN
ejpam-2553	345	8	a.1	a.1	PUNCT
ejpam-2553	345	9	⊆	⊆	NUM
ejpam-2553	345	10	a(uv	a(uv	NOUN
ejpam-2553	345	11	)	)	PUNCT
ejpam-2553	345	12	=	=	PUNCT
ejpam-2553	345	13	(	(	PUNCT
ejpam-2553	346	1	au)v	au)v	PROPN
ejpam-2553	346	2	⊆	⊆	NUM
ejpam-2553	346	3	a2v	a2v	ADP
ejpam-2553	346	4	=	=	SYM
ejpam-2553	346	5	ava	ava	PROPN
ejpam-2553	346	6	.	.	PUNCT
ejpam-2553	347	1	hence	hence	ADV
ejpam-2553	347	2	a	a	PRON
ejpam-2553	347	3	is	be	AUX
ejpam-2553	347	4	regular	regular	ADJ
ejpam-2553	347	5	.	.	PUNCT
ejpam-2553	348	1	(	(	PUNCT
ejpam-2553	348	2	4	4	X
ejpam-2553	348	3	)	)	PUNCT
ejpam-2553	348	4	assume	assume	VERB
ejpam-2553	348	5	that	that	SCONJ
ejpam-2553	348	6	η	η	PROPN
ejpam-2553	348	7	=	=	PROPN
ejpam-2553	348	8	ue	ue	PROPN
ejpam-2553	348	9	for	for	ADP
ejpam-2553	348	10	some	some	DET
ejpam-2553	348	11	u	u	PROPN
ejpam-2553	348	12	∈	∈	PROPN
ejpam-2553	348	13	u(r	u(r	PROPN
ejpam-2553	348	14	)	)	PUNCT
ejpam-2553	348	15	and	and	CCONJ
ejpam-2553	348	16	e	e	PROPN
ejpam-2553	348	17	∈	∈	PROPN
ejpam-2553	348	18	idem(r	idem(r	PROPN
ejpam-2553	348	19	)	)	PUNCT
ejpam-2553	348	20	.	.	PUNCT
ejpam-2553	349	1	let	let	VERB
ejpam-2553	349	2	τ	τ	X
ejpam-2553	349	3	=	=	PUNCT
ejpam-2553	349	4	u(1	u(1	PROPN
ejpam-2553	349	5	−	−	PROPN
ejpam-2553	349	6	e	e	NOUN
ejpam-2553	349	7	)	)	PUNCT
ejpam-2553	349	8	.	.	PUNCT
ejpam-2553	350	1	then	then	ADV
ejpam-2553	350	2	ητ	ητ	VERB
ejpam-2553	350	3	=	=	SYM
ejpam-2553	350	4	ue(u(1	ue(u(1	NOUN
ejpam-2553	350	5	−	−	PROPN
ejpam-2553	350	6	e	e	NOUN
ejpam-2553	350	7	)	)	PUNCT
ejpam-2553	350	8	)	)	PUNCT
ejpam-2553	351	1	⊆	⊆	NUM
ejpam-2553	351	2	ueu1	ueu1	NOUN
ejpam-2553	351	3	−	−	NOUN
ejpam-2553	351	4	ueue	ueue	NOUN
ejpam-2553	351	5	=	=	PUNCT
ejpam-2553	351	6	ueu	ueu	NOUN
ejpam-2553	351	7	−	−	PROPN
ejpam-2553	351	8	ueue	ueue	NOUN
ejpam-2553	351	9	⊆	⊆	NUM
ejpam-2553	351	10	ueue	ueue	NOUN
ejpam-2553	351	11	−	−	NOUN
ejpam-2553	351	12	ueue	ueue	NOUN
ejpam-2553	351	13	=	=	PUNCT
ejpam-2553	351	14	η2	η2	NOUN
ejpam-2553	351	15	−	−	NOUN
ejpam-2553	351	16	η2	η2	ADJ
ejpam-2553	351	17	=	=	SYM
ejpam-2553	351	18	0	0	NUM
ejpam-2553	351	19	,	,	PUNCT
ejpam-2553	351	20	i.e.	i.e.	X
ejpam-2553	351	21	,	,	PUNCT
ejpam-2553	351	22	for	for	ADP
ejpam-2553	351	23	all	all	DET
ejpam-2553	351	24	`	`	PUNCT
ejpam-2553	351	25	∈	∈	PROPN
ejpam-2553	351	26	η	η	PROPN
ejpam-2553	351	27	,	,	PUNCT
ejpam-2553	351	28	b	b	PROPN
ejpam-2553	351	29	∈	∈	PROPN
ejpam-2553	351	30	τ	τ	PROPN
ejpam-2553	351	31	,	,	PUNCT
ejpam-2553	351	32	`	`	PUNCT
ejpam-2553	351	33	b	b	X
ejpam-2553	351	34	=	=	SYM
ejpam-2553	351	35	0	0	PROPN
ejpam-2553	351	36	.	.	PUNCT
ejpam-2553	352	1	now	now	ADV
ejpam-2553	352	2	,	,	PUNCT
ejpam-2553	352	3	we	we	PRON
ejpam-2553	352	4	need	need	VERB
ejpam-2553	352	5	to	to	PART
ejpam-2553	352	6	show	show	VERB
ejpam-2553	352	7	that	that	SCONJ
ejpam-2553	352	8	η+	η+	ADP
ejpam-2553	352	9	b	b	NOUN
ejpam-2553	352	10	∩	∩	ADJ
ejpam-2553	352	11	u(r	u(r	NOUN
ejpam-2553	352	12	)	)	PUNCT
ejpam-2553	352	13	6=	6=	PUNCT
ejpam-2553	352	14	;	;	PUNCT
ejpam-2553	352	15	.	.	PUNCT
ejpam-2553	353	1	we	we	PRON
ejpam-2553	353	2	have	have	VERB
ejpam-2553	353	3	1	1	NUM
ejpam-2553	353	4	∈	∈	PROPN
ejpam-2553	353	5	u.1.u	u.1.u	X
ejpam-2553	353	6	⊆	⊆	NUM
ejpam-2553	353	7	u(e+	u(e+	NOUN
ejpam-2553	353	8	(	(	PUNCT
ejpam-2553	353	9	1−	1−	NUM
ejpam-2553	353	10	e))u	e))u	NOUN
ejpam-2553	353	11	⊆	⊆	NUM
ejpam-2553	353	12	(	(	PUNCT
ejpam-2553	353	13	ue+	ue+	NOUN
ejpam-2553	353	14	u(1−	u(1−	VERB
ejpam-2553	353	15	e))u=	e))u=	NOUN
ejpam-2553	353	16	(	(	PUNCT
ejpam-2553	353	17	η+	η+	NOUN
ejpam-2553	353	18	b)u	b)u	NOUN
ejpam-2553	353	19	,	,	PUNCT
ejpam-2553	353	20	i.e.	i.e.	X
ejpam-2553	353	21	,	,	PUNCT
ejpam-2553	353	22	1	1	NUM
ejpam-2553	353	23	∈	∈	NOUN
ejpam-2553	353	24	(	(	PUNCT
ejpam-2553	353	25	η+	η+	NOUN
ejpam-2553	353	26	b)u	b)u	NOUN
ejpam-2553	353	27	,	,	PUNCT
ejpam-2553	353	28	so	so	SCONJ
ejpam-2553	353	29	there	there	PRON
ejpam-2553	353	30	exist	exist	VERB
ejpam-2553	353	31	b	b	PROPN
ejpam-2553	353	32	∈	∈	PROPN
ejpam-2553	353	33	τ	τ	X
ejpam-2553	353	34	,	,	PUNCT
ejpam-2553	353	35	`	`	PUNCT
ejpam-2553	353	36	∈	∈	PROPN
ejpam-2553	353	37	η	η	PROPN
ejpam-2553	353	38	such	such	ADJ
ejpam-2553	353	39	that	that	SCONJ
ejpam-2553	353	40	1	1	NUM
ejpam-2553	353	41	∈	∈	NOUN
ejpam-2553	353	42	b+	b+	NOUN
ejpam-2553	353	43	`	`	PUNCT
ejpam-2553	353	44	.	.	PUNCT
ejpam-2553	354	1	therefore	therefore	ADV
ejpam-2553	354	2	`	`	PUNCT
ejpam-2553	354	3	+	+	CCONJ
ejpam-2553	354	4	b	b	X
ejpam-2553	354	5	∈	∈	PROPN
ejpam-2553	354	6	u(r	u(r	NOUN
ejpam-2553	354	7	)	)	PUNCT
ejpam-2553	354	8	.	.	PUNCT
ejpam-2553	355	1	definition	definition	NOUN
ejpam-2553	355	2	9	9	NUM
ejpam-2553	355	3	.	.	PUNCT
ejpam-2553	356	1	let	let	VERB
ejpam-2553	356	2	r	r	PRON
ejpam-2553	356	3	be	be	AUX
ejpam-2553	356	4	a	a	DET
ejpam-2553	356	5	multiplicative	multiplicative	ADJ
ejpam-2553	356	6	hyperring	hyperring	NOUN
ejpam-2553	356	7	.	.	PUNCT
ejpam-2553	357	1	then	then	ADV
ejpam-2553	357	2	r	r	NOUN
ejpam-2553	357	3	is	be	AUX
ejpam-2553	357	4	said	say	VERB
ejpam-2553	357	5	to	to	PART
ejpam-2553	357	6	be	be	AUX
ejpam-2553	357	7	π	π	NOUN
ejpam-2553	357	8	-	-	NOUN
ejpam-2553	357	9	regular	regular	ADJ
ejpam-2553	357	10	if	if	SCONJ
ejpam-2553	357	11	for	for	ADP
ejpam-2553	357	12	all	all	DET
ejpam-2553	357	13	a	a	DET
ejpam-2553	357	14	∈	∈	NOUN
ejpam-2553	357	15	r	r	NOUN
ejpam-2553	357	16	there	there	PRON
ejpam-2553	357	17	are	be	VERB
ejpam-2553	357	18	r	r	NOUN
ejpam-2553	357	19	∈	∈	NOUN
ejpam-2553	357	20	r	r	NOUN
ejpam-2553	357	21	and	and	CCONJ
ejpam-2553	357	22	an	an	DET
ejpam-2553	357	23	integer	integer	NOUN
ejpam-2553	357	24	n≥	n≥	NOUN
ejpam-2553	357	25	1	1	NUM
ejpam-2553	357	26	,	,	PUNCT
ejpam-2553	357	27	such	such	ADJ
ejpam-2553	357	28	that	that	SCONJ
ejpam-2553	357	29	an	an	DET
ejpam-2553	357	30	⊆	⊆	NUM
ejpam-2553	357	31	anran	anran	NOUN
ejpam-2553	357	32	.	.	PUNCT
ejpam-2553	358	1	clearly	clearly	ADV
ejpam-2553	358	2	,	,	PUNCT
ejpam-2553	358	3	a	a	DET
ejpam-2553	358	4	regular	regular	ADJ
ejpam-2553	358	5	multiplicative	multiplicative	ADJ
ejpam-2553	358	6	hyperring	hyperring	NOUN
ejpam-2553	358	7	is	be	AUX
ejpam-2553	358	8	π	π	ADJ
ejpam-2553	358	9	-	-	ADJ
ejpam-2553	358	10	regular	regular	ADJ
ejpam-2553	358	11	multiplicative	multiplicative	ADJ
ejpam-2553	358	12	hyperring	hyperring	NOUN
ejpam-2553	358	13	.	.	PUNCT
ejpam-2553	359	1	also	also	ADV
ejpam-2553	359	2	,	,	PUNCT
ejpam-2553	359	3	if	if	SCONJ
ejpam-2553	359	4	a	a	DET
ejpam-2553	359	5	∈	∈	NOUN
ejpam-2553	359	6	r	r	NOUN
ejpam-2553	359	7	is	be	AUX
ejpam-2553	359	8	π	π	ADJ
ejpam-2553	359	9	-	-	ADJ
ejpam-2553	359	10	regular	regular	ADJ
ejpam-2553	359	11	multiplicative	multiplicative	ADJ
ejpam-2553	359	12	hyperring	hyperring	NOUN
ejpam-2553	359	13	then	then	ADV
ejpam-2553	359	14	for	for	ADP
ejpam-2553	359	15	an	an	DET
ejpam-2553	359	16	n≥	n≥	PROPN
ejpam-2553	359	17	1	1	NUM
ejpam-2553	359	18	,	,	PUNCT
ejpam-2553	359	19	an	an	PRON
ejpam-2553	359	20	is	be	AUX
ejpam-2553	359	21	regular	regular	ADJ
ejpam-2553	359	22	.	.	PUNCT
ejpam-2553	360	1	theorem	theorem	ADJ
ejpam-2553	360	2	10	10	NUM
ejpam-2553	360	3	.	.	PUNCT
ejpam-2553	361	1	let	let	VERB
ejpam-2553	361	2	r	r	PRON
ejpam-2553	361	3	be	be	AUX
ejpam-2553	361	4	a	a	DET
ejpam-2553	361	5	commutative	commutative	ADJ
ejpam-2553	361	6	multiplicative	multiplicative	ADJ
ejpam-2553	361	7	hyperring	hyperring	NOUN
ejpam-2553	361	8	with	with	ADP
ejpam-2553	361	9	a	a	DET
ejpam-2553	361	10	scalar	scalar	ADJ
ejpam-2553	361	11	identity	identity	NOUN
ejpam-2553	361	12	1	1	NUM
ejpam-2553	361	13	.	.	PUNCT
ejpam-2553	362	1	then	then	ADV
ejpam-2553	362	2	for	for	ADP
ejpam-2553	362	3	a	a	DET
ejpam-2553	362	4	∈	∈	PROPN
ejpam-2553	362	5	r	r	NOUN
ejpam-2553	362	6	,	,	PUNCT
ejpam-2553	362	7	the	the	DET
ejpam-2553	362	8	following	follow	VERB
ejpam-2553	362	9	hold	hold	NOUN
ejpam-2553	362	10	:	:	PUNCT
ejpam-2553	362	11	(	(	PUNCT
ejpam-2553	362	12	1	1	X
ejpam-2553	362	13	)	)	PUNCT
ejpam-2553	362	14	a	a	PRON
ejpam-2553	362	15	is	be	AUX
ejpam-2553	362	16	a	a	DET
ejpam-2553	362	17	π	π	NOUN
ejpam-2553	362	18	-	-	NOUN
ejpam-2553	362	19	regular	regular	ADJ
ejpam-2553	362	20	if	if	SCONJ
ejpam-2553	362	21	and	and	CCONJ
ejpam-2553	362	22	only	only	ADV
ejpam-2553	362	23	if	if	SCONJ
ejpam-2553	362	24	for	for	ADP
ejpam-2553	362	25	some	some	DET
ejpam-2553	362	26	n≥	n≥	NOUN
ejpam-2553	362	27	1	1	NUM
ejpam-2553	362	28	,	,	PUNCT
ejpam-2553	362	29	an	an	PRON
ejpam-2553	362	30	is	be	AUX
ejpam-2553	362	31	regular	regular	ADJ
ejpam-2553	362	32	.	.	PUNCT
ejpam-2553	363	1	(	(	PUNCT
ejpam-2553	363	2	2	2	X
ejpam-2553	363	3	)	)	PUNCT
ejpam-2553	363	4	if	if	SCONJ
ejpam-2553	363	5	an	an	DET
ejpam-2553	363	6	⊆	⊆	NUM
ejpam-2553	363	7	anran	anran	NOUN
ejpam-2553	363	8	for	for	ADP
ejpam-2553	363	9	r	r	NOUN
ejpam-2553	363	10	∈	∈	PROPN
ejpam-2553	363	11	r	r	NOUN
ejpam-2553	363	12	and	and	CCONJ
ejpam-2553	363	13	n≥	n≥	NOUN
ejpam-2553	363	14	1	1	NUM
ejpam-2553	363	15	,	,	PUNCT
ejpam-2553	363	16	then	then	ADV
ejpam-2553	363	17	anr	anr	VERB
ejpam-2553	363	18	⊆	⊆	NUM
ejpam-2553	363	19	idem(r	idem(r	NOUN
ejpam-2553	363	20	)	)	PUNCT
ejpam-2553	363	21	.	.	PUNCT
ejpam-2553	364	1	(	(	PUNCT
ejpam-2553	364	2	3	3	X
ejpam-2553	364	3	)	)	PUNCT
ejpam-2553	364	4	if	if	SCONJ
ejpam-2553	364	5	an	an	PRON
ejpam-2553	364	6	=	=	X
ejpam-2553	364	7	ue	ue	PROPN
ejpam-2553	364	8	for	for	ADP
ejpam-2553	364	9	some	some	DET
ejpam-2553	364	10	u	u	PROPN
ejpam-2553	364	11	∈	∈	PROPN
ejpam-2553	364	12	u(r	u(r	PROPN
ejpam-2553	364	13	)	)	PUNCT
ejpam-2553	364	14	and	and	CCONJ
ejpam-2553	364	15	e	e	X
ejpam-2553	364	16	∈	∈	PROPN
ejpam-2553	364	17	idem(r	idem(r	PROPN
ejpam-2553	364	18	)	)	PUNCT
ejpam-2553	364	19	and	and	CCONJ
ejpam-2553	364	20	|a2n|=	|a2n|=	PROPN
ejpam-2553	364	21	1	1	NUM
ejpam-2553	364	22	then	then	ADV
ejpam-2553	364	23	there	there	PRON
ejpam-2553	364	24	exists	exist	VERB
ejpam-2553	364	25	`	`	PUNCT
ejpam-2553	364	26	∈	∈	PROPN
ejpam-2553	364	27	an	an	DET
ejpam-2553	364	28	such	such	ADJ
ejpam-2553	364	29	that	that	SCONJ
ejpam-2553	364	30	`	`	PUNCT
ejpam-2553	364	31	is	be	AUX
ejpam-2553	364	32	π	π	PROPN
ejpam-2553	364	33	-	-	ADJ
ejpam-2553	364	34	regular	regular	ADJ
ejpam-2553	364	35	.	.	PUNCT
ejpam-2553	365	1	proof	proof	NOUN
ejpam-2553	365	2	.	.	PUNCT
ejpam-2553	366	1	it	it	PRON
ejpam-2553	366	2	’s	’	VERB
ejpam-2553	366	3	straightforward	straightforward	ADJ
ejpam-2553	366	4	by	by	ADP
ejpam-2553	366	5	theorem	theorem	NOUN
ejpam-2553	366	6	9	9	NUM
ejpam-2553	366	7	.	.	PUNCT
ejpam-2553	367	1	for	for	ADP
ejpam-2553	367	2	each	each	DET
ejpam-2553	367	3	hyperideal	hyperideal	NOUN
ejpam-2553	367	4	i	i	PRON
ejpam-2553	367	5	of	of	ADP
ejpam-2553	367	6	multiplicative	multiplicative	ADJ
ejpam-2553	367	7	hyperring	hyperring	NOUN
ejpam-2553	367	8	r	r	NOUN
ejpam-2553	367	9	,	,	PUNCT
ejpam-2553	367	10	letting	let	VERB
ejpam-2553	367	11	oi	oi	X
ejpam-2553	367	12	=	=	X
ejpam-2553	367	13	{	{	PUNCT
ejpam-2553	367	14	a	a	DET
ejpam-2553	367	15	∈	∈	X
ejpam-2553	367	16	i	i	PRON
ejpam-2553	367	17	:	:	PUNCT
ejpam-2553	367	18	a	a	DET
ejpam-2553	367	19	∈	∈	NOUN
ejpam-2553	367	20	ai	ai	VERB
ejpam-2553	367	21	}	}	PUNCT
ejpam-2553	367	22	.	.	PUNCT
ejpam-2553	368	1	then	then	ADV
ejpam-2553	368	2	oi	oi	INTJ
ejpam-2553	368	3	is	be	AUX
ejpam-2553	368	4	called	call	VERB
ejpam-2553	368	5	a	a	DET
ejpam-2553	368	6	pure	pure	ADJ
ejpam-2553	368	7	part	part	NOUN
ejpam-2553	368	8	of	of	ADP
ejpam-2553	368	9	i	i	PRON
ejpam-2553	368	10	.	.	PUNCT
ejpam-2553	369	1	an	an	DET
ejpam-2553	369	2	hyperideal	hyperideal	NOUN
ejpam-2553	369	3	i	i	PRON
ejpam-2553	369	4	is	be	AUX
ejpam-2553	369	5	called	call	VERB
ejpam-2553	369	6	a	a	DET
ejpam-2553	369	7	pure	pure	ADJ
ejpam-2553	369	8	hyperideal	hyperideal	NOUN
ejpam-2553	369	9	if	if	SCONJ
ejpam-2553	369	10	i	i	PRON
ejpam-2553	369	11	=	=	NOUN
ejpam-2553	369	12	oi	oi	INTJ
ejpam-2553	369	13	.	.	PUNCT
ejpam-2553	370	1	theorem	theorem	VERB
ejpam-2553	370	2	11	11	NUM
ejpam-2553	370	3	.	.	PUNCT
ejpam-2553	371	1	let	let	VERB
ejpam-2553	371	2	r	r	PRON
ejpam-2553	371	3	be	be	AUX
ejpam-2553	371	4	a	a	DET
ejpam-2553	371	5	commutative	commutative	ADJ
ejpam-2553	371	6	multiplicative	multiplicative	ADJ
ejpam-2553	371	7	hyperring	hyperring	NOUN
ejpam-2553	371	8	with	with	ADP
ejpam-2553	371	9	a	a	DET
ejpam-2553	371	10	scalar	scalar	ADJ
ejpam-2553	371	11	identity	identity	NOUN
ejpam-2553	371	12	1	1	NUM
ejpam-2553	371	13	and	and	CCONJ
ejpam-2553	371	14	a	a	DET
ejpam-2553	371	15	∈	∈	NOUN
ejpam-2553	371	16	r	r	NOUN
ejpam-2553	371	17	and	and	CCONJ
ejpam-2553	371	18	also	also	ADV
ejpam-2553	371	19	for	for	ADP
ejpam-2553	371	20	r	r	NOUN
ejpam-2553	371	21	∈	∈	NOUN
ejpam-2553	371	22	r	r	NOUN
ejpam-2553	371	23	there	there	PRON
ejpam-2553	371	24	exists	exist	VERB
ejpam-2553	371	25	n	n	PRON
ejpam-2553	371	26	∈	∈	PROPN
ejpam-2553	371	27	n	n	PRON
ejpam-2553	371	28	such	such	ADJ
ejpam-2553	371	29	that	that	SCONJ
ejpam-2553	371	30	|arn|	|arn|	PROPN
ejpam-2553	371	31	=	=	PROPN
ejpam-2553	372	1	1	1	X
ejpam-2553	372	2	.	.	X
ejpam-2553	373	1	if	if	SCONJ
ejpam-2553	373	2	hyperideal	hyperideal	VERB
ejpam-2553	373	3	<	<	X
ejpam-2553	373	4	a	a	X
ejpam-2553	373	5	>	>	X
ejpam-2553	373	6	is	be	AUX
ejpam-2553	373	7	pure	pure	ADJ
ejpam-2553	373	8	hyperideal	hyperideal	NOUN
ejpam-2553	373	9	,	,	PUNCT
ejpam-2553	373	10	then	then	ADV
ejpam-2553	373	11	r=	r=	VERB
ejpam-2553	373	12	<	<	X
ejpam-2553	373	13	a	a	X
ejpam-2553	373	14	>	>	X
ejpam-2553	373	15	+	+	PROPN
ejpam-2553	373	16	ann(a	ann(a	PROPN
ejpam-2553	373	17	)	)	PUNCT
ejpam-2553	373	18	.	.	PUNCT
ejpam-2553	374	1	r.	r.	PROPN
ejpam-2553	374	2	ameri	ameri	PROPN
ejpam-2553	374	3	,	,	PUNCT
ejpam-2553	374	4	a.	a.	PROPN
ejpam-2553	374	5	kordi	kordi	PROPN
ejpam-2553	374	6	/	/	PUNCT
ejpam-2553	374	7	eur	eur	PROPN
ejpam-2553	374	8	.	.	PUNCT
ejpam-2553	375	1	j.	j.	PROPN
ejpam-2553	375	2	pure	pure	PROPN
ejpam-2553	375	3	appl	appl	PROPN
ejpam-2553	375	4	.	.	PROPN
ejpam-2553	375	5	math	math	PROPN
ejpam-2553	375	6	,	,	PUNCT
ejpam-2553	375	7	9	9	NUM
ejpam-2553	375	8	(	(	PUNCT
ejpam-2553	375	9	2016	2016	NUM
ejpam-2553	375	10	)	)	PUNCT
ejpam-2553	375	11	,	,	PUNCT
ejpam-2553	375	12	402	402	NUM
ejpam-2553	375	13	-	-	SYM
ejpam-2553	375	14	418	418	NUM
ejpam-2553	375	15	413	413	NUM
ejpam-2553	375	16	proof	proof	NOUN
ejpam-2553	375	17	.	.	PUNCT
ejpam-2553	375	18	suppose	suppose	VERB
ejpam-2553	375	19	that	that	SCONJ
ejpam-2553	375	20	<	<	X
ejpam-2553	375	21	a	a	X
ejpam-2553	375	22	>	>	X
ejpam-2553	375	23	is	be	AUX
ejpam-2553	375	24	pure	pure	ADJ
ejpam-2553	375	25	hyperideal	hyperideal	NOUN
ejpam-2553	375	26	,	,	PUNCT
ejpam-2553	375	27	then	then	ADV
ejpam-2553	375	28	there	there	PRON
ejpam-2553	375	29	exists	exist	VERB
ejpam-2553	375	30	x	x	PUNCT
ejpam-2553	375	31	=	=	SYM
ejpam-2553	375	32	sa+	sa+	X
ejpam-2553	376	1	∑n	∑n	PROPN
ejpam-2553	376	2	i=1	i=1	PROPN
ejpam-2553	376	3	a	a	PRON
ejpam-2553	376	4	·	·	PUNCT
ejpam-2553	376	5	x	x	SYM
ejpam-2553	376	6	i	i	PRON
ejpam-2553	376	7	⊆	⊆	X
ejpam-2553	376	8	<	<	X
ejpam-2553	376	9	a	a	DET
ejpam-2553	376	10	>	>	X
ejpam-2553	376	11	,	,	PUNCT
ejpam-2553	376	12	such	such	ADJ
ejpam-2553	376	13	that	that	SCONJ
ejpam-2553	376	14	a	a	DET
ejpam-2553	376	15	∈	∈	NOUN
ejpam-2553	376	16	ax	ax	NOUN
ejpam-2553	376	17	,	,	PUNCT
ejpam-2553	376	18	where	where	SCONJ
ejpam-2553	376	19	s	s	VERB
ejpam-2553	376	20	∈	∈	PROPN
ejpam-2553	376	21	n	n	CCONJ
ejpam-2553	376	22	,	,	PUNCT
ejpam-2553	376	23	r	r	NOUN
ejpam-2553	376	24	,	,	PUNCT
ejpam-2553	376	25	x	x	VERB
ejpam-2553	376	26	i	i	PROPN
ejpam-2553	376	27	∈	∈	PROPN
ejpam-2553	376	28	r.	r.	PROPN
ejpam-2553	376	29	so	so	ADV
ejpam-2553	376	30	,	,	PUNCT
ejpam-2553	376	31	there	there	PRON
ejpam-2553	376	32	exists	exist	VERB
ejpam-2553	376	33	`	`	PUNCT
ejpam-2553	376	34	∈	∈	PROPN
ejpam-2553	376	35	x	x	X
ejpam-2553	376	36	such	such	ADJ
ejpam-2553	376	37	that	that	SCONJ
ejpam-2553	376	38	a	a	DET
ejpam-2553	376	39	∈	∈	PROPN
ejpam-2553	376	40	a	a	DET
ejpam-2553	376	41	`	`	PUNCT
ejpam-2553	376	42	.	.	PUNCT
ejpam-2553	377	1	thus	thus	ADV
ejpam-2553	377	2	we	we	PRON
ejpam-2553	377	3	have	have	VERB
ejpam-2553	377	4	a(1−	a(1−	NOUN
ejpam-2553	377	5	`	`	PUNCT
ejpam-2553	377	6	)	)	PUNCT
ejpam-2553	377	7	⊆	⊆	X
ejpam-2553	377	8	a	a	DET
ejpam-2553	377	9	−	−	NOUN
ejpam-2553	377	10	a	a	DET
ejpam-2553	377	11	`	`	PUNCT
ejpam-2553	377	12	⊆	⊆	NUM
ejpam-2553	377	13	a	a	DET
ejpam-2553	377	14	−	−	NOUN
ejpam-2553	377	15	a`2	a`2	PROPN
ejpam-2553	377	16	⊆	⊆	NUM
ejpam-2553	377	17	.	.	PUNCT
ejpam-2553	377	18	.	.	PUNCT
ejpam-2553	377	19	.	.	PUNCT
ejpam-2553	378	1	⊆	⊆	X
ejpam-2553	378	2	a	a	DET
ejpam-2553	378	3	−	−	NOUN
ejpam-2553	378	4	a`n	a`n	NOUN
ejpam-2553	378	5	=	=	SYM
ejpam-2553	378	6	0	0	NUM
ejpam-2553	378	7	,	,	PUNCT
ejpam-2553	378	8	i.e.	i.e.	X
ejpam-2553	378	9	,	,	PUNCT
ejpam-2553	378	10	a(1−	a(1−	NOUN
ejpam-2553	378	11	`	`	PUNCT
ejpam-2553	378	12	)	)	PUNCT
ejpam-2553	378	13	=	=	SYM
ejpam-2553	378	14	0	0	NUM
ejpam-2553	378	15	,	,	PUNCT
ejpam-2553	378	16	which	which	PRON
ejpam-2553	378	17	implies	imply	VERB
ejpam-2553	378	18	that	that	SCONJ
ejpam-2553	378	19	1−	1−	NUM
ejpam-2553	378	20	`	`	PUNCT
ejpam-2553	378	21	∈	∈	PROPN
ejpam-2553	378	22	ann(a	ann(a	PROPN
ejpam-2553	378	23	)	)	PUNCT
ejpam-2553	378	24	.	.	PUNCT
ejpam-2553	379	1	therefore	therefore	ADV
ejpam-2553	379	2	1=	1=	NUM
ejpam-2553	379	3	`	`	PUNCT
ejpam-2553	379	4	+	+	CCONJ
ejpam-2553	379	5	(	(	PUNCT
ejpam-2553	379	6	1−	1−	NUM
ejpam-2553	379	7	`	`	PUNCT
ejpam-2553	379	8	)	)	PUNCT
ejpam-2553	379	9	∈	∈	PROPN
ejpam-2553	379	10	<	<	X
ejpam-2553	379	11	a	a	X
ejpam-2553	379	12	>	>	X
ejpam-2553	379	13	+	+	PROPN
ejpam-2553	379	14	ann(a	ann(a	PROPN
ejpam-2553	379	15	)	)	PUNCT
ejpam-2553	379	16	.	.	PUNCT
ejpam-2553	380	1	hence	hence	ADV
ejpam-2553	380	2	r=	r=	VERB
ejpam-2553	380	3	<	<	X
ejpam-2553	380	4	a	a	DET
ejpam-2553	380	5	>	>	X
ejpam-2553	380	6	+	+	PROPN
ejpam-2553	380	7	ann(a	ann(a	PROPN
ejpam-2553	380	8	)	)	PUNCT
ejpam-2553	380	9	.	.	PUNCT
ejpam-2553	381	1	theorem	theorem	NOUN
ejpam-2553	381	2	12	12	NUM
ejpam-2553	381	3	.	.	PUNCT
ejpam-2553	382	1	let	let	VERB
ejpam-2553	382	2	r	r	PRON
ejpam-2553	382	3	be	be	AUX
ejpam-2553	382	4	a	a	DET
ejpam-2553	382	5	multiplicative	multiplicative	ADJ
ejpam-2553	382	6	hyperring	hyperring	NOUN
ejpam-2553	382	7	with	with	ADP
ejpam-2553	382	8	a	a	DET
ejpam-2553	382	9	scalar	scalar	ADJ
ejpam-2553	382	10	identity	identity	NOUN
ejpam-2553	382	11	1	1	NUM
ejpam-2553	382	12	and	and	CCONJ
ejpam-2553	382	13	m	m	AUX
ejpam-2553	382	14	be	be	AUX
ejpam-2553	382	15	a	a	DET
ejpam-2553	382	16	maximal	maximal	ADJ
ejpam-2553	382	17	hyperideal	hyperideal	NOUN
ejpam-2553	382	18	of	of	ADP
ejpam-2553	382	19	r.	r.	PROPN
ejpam-2553	382	20	also	also	ADV
ejpam-2553	382	21	,	,	PUNCT
ejpam-2553	382	22	for	for	ADP
ejpam-2553	382	23	a	a	DET
ejpam-2553	382	24	∈	∈	NOUN
ejpam-2553	382	25	m	m	NOUN
ejpam-2553	382	26	and	and	CCONJ
ejpam-2553	382	27	r	r	NOUN
ejpam-2553	382	28	∈	∈	NOUN
ejpam-2553	382	29	r	r	NOUN
ejpam-2553	382	30	there	there	PRON
ejpam-2553	382	31	exists	exist	VERB
ejpam-2553	382	32	s	s	PROPN
ejpam-2553	382	33	∈	∈	PROPN
ejpam-2553	382	34	n	n	CCONJ
ejpam-2553	382	35	,	,	PUNCT
ejpam-2553	382	36	such	such	ADJ
ejpam-2553	382	37	that	that	DET
ejpam-2553	382	38	|ars|	|ars|	PROPN
ejpam-2553	382	39	=	=	SYM
ejpam-2553	383	1	1	1	X
ejpam-2553	383	2	.	.	PUNCT
ejpam-2553	384	1	then	then	ADV
ejpam-2553	384	2	we	we	PRON
ejpam-2553	384	3	have	have	VERB
ejpam-2553	384	4	the	the	DET
ejpam-2553	384	5	following	following	ADJ
ejpam-2553	384	6	statements	statement	NOUN
ejpam-2553	384	7	:	:	PUNCT
ejpam-2553	384	8	(	(	PUNCT
ejpam-2553	384	9	1	1	X
ejpam-2553	384	10	)	)	PUNCT
ejpam-2553	384	11	if	if	SCONJ
ejpam-2553	384	12	a	a	DET
ejpam-2553	384	13	∈	∈	PROPN
ejpam-2553	384	14	v	v	NOUN
ejpam-2553	384	15	(	(	PUNCT
ejpam-2553	384	16	r	r	NOUN
ejpam-2553	384	17	)	)	PUNCT
ejpam-2553	384	18	then	then	ADV
ejpam-2553	384	19	for	for	ADP
ejpam-2553	384	20	a	a	DET
ejpam-2553	384	21	∈	∈	PROPN
ejpam-2553	384	22	m	m	NOUN
ejpam-2553	384	23	,	,	PUNCT
ejpam-2553	384	24	a	a	DET
ejpam-2553	384	25	∈	∈	NOUN
ejpam-2553	384	26	om	om	NOUN
ejpam-2553	384	27	,	,	PUNCT
ejpam-2553	384	28	(	(	PUNCT
ejpam-2553	384	29	2	2	X
ejpam-2553	384	30	)	)	PUNCT
ejpam-2553	384	31	a	a	DET
ejpam-2553	384	32	∈	∈	NOUN
ejpam-2553	384	33	om	om	NOUN
ejpam-2553	384	34	for	for	ADP
ejpam-2553	384	35	a	a	DET
ejpam-2553	384	36	∈	∈	NOUN
ejpam-2553	384	37	m	m	NOUN
ejpam-2553	384	38	if	if	SCONJ
ejpam-2553	385	1	and	and	CCONJ
ejpam-2553	385	2	only	only	ADV
ejpam-2553	385	3	if	if	SCONJ
ejpam-2553	385	4	ann(a	ann(a	NOUN
ejpam-2553	385	5	)	)	PUNCT
ejpam-2553	385	6	is	be	AUX
ejpam-2553	385	7	not	not	PART
ejpam-2553	385	8	contained	contain	VERB
ejpam-2553	385	9	in	in	ADP
ejpam-2553	385	10	m.	m.	NOUN
ejpam-2553	385	11	proof	proof	NOUN
ejpam-2553	385	12	.	.	PUNCT
ejpam-2553	386	1	(	(	PUNCT
ejpam-2553	386	2	1	1	X
ejpam-2553	386	3	)	)	PUNCT
ejpam-2553	386	4	since	since	SCONJ
ejpam-2553	386	5	a	a	DET
ejpam-2553	386	6	∈	∈	PROPN
ejpam-2553	386	7	v	v	NOUN
ejpam-2553	386	8	(	(	PUNCT
ejpam-2553	386	9	r	r	NOUN
ejpam-2553	386	10	)	)	PUNCT
ejpam-2553	386	11	,	,	PUNCT
ejpam-2553	386	12	then	then	ADV
ejpam-2553	386	13	for	for	ADP
ejpam-2553	386	14	some	some	DET
ejpam-2553	386	15	r	r	NOUN
ejpam-2553	386	16	∈	∈	NOUN
ejpam-2553	386	17	r	r	NOUN
ejpam-2553	386	18	,	,	PUNCT
ejpam-2553	386	19	a	a	DET
ejpam-2553	386	20	∈	∈	PROPN
ejpam-2553	386	21	ara	ara	NOUN
ejpam-2553	386	22	.	.	PUNCT
ejpam-2553	387	1	now	now	ADV
ejpam-2553	387	2	,	,	PUNCT
ejpam-2553	387	3	for	for	ADP
ejpam-2553	387	4	a	a	DET
ejpam-2553	387	5	maximal	maximal	ADJ
ejpam-2553	387	6	hyperideal	hyperideal	NOUN
ejpam-2553	387	7	m	m	VERB
ejpam-2553	387	8	such	such	ADJ
ejpam-2553	387	9	that	that	SCONJ
ejpam-2553	387	10	a	a	DET
ejpam-2553	387	11	∈	∈	NOUN
ejpam-2553	387	12	m	m	VERB
ejpam-2553	387	13	,	,	PUNCT
ejpam-2553	387	14	we	we	PRON
ejpam-2553	387	15	have	have	VERB
ejpam-2553	387	16	a	a	DET
ejpam-2553	387	17	∈	∈	PROPN
ejpam-2553	387	18	ara	ara	NOUN
ejpam-2553	387	19	=	=	SYM
ejpam-2553	387	20	a(ra	a(ra	PROPN
ejpam-2553	387	21	)	)	PUNCT
ejpam-2553	387	22	⊆	⊆	NUM
ejpam-2553	387	23	am	am	NOUN
ejpam-2553	387	24	.	.	PUNCT
ejpam-2553	388	1	hence	hence	ADV
ejpam-2553	388	2	a	a	DET
ejpam-2553	388	3	∈	∈	NOUN
ejpam-2553	388	4	om	om	NOUN
ejpam-2553	388	5	.	.	PUNCT
ejpam-2553	389	1	(	(	PUNCT
ejpam-2553	389	2	2	2	NUM
ejpam-2553	389	3	):	):	PUNCT
ejpam-2553	389	4	(	(	PUNCT
ejpam-2553	389	5	⇒	⇒	NOUN
ejpam-2553	389	6	)	)	PUNCT
ejpam-2553	389	7	suppose	suppose	VERB
ejpam-2553	389	8	that	that	SCONJ
ejpam-2553	389	9	a	a	DET
ejpam-2553	389	10	∈	∈	PROPN
ejpam-2553	389	11	om	om	NOUN
ejpam-2553	389	12	and	and	CCONJ
ejpam-2553	389	13	ann(a	ann(a	PROPN
ejpam-2553	389	14	)	)	PUNCT
ejpam-2553	389	15	⊆	⊆	NUM
ejpam-2553	389	16	m	m	NOUN
ejpam-2553	389	17	.	.	PUNCT
ejpam-2553	390	1	then	then	ADV
ejpam-2553	390	2	there	there	PRON
ejpam-2553	390	3	is	be	VERB
ejpam-2553	390	4	a	a	DET
ejpam-2553	390	5	m	m	NOUN
ejpam-2553	390	6	∈	∈	NOUN
ejpam-2553	390	7	m	m	NOUN
ejpam-2553	390	8	such	such	ADJ
ejpam-2553	390	9	that	that	SCONJ
ejpam-2553	390	10	a	a	DET
ejpam-2553	390	11	∈	∈	NOUN
ejpam-2553	390	12	am	be	AUX
ejpam-2553	390	13	.	.	PUNCT
ejpam-2553	391	1	thus	thus	ADV
ejpam-2553	391	2	a(1	a(1	ADJ
ejpam-2553	391	3	−	−	PROPN
ejpam-2553	391	4	m	m	NOUN
ejpam-2553	391	5	)	)	PUNCT
ejpam-2553	391	6	⊆	⊆	X
ejpam-2553	391	7	a	a	DET
ejpam-2553	391	8	−	−	PROPN
ejpam-2553	391	9	am	be	AUX
ejpam-2553	391	10	⊆	⊆	NUM
ejpam-2553	391	11	a	a	DET
ejpam-2553	391	12	−	−	NOUN
ejpam-2553	391	13	am2	am2	DET
ejpam-2553	391	14	⊆	⊆	NUM
ejpam-2553	391	15	.	.	PUNCT
ejpam-2553	391	16	.	.	PUNCT
ejpam-2553	391	17	.	.	PUNCT
ejpam-2553	392	1	⊆	⊆	X
ejpam-2553	392	2	a	a	DET
ejpam-2553	392	3	−	−	NOUN
ejpam-2553	392	4	ams	am	NOUN
ejpam-2553	392	5	=	=	SYM
ejpam-2553	392	6	0	0	NUM
ejpam-2553	392	7	,	,	PUNCT
ejpam-2553	392	8	i.e.	i.e.	X
ejpam-2553	392	9	,	,	PUNCT
ejpam-2553	392	10	a(1	a(1	PROPN
ejpam-2553	392	11	−	−	PROPN
ejpam-2553	392	12	m	m	NOUN
ejpam-2553	392	13	)	)	PUNCT
ejpam-2553	393	1	=	=	SYM
ejpam-2553	393	2	0	0	X
ejpam-2553	393	3	.	.	PUNCT
ejpam-2553	394	1	therefore	therefore	ADV
ejpam-2553	394	2	,	,	PUNCT
ejpam-2553	394	3	1−m	1−m	NUM
ejpam-2553	394	4	∈	∈	NOUN
ejpam-2553	394	5	ann(a	ann(a	PROPN
ejpam-2553	394	6	)	)	PUNCT
ejpam-2553	394	7	⊆	⊆	NUM
ejpam-2553	394	8	m	m	NOUN
ejpam-2553	394	9	,	,	PUNCT
ejpam-2553	394	10	i.e.	i.e.	X
ejpam-2553	394	11	,	,	PUNCT
ejpam-2553	394	12	1−m	1−m	NUM
ejpam-2553	394	13	∈	∈	NOUN
ejpam-2553	394	14	m	m	VERB
ejpam-2553	394	15	and	and	CCONJ
ejpam-2553	394	16	it	it	PRON
ejpam-2553	394	17	’s	’	VERB
ejpam-2553	394	18	contradiction	contradiction	NOUN
ejpam-2553	394	19	.	.	PUNCT
ejpam-2553	395	1	hence	hence	ADV
ejpam-2553	395	2	ann(a	ann(a	PROPN
ejpam-2553	395	3	)	)	PUNCT
ejpam-2553	395	4	6⊆	6⊆	NUM
ejpam-2553	395	5	m	m	PROPN
ejpam-2553	395	6	.	.	PUNCT
ejpam-2553	396	1	(	(	PUNCT
ejpam-2553	396	2	⇐	⇐	ADJ
ejpam-2553	396	3	)	)	PUNCT
ejpam-2553	396	4	assume	assume	VERB
ejpam-2553	396	5	that	that	SCONJ
ejpam-2553	396	6	ann(a	ann(a	PROPN
ejpam-2553	396	7	)	)	PUNCT
ejpam-2553	396	8	is	be	AUX
ejpam-2553	396	9	not	not	PART
ejpam-2553	396	10	contained	contain	VERB
ejpam-2553	396	11	in	in	ADP
ejpam-2553	396	12	m	m	PROPN
ejpam-2553	396	13	.	.	PUNCT
ejpam-2553	397	1	then	then	ADV
ejpam-2553	397	2	r	r	NOUN
ejpam-2553	397	3	=	=	PUNCT
ejpam-2553	397	4	m	m	VERB
ejpam-2553	397	5	+	+	ADJ
ejpam-2553	397	6	ann(a	ann(a	PROPN
ejpam-2553	397	7	)	)	PUNCT
ejpam-2553	397	8	.	.	PUNCT
ejpam-2553	398	1	so	so	ADV
ejpam-2553	398	2	,	,	PUNCT
ejpam-2553	398	3	there	there	PRON
ejpam-2553	398	4	exist	exist	VERB
ejpam-2553	398	5	m	m	PROPN
ejpam-2553	398	6	∈	∈	NOUN
ejpam-2553	398	7	m	m	NOUN
ejpam-2553	398	8	,	,	PUNCT
ejpam-2553	398	9	x	x	SYM
ejpam-2553	398	10	∈	∈	PROPN
ejpam-2553	398	11	ann(a	ann(a	PROPN
ejpam-2553	398	12	)	)	PUNCT
ejpam-2553	398	13	such	such	ADJ
ejpam-2553	398	14	that	that	SCONJ
ejpam-2553	398	15	1	1	NUM
ejpam-2553	398	16	=	=	SYM
ejpam-2553	398	17	m	m	VERB
ejpam-2553	398	18	+	+	ADJ
ejpam-2553	398	19	x	x	SYM
ejpam-2553	398	20	,	,	PUNCT
ejpam-2553	398	21	then	then	ADV
ejpam-2553	398	22	a	a	DET
ejpam-2553	398	23	∈	∈	NOUN
ejpam-2553	398	24	a.1	a.1	PUNCT
ejpam-2553	399	1	=	=	PUNCT
ejpam-2553	399	2	a(m	a(m	PROPN
ejpam-2553	399	3	+	+	SYM
ejpam-2553	399	4	x	x	X
ejpam-2553	399	5	)	)	PUNCT
ejpam-2553	399	6	⊆	⊆	NUM
ejpam-2553	399	7	am	am	NOUN
ejpam-2553	399	8	+	+	CCONJ
ejpam-2553	399	9	ax	ax	NOUN
ejpam-2553	399	10	=	=	SYM
ejpam-2553	399	11	am	be	AUX
ejpam-2553	399	12	,	,	PUNCT
ejpam-2553	399	13	i.e.	i.e.	X
ejpam-2553	399	14	,	,	PUNCT
ejpam-2553	399	15	a	a	DET
ejpam-2553	399	16	∈	∈	NOUN
ejpam-2553	399	17	am	be	AUX
ejpam-2553	399	18	.	.	PUNCT
ejpam-2553	400	1	hence	hence	ADV
ejpam-2553	400	2	a	a	DET
ejpam-2553	400	3	∈	∈	NOUN
ejpam-2553	400	4	om	om	NOUN
ejpam-2553	400	5	.	.	PUNCT
ejpam-2553	401	1	3	3	X
ejpam-2553	401	2	.	.	X
ejpam-2553	401	3	properties	property	NOUN
ejpam-2553	401	4	of	of	ADP
ejpam-2553	401	5	m(r	m(r	PROPN
ejpam-2553	401	6	)	)	PUNCT
ejpam-2553	401	7	definition	definition	NOUN
ejpam-2553	401	8	10	10	NUM
ejpam-2553	401	9	.	.	PUNCT
ejpam-2553	402	1	let	let	VERB
ejpam-2553	402	2	r	r	PRON
ejpam-2553	402	3	be	be	AUX
ejpam-2553	402	4	a	a	DET
ejpam-2553	402	5	multiplicative	multiplicative	ADJ
ejpam-2553	402	6	hyperring	hyperring	NOUN
ejpam-2553	402	7	.	.	PUNCT
ejpam-2553	403	1	denote	denote	VERB
ejpam-2553	403	2	by	by	ADP
ejpam-2553	403	3	m(r	m(r	PROPN
ejpam-2553	403	4	)	)	PUNCT
ejpam-2553	403	5	the	the	DET
ejpam-2553	403	6	set	set	NOUN
ejpam-2553	403	7	of	of	ADP
ejpam-2553	403	8	all	all	DET
ejpam-2553	403	9	elements	element	NOUN
ejpam-2553	403	10	in	in	ADP
ejpam-2553	403	11	r	r	NOUN
ejpam-2553	403	12	such	such	ADJ
ejpam-2553	403	13	that	that	SCONJ
ejpam-2553	403	14	the	the	DET
ejpam-2553	403	15	generated	generate	VERB
ejpam-2553	403	16	hyperideals	hyperideal	NOUN
ejpam-2553	403	17	by	by	ADP
ejpam-2553	403	18	each	each	PRON
ejpam-2553	403	19	of	of	ADP
ejpam-2553	403	20	these	these	DET
ejpam-2553	403	21	elements	element	NOUN
ejpam-2553	403	22	are	be	AUX
ejpam-2553	403	23	regular	regular	ADJ
ejpam-2553	403	24	.	.	PUNCT
ejpam-2553	404	1	clearly	clearly	ADV
ejpam-2553	404	2	,	,	PUNCT
ejpam-2553	404	3	m(r	m(r	PROPN
ejpam-2553	404	4	)	)	PUNCT
ejpam-2553	404	5	is	be	AUX
ejpam-2553	404	6	a	a	DET
ejpam-2553	404	7	regular	regular	ADJ
ejpam-2553	404	8	hyperring	hyperring	NOUN
ejpam-2553	404	9	.	.	PUNCT
ejpam-2553	405	1	lemma	lemma	PROPN
ejpam-2553	405	2	1	1	NUM
ejpam-2553	405	3	.	.	PUNCT
ejpam-2553	406	1	if	if	SCONJ
ejpam-2553	406	2	a	a	PRON
ejpam-2553	406	3	and	and	CCONJ
ejpam-2553	406	4	b	b	NOUN
ejpam-2553	406	5	are	be	AUX
ejpam-2553	406	6	two	two	NUM
ejpam-2553	406	7	regular	regular	ADJ
ejpam-2553	406	8	elements	element	NOUN
ejpam-2553	406	9	in	in	ADP
ejpam-2553	406	10	a	a	DET
ejpam-2553	406	11	commutative	commutative	ADJ
ejpam-2553	406	12	multiplicative	multiplicative	ADJ
ejpam-2553	406	13	hyperring	hyperring	NOUN
ejpam-2553	406	14	r.	r.	PROPN
ejpam-2553	406	15	then	then	ADV
ejpam-2553	406	16	the	the	DET
ejpam-2553	406	17	following	follow	VERB
ejpam-2553	406	18	statements	statement	NOUN
ejpam-2553	406	19	hold	hold	VERB
ejpam-2553	406	20	:	:	PUNCT
ejpam-2553	406	21	(	(	PUNCT
ejpam-2553	406	22	1	1	X
ejpam-2553	406	23	)	)	PUNCT
ejpam-2553	406	24	γ∗(ab	γ∗(ab	PROPN
ejpam-2553	406	25	)	)	PUNCT
ejpam-2553	406	26	is	be	AUX
ejpam-2553	406	27	regular	regular	ADJ
ejpam-2553	406	28	in	in	ADP
ejpam-2553	406	29	r	r	NOUN
ejpam-2553	406	30	/	/	SYM
ejpam-2553	406	31	γ∗	γ∗	NOUN
ejpam-2553	406	32	,	,	PUNCT
ejpam-2553	406	33	(	(	PUNCT
ejpam-2553	406	34	2	2	X
ejpam-2553	406	35	)	)	PUNCT
ejpam-2553	406	36	moreover	moreover	ADV
ejpam-2553	406	37	,	,	PUNCT
ejpam-2553	406	38	if	if	SCONJ
ejpam-2553	406	39	|ab|=	|ab|=	PROPN
ejpam-2553	406	40	1	1	NUM
ejpam-2553	406	41	,	,	PUNCT
ejpam-2553	406	42	then	then	ADV
ejpam-2553	406	43	so	so	ADV
ejpam-2553	406	44	is	be	AUX
ejpam-2553	406	45	ab	ab	PROPN
ejpam-2553	406	46	.	.	PUNCT
ejpam-2553	406	47	proof	proof	NOUN
ejpam-2553	406	48	.	.	PUNCT
ejpam-2553	407	1	(	(	PUNCT
ejpam-2553	407	2	1	1	X
ejpam-2553	407	3	)	)	PUNCT
ejpam-2553	407	4	since	since	SCONJ
ejpam-2553	407	5	a	a	DET
ejpam-2553	407	6	,	,	PUNCT
ejpam-2553	407	7	b	b	NOUN
ejpam-2553	407	8	are	be	AUX
ejpam-2553	407	9	regular	regular	ADJ
ejpam-2553	407	10	in	in	ADP
ejpam-2553	407	11	r	r	NOUN
ejpam-2553	407	12	,	,	PUNCT
ejpam-2553	407	13	then	then	ADV
ejpam-2553	407	14	there	there	PRON
ejpam-2553	407	15	exist	exist	VERB
ejpam-2553	407	16	r1	r1	NOUN
ejpam-2553	407	17	,	,	PUNCT
ejpam-2553	407	18	r2	r2	PROPN
ejpam-2553	407	19	∈	∈	PROPN
ejpam-2553	407	20	r	r	NOUN
ejpam-2553	407	21	,	,	PUNCT
ejpam-2553	407	22	such	such	ADJ
ejpam-2553	407	23	that	that	SCONJ
ejpam-2553	407	24	a	a	DET
ejpam-2553	407	25	∈	∈	PROPN
ejpam-2553	407	26	ar1a	ar1a	PROPN
ejpam-2553	407	27	,	,	PUNCT
ejpam-2553	407	28	b	b	X
ejpam-2553	407	29	∈	∈	PROPN
ejpam-2553	407	30	br2	br2	PROPN
ejpam-2553	407	31	b.	b.	PROPN
ejpam-2553	407	32	so	so	SCONJ
ejpam-2553	407	33	γ∗(ab	γ∗(ab	PROPN
ejpam-2553	407	34	)	)	PUNCT
ejpam-2553	407	35	=	=	SYM
ejpam-2553	407	36	γ∗(ab	γ∗(ab	PROPN
ejpam-2553	407	37	)	)	PUNCT
ejpam-2553	407	38	�	�	PROPN
ejpam-2553	407	39	γ∗(r1r2	γ∗(r1r2	NUM
ejpam-2553	407	40	)	)	PUNCT
ejpam-2553	407	41	�	�	PROPN
ejpam-2553	407	42	γ∗(ab	γ∗(ab	PROPN
ejpam-2553	407	43	)	)	PUNCT
ejpam-2553	407	44	.	.	PUNCT
ejpam-2553	408	1	hence	hence	ADV
ejpam-2553	408	2	γ∗(ab	γ∗(ab	PROPN
ejpam-2553	408	3	)	)	PUNCT
ejpam-2553	408	4	is	be	AUX
ejpam-2553	408	5	regular	regular	ADJ
ejpam-2553	408	6	in	in	ADP
ejpam-2553	408	7	r	r	NOUN
ejpam-2553	408	8	/	/	SYM
ejpam-2553	408	9	γ∗.	γ∗.	ADJ
ejpam-2553	408	10	(	(	PUNCT
ejpam-2553	408	11	2	2	NUM
ejpam-2553	408	12	)	)	PUNCT
ejpam-2553	408	13	by	by	ADP
ejpam-2553	408	14	definition	definition	NOUN
ejpam-2553	408	15	of	of	ADP
ejpam-2553	408	16	regular	regular	ADJ
ejpam-2553	408	17	element	element	NOUN
ejpam-2553	408	18	,	,	PUNCT
ejpam-2553	408	19	there	there	PRON
ejpam-2553	408	20	exist	exist	VERB
ejpam-2553	408	21	r1	r1	NOUN
ejpam-2553	408	22	,	,	PUNCT
ejpam-2553	408	23	r2	r2	PROPN
ejpam-2553	408	24	∈	∈	PROPN
ejpam-2553	408	25	r	r	NOUN
ejpam-2553	408	26	,	,	PUNCT
ejpam-2553	408	27	such	such	ADJ
ejpam-2553	408	28	that	that	SCONJ
ejpam-2553	408	29	a	a	DET
ejpam-2553	408	30	∈	∈	PROPN
ejpam-2553	408	31	ar1a	ar1a	PROPN
ejpam-2553	408	32	and	and	CCONJ
ejpam-2553	408	33	b	b	PROPN
ejpam-2553	408	34	∈	∈	PROPN
ejpam-2553	408	35	br2	br2	PROPN
ejpam-2553	408	36	b.	b.	PROPN
ejpam-2553	408	37	since	since	SCONJ
ejpam-2553	408	38	|ab|=	|ab|=	PROPN
ejpam-2553	408	39	1	1	NUM
ejpam-2553	408	40	,	,	PUNCT
ejpam-2553	408	41	we	we	PRON
ejpam-2553	408	42	have	have	VERB
ejpam-2553	408	43	ab	ab	PROPN
ejpam-2553	408	44	∈	∈	PROPN
ejpam-2553	408	45	(	(	PUNCT
ejpam-2553	408	46	ar1a)(br2	ar1a)(br2	PROPN
ejpam-2553	408	47	b	b	X
ejpam-2553	408	48	)	)	PUNCT
ejpam-2553	408	49	=	=	SYM
ejpam-2553	408	50	(	(	PUNCT
ejpam-2553	408	51	ab)r1r2(ab	ab)r1r2(ab	PROPN
ejpam-2553	408	52	)	)	PUNCT
ejpam-2553	408	53	.	.	PUNCT
ejpam-2553	409	1	theorem	theorem	VERB
ejpam-2553	409	2	13	13	NUM
ejpam-2553	409	3	.	.	PUNCT
ejpam-2553	410	1	let	let	VERB
ejpam-2553	410	2	r	r	PRON
ejpam-2553	410	3	be	be	AUX
ejpam-2553	410	4	a	a	DET
ejpam-2553	410	5	strongly	strongly	ADV
ejpam-2553	410	6	distributive	distributive	ADJ
ejpam-2553	410	7	multiplicative	multiplicative	ADJ
ejpam-2553	410	8	hyperring	hyperring	NOUN
ejpam-2553	410	9	and	and	CCONJ
ejpam-2553	410	10	a	a	DET
ejpam-2553	410	11	∈	∈	PROPN
ejpam-2553	410	12	r.	r.	NOUN
ejpam-2553	410	13	thus	thus	ADV
ejpam-2553	410	14	we	we	PRON
ejpam-2553	410	15	have	have	VERB
ejpam-2553	410	16	the	the	DET
ejpam-2553	410	17	following	following	ADJ
ejpam-2553	410	18	statements	statement	NOUN
ejpam-2553	410	19	:	:	PUNCT
ejpam-2553	410	20	r.	r.	PROPN
ejpam-2553	410	21	ameri	ameri	PROPN
ejpam-2553	410	22	,	,	PUNCT
ejpam-2553	410	23	a.	a.	PROPN
ejpam-2553	410	24	kordi	kordi	PROPN
ejpam-2553	410	25	/	/	PUNCT
ejpam-2553	410	26	eur	eur	PROPN
ejpam-2553	410	27	.	.	PUNCT
ejpam-2553	411	1	j.	j.	PROPN
ejpam-2553	411	2	pure	pure	PROPN
ejpam-2553	411	3	appl	appl	PROPN
ejpam-2553	411	4	.	.	PROPN
ejpam-2553	411	5	math	math	PROPN
ejpam-2553	411	6	,	,	PUNCT
ejpam-2553	411	7	9	9	NUM
ejpam-2553	411	8	(	(	PUNCT
ejpam-2553	411	9	2016	2016	NUM
ejpam-2553	411	10	)	)	PUNCT
ejpam-2553	411	11	,	,	PUNCT
ejpam-2553	411	12	402	402	NUM
ejpam-2553	411	13	-	-	SYM
ejpam-2553	411	14	418	418	NUM
ejpam-2553	411	15	414	414	NUM
ejpam-2553	411	16	(	(	PUNCT
ejpam-2553	411	17	1	1	NUM
ejpam-2553	411	18	)	)	PUNCT
ejpam-2553	411	19	if	if	SCONJ
ejpam-2553	411	20	there	there	PRON
ejpam-2553	411	21	is	be	VERB
ejpam-2553	411	22	a	a	DET
ejpam-2553	411	23	regular	regular	ADJ
ejpam-2553	411	24	element	element	NOUN
ejpam-2553	411	25	c	c	PROPN
ejpam-2553	411	26	in	in	ADP
ejpam-2553	411	27	a2	a2	PROPN
ejpam-2553	411	28	−	−	PROPN
ejpam-2553	411	29	a	a	PROPN
ejpam-2553	411	30	,	,	PUNCT
ejpam-2553	411	31	then	then	ADV
ejpam-2553	411	32	a	a	PRON
ejpam-2553	411	33	and	and	CCONJ
ejpam-2553	411	34	1−	1−	NUM
ejpam-2553	411	35	a	a	PRON
ejpam-2553	411	36	are	be	AUX
ejpam-2553	411	37	regular	regular	ADJ
ejpam-2553	411	38	.	.	PUNCT
ejpam-2553	412	1	(	(	PUNCT
ejpam-2553	412	2	2	2	X
ejpam-2553	412	3	)	)	PUNCT
ejpam-2553	412	4	if	if	SCONJ
ejpam-2553	412	5	a	a	PRON
ejpam-2553	412	6	and	and	CCONJ
ejpam-2553	412	7	1−	1−	NUM
ejpam-2553	412	8	a	a	PRON
ejpam-2553	412	9	are	be	AUX
ejpam-2553	412	10	regular	regular	ADJ
ejpam-2553	412	11	in	in	ADP
ejpam-2553	412	12	r	r	NOUN
ejpam-2553	412	13	such	such	ADJ
ejpam-2553	412	14	that	that	SCONJ
ejpam-2553	412	15	|a(1−	|a(1−	NOUN
ejpam-2553	412	16	a)|=	a)|=	PROPN
ejpam-2553	412	17	1	1	NUM
ejpam-2553	412	18	,	,	PUNCT
ejpam-2553	412	19	then	then	ADV
ejpam-2553	412	20	so	so	ADV
ejpam-2553	412	21	is	be	AUX
ejpam-2553	412	22	a(1−	a(1−	NOUN
ejpam-2553	412	23	a	a	PRON
ejpam-2553	412	24	)	)	PUNCT
ejpam-2553	412	25	.	.	PUNCT
ejpam-2553	413	1	proof	proof	NOUN
ejpam-2553	413	2	.	.	PUNCT
ejpam-2553	414	1	(	(	PUNCT
ejpam-2553	414	2	1	1	X
ejpam-2553	414	3	)	)	PUNCT
ejpam-2553	414	4	it	it	PRON
ejpam-2553	414	5	immediately	immediately	ADV
ejpam-2553	414	6	follows	follow	VERB
ejpam-2553	414	7	by	by	ADP
ejpam-2553	414	8	proposition	proposition	NOUN
ejpam-2553	414	9	2	2	NUM
ejpam-2553	414	10	.	.	PUNCT
ejpam-2553	415	1	(	(	PUNCT
ejpam-2553	415	2	2	2	NUM
ejpam-2553	415	3	)	)	PUNCT
ejpam-2553	415	4	by	by	ADP
ejpam-2553	415	5	lemma	lemma	PROPN
ejpam-2553	415	6	1(2	1(2	NUM
ejpam-2553	415	7	)	)	PUNCT
ejpam-2553	416	1	it	it	PRON
ejpam-2553	416	2	is	be	AUX
ejpam-2553	416	3	clear	clear	ADJ
ejpam-2553	416	4	.	.	PUNCT
ejpam-2553	417	1	theorem	theorem	ADJ
ejpam-2553	417	2	14	14	NUM
ejpam-2553	417	3	.	.	PUNCT
ejpam-2553	418	1	if	if	SCONJ
ejpam-2553	418	2	r	r	PRON
ejpam-2553	418	3	be	be	VERB
ejpam-2553	418	4	a	a	DET
ejpam-2553	418	5	strongly	strongly	ADV
ejpam-2553	418	6	distributive	distributive	ADJ
ejpam-2553	418	7	multiplicative	multiplicative	ADJ
ejpam-2553	418	8	hyperring	hyperring	NOUN
ejpam-2553	418	9	,	,	PUNCT
ejpam-2553	418	10	then	then	ADV
ejpam-2553	418	11	a	a	DET
ejpam-2553	418	12	right	right	ADJ
ejpam-2553	418	13	hyperideal	hyperideal	NOUN
ejpam-2553	418	14	i	i	PRON
ejpam-2553	418	15	in	in	ADP
ejpam-2553	418	16	the	the	DET
ejpam-2553	418	17	hyperring	hyperring	NOUN
ejpam-2553	418	18	m(r	m(r	PROPN
ejpam-2553	418	19	)	)	PUNCT
ejpam-2553	418	20	is	be	AUX
ejpam-2553	418	21	a	a	DET
ejpam-2553	418	22	right	right	ADJ
ejpam-2553	418	23	hyperideal	hyperideal	NOUN
ejpam-2553	418	24	in	in	ADP
ejpam-2553	418	25	r.	r.	PROPN
ejpam-2553	418	26	proof	proof	NOUN
ejpam-2553	418	27	.	.	PUNCT
ejpam-2553	419	1	suppose	suppose	VERB
ejpam-2553	419	2	a	a	DET
ejpam-2553	419	3	∈	∈	PROPN
ejpam-2553	419	4	i	i	PRON
ejpam-2553	419	5	,	,	PUNCT
ejpam-2553	419	6	r	r	NOUN
ejpam-2553	419	7	∈	∈	PROPN
ejpam-2553	419	8	r	r	NOUN
ejpam-2553	419	9	,	,	PUNCT
ejpam-2553	419	10	then	then	ADV
ejpam-2553	419	11	ar	ar	VERB
ejpam-2553	419	12	⊆	⊆	NUM
ejpam-2553	419	13	m(r	m(r	PROPN
ejpam-2553	419	14	)	)	PUNCT
ejpam-2553	419	15	,	,	PUNCT
ejpam-2553	419	16	hence	hence	ADV
ejpam-2553	419	17	for	for	ADP
ejpam-2553	419	18	some	some	DET
ejpam-2553	419	19	element	element	NOUN
ejpam-2553	419	20	r	r	NOUN
ejpam-2553	419	21	′	′	NOUN
ejpam-2553	419	22	∈	∈	NOUN
ejpam-2553	419	23	r	r	NOUN
ejpam-2553	419	24	,	,	PUNCT
ejpam-2553	419	25	ar	ar	NOUN
ejpam-2553	419	26	⊆	⊆	NUM
ejpam-2553	419	27	arr	arr	NOUN
ejpam-2553	419	28	′ar	′ar	NOUN
ejpam-2553	419	29	.	.	PUNCT
ejpam-2553	420	1	but	but	CCONJ
ejpam-2553	420	2	r	r	NOUN
ejpam-2553	420	3	r	r	NOUN
ejpam-2553	420	4	′ar	′ar	NOUN
ejpam-2553	420	5	⊆	⊆	NUM
ejpam-2553	420	6	m(r	m(r	NOUN
ejpam-2553	420	7	)	)	PUNCT
ejpam-2553	420	8	,	,	PUNCT
ejpam-2553	420	9	so	so	CCONJ
ejpam-2553	420	10	ar	ar	PROPN
ejpam-2553	420	11	⊆	⊆	NUM
ejpam-2553	420	12	i	i	PRON
ejpam-2553	420	13	.	.	PUNCT
ejpam-2553	421	1	thus	thus	ADV
ejpam-2553	421	2	i	i	PRON
ejpam-2553	421	3	is	be	AUX
ejpam-2553	421	4	a	a	DET
ejpam-2553	421	5	right	right	ADJ
ejpam-2553	421	6	hyperideal	hyperideal	NOUN
ejpam-2553	421	7	in	in	ADP
ejpam-2553	421	8	r.	r.	PROPN
ejpam-2553	421	9	theorem	theorem	PROPN
ejpam-2553	421	10	15	15	NUM
ejpam-2553	421	11	.	.	PUNCT
ejpam-2553	422	1	let	let	VERB
ejpam-2553	422	2	r	r	PRON
ejpam-2553	422	3	be	be	AUX
ejpam-2553	422	4	a	a	DET
ejpam-2553	422	5	strongly	strongly	ADV
ejpam-2553	422	6	distributive	distributive	ADJ
ejpam-2553	422	7	multiplicative	multiplicative	ADJ
ejpam-2553	422	8	hyperring	hyperring	NOUN
ejpam-2553	422	9	.	.	PUNCT
ejpam-2553	423	1	then	then	ADV
ejpam-2553	423	2	m(r	m(r	PROPN
ejpam-2553	423	3	)	)	PUNCT
ejpam-2553	423	4	is	be	AUX
ejpam-2553	423	5	a	a	DET
ejpam-2553	423	6	hyperideal	hyperideal	NOUN
ejpam-2553	423	7	of	of	ADP
ejpam-2553	423	8	r.	r.	PROPN
ejpam-2553	423	9	proof	proof	NOUN
ejpam-2553	423	10	.	.	PUNCT
ejpam-2553	424	1	let	let	VERB
ejpam-2553	424	2	z	z	NOUN
ejpam-2553	424	3	∈	∈	PROPN
ejpam-2553	424	4	m(r	m(r	PROPN
ejpam-2553	424	5	)	)	PUNCT
ejpam-2553	424	6	and	and	CCONJ
ejpam-2553	424	7	r	r	PROPN
ejpam-2553	424	8	∈	∈	PROPN
ejpam-2553	424	9	r.	r.	NOUN
ejpam-2553	424	10	since	since	SCONJ
ejpam-2553	424	11	<	<	X
ejpam-2553	424	12	zr	zr	X
ejpam-2553	424	13	>	>	X
ejpam-2553	424	14	⊆	⊆	X
ejpam-2553	424	15	<	<	X
ejpam-2553	424	16	z	z	X
ejpam-2553	424	17	>	>	X
ejpam-2553	424	18	and	and	CCONJ
ejpam-2553	424	19	<	<	X
ejpam-2553	424	20	rz	rz	NOUN
ejpam-2553	424	21	>	>	X
ejpam-2553	424	22	⊆	⊆	X
ejpam-2553	424	23	<	<	X
ejpam-2553	424	24	z	z	X
ejpam-2553	424	25	>	>	PUNCT
ejpam-2553	424	26	,	,	PUNCT
ejpam-2553	424	27	we	we	PRON
ejpam-2553	424	28	have	have	VERB
ejpam-2553	424	29	zr	zr	PROPN
ejpam-2553	424	30	⊆	⊆	NUM
ejpam-2553	424	31	m(r	m(r	NOUN
ejpam-2553	424	32	)	)	PUNCT
ejpam-2553	424	33	and	and	CCONJ
ejpam-2553	424	34	rz	rz	NOUN
ejpam-2553	424	35	⊆	⊆	NUM
ejpam-2553	424	36	m(r	m(r	PROPN
ejpam-2553	424	37	)	)	PUNCT
ejpam-2553	424	38	,	,	PUNCT
ejpam-2553	424	39	thus	thus	ADV
ejpam-2553	424	40	zr	zr	X
ejpam-2553	424	41	∪	∪	ADP
ejpam-2553	424	42	rz	rz	PROPN
ejpam-2553	424	43	⊆	⊆	NUM
ejpam-2553	424	44	m(r	m(r	PROPN
ejpam-2553	424	45	)	)	PUNCT
ejpam-2553	424	46	.	.	PUNCT
ejpam-2553	425	1	now	now	ADV
ejpam-2553	425	2	,	,	PUNCT
ejpam-2553	425	3	assume	assume	VERB
ejpam-2553	425	4	that	that	SCONJ
ejpam-2553	425	5	t1	t1	NOUN
ejpam-2553	425	6	,	,	PUNCT
ejpam-2553	425	7	t2	t2	PROPN
ejpam-2553	425	8	∈	∈	PROPN
ejpam-2553	425	9	m(r	m(r	PROPN
ejpam-2553	425	10	)	)	PUNCT
ejpam-2553	425	11	.	.	PUNCT
ejpam-2553	426	1	we	we	PRON
ejpam-2553	426	2	need	need	VERB
ejpam-2553	426	3	to	to	PART
ejpam-2553	426	4	prove	prove	VERB
ejpam-2553	426	5	that	that	SCONJ
ejpam-2553	426	6	all	all	PRON
ejpam-2553	426	7	of	of	ADP
ejpam-2553	426	8	elements	element	NOUN
ejpam-2553	426	9	in	in	ADP
ejpam-2553	426	10	<	<	X
ejpam-2553	426	11	t1	t1	NOUN
ejpam-2553	426	12	−	−	PROPN
ejpam-2553	426	13	t2	t2	PROPN
ejpam-2553	426	14	>	>	PUNCT
ejpam-2553	426	15	is	be	AUX
ejpam-2553	426	16	regular	regular	ADJ
ejpam-2553	426	17	.	.	PUNCT
ejpam-2553	427	1	for	for	ADP
ejpam-2553	427	2	achieving	achieve	VERB
ejpam-2553	427	3	to	to	ADP
ejpam-2553	427	4	it	it	PRON
ejpam-2553	427	5	,	,	PUNCT
ejpam-2553	427	6	let	let	VERB
ejpam-2553	427	7	a	a	DET
ejpam-2553	427	8	∈	∈	PROPN
ejpam-2553	427	9	<	<	X
ejpam-2553	427	10	t1	t1	NOUN
ejpam-2553	427	11	−	−	PROPN
ejpam-2553	427	12	t2	t2	PROPN
ejpam-2553	427	13	>	>	PUNCT
ejpam-2553	427	14	.	.	PUNCT
ejpam-2553	428	1	then	then	ADV
ejpam-2553	428	2	for	for	ADP
ejpam-2553	428	3	some	some	DET
ejpam-2553	428	4	u	u	NOUN
ejpam-2553	428	5	∈	∈	PROPN
ejpam-2553	428	6	<	<	X
ejpam-2553	428	7	t1	t1	NOUN
ejpam-2553	428	8	>	>	X
ejpam-2553	428	9	and	and	CCONJ
ejpam-2553	428	10	v	v	ADP
ejpam-2553	428	11	∈	∈	PROPN
ejpam-2553	428	12	<	<	X
ejpam-2553	428	13	t2	t2	PROPN
ejpam-2553	428	14	>	>	X
ejpam-2553	428	15	we	we	PRON
ejpam-2553	428	16	have	have	VERB
ejpam-2553	428	17	a	a	DET
ejpam-2553	428	18	=	=	SYM
ejpam-2553	428	19	u−	u−	PROPN
ejpam-2553	429	1	v.	v.	CCONJ
ejpam-2553	429	2	as	as	SCONJ
ejpam-2553	429	3	<	<	X
ejpam-2553	429	4	t1	t1	NOUN
ejpam-2553	429	5	>	>	X
ejpam-2553	429	6	is	be	AUX
ejpam-2553	429	7	regular	regular	ADJ
ejpam-2553	429	8	then	then	ADV
ejpam-2553	429	9	there	there	PRON
ejpam-2553	429	10	exists	exist	VERB
ejpam-2553	429	11	r	r	NOUN
ejpam-2553	429	12	∈	∈	PROPN
ejpam-2553	429	13	r	r	NOUN
ejpam-2553	429	14	such	such	ADJ
ejpam-2553	429	15	that	that	SCONJ
ejpam-2553	429	16	u	u	PROPN
ejpam-2553	429	17	∈	∈	PROPN
ejpam-2553	429	18	uru	uru	X
ejpam-2553	429	19	.	.	PUNCT
ejpam-2553	430	1	thus	thus	ADV
ejpam-2553	430	2	by	by	ADP
ejpam-2553	430	3	distributive	distributive	ADJ
ejpam-2553	430	4	property	property	NOUN
ejpam-2553	430	5	of	of	ADP
ejpam-2553	430	6	r	r	NOUN
ejpam-2553	430	7	we	we	PRON
ejpam-2553	430	8	have	have	VERB
ejpam-2553	430	9	ara	ara	PROPN
ejpam-2553	430	10	=(	=(	PROPN
ejpam-2553	430	11	u−	u−	PROPN
ejpam-2553	430	12	v)r(u−	v)r(u−	PRON
ejpam-2553	430	13	v)−	v)−	PROPN
ejpam-2553	430	14	u+	u+	NOUN
ejpam-2553	430	15	v	v	ADP
ejpam-2553	430	16	=	=	PROPN
ejpam-2553	430	17	uru−	uru−	PROPN
ejpam-2553	430	18	urv	urv	ADP
ejpam-2553	430	19	−	−	PROPN
ejpam-2553	430	20	vru+	vru+	ADV
ejpam-2553	430	21	vrv	vrv	PROPN
ejpam-2553	431	1	−	−	PROPN
ejpam-2553	431	2	u+	u+	NOUN
ejpam-2553	431	3	v	v	ADP
ejpam-2553	431	4	⊆uru−	⊆uru−	PROPN
ejpam-2553	431	5	uru+	uru+	ADJ
ejpam-2553	431	6	v	v	ADP
ejpam-2553	431	7	−	−	PROPN
ejpam-2553	431	8	urv	urv	ADP
ejpam-2553	431	9	−	−	PROPN
ejpam-2553	431	10	vru+	vru+	ADV
ejpam-2553	431	11	vrv	vrv	PROPN
ejpam-2553	432	1	=	=	NOUN
ejpam-2553	432	2	u(r	u(r	ADV
ejpam-2553	432	3	−	−	ADP
ejpam-2553	432	4	r)u+	r)u+	NOUN
ejpam-2553	432	5	v	v	ADP
ejpam-2553	432	6	−	−	NOUN
ejpam-2553	432	7	urv	urv	ADP
ejpam-2553	432	8	−	−	PROPN
ejpam-2553	432	9	vru+	vru+	ADV
ejpam-2553	432	10	vrv	vrv	PROPN
ejpam-2553	433	1	=	=	NOUN
ejpam-2553	433	2	u0u+	u0u+	NOUN
ejpam-2553	433	3	v	v	ADP
ejpam-2553	433	4	−	−	PROPN
ejpam-2553	433	5	urv	urv	ADP
ejpam-2553	433	6	−	−	PROPN
ejpam-2553	433	7	vru+	vru+	ADV
ejpam-2553	433	8	vrv	vrv	PROPN
ejpam-2553	433	9	=	=	NUM
ejpam-2553	433	10	u(v	u(v	NOUN
ejpam-2553	433	11	−	−	PROPN
ejpam-2553	433	12	v)u+	v)u+	PROPN
ejpam-2553	433	13	v	v	ADP
ejpam-2553	433	14	−	−	NOUN
ejpam-2553	433	15	urv	urv	ADP
ejpam-2553	433	16	−	−	PROPN
ejpam-2553	433	17	vru+	vru+	ADV
ejpam-2553	433	18	vrv	vrv	PROPN
ejpam-2553	434	1	=	=	ADJ
ejpam-2553	434	2	uvu−	uvu−	PROPN
ejpam-2553	434	3	uvu+	uvu+	PROPN
ejpam-2553	434	4	v	v	NOUN
ejpam-2553	434	5	−	−	PROPN
ejpam-2553	434	6	urv	urv	ADP
ejpam-2553	434	7	−	−	PROPN
ejpam-2553	434	8	vru+	vru+	ADV
ejpam-2553	434	9	vrv	vrv	PROPN
ejpam-2553	434	10	,	,	PUNCT
ejpam-2553	434	11	since	since	SCONJ
ejpam-2553	434	12	the	the	DET
ejpam-2553	434	13	right	right	ADJ
ejpam-2553	434	14	side	side	NOUN
ejpam-2553	434	15	is	be	AUX
ejpam-2553	434	16	in	in	ADP
ejpam-2553	434	17	<	<	X
ejpam-2553	434	18	t2	t2	PROPN
ejpam-2553	434	19	>	>	PUNCT
ejpam-2553	434	20	,	,	PUNCT
ejpam-2553	434	21	then	then	ADV
ejpam-2553	434	22	ara	ara	PROPN
ejpam-2553	434	23	−	−	PROPN
ejpam-2553	434	24	a	a	DET
ejpam-2553	434	25	⊆	⊆	NOUN
ejpam-2553	434	26	<	<	X
ejpam-2553	434	27	t2	t2	NOUN
ejpam-2553	434	28	>	>	X
ejpam-2553	434	29	,	,	PUNCT
ejpam-2553	434	30	i.e.	i.e.	X
ejpam-2553	434	31	,	,	PUNCT
ejpam-2553	434	32	ara	ara	PROPN
ejpam-2553	434	33	−	−	PROPN
ejpam-2553	434	34	a	a	PRON
ejpam-2553	434	35	is	be	AUX
ejpam-2553	434	36	regular	regular	ADJ
ejpam-2553	434	37	and	and	CCONJ
ejpam-2553	434	38	by	by	ADP
ejpam-2553	434	39	proposition	proposition	NOUN
ejpam-2553	434	40	2	2	NUM
ejpam-2553	434	41	,	,	PUNCT
ejpam-2553	434	42	a	a	PRON
ejpam-2553	434	43	is	be	AUX
ejpam-2553	434	44	regular	regular	ADJ
ejpam-2553	434	45	.	.	PUNCT
ejpam-2553	435	1	hence	hence	ADV
ejpam-2553	435	2	<	<	X
ejpam-2553	435	3	t1	t1	PROPN
ejpam-2553	435	4	−	−	PROPN
ejpam-2553	435	5	t2	t2	PROPN
ejpam-2553	435	6	>	>	PUNCT
ejpam-2553	435	7	is	be	AUX
ejpam-2553	435	8	a	a	DET
ejpam-2553	435	9	regular	regular	ADJ
ejpam-2553	435	10	hyperideal	hyperideal	NOUN
ejpam-2553	435	11	and	and	CCONJ
ejpam-2553	435	12	t1	t1	VERB
ejpam-2553	435	13	−	−	PROPN
ejpam-2553	435	14	t2	t2	PROPN
ejpam-2553	435	15	∈	∈	PROPN
ejpam-2553	435	16	m(r	m(r	PROPN
ejpam-2553	435	17	)	)	PUNCT
ejpam-2553	435	18	,	,	PUNCT
ejpam-2553	435	19	i.e.	i.e.	X
ejpam-2553	435	20	,	,	PUNCT
ejpam-2553	435	21	m(r)−m(r	m(r)−m(r	NOUN
ejpam-2553	435	22	)	)	PUNCT
ejpam-2553	435	23	⊆	⊆	NUM
ejpam-2553	435	24	m(r	m(r	NOUN
ejpam-2553	435	25	)	)	PUNCT
ejpam-2553	435	26	.	.	PUNCT
ejpam-2553	436	1	let	let	VERB
ejpam-2553	436	2	(	(	PUNCT
ejpam-2553	436	3	r,+	r,+	NUM
ejpam-2553	436	4	,	,	PUNCT
ejpam-2553	436	5	.	.	PUNCT
ejpam-2553	436	6	)	)	PUNCT
ejpam-2553	437	1	be	be	AUX
ejpam-2553	437	2	a	a	DET
ejpam-2553	437	3	multiplicative	multiplicative	ADJ
ejpam-2553	437	4	hyperring	hyperring	NOUN
ejpam-2553	438	1	and	and	CCONJ
ejpam-2553	438	2	i	i	PRON
ejpam-2553	438	3	be	be	VERB
ejpam-2553	438	4	a	a	DET
ejpam-2553	438	5	hyperideal	hyperideal	NOUN
ejpam-2553	438	6	of	of	ADP
ejpam-2553	438	7	it	it	PRON
ejpam-2553	438	8	.	.	PUNCT
ejpam-2553	439	1	we	we	PRON
ejpam-2553	439	2	consider	consider	VERB
ejpam-2553	439	3	the	the	DET
ejpam-2553	439	4	usual	usual	ADJ
ejpam-2553	439	5	addition	addition	NOUN
ejpam-2553	439	6	of	of	ADP
ejpam-2553	439	7	cosets	coset	NOUN
ejpam-2553	439	8	and	and	CCONJ
ejpam-2553	439	9	the	the	DET
ejpam-2553	439	10	multiplication	multiplication	NOUN
ejpam-2553	439	11	defined	define	VERB
ejpam-2553	439	12	as	as	SCONJ
ejpam-2553	439	13	follows	follow	VERB
ejpam-2553	439	14	:	:	PUNCT
ejpam-2553	439	15	(	(	PUNCT
ejpam-2553	439	16	a+	a+	PUNCT
ejpam-2553	439	17	i	i	NOUN
ejpam-2553	439	18	)	)	PUNCT
ejpam-2553	439	19	∗	∗	NOUN
ejpam-2553	439	20	(	(	PUNCT
ejpam-2553	439	21	b+	b+	X
ejpam-2553	439	22	i	i	X
ejpam-2553	439	23	)	)	PUNCT
ejpam-2553	439	24	=	=	PUNCT
ejpam-2553	440	1	{	{	PUNCT
ejpam-2553	440	2	c	c	NOUN
ejpam-2553	440	3	+	+	NOUN
ejpam-2553	440	4	i	i	PRON
ejpam-2553	440	5	|c	|c	VERB
ejpam-2553	440	6	∈	∈	PROPN
ejpam-2553	440	7	a.b	a.b	NOUN
ejpam-2553	440	8	}	}	PUNCT
ejpam-2553	440	9	.	.	PUNCT
ejpam-2553	441	1	on	on	ADP
ejpam-2553	441	2	the	the	DET
ejpam-2553	441	3	set	set	NOUN
ejpam-2553	441	4	r	r	NOUN
ejpam-2553	441	5	/	/	SYM
ejpam-2553	441	6	i	i	NOUN
ejpam-2553	441	7	=	=	PUNCT
ejpam-2553	441	8	{	{	PUNCT
ejpam-2553	441	9	r	r	NOUN
ejpam-2553	442	1	+	+	NOUN
ejpam-2553	442	2	i	i	PRON
ejpam-2553	442	3	|r	|r	X
ejpam-2553	442	4	∈	∈	PROPN
ejpam-2553	442	5	r	r	NOUN
ejpam-2553	442	6	}	}	PUNCT
ejpam-2553	442	7	of	of	ADP
ejpam-2553	442	8	all	all	DET
ejpam-2553	442	9	cosets	coset	NOUN
ejpam-2553	442	10	of	of	ADP
ejpam-2553	442	11	i	i	PRON
ejpam-2553	442	12	.	.	PUNCT
ejpam-2553	443	1	then	then	ADV
ejpam-2553	443	2	(	(	PUNCT
ejpam-2553	443	3	r	r	X
ejpam-2553	443	4	/	/	SYM
ejpam-2553	443	5	i	i	PROPN
ejpam-2553	443	6	,	,	PUNCT
ejpam-2553	443	7	+	+	ADJ
ejpam-2553	443	8	,	,	PUNCT
ejpam-2553	443	9	∗	∗	NOUN
ejpam-2553	443	10	)	)	PUNCT
ejpam-2553	443	11	is	be	AUX
ejpam-2553	443	12	a	a	DET
ejpam-2553	443	13	multiplicative	multiplicative	ADJ
ejpam-2553	443	14	hyperring	hyperring	NOUN
ejpam-2553	443	15	.	.	PUNCT
ejpam-2553	444	1	here	here	ADV
ejpam-2553	444	2	here	here	ADV
ejpam-2553	444	3	here	here	ADV
ejpam-2553	444	4	proposition	proposition	NOUN
ejpam-2553	444	5	3	3	NUM
ejpam-2553	444	6	.	.	PUNCT
ejpam-2553	445	1	(	(	PUNCT
ejpam-2553	445	2	[	[	X
ejpam-2553	445	3	19	19	NUM
ejpam-2553	445	4	]	]	PUNCT
ejpam-2553	445	5	)	)	PUNCT
ejpam-2553	445	6	if	if	SCONJ
ejpam-2553	445	7	(	(	PUNCT
ejpam-2553	445	8	r,+	r,+	NUM
ejpam-2553	445	9	,	,	PUNCT
ejpam-2553	445	10	.	.	PUNCT
ejpam-2553	445	11	)	)	PUNCT
ejpam-2553	445	12	is	be	AUX
ejpam-2553	445	13	a	a	DET
ejpam-2553	445	14	strongly	strongly	ADV
ejpam-2553	445	15	distributive	distributive	ADJ
ejpam-2553	445	16	multiplicative	multiplicative	ADJ
ejpam-2553	445	17	hyperring	hyperring	NOUN
ejpam-2553	446	1	and	and	CCONJ
ejpam-2553	446	2	i	i	PRON
ejpam-2553	446	3	is	be	AUX
ejpam-2553	446	4	a	a	DET
ejpam-2553	446	5	hyperideal	hyperideal	NOUN
ejpam-2553	446	6	of	of	ADP
ejpam-2553	446	7	r	r	NOUN
ejpam-2553	446	8	,	,	PUNCT
ejpam-2553	446	9	then	then	ADV
ejpam-2553	446	10	(	(	PUNCT
ejpam-2553	446	11	r	r	X
ejpam-2553	446	12	/	/	SYM
ejpam-2553	446	13	i	i	PROPN
ejpam-2553	446	14	,	,	PUNCT
ejpam-2553	446	15	+	+	ADJ
ejpam-2553	446	16	,	,	PUNCT
ejpam-2553	446	17	∗	∗	NOUN
ejpam-2553	446	18	)	)	PUNCT
ejpam-2553	446	19	is	be	AUX
ejpam-2553	446	20	ring	ring	NOUN
ejpam-2553	446	21	.	.	PUNCT
ejpam-2553	447	1	r.	r.	PROPN
ejpam-2553	447	2	ameri	ameri	PROPN
ejpam-2553	447	3	,	,	PUNCT
ejpam-2553	447	4	a.	a.	PROPN
ejpam-2553	447	5	kordi	kordi	PROPN
ejpam-2553	447	6	/	/	PUNCT
ejpam-2553	447	7	eur	eur	PROPN
ejpam-2553	447	8	.	.	PUNCT
ejpam-2553	448	1	j.	j.	PROPN
ejpam-2553	448	2	pure	pure	PROPN
ejpam-2553	448	3	appl	appl	PROPN
ejpam-2553	448	4	.	.	PROPN
ejpam-2553	448	5	math	math	PROPN
ejpam-2553	448	6	,	,	PUNCT
ejpam-2553	448	7	9	9	NUM
ejpam-2553	448	8	(	(	PUNCT
ejpam-2553	448	9	2016	2016	NUM
ejpam-2553	448	10	)	)	PUNCT
ejpam-2553	448	11	,	,	PUNCT
ejpam-2553	448	12	402	402	NUM
ejpam-2553	448	13	-	-	SYM
ejpam-2553	448	14	418	418	NUM
ejpam-2553	448	15	415	415	NUM
ejpam-2553	448	16	corollary	corollary	ADJ
ejpam-2553	448	17	1	1	NUM
ejpam-2553	448	18	.	.	PUNCT
ejpam-2553	449	1	if	if	SCONJ
ejpam-2553	449	2	(	(	PUNCT
ejpam-2553	449	3	r,+	r,+	NUM
ejpam-2553	449	4	,	,	PUNCT
ejpam-2553	449	5	.	.	PUNCT
ejpam-2553	449	6	)	)	PUNCT
ejpam-2553	449	7	is	be	AUX
ejpam-2553	449	8	a	a	DET
ejpam-2553	449	9	strongly	strongly	ADV
ejpam-2553	449	10	distributive	distributive	ADJ
ejpam-2553	449	11	multiplicative	multiplicative	ADJ
ejpam-2553	449	12	hyperring	hyperring	NOUN
ejpam-2553	449	13	,	,	PUNCT
ejpam-2553	449	14	then	then	ADV
ejpam-2553	449	15	(	(	PUNCT
ejpam-2553	449	16	r	r	X
ejpam-2553	449	17	/	/	SYM
ejpam-2553	449	18	m(r),+,∗	m(r),+,∗	PROPN
ejpam-2553	449	19	)	)	PUNCT
ejpam-2553	449	20	is	be	AUX
ejpam-2553	449	21	ring	ring	NOUN
ejpam-2553	449	22	.	.	PUNCT
ejpam-2553	450	1	theorem	theorem	VERB
ejpam-2553	450	2	16	16	NUM
ejpam-2553	450	3	.	.	PUNCT
ejpam-2553	451	1	if	if	SCONJ
ejpam-2553	451	2	(	(	PUNCT
ejpam-2553	451	3	r,+	r,+	NUM
ejpam-2553	451	4	,	,	PUNCT
ejpam-2553	451	5	.	.	PUNCT
ejpam-2553	451	6	)	)	PUNCT
ejpam-2553	451	7	is	be	AUX
ejpam-2553	451	8	a	a	DET
ejpam-2553	451	9	strongly	strongly	ADV
ejpam-2553	451	10	distributive	distributive	ADJ
ejpam-2553	451	11	multiplicative	multiplicative	ADJ
ejpam-2553	451	12	hyperring	hyperring	NOUN
ejpam-2553	451	13	,	,	PUNCT
ejpam-2553	451	14	then	then	ADV
ejpam-2553	451	15	m(r	m(r	PROPN
ejpam-2553	451	16	/	/	SYM
ejpam-2553	451	17	m(r	m(r	PROPN
ejpam-2553	451	18	)	)	PUNCT
ejpam-2553	451	19	)	)	PUNCT
ejpam-2553	452	1	=	=	PRON
ejpam-2553	452	2	{	{	PUNCT
ejpam-2553	452	3	0	0	NUM
ejpam-2553	452	4	}	}	PUNCT
ejpam-2553	452	5	.	.	PUNCT
ejpam-2553	453	1	proof	proof	NOUN
ejpam-2553	453	2	.	.	PUNCT
ejpam-2553	454	1	assume	assume	VERB
ejpam-2553	454	2	that	that	SCONJ
ejpam-2553	454	3	a	a	DET
ejpam-2553	454	4	+	+	PROPN
ejpam-2553	454	5	m(r	m(r	NOUN
ejpam-2553	454	6	)	)	PUNCT
ejpam-2553	454	7	denote	denote	VERB
ejpam-2553	454	8	the	the	DET
ejpam-2553	454	9	residue	residue	NOUN
ejpam-2553	454	10	class	class	NOUN
ejpam-2553	454	11	modulo	modulo	NOUN
ejpam-2553	454	12	m(r	m(r	PROPN
ejpam-2553	454	13	)	)	PUNCT
ejpam-2553	454	14	which	which	PRON
ejpam-2553	454	15	contains	contain	VERB
ejpam-2553	454	16	the	the	DET
ejpam-2553	454	17	element	element	NOUN
ejpam-2553	454	18	a	a	PRON
ejpam-2553	454	19	of	of	ADP
ejpam-2553	454	20	r.	r.	PROPN
ejpam-2553	454	21	if	if	SCONJ
ejpam-2553	454	22	b	b	PROPN
ejpam-2553	454	23	+	+	CCONJ
ejpam-2553	454	24	m(r	m(r	PROPN
ejpam-2553	454	25	)	)	PUNCT
ejpam-2553	454	26	∈	∈	PROPN
ejpam-2553	454	27	m(r	m(r	PROPN
ejpam-2553	454	28	/	/	SYM
ejpam-2553	454	29	m(r	m(r	PROPN
ejpam-2553	454	30	)	)	PUNCT
ejpam-2553	454	31	)	)	PUNCT
ejpam-2553	454	32	and	and	CCONJ
ejpam-2553	454	33	a	a	PRON
ejpam-2553	454	34	∈	∈	PROPN
ejpam-2553	454	35	<	<	X
ejpam-2553	454	36	b	b	X
ejpam-2553	454	37	>	>	PUNCT
ejpam-2553	454	38	,	,	PUNCT
ejpam-2553	454	39	then	then	ADV
ejpam-2553	454	40	a	a	DET
ejpam-2553	454	41	+	+	PROPN
ejpam-2553	454	42	m(r	m(r	NOUN
ejpam-2553	454	43	)	)	PUNCT
ejpam-2553	454	44	∈	∈	PROPN
ejpam-2553	454	45	<	<	X
ejpam-2553	454	46	b	b	PROPN
ejpam-2553	454	47	+	+	PROPN
ejpam-2553	454	48	m(r	m(r	PROPN
ejpam-2553	454	49	)	)	PUNCT
ejpam-2553	454	50	>	>	PUNCT
ejpam-2553	454	51	.	.	PUNCT
ejpam-2553	455	1	since	since	SCONJ
ejpam-2553	455	2	<	<	X
ejpam-2553	455	3	b	b	PROPN
ejpam-2553	455	4	+	+	CCONJ
ejpam-2553	455	5	m(r	m(r	PROPN
ejpam-2553	455	6	)	)	PUNCT
ejpam-2553	455	7	>	>	X
ejpam-2553	455	8	is	be	AUX
ejpam-2553	455	9	a	a	DET
ejpam-2553	455	10	regular	regular	ADJ
ejpam-2553	455	11	ideal	ideal	NOUN
ejpam-2553	455	12	in	in	ADP
ejpam-2553	455	13	r	r	PROPN
ejpam-2553	455	14	/	/	SYM
ejpam-2553	455	15	m(r	m(r	PROPN
ejpam-2553	455	16	)	)	PUNCT
ejpam-2553	455	17	,	,	PUNCT
ejpam-2553	455	18	then	then	ADV
ejpam-2553	455	19	a	a	DET
ejpam-2553	455	20	+	+	ADJ
ejpam-2553	455	21	m(r	m(r	NOUN
ejpam-2553	455	22	)	)	PUNCT
ejpam-2553	455	23	is	be	AUX
ejpam-2553	455	24	regular	regular	ADJ
ejpam-2553	455	25	.	.	PUNCT
ejpam-2553	456	1	thus	thus	ADV
ejpam-2553	456	2	,	,	PUNCT
ejpam-2553	456	3	for	for	ADP
ejpam-2553	456	4	some	some	DET
ejpam-2553	456	5	x+m(r	x+m(r	NOUN
ejpam-2553	456	6	)	)	PUNCT
ejpam-2553	456	7	∈	∈	PROPN
ejpam-2553	456	8	r	r	X
ejpam-2553	456	9	/	/	SYM
ejpam-2553	456	10	m(r	m(r	PROPN
ejpam-2553	456	11	)	)	PUNCT
ejpam-2553	456	12	,	,	PUNCT
ejpam-2553	456	13	a+m(r	a+m(r	NUM
ejpam-2553	456	14	)	)	PUNCT
ejpam-2553	456	15	=	=	SYM
ejpam-2553	456	16	(	(	PUNCT
ejpam-2553	456	17	a+m(r))∗(x+m(r))∗(a+m(r	a+m(r))∗(x+m(r))∗(a+m(r	PROPN
ejpam-2553	456	18	)	)	PUNCT
ejpam-2553	456	19	)	)	PUNCT
ejpam-2553	456	20	or	or	CCONJ
ejpam-2553	456	21	a+m(r	a+m(r	X
ejpam-2553	456	22	)	)	PUNCT
ejpam-2553	456	23	=	=	SYM
ejpam-2553	456	24	axa+m(r	axa+m(r	NOUN
ejpam-2553	456	25	)	)	PUNCT
ejpam-2553	456	26	,	,	PUNCT
ejpam-2553	456	27	i.e.	i.e.	X
ejpam-2553	456	28	,	,	PUNCT
ejpam-2553	456	29	axa−a	axa−a	VERB
ejpam-2553	456	30	⊆	⊆	NUM
ejpam-2553	456	31	m(r	m(r	NOUN
ejpam-2553	456	32	)	)	PUNCT
ejpam-2553	456	33	.	.	PUNCT
ejpam-2553	457	1	therefore	therefore	ADV
ejpam-2553	457	2	axa−a	axa−a	PROPN
ejpam-2553	457	3	is	be	AUX
ejpam-2553	457	4	regular	regular	ADJ
ejpam-2553	457	5	element	element	NOUN
ejpam-2553	457	6	of	of	ADP
ejpam-2553	457	7	strongly	strongly	ADV
ejpam-2553	457	8	distributive	distributive	ADJ
ejpam-2553	457	9	multiplicative	multiplicative	ADJ
ejpam-2553	457	10	hyperring	hyperring	NOUN
ejpam-2553	457	11	r	r	NOUN
ejpam-2553	457	12	and	and	CCONJ
ejpam-2553	457	13	by	by	ADP
ejpam-2553	457	14	proposition	proposition	NOUN
ejpam-2553	457	15	2	2	NUM
ejpam-2553	457	16	,	,	PUNCT
ejpam-2553	457	17	a	a	PRON
ejpam-2553	457	18	is	be	AUX
ejpam-2553	457	19	regular	regular	ADJ
ejpam-2553	457	20	.	.	PUNCT
ejpam-2553	458	1	thus	thus	ADV
ejpam-2553	458	2	<	<	X
ejpam-2553	458	3	b	b	X
ejpam-2553	458	4	>	>	X
ejpam-2553	458	5	is	be	AUX
ejpam-2553	458	6	regular	regular	ADJ
ejpam-2553	458	7	hyperideal	hyperideal	NOUN
ejpam-2553	458	8	and	and	CCONJ
ejpam-2553	458	9	hence	hence	ADV
ejpam-2553	458	10	b	b	PROPN
ejpam-2553	458	11	∈	∈	PROPN
ejpam-2553	458	12	m(r	m(r	PROPN
ejpam-2553	458	13	)	)	PUNCT
ejpam-2553	458	14	.	.	PUNCT
ejpam-2553	459	1	so	so	ADV
ejpam-2553	459	2	,	,	PUNCT
ejpam-2553	459	3	b+m(r	b+m(r	NOUN
ejpam-2553	459	4	)	)	PUNCT
ejpam-2553	459	5	=	=	SYM
ejpam-2553	459	6	0r	0r	NUM
ejpam-2553	459	7	/	/	SYM
ejpam-2553	459	8	m(r	m(r	PROPN
ejpam-2553	459	9	)	)	PUNCT
ejpam-2553	459	10	.	.	PUNCT
ejpam-2553	460	1	theorem	theorem	NOUN
ejpam-2553	460	2	17	17	NUM
ejpam-2553	460	3	.	.	PUNCT
ejpam-2553	461	1	let	let	VERB
ejpam-2553	461	2	b	b	X
ejpam-2553	461	3	be	be	AUX
ejpam-2553	461	4	a	a	DET
ejpam-2553	461	5	hyperideal	hyperideal	NOUN
ejpam-2553	461	6	of	of	ADP
ejpam-2553	461	7	a	a	DET
ejpam-2553	461	8	strongly	strongly	ADV
ejpam-2553	461	9	distributive	distributive	ADJ
ejpam-2553	461	10	multiplicative	multiplicative	ADJ
ejpam-2553	461	11	hyperring	hyperring	NOUN
ejpam-2553	461	12	r.	r.	PROPN
ejpam-2553	461	13	then	then	ADV
ejpam-2553	461	14	m(b	m(b	PROPN
ejpam-2553	461	15	)	)	PUNCT
ejpam-2553	461	16	=	=	SYM
ejpam-2553	461	17	b	b	PROPN
ejpam-2553	461	18	∩m(r	∩m(r	PROPN
ejpam-2553	461	19	)	)	PUNCT
ejpam-2553	461	20	.	.	PUNCT
ejpam-2553	462	1	proof	proof	NOUN
ejpam-2553	462	2	.	.	PUNCT
ejpam-2553	463	1	assume	assume	VERB
ejpam-2553	463	2	that	that	SCONJ
ejpam-2553	463	3	b	b	PROPN
ejpam-2553	463	4	is	be	AUX
ejpam-2553	463	5	a	a	DET
ejpam-2553	463	6	hyperideal	hyperideal	NOUN
ejpam-2553	463	7	of	of	ADP
ejpam-2553	463	8	r	r	NOUN
ejpam-2553	463	9	and	and	CCONJ
ejpam-2553	463	10	let	let	VERB
ejpam-2553	463	11	b	b	X
ejpam-2553	463	12	be	be	AUX
ejpam-2553	463	13	a	a	DET
ejpam-2553	463	14	element	element	NOUN
ejpam-2553	463	15	of	of	ADP
ejpam-2553	463	16	b	b	NOUN
ejpam-2553	463	17	such	such	ADJ
ejpam-2553	463	18	that	that	PRON
ejpam-2553	463	19	generated	generate	VERB
ejpam-2553	463	20	a	a	DET
ejpam-2553	463	21	regular	regular	ADJ
ejpam-2553	463	22	hyperideal	hyperideal	NOUN
ejpam-2553	463	23	<	<	X
ejpam-2553	463	24	b	b	X
ejpam-2553	463	25	>	>	X
ejpam-2553	463	26	∗	∗	NOUN
ejpam-2553	463	27	in	in	ADP
ejpam-2553	463	28	the	the	DET
ejpam-2553	463	29	strongly	strongly	ADV
ejpam-2553	463	30	distributive	distributive	ADJ
ejpam-2553	463	31	multiplicative	multiplicative	ADJ
ejpam-2553	463	32	hyperring	hyperring	NOUN
ejpam-2553	463	33	b.	b.	PROPN
ejpam-2553	463	34	suppose	suppose	VERB
ejpam-2553	463	35	<	<	X
ejpam-2553	463	36	b	b	AUX
ejpam-2553	463	37	>	>	X
ejpam-2553	463	38	be	be	AUX
ejpam-2553	463	39	the	the	DET
ejpam-2553	463	40	hyperideal	hyperideal	NOUN
ejpam-2553	463	41	in	in	ADP
ejpam-2553	463	42	r	r	NOUN
ejpam-2553	463	43	generated	generate	VERB
ejpam-2553	463	44	by	by	ADP
ejpam-2553	463	45	the	the	DET
ejpam-2553	463	46	element	element	NOUN
ejpam-2553	463	47	b	b	PROPN
ejpam-2553	463	48	,	,	PUNCT
ejpam-2553	463	49	and	and	CCONJ
ejpam-2553	463	50	let	let	VERB
ejpam-2553	463	51	c	c	NOUN
ejpam-2553	463	52	=	=	VERB
ejpam-2553	463	53	nb+	nb+	PROPN
ejpam-2553	463	54	r	r	NOUN
ejpam-2553	463	55	b+	b+	X
ejpam-2553	463	56	bs+σri	bs+σri	ADP
ejpam-2553	463	57	bsi	bsi	PROPN
ejpam-2553	463	58	,	,	PUNCT
ejpam-2553	463	59	where	where	SCONJ
ejpam-2553	463	60	n	n	X
ejpam-2553	463	61	∈	∈	PROPN
ejpam-2553	463	62	z	z	PROPN
ejpam-2553	463	63	,	,	PUNCT
ejpam-2553	463	64	r	r	NOUN
ejpam-2553	463	65	,	,	PUNCT
ejpam-2553	463	66	s	s	PROPN
ejpam-2553	463	67	,	,	PUNCT
ejpam-2553	463	68	ri	ri	PROPN
ejpam-2553	463	69	,	,	PUNCT
ejpam-2553	463	70	si	si	PROPN
ejpam-2553	463	71	∈	∈	PROPN
ejpam-2553	463	72	r	r	NOUN
ejpam-2553	463	73	be	be	VERB
ejpam-2553	463	74	any	any	DET
ejpam-2553	463	75	element	element	NOUN
ejpam-2553	463	76	of	of	ADP
ejpam-2553	463	77	<	<	X
ejpam-2553	463	78	b	b	X
ejpam-2553	463	79	>	>	X
ejpam-2553	463	80	.	.	PUNCT
ejpam-2553	464	1	since	since	SCONJ
ejpam-2553	464	2	b	b	PROPN
ejpam-2553	464	3	is	be	AUX
ejpam-2553	464	4	regular	regular	ADJ
ejpam-2553	464	5	in	in	ADP
ejpam-2553	464	6	b	b	NOUN
ejpam-2553	464	7	we	we	PRON
ejpam-2553	464	8	have	have	VERB
ejpam-2553	464	9	for	for	ADP
ejpam-2553	464	10	some	some	DET
ejpam-2553	464	11	x	x	SYM
ejpam-2553	464	12	∈	∈	PROPN
ejpam-2553	464	13	b	b	PROPN
ejpam-2553	464	14	,	,	PUNCT
ejpam-2553	464	15	b	b	PROPN
ejpam-2553	464	16	∈	∈	PROPN
ejpam-2553	464	17	bx	bx	X
ejpam-2553	464	18	b.	b.	PROPN
ejpam-2553	464	19	hence	hence	ADV
ejpam-2553	464	20	c	c	PROPN
ejpam-2553	464	21	⊆nb+	⊆nb+	VERB
ejpam-2553	464	22	r(bx	r(bx	DET
ejpam-2553	464	23	b	b	NOUN
ejpam-2553	464	24	)	)	PUNCT
ejpam-2553	465	1	+	+	CCONJ
ejpam-2553	465	2	(	(	PUNCT
ejpam-2553	465	3	bx	bx	NOUN
ejpam-2553	465	4	b)s+σri(bx	b)s+σri(bx	VERB
ejpam-2553	465	5	b)si	b)si	PROPN
ejpam-2553	465	6	=	=	PRON
ejpam-2553	465	7	nb+	nb+	VERB
ejpam-2553	465	8	r(bx	r(bx	PRON
ejpam-2553	465	9	b	b	NOUN
ejpam-2553	465	10	)	)	PUNCT
ejpam-2553	466	1	+	+	CCONJ
ejpam-2553	466	2	(	(	PUNCT
ejpam-2553	466	3	bx	bx	NOUN
ejpam-2553	466	4	b)s+σri(bx	b)s+σri(bx	PROPN
ejpam-2553	466	5	bx	bx	NOUN
ejpam-2553	466	6	b)si	b)si	NOUN
ejpam-2553	466	7	=	=	PRON
ejpam-2553	466	8	nb+	nb+	VERB
ejpam-2553	466	9	r(bx	r(bx	DET
ejpam-2553	466	10	b	b	NOUN
ejpam-2553	466	11	)	)	PUNCT
ejpam-2553	467	1	+	+	CCONJ
ejpam-2553	467	2	(	(	PUNCT
ejpam-2553	467	3	bx	bx	PROPN
ejpam-2553	467	4	b)s+σ(ri	b)s+σ(ri	PROPN
ejpam-2553	467	5	bx)b(x	bx)b(x	PROPN
ejpam-2553	467	6	bsi	bsi	PROPN
ejpam-2553	467	7	)	)	PUNCT
ejpam-2553	467	8	,	,	PUNCT
ejpam-2553	467	9	and	and	CCONJ
ejpam-2553	467	10	thus	thus	ADV
ejpam-2553	467	11	c	c	NOUN
ejpam-2553	467	12	⊆	⊆	X
ejpam-2553	467	13	<	<	X
ejpam-2553	467	14	b	b	X
ejpam-2553	467	15	>	>	X
ejpam-2553	467	16	∗.	∗.	PROPN
ejpam-2553	467	17	as	as	ADP
ejpam-2553	467	18	<	<	X
ejpam-2553	467	19	b	b	X
ejpam-2553	467	20	>	>	X
ejpam-2553	467	21	coincides	coincide	VERB
ejpam-2553	467	22	with	with	ADP
ejpam-2553	467	23	<	<	X
ejpam-2553	467	24	b	b	X
ejpam-2553	467	25	>	>	X
ejpam-2553	467	26	∗	∗	NOUN
ejpam-2553	467	27	,	,	PUNCT
ejpam-2553	467	28	then	then	ADV
ejpam-2553	467	29	<	<	X
ejpam-2553	467	30	b	b	X
ejpam-2553	467	31	>	>	X
ejpam-2553	467	32	is	be	AUX
ejpam-2553	467	33	regular	regular	ADJ
ejpam-2553	467	34	.	.	PUNCT
ejpam-2553	468	1	hence	hence	ADV
ejpam-2553	468	2	,	,	PUNCT
ejpam-2553	468	3	for	for	ADP
ejpam-2553	468	4	b	b	PROPN
ejpam-2553	468	5	∈	∈	PROPN
ejpam-2553	468	6	m(b	m(b	PROPN
ejpam-2553	468	7	)	)	PUNCT
ejpam-2553	468	8	we	we	PRON
ejpam-2553	468	9	have	have	VERB
ejpam-2553	468	10	b	b	NUM
ejpam-2553	468	11	∈	∈	PROPN
ejpam-2553	468	12	b	b	PROPN
ejpam-2553	468	13	∩m(r	∩m(r	PROPN
ejpam-2553	468	14	)	)	PUNCT
ejpam-2553	468	15	,	,	PUNCT
ejpam-2553	468	16	i.e.	i.e.	X
ejpam-2553	468	17	,	,	PUNCT
ejpam-2553	468	18	m(b	m(b	NOUN
ejpam-2553	468	19	)	)	PUNCT
ejpam-2553	468	20	⊆	⊆	NUM
ejpam-2553	468	21	b	b	PROPN
ejpam-2553	468	22	∩m(r	∩m(r	PROPN
ejpam-2553	468	23	)	)	PUNCT
ejpam-2553	468	24	.	.	PUNCT
ejpam-2553	469	1	conversely	conversely	ADV
ejpam-2553	469	2	,	,	PUNCT
ejpam-2553	469	3	let	let	VERB
ejpam-2553	469	4	b	b	X
ejpam-2553	469	5	∈	∈	PROPN
ejpam-2553	469	6	b	b	PROPN
ejpam-2553	469	7	∩m(r	∩m(r	PROPN
ejpam-2553	469	8	)	)	PUNCT
ejpam-2553	469	9	,	,	PUNCT
ejpam-2553	469	10	then	then	ADV
ejpam-2553	469	11	b	b	X
ejpam-2553	469	12	∈	∈	PROPN
ejpam-2553	469	13	b	b	PROPN
ejpam-2553	469	14	,	,	PUNCT
ejpam-2553	469	15	b	b	PROPN
ejpam-2553	469	16	∈	∈	PROPN
ejpam-2553	469	17	m(r	m(r	PROPN
ejpam-2553	469	18	)	)	PUNCT
ejpam-2553	469	19	.	.	PUNCT
ejpam-2553	470	1	as	as	SCONJ
ejpam-2553	470	2	b	b	PROPN
ejpam-2553	470	3	is	be	AUX
ejpam-2553	470	4	regular	regular	ADJ
ejpam-2553	470	5	in	in	ADP
ejpam-2553	470	6	r	r	NOUN
ejpam-2553	470	7	,	,	PUNCT
ejpam-2553	470	8	then	then	ADV
ejpam-2553	470	9	there	there	PRON
ejpam-2553	470	10	exists	exist	VERB
ejpam-2553	470	11	r	r	NOUN
ejpam-2553	470	12	∈	∈	PROPN
ejpam-2553	470	13	r	r	NOUN
ejpam-2553	471	1	such	such	DET
ejpam-2553	471	2	that	that	DET
ejpam-2553	471	3	b	b	PROPN
ejpam-2553	471	4	∈	∈	PROPN
ejpam-2553	471	5	br	br	NOUN
ejpam-2553	471	6	b	b	PROPN
ejpam-2553	472	1	and	and	CCONJ
ejpam-2553	472	2	since	since	SCONJ
ejpam-2553	472	3	b	b	PROPN
ejpam-2553	472	4	is	be	AUX
ejpam-2553	472	5	a	a	DET
ejpam-2553	472	6	hyperideal	hyperideal	NOUN
ejpam-2553	472	7	and	and	CCONJ
ejpam-2553	472	8	b	b	NOUN
ejpam-2553	472	9	∈	∈	ADP
ejpam-2553	472	10	b	b	NOUN
ejpam-2553	472	11	we	we	PRON
ejpam-2553	472	12	have	have	VERB
ejpam-2553	472	13	r	r	NOUN
ejpam-2553	472	14	b	b	PROPN
ejpam-2553	472	15	⊆	⊆	NUM
ejpam-2553	472	16	b	b	NOUN
ejpam-2553	472	17	then	then	ADV
ejpam-2553	472	18	b	b	PROPN
ejpam-2553	472	19	∈	∈	PROPN
ejpam-2553	472	20	br	br	NOUN
ejpam-2553	472	21	b	b	PROPN
ejpam-2553	472	22	⊆	⊆	NUM
ejpam-2553	472	23	b(r	b(r	PROPN
ejpam-2553	472	24	br)b	br)b	PROPN
ejpam-2553	472	25	,	,	PUNCT
ejpam-2553	472	26	where	where	SCONJ
ejpam-2553	472	27	r	r	NOUN
ejpam-2553	472	28	br	br	NOUN
ejpam-2553	472	29	⊆	⊆	NUM
ejpam-2553	472	30	b.	b.	NOUN
ejpam-2553	472	31	it	it	PRON
ejpam-2553	472	32	shows	show	VERB
ejpam-2553	472	33	that	that	SCONJ
ejpam-2553	472	34	b	b	NOUN
ejpam-2553	472	35	is	be	AUX
ejpam-2553	472	36	regular	regular	ADJ
ejpam-2553	472	37	in	in	ADP
ejpam-2553	472	38	b.	b.	PROPN
ejpam-2553	472	39	therefore	therefore	ADV
ejpam-2553	472	40	b	b	PROPN
ejpam-2553	472	41	∩	∩	PROPN
ejpam-2553	472	42	m(r	m(r	PROPN
ejpam-2553	472	43	)	)	PUNCT
ejpam-2553	472	44	is	be	AUX
ejpam-2553	472	45	regular	regular	ADJ
ejpam-2553	472	46	hyperideal	hyperideal	NOUN
ejpam-2553	472	47	in	in	ADP
ejpam-2553	472	48	the	the	DET
ejpam-2553	472	49	strongly	strongly	ADV
ejpam-2553	472	50	distributive	distributive	ADJ
ejpam-2553	472	51	multiplicative	multiplicative	ADJ
ejpam-2553	472	52	hyperring	hyperring	NOUN
ejpam-2553	472	53	b.	b.	PROPN
ejpam-2553	472	54	thus	thus	ADV
ejpam-2553	472	55	b	b	PROPN
ejpam-2553	472	56	∩m(r	∩m(r	PROPN
ejpam-2553	472	57	)	)	PUNCT
ejpam-2553	472	58	⊆	⊆	NUM
ejpam-2553	472	59	m(b	m(b	NOUN
ejpam-2553	472	60	)	)	PUNCT
ejpam-2553	472	61	.	.	PUNCT
ejpam-2553	473	1	this	this	PRON
ejpam-2553	473	2	completes	complete	VERB
ejpam-2553	473	3	the	the	DET
ejpam-2553	473	4	proof	proof	NOUN
ejpam-2553	473	5	.	.	PUNCT
ejpam-2553	474	1	theorem	theorem	VERB
ejpam-2553	474	2	18	18	NUM
ejpam-2553	474	3	.	.	PUNCT
ejpam-2553	475	1	if	if	SCONJ
ejpam-2553	475	2	r	r	NOUN
ejpam-2553	475	3	is	be	AUX
ejpam-2553	475	4	a	a	DET
ejpam-2553	475	5	multiplicative	multiplicative	ADJ
ejpam-2553	475	6	hyperring	hyperring	NOUN
ejpam-2553	475	7	and	and	CCONJ
ejpam-2553	475	8	it	it	PRON
ejpam-2553	475	9	has	have	VERB
ejpam-2553	475	10	zero	zero	NUM
ejpam-2553	475	11	absorbing	absorbing	NOUN
ejpam-2553	475	12	property	property	NOUN
ejpam-2553	475	13	,	,	PUNCT
ejpam-2553	475	14	then	then	ADV
ejpam-2553	475	15	m(r)∩	m(r)∩	VERB
ejpam-2553	475	16	ann(m(r	ann(m(r	NOUN
ejpam-2553	475	17	)	)	PUNCT
ejpam-2553	475	18	)	)	PUNCT
ejpam-2553	476	1	=	=	PRON
ejpam-2553	476	2	{	{	PUNCT
ejpam-2553	476	3	0	0	NUM
ejpam-2553	476	4	}	}	PUNCT
ejpam-2553	476	5	.	.	PUNCT
ejpam-2553	477	1	proof	proof	NOUN
ejpam-2553	477	2	.	.	PUNCT
ejpam-2553	478	1	for	for	ADP
ejpam-2553	478	2	all	all	DET
ejpam-2553	478	3	a	a	DET
ejpam-2553	478	4	∈	∈	PROPN
ejpam-2553	478	5	m(r)∩ann(m(r	m(r)∩ann(m(r	NOUN
ejpam-2553	478	6	)	)	PUNCT
ejpam-2553	478	7	)	)	PUNCT
ejpam-2553	478	8	we	we	PRON
ejpam-2553	478	9	have	have	VERB
ejpam-2553	478	10	a	a	DET
ejpam-2553	478	11	∈	∈	PROPN
ejpam-2553	478	12	m(r	m(r	NOUN
ejpam-2553	478	13	)	)	PUNCT
ejpam-2553	478	14	,	,	PUNCT
ejpam-2553	478	15	a	a	DET
ejpam-2553	478	16	∈	∈	PROPN
ejpam-2553	478	17	ann(m(r	ann(m(r	NOUN
ejpam-2553	478	18	)	)	PUNCT
ejpam-2553	478	19	)	)	PUNCT
ejpam-2553	478	20	.	.	PUNCT
ejpam-2553	479	1	as	as	ADP
ejpam-2553	479	2	a	a	DET
ejpam-2553	479	3	∈	∈	PROPN
ejpam-2553	479	4	m(r	m(r	NOUN
ejpam-2553	479	5	)	)	PUNCT
ejpam-2553	479	6	thus	thus	ADV
ejpam-2553	479	7	there	there	PRON
ejpam-2553	479	8	exists	exist	VERB
ejpam-2553	479	9	r	r	NOUN
ejpam-2553	479	10	∈	∈	PROPN
ejpam-2553	479	11	r	r	NOUN
ejpam-2553	479	12	such	such	ADJ
ejpam-2553	479	13	that	that	SCONJ
ejpam-2553	479	14	a	a	DET
ejpam-2553	479	15	∈	∈	PROPN
ejpam-2553	479	16	ara	ara	NOUN
ejpam-2553	479	17	.	.	PUNCT
ejpam-2553	480	1	since	since	SCONJ
ejpam-2553	480	2	ra	ra	PROPN
ejpam-2553	480	3	⊆	⊆	NUM
ejpam-2553	480	4	m(r	m(r	PROPN
ejpam-2553	480	5	)	)	PUNCT
ejpam-2553	480	6	,	,	PUNCT
ejpam-2553	480	7	we	we	PRON
ejpam-2553	480	8	have	have	VERB
ejpam-2553	480	9	a	a	DET
ejpam-2553	480	10	∈	∈	PROPN
ejpam-2553	480	11	a(ra	a(ra	NOUN
ejpam-2553	480	12	)	)	PUNCT
ejpam-2553	480	13	=	=	PUNCT
ejpam-2553	480	14	{	{	PUNCT
ejpam-2553	480	15	0	0	NUM
ejpam-2553	480	16	}	}	PUNCT
ejpam-2553	480	17	.	.	PUNCT
ejpam-2553	481	1	therefore	therefore	ADV
ejpam-2553	481	2	a	a	PRON
ejpam-2553	481	3	=	=	ADJ
ejpam-2553	481	4	0	0	X
ejpam-2553	481	5	.	.	PUNCT
ejpam-2553	482	1	proposition	proposition	NOUN
ejpam-2553	482	2	4	4	NUM
ejpam-2553	482	3	.	.	PUNCT
ejpam-2553	483	1	(	(	PUNCT
ejpam-2553	483	2	[	[	X
ejpam-2553	483	3	19	19	NUM
ejpam-2553	483	4	]	]	SYM
ejpam-2553	483	5	)	)	PUNCT
ejpam-2553	483	6	every	every	DET
ejpam-2553	483	7	strongly	strongly	ADV
ejpam-2553	483	8	distributive	distributive	ADJ
ejpam-2553	483	9	hyperring	hyperring	NOUN
ejpam-2553	483	10	(	(	PUNCT
ejpam-2553	483	11	r,+	r,+	NUM
ejpam-2553	483	12	,	,	PUNCT
ejpam-2553	483	13	.	.	PUNCT
ejpam-2553	483	14	)	)	PUNCT
ejpam-2553	484	1	with	with	ADP
ejpam-2553	484	2	a	a	DET
ejpam-2553	484	3	scalar	scalar	ADJ
ejpam-2553	484	4	identity	identity	NOUN
ejpam-2553	484	5	1	1	NUM
ejpam-2553	484	6	is	be	AUX
ejpam-2553	484	7	a	a	DET
ejpam-2553	484	8	ring	ring	NOUN
ejpam-2553	484	9	.	.	PUNCT
ejpam-2553	485	1	references	reference	NOUN
ejpam-2553	485	2	416	416	NUM
ejpam-2553	485	3	theorem	theorem	NOUN
ejpam-2553	485	4	19	19	NUM
ejpam-2553	485	5	.	.	PUNCT
ejpam-2553	486	1	let	let	VERB
ejpam-2553	486	2	r	r	PRON
ejpam-2553	486	3	be	be	AUX
ejpam-2553	486	4	a	a	DET
ejpam-2553	486	5	multiplicative	multiplicative	ADJ
ejpam-2553	486	6	hyperring	hyperring	NOUN
ejpam-2553	487	1	and	and	CCONJ
ejpam-2553	487	2	i	i	PRON
ejpam-2553	487	3	be	be	VERB
ejpam-2553	487	4	a	a	DET
ejpam-2553	487	5	strongly	strongly	ADV
ejpam-2553	487	6	distributive	distributive	ADJ
ejpam-2553	487	7	hyperideal	hyperideal	NOUN
ejpam-2553	487	8	of	of	ADP
ejpam-2553	487	9	r	r	NOUN
ejpam-2553	487	10	such	such	ADJ
ejpam-2553	487	11	that	that	SCONJ
ejpam-2553	487	12	it	it	PRON
ejpam-2553	487	13	has	have	VERB
ejpam-2553	487	14	a	a	DET
ejpam-2553	487	15	scalar	scalar	ADJ
ejpam-2553	487	16	identity	identity	NOUN
ejpam-2553	487	17	e	e	NOUN
ejpam-2553	487	18	and	and	CCONJ
ejpam-2553	487	19	for	for	ADP
ejpam-2553	487	20	all	all	DET
ejpam-2553	487	21	x	x	SYM
ejpam-2553	487	22	∈	∈	PROPN
ejpam-2553	487	23	r	r	NOUN
ejpam-2553	487	24	,	,	PUNCT
ejpam-2553	487	25	|xe|=	|xe|=	PROPN
ejpam-2553	487	26	1	1	NUM
ejpam-2553	487	27	,	,	PUNCT
ejpam-2553	487	28	then	then	ADV
ejpam-2553	487	29	r=	r=	VERB
ejpam-2553	487	30	i	i	PRON
ejpam-2553	487	31	+	+	CCONJ
ejpam-2553	487	32	ann(i	ann(i	NUM
ejpam-2553	487	33	)	)	PUNCT
ejpam-2553	487	34	.	.	PUNCT
ejpam-2553	488	1	proof	proof	NOUN
ejpam-2553	488	2	.	.	PUNCT
ejpam-2553	489	1	let	let	VERB
ejpam-2553	489	2	x	x	PRON
ejpam-2553	489	3	be	be	AUX
ejpam-2553	489	4	an	an	DET
ejpam-2553	489	5	arbitrary	arbitrary	ADJ
ejpam-2553	489	6	element	element	NOUN
ejpam-2553	489	7	of	of	ADP
ejpam-2553	489	8	r	r	NOUN
ejpam-2553	489	9	,	,	PUNCT
ejpam-2553	489	10	then	then	ADV
ejpam-2553	489	11	xe∪ex	xe∪ex	PROPN
ejpam-2553	490	1	⊆	⊆	NUM
ejpam-2553	490	2	i	i	PRON
ejpam-2553	490	3	.	.	PUNCT
ejpam-2553	491	1	so	so	ADV
ejpam-2553	491	2	(	(	PUNCT
ejpam-2553	491	3	xe)e	xe)e	PROPN
ejpam-2553	491	4	=	=	SYM
ejpam-2553	491	5	e(xe	e(xe	PROPN
ejpam-2553	491	6	)	)	PUNCT
ejpam-2553	491	7	,	,	PUNCT
ejpam-2553	491	8	i.e.	i.e.	X
ejpam-2553	491	9	,	,	PUNCT
ejpam-2553	491	10	xe	xe	PROPN
ejpam-2553	491	11	=	=	PROPN
ejpam-2553	491	12	exe	exe	PROPN
ejpam-2553	491	13	and	and	CCONJ
ejpam-2553	491	14	similarly	similarly	ADV
ejpam-2553	491	15	ex	ex	X
ejpam-2553	491	16	=	=	PROPN
ejpam-2553	491	17	exe	exe	NOUN
ejpam-2553	491	18	.	.	PUNCT
ejpam-2553	492	1	thus	thus	ADV
ejpam-2553	492	2	xe	xe	PROPN
ejpam-2553	492	3	=	=	SYM
ejpam-2553	492	4	ex	ex	PROPN
ejpam-2553	492	5	and	and	CCONJ
ejpam-2553	492	6	e	e	NOUN
ejpam-2553	492	7	is	be	AUX
ejpam-2553	492	8	in	in	ADP
ejpam-2553	492	9	center	center	NOUN
ejpam-2553	492	10	of	of	ADP
ejpam-2553	492	11	r.	r.	PROPN
ejpam-2553	492	12	as	as	ADP
ejpam-2553	492	13	,	,	PUNCT
ejpam-2553	492	14	by	by	ADP
ejpam-2553	492	15	proposition	proposition	NOUN
ejpam-2553	492	16	4	4	NUM
ejpam-2553	492	17	,	,	PUNCT
ejpam-2553	492	18	ex	ex	PRON
ejpam-2553	492	19	∈	∈	NOUN
ejpam-2553	492	20	i	i	PRON
ejpam-2553	492	21	and	and	CCONJ
ejpam-2553	492	22	for	for	ADP
ejpam-2553	492	23	all	all	DET
ejpam-2553	492	24	y	y	PROPN
ejpam-2553	492	25	∈	∈	PROPN
ejpam-2553	492	26	i	i	PRON
ejpam-2553	492	27	,	,	PUNCT
ejpam-2553	492	28	y(x−ex	y(x−ex	NOUN
ejpam-2553	492	29	)	)	PUNCT
ejpam-2553	492	30	=	=	SYM
ejpam-2553	492	31	ye(x−ex	ye(x−ex	NOUN
ejpam-2553	492	32	)	)	PUNCT
ejpam-2553	492	33	=	=	SYM
ejpam-2553	492	34	y(ex−e2	y(ex−e2	NOUN
ejpam-2553	492	35	x	x	X
ejpam-2553	492	36	)	)	PUNCT
ejpam-2553	492	37	=	=	SYM
ejpam-2553	493	1	y(ex−ex	y(ex−ex	PROPN
ejpam-2553	493	2	)	)	PUNCT
ejpam-2553	493	3	=	=	VERB
ejpam-2553	494	1	y0=	y0=	NOUN
ejpam-2553	494	2	0	0	NUM
ejpam-2553	494	3	,	,	PUNCT
ejpam-2553	494	4	i.e.	i.e.	X
ejpam-2553	494	5	,	,	PUNCT
ejpam-2553	494	6	x−ex	x−ex	PROPN
ejpam-2553	494	7	∈	∈	PROPN
ejpam-2553	494	8	ann(i	ann(i	PROPN
ejpam-2553	494	9	)	)	PUNCT
ejpam-2553	494	10	.	.	PUNCT
ejpam-2553	495	1	thus	thus	ADV
ejpam-2553	495	2	we	we	PRON
ejpam-2553	495	3	have	have	VERB
ejpam-2553	495	4	,	,	PUNCT
ejpam-2553	495	5	x	x	PUNCT
ejpam-2553	495	6	=	=	PUNCT
ejpam-2553	495	7	ex	ex	X
ejpam-2553	496	1	+	+	NOUN
ejpam-2553	496	2	(	(	PUNCT
ejpam-2553	496	3	x	x	PART
ejpam-2553	496	4	−	−	PROPN
ejpam-2553	496	5	ex	ex	NOUN
ejpam-2553	496	6	)	)	PUNCT
ejpam-2553	496	7	.	.	PUNCT
ejpam-2553	497	1	hence	hence	ADV
ejpam-2553	497	2	for	for	ADP
ejpam-2553	497	3	all	all	DET
ejpam-2553	497	4	x	x	SYM
ejpam-2553	497	5	∈	∈	NOUN
ejpam-2553	497	6	r	r	NOUN
ejpam-2553	497	7	as	as	ADP
ejpam-2553	497	8	a	a	DET
ejpam-2553	497	9	sum	sum	NOUN
ejpam-2553	497	10	of	of	ADP
ejpam-2553	497	11	elements	element	NOUN
ejpam-2553	497	12	ex	ex	X
ejpam-2553	497	13	∈	∈	NOUN
ejpam-2553	498	1	i	i	PRON
ejpam-2553	498	2	and	and	CCONJ
ejpam-2553	498	3	x	x	SYM
ejpam-2553	498	4	−	−	X
ejpam-2553	498	5	ex	ex	PRON
ejpam-2553	498	6	∈	∈	NOUN
ejpam-2553	498	7	i	i	PRON
ejpam-2553	498	8	of	of	ADP
ejpam-2553	498	9	ann(i	ann(i	PROPN
ejpam-2553	498	10	)	)	PUNCT
ejpam-2553	498	11	.	.	PUNCT
ejpam-2553	499	1	this	this	PRON
ejpam-2553	499	2	completes	complete	VERB
ejpam-2553	499	3	the	the	DET
ejpam-2553	499	4	proof	proof	NOUN
ejpam-2553	499	5	.	.	PUNCT
ejpam-2553	500	1	the	the	DET
ejpam-2553	500	2	following	follow	VERB
ejpam-2553	500	3	example	example	NOUN
ejpam-2553	500	4	show	show	VERB
ejpam-2553	500	5	that	that	SCONJ
ejpam-2553	500	6	under	under	ADP
ejpam-2553	500	7	theorem	theorem	NOUN
ejpam-2553	500	8	19	19	NUM
ejpam-2553	500	9	,	,	PUNCT
ejpam-2553	500	10	r	r	NOUN
ejpam-2553	500	11	is	be	AUX
ejpam-2553	500	12	not	not	PART
ejpam-2553	500	13	a	a	DET
ejpam-2553	500	14	ring	ring	NOUN
ejpam-2553	500	15	:	:	PUNCT
ejpam-2553	500	16	example	example	NOUN
ejpam-2553	500	17	3	3	X
ejpam-2553	500	18	.	.	PUNCT
ejpam-2553	500	19	let	let	VERB
ejpam-2553	500	20	r=	r=	PROPN
ejpam-2553	500	21	z	z	PROPN
ejpam-2553	500	22	⊕	⊕	PROPN
ejpam-2553	500	23	z	z	PROPN
ejpam-2553	500	24	and	and	CCONJ
ejpam-2553	500	25	define	define	VERB
ejpam-2553	500	26	(	(	PUNCT
ejpam-2553	500	27	a	a	PRON
ejpam-2553	500	28	,	,	PUNCT
ejpam-2553	500	29	b	b	NOUN
ejpam-2553	500	30	)	)	PUNCT
ejpam-2553	500	31	◦	◦	NOUN
ejpam-2553	500	32	(	(	PUNCT
ejpam-2553	500	33	c	c	X
ejpam-2553	500	34	,	,	PUNCT
ejpam-2553	500	35	d	d	NOUN
ejpam-2553	500	36	)	)	PUNCT
ejpam-2553	501	1	=	=	SYM
ejpam-2553	501	2	¨	¨	X
ejpam-2553	501	3	(	(	PUNCT
ejpam-2553	501	4	ac	ac	PROPN
ejpam-2553	501	5	,	,	PUNCT
ejpam-2553	501	6	z	z	NOUN
ejpam-2553	501	7	)	)	PUNCT
ejpam-2553	501	8	bd	bd	PROPN
ejpam-2553	501	9	6=	6=	NUM
ejpam-2553	501	10	0	0	NUM
ejpam-2553	501	11	(	(	PUNCT
ejpam-2553	501	12	ac	ac	PROPN
ejpam-2553	501	13	,	,	PUNCT
ejpam-2553	501	14	0	0	NUM
ejpam-2553	501	15	)	)	PUNCT
ejpam-2553	501	16	bd	bd	NOUN
ejpam-2553	502	1	=	=	NOUN
ejpam-2553	502	2	0	0	PROPN
ejpam-2553	503	1	then	then	ADV
ejpam-2553	503	2	(	(	PUNCT
ejpam-2553	503	3	r,+	r,+	NUM
ejpam-2553	503	4	,	,	PUNCT
ejpam-2553	503	5	◦	◦	NOUN
ejpam-2553	503	6	)	)	PUNCT
ejpam-2553	503	7	is	be	AUX
ejpam-2553	503	8	a	a	DET
ejpam-2553	503	9	multiplicative	multiplicative	ADJ
ejpam-2553	503	10	hyperring	hyperring	NOUN
ejpam-2553	503	11	such	such	ADJ
ejpam-2553	503	12	that	that	PRON
ejpam-2553	503	13	is	be	AUX
ejpam-2553	503	14	not	not	PART
ejpam-2553	503	15	strongly	strongly	ADV
ejpam-2553	503	16	distributive	distributive	ADJ
ejpam-2553	503	17	,	,	PUNCT
ejpam-2553	503	18	because	because	SCONJ
ejpam-2553	503	19	by	by	ADP
ejpam-2553	503	20	considering	consider	VERB
ejpam-2553	503	21	a	a	DET
ejpam-2553	503	22	=	=	SYM
ejpam-2553	503	23	(	(	PUNCT
ejpam-2553	503	24	1,1	1,1	NUM
ejpam-2553	503	25	)	)	PUNCT
ejpam-2553	503	26	,	,	PUNCT
ejpam-2553	503	27	b	b	X
ejpam-2553	503	28	=	=	SYM
ejpam-2553	503	29	(	(	PUNCT
ejpam-2553	503	30	0,2	0,2	NUM
ejpam-2553	503	31	)	)	PUNCT
ejpam-2553	503	32	,	,	PUNCT
ejpam-2553	503	33	c	c	NOUN
ejpam-2553	503	34	=	=	SYM
ejpam-2553	503	35	(	(	PUNCT
ejpam-2553	503	36	0,−2	0,−2	NUM
ejpam-2553	503	37	)	)	PUNCT
ejpam-2553	503	38	,	,	PUNCT
ejpam-2553	503	39	we	we	PRON
ejpam-2553	503	40	have	have	VERB
ejpam-2553	503	41	a	a	DET
ejpam-2553	503	42	◦	◦	NOUN
ejpam-2553	503	43	(	(	PUNCT
ejpam-2553	503	44	b	b	X
ejpam-2553	503	45	+	+	CCONJ
ejpam-2553	503	46	c	c	NOUN
ejpam-2553	503	47	)	)	PUNCT
ejpam-2553	503	48	=	=	SYM
ejpam-2553	503	49	(	(	PUNCT
ejpam-2553	503	50	1	1	NUM
ejpam-2553	503	51	,	,	PUNCT
ejpam-2553	503	52	1	1	NUM
ejpam-2553	503	53	)	)	PUNCT
ejpam-2553	503	54	◦	◦	NOUN
ejpam-2553	503	55	(	(	PUNCT
ejpam-2553	503	56	0	0	NUM
ejpam-2553	503	57	,	,	PUNCT
ejpam-2553	503	58	0	0	NUM
ejpam-2553	503	59	)	)	PUNCT
ejpam-2553	503	60	=	=	SYM
ejpam-2553	503	61	(	(	PUNCT
ejpam-2553	503	62	0,0	0,0	NOUN
ejpam-2553	503	63	)	)	PUNCT
ejpam-2553	503	64	but	but	CCONJ
ejpam-2553	503	65	a	a	DET
ejpam-2553	503	66	◦	◦	NOUN
ejpam-2553	503	67	b+a	b+a	NOUN
ejpam-2553	503	68	◦	◦	NOUN
ejpam-2553	503	69	c	c	NOUN
ejpam-2553	504	1	=	=	PUNCT
ejpam-2553	504	2	(	(	PUNCT
ejpam-2553	504	3	0,z)+(0,z	0,z)+(0,z	NUM
ejpam-2553	504	4	)	)	PUNCT
ejpam-2553	504	5	=	=	SYM
ejpam-2553	504	6	(	(	PUNCT
ejpam-2553	504	7	0,z+z	0,z+z	NUM
ejpam-2553	504	8	)	)	PUNCT
ejpam-2553	504	9	.	.	PUNCT
ejpam-2553	505	1	now	now	ADV
ejpam-2553	505	2	,	,	PUNCT
ejpam-2553	505	3	let	let	VERB
ejpam-2553	505	4	i	i	PRON
ejpam-2553	505	5	=	=	PUNCT
ejpam-2553	505	6	(	(	PUNCT
ejpam-2553	505	7	z	z	NOUN
ejpam-2553	505	8	,	,	PUNCT
ejpam-2553	505	9	0	0	NUM
ejpam-2553	505	10	)	)	PUNCT
ejpam-2553	505	11	be	be	AUX
ejpam-2553	505	12	a	a	DET
ejpam-2553	505	13	strongly	strongly	ADV
ejpam-2553	505	14	distributive	distributive	ADJ
ejpam-2553	505	15	hyperideal	hyperideal	NOUN
ejpam-2553	505	16	of	of	ADP
ejpam-2553	505	17	r	r	NOUN
ejpam-2553	505	18	and	and	CCONJ
ejpam-2553	505	19	e	e	NOUN
ejpam-2553	505	20	=	=	PUNCT
ejpam-2553	505	21	(	(	PUNCT
ejpam-2553	505	22	1	1	NUM
ejpam-2553	505	23	,	,	PUNCT
ejpam-2553	505	24	0	0	NUM
ejpam-2553	505	25	)	)	PUNCT
ejpam-2553	505	26	is	be	AUX
ejpam-2553	505	27	scalar	scalar	ADJ
ejpam-2553	505	28	identity	identity	NOUN
ejpam-2553	505	29	element	element	NOUN
ejpam-2553	505	30	of	of	ADP
ejpam-2553	505	31	i	i	PRON
ejpam-2553	505	32	such	such	ADJ
ejpam-2553	505	33	that	that	SCONJ
ejpam-2553	505	34	for	for	ADP
ejpam-2553	505	35	all	all	DET
ejpam-2553	505	36	r	r	NOUN
ejpam-2553	505	37	∈	∈	NOUN
ejpam-2553	505	38	r	r	NOUN
ejpam-2553	505	39	we	we	PRON
ejpam-2553	505	40	have	have	VERB
ejpam-2553	505	41	|xe|	|xe|	VERB
ejpam-2553	505	42	=	=	SYM
ejpam-2553	506	1	1	1	X
ejpam-2553	506	2	.	.	PUNCT
ejpam-2553	507	1	it	it	PRON
ejpam-2553	507	2	is	be	AUX
ejpam-2553	507	3	clear	clear	ADJ
ejpam-2553	507	4	that	that	SCONJ
ejpam-2553	507	5	ann(i	ann(i	X
ejpam-2553	507	6	)	)	PUNCT
ejpam-2553	507	7	=	=	SYM
ejpam-2553	508	1	(	(	PUNCT
ejpam-2553	508	2	0,z	0,z	ADV
ejpam-2553	508	3	)	)	PUNCT
ejpam-2553	508	4	and	and	CCONJ
ejpam-2553	508	5	r=	r=	ADJ
ejpam-2553	509	1	i	i	PRON
ejpam-2553	509	2	+	+	CCONJ
ejpam-2553	509	3	ann(i	ann(i	NUM
ejpam-2553	509	4	)	)	PUNCT
ejpam-2553	509	5	but	but	CCONJ
ejpam-2553	509	6	(	(	PUNCT
ejpam-2553	509	7	r,+	r,+	NUM
ejpam-2553	509	8	,	,	PUNCT
ejpam-2553	509	9	◦	◦	NOUN
ejpam-2553	509	10	)	)	PUNCT
ejpam-2553	509	11	is	be	AUX
ejpam-2553	509	12	not	not	PART
ejpam-2553	509	13	a	a	DET
ejpam-2553	509	14	ring	ring	NOUN
ejpam-2553	509	15	.	.	PUNCT
ejpam-2553	510	1	corollary	corollary	ADJ
ejpam-2553	510	2	2	2	NUM
ejpam-2553	510	3	.	.	PUNCT
ejpam-2553	511	1	(	(	PUNCT
ejpam-2553	511	2	[	[	X
ejpam-2553	511	3	7	7	NUM
ejpam-2553	511	4	]	]	PUNCT
ejpam-2553	511	5	)	)	PUNCT
ejpam-2553	511	6	if	if	SCONJ
ejpam-2553	511	7	an	an	DET
ejpam-2553	511	8	ideal	ideal	NOUN
ejpam-2553	511	9	i	i	PRON
ejpam-2553	511	10	in	in	ADP
ejpam-2553	511	11	a	a	DET
ejpam-2553	511	12	ring	ring	NOUN
ejpam-2553	511	13	r	r	NOUN
ejpam-2553	511	14	has	have	VERB
ejpam-2553	511	15	a	a	DET
ejpam-2553	511	16	unit	unit	NOUN
ejpam-2553	511	17	element	element	NOUN
ejpam-2553	511	18	e	e	NOUN
ejpam-2553	511	19	,	,	PUNCT
ejpam-2553	511	20	then	then	ADV
ejpam-2553	511	21	r=	r=	VERB
ejpam-2553	511	22	i	i	PRON
ejpam-2553	511	23	+	+	CCONJ
ejpam-2553	511	24	ann(i	ann(i	NUM
ejpam-2553	511	25	)	)	PUNCT
ejpam-2553	511	26	.	.	PUNCT
ejpam-2553	512	1	acknowledgements	acknowledgement	NOUN
ejpam-2553	512	2	the	the	DET
ejpam-2553	512	3	first	first	ADJ
ejpam-2553	512	4	author	author	NOUN
ejpam-2553	512	5	partially	partially	ADV
ejpam-2553	512	6	has	have	AUX
ejpam-2553	512	7	been	be	AUX
ejpam-2553	512	8	supported	support	VERB
ejpam-2553	512	9	by	by	ADP
ejpam-2553	512	10	"	"	PUNCT
ejpam-2553	512	11	algebraic	algebraic	PROPN
ejpam-2553	512	12	hyperstructure	hyperstructure	PROPN
ejpam-2553	512	13	excellence	excellence	PROPN
ejpam-2553	512	14	,	,	PUNCT
ejpam-2553	512	15	tarbiat	tarbiat	PROPN
ejpam-2553	512	16	modares	modares	PROPN
ejpam-2553	512	17	university	university	PROPN
ejpam-2553	512	18	,	,	PUNCT
ejpam-2553	512	19	tehran	tehran	PROPN
ejpam-2553	512	20	,	,	PUNCT
ejpam-2553	512	21	iran	iran	PROPN
ejpam-2553	512	22	"	"	PUNCT
ejpam-2553	512	23	and	and	CCONJ
ejpam-2553	512	24	"	"	PUNCT
ejpam-2553	512	25	research	research	NOUN
ejpam-2553	512	26	center	center	NOUN
ejpam-2553	512	27	in	in	ADP
ejpam-2553	512	28	algebraic	algebraic	ADJ
ejpam-2553	512	29	hyperstructures	hyperstructure	NOUN
ejpam-2553	512	30	and	and	CCONJ
ejpam-2553	512	31	fuzzy	fuzzy	ADJ
ejpam-2553	512	32	mathematics	mathematic	NOUN
ejpam-2553	512	33	,	,	PUNCT
ejpam-2553	512	34	university	university	NOUN
ejpam-2553	512	35	of	of	ADP
ejpam-2553	512	36	mazandaran	mazandaran	PROPN
ejpam-2553	512	37	,	,	PUNCT
ejpam-2553	512	38	babolsar	babolsar	PROPN
ejpam-2553	512	39	,	,	PUNCT
ejpam-2553	512	40	iran	iran	PROPN
ejpam-2553	512	41	"	"	PUNCT
ejpam-2553	512	42	.	.	PUNCT
ejpam-2553	513	1	references	reference	NOUN
ejpam-2553	513	2	[	[	X
ejpam-2553	513	3	1	1	NUM
ejpam-2553	513	4	]	]	X
ejpam-2553	513	5	r.	r.	PROPN
ejpam-2553	513	6	ameri	ameri	PROPN
ejpam-2553	513	7	and	and	CCONJ
ejpam-2553	513	8	m.	m.	PROPN
ejpam-2553	513	9	norouzi	norouzi	PROPN
ejpam-2553	513	10	.	.	PUNCT
ejpam-2553	514	1	new	new	ADJ
ejpam-2553	514	2	fundamental	fundamental	ADJ
ejpam-2553	514	3	relation	relation	NOUN
ejpam-2553	514	4	of	of	ADP
ejpam-2553	514	5	hyperrings	hyperring	NOUN
ejpam-2553	514	6	,	,	PUNCT
ejpam-2553	514	7	european	european	PROPN
ejpam-2553	514	8	journal	journal	PROPN
ejpam-2553	514	9	of	of	ADP
ejpam-2553	514	10	combinatorics	combinatorics	PROPN
ejpam-2553	514	11	,	,	PUNCT
ejpam-2553	514	12	34	34	NUM
ejpam-2553	514	13	,	,	PUNCT
ejpam-2553	514	14	884–891	884–891	NUM
ejpam-2553	514	15	.	.	PUNCT
ejpam-2553	514	16	2013	2013	NUM
ejpam-2553	514	17	.	.	PUNCT
ejpam-2553	515	1	[	[	X
ejpam-2553	515	2	2	2	NUM
ejpam-2553	515	3	]	]	X
ejpam-2553	515	4	r.	r.	PROPN
ejpam-2553	515	5	ameri	ameri	PROPN
ejpam-2553	515	6	and	and	CCONJ
ejpam-2553	515	7	m.	m.	PROPN
ejpam-2553	515	8	norouzi	norouzi	PROPN
ejpam-2553	515	9	.	.	PUNCT
ejpam-2553	516	1	prime	prime	ADJ
ejpam-2553	516	2	and	and	CCONJ
ejpam-2553	516	3	primary	primary	ADJ
ejpam-2553	516	4	hyperideals	hyperideal	NOUN
ejpam-2553	516	5	in	in	ADP
ejpam-2553	516	6	krasner	krasner	NOUN
ejpam-2553	516	7	,	,	PUNCT
ejpam-2553	516	8	european	european	PROPN
ejpam-2553	516	9	journal	journal	PROPN
ejpam-2553	516	10	of	of	ADP
ejpam-2553	516	11	combinatorics	combinatorics	PROPN
ejpam-2553	516	12	,	,	PUNCT
ejpam-2553	516	13	34	34	NUM
ejpam-2553	516	14	,	,	PUNCT
ejpam-2553	516	15	379–390	379–390	NUM
ejpam-2553	516	16	.	.	PUNCT
ejpam-2553	516	17	2013	2013	NUM
ejpam-2553	516	18	.	.	PUNCT
ejpam-2553	517	1	[	[	X
ejpam-2553	517	2	3	3	NUM
ejpam-2553	517	3	]	]	X
ejpam-2553	517	4	r.	r.	PROPN
ejpam-2553	517	5	ameri	ameri	PROPN
ejpam-2553	517	6	and	and	CCONJ
ejpam-2553	517	7	m.m	m.m	PROPN
ejpam-2553	517	8	.	.	PROPN
ejpam-2553	517	9	zahedi	zahedi	PROPN
ejpam-2553	517	10	.	.	PUNCT
ejpam-2553	518	1	hyperalgebraic	hyperalgebraic	PROPN
ejpam-2553	518	2	systems	system	NOUN
ejpam-2553	518	3	,	,	PUNCT
ejpam-2553	518	4	italian	italian	ADJ
ejpam-2553	518	5	journal	journal	NOUN
ejpam-2553	518	6	of	of	ADP
ejpam-2553	518	7	pure	pure	ADJ
ejpam-2553	518	8	and	and	CCONJ
ejpam-2553	518	9	applied	applied	ADJ
ejpam-2553	518	10	mathematics	mathematic	NOUN
ejpam-2553	518	11	,	,	PUNCT
ejpam-2553	518	12	6	6	NUM
ejpam-2553	518	13	,	,	PUNCT
ejpam-2553	518	14	21	21	NUM
ejpam-2553	518	15	-	-	SYM
ejpam-2553	518	16	32	32	NUM
ejpam-2553	518	17	.	.	PUNCT
ejpam-2553	518	18	1999	1999	NUM
ejpam-2553	518	19	.	.	PUNCT
ejpam-2553	519	1	[	[	X
ejpam-2553	519	2	4	4	NUM
ejpam-2553	519	3	]	]	PUNCT
ejpam-2553	519	4	a.	a.	NOUN
ejpam-2553	519	5	asokkumar	asokkumar	PROPN
ejpam-2553	519	6	and	and	CCONJ
ejpam-2553	519	7	m.	m.	PROPN
ejpam-2553	519	8	velrajan	velrajan	NOUN
ejpam-2553	519	9	.	.	PUNCT
ejpam-2553	520	1	characterizations	characterization	NOUN
ejpam-2553	520	2	of	of	ADP
ejpam-2553	520	3	regular	regular	ADJ
ejpam-2553	520	4	hyperrings	hyperring	NOUN
ejpam-2553	520	5	,	,	PUNCT
ejpam-2553	520	6	italian	italian	ADJ
ejpam-2553	520	7	journal	journal	NOUN
ejpam-2553	520	8	of	of	ADP
ejpam-2553	520	9	pure	pure	ADJ
ejpam-2553	520	10	and	and	CCONJ
ejpam-2553	520	11	applied	applied	ADJ
ejpam-2553	520	12	mathematics	mathematic	NOUN
ejpam-2553	520	13	,	,	PUNCT
ejpam-2553	520	14	22	22	NUM
ejpam-2553	520	15	,	,	PUNCT
ejpam-2553	520	16	115	115	NUM
ejpam-2553	520	17	-	-	SYM
ejpam-2553	520	18	124	124	NUM
ejpam-2553	520	19	.	.	PUNCT
ejpam-2553	520	20	2007	2007	NUM
ejpam-2553	520	21	.	.	PUNCT
ejpam-2553	521	1	references	reference	NOUN
ejpam-2553	521	2	417	417	NUM
ejpam-2553	521	3	[	[	X
ejpam-2553	521	4	5	5	NUM
ejpam-2553	521	5	]	]	PUNCT
ejpam-2553	521	6	a.	a.	NOUN
ejpam-2553	521	7	asokkumar	asokkumar	PROPN
ejpam-2553	521	8	and	and	CCONJ
ejpam-2553	521	9	m.	m.	PROPN
ejpam-2553	521	10	velrajan	velrajan	NOUN
ejpam-2553	521	11	.	.	PUNCT
ejpam-2553	522	1	hyperring	hyperring	NOUN
ejpam-2553	522	2	of	of	ADP
ejpam-2553	522	3	matrices	matrix	NOUN
ejpam-2553	522	4	over	over	ADP
ejpam-2553	522	5	a	a	DET
ejpam-2553	522	6	regular	regular	ADJ
ejpam-2553	522	7	hyperring	hyperring	NOUN
ejpam-2553	522	8	,	,	PUNCT
ejpam-2553	522	9	italian	italian	ADJ
ejpam-2553	522	10	journal	journal	NOUN
ejpam-2553	522	11	of	of	ADP
ejpam-2553	522	12	pure	pure	ADJ
ejpam-2553	522	13	and	and	CCONJ
ejpam-2553	522	14	applied	applied	ADJ
ejpam-2553	522	15	mathematics	mathematic	NOUN
ejpam-2553	522	16	,	,	PUNCT
ejpam-2553	522	17	23	23	NUM
ejpam-2553	522	18	,	,	PUNCT
ejpam-2553	522	19	13	13	NUM
ejpam-2553	522	20	-	-	SYM
ejpam-2553	522	21	120	120	NUM
ejpam-2553	522	22	.	.	PUNCT
ejpam-2553	522	23	2008	2008	NUM
ejpam-2553	522	24	.	.	PUNCT
ejpam-2553	523	1	[	[	X
ejpam-2553	523	2	6	6	NUM
ejpam-2553	523	3	]	]	PUNCT
ejpam-2553	523	4	a.	a.	NOUN
ejpam-2553	523	5	asokkumar	asokkumar	PROPN
ejpam-2553	523	6	and	and	CCONJ
ejpam-2553	523	7	m.	m.	PROPN
ejpam-2553	523	8	velrajan	velrajan	NOUN
ejpam-2553	523	9	.	.	PUNCT
ejpam-2553	524	1	a	a	DET
ejpam-2553	524	2	radical	radical	ADJ
ejpam-2553	524	3	property	property	NOUN
ejpam-2553	524	4	of	of	ADP
ejpam-2553	524	5	hyperrings	hyperring	NOUN
ejpam-2553	524	6	,	,	PUNCT
ejpam-2553	524	7	italian	italian	ADJ
ejpam-2553	524	8	journal	journal	NOUN
ejpam-2553	524	9	of	of	ADP
ejpam-2553	524	10	pure	pure	ADJ
ejpam-2553	524	11	and	and	CCONJ
ejpam-2553	524	12	applied	applied	ADJ
ejpam-2553	524	13	mathematics	mathematic	NOUN
ejpam-2553	524	14	,	,	PUNCT
ejpam-2553	524	15	29	29	NUM
ejpam-2553	524	16	,	,	PUNCT
ejpam-2553	524	17	301	301	NUM
ejpam-2553	524	18	-	-	SYM
ejpam-2553	524	19	308	308	NUM
ejpam-2553	524	20	.	.	PUNCT
ejpam-2553	524	21	2012	2012	NUM
ejpam-2553	524	22	.	.	PUNCT
ejpam-2553	525	1	[	[	X
ejpam-2553	525	2	7	7	X
ejpam-2553	525	3	]	]	X
ejpam-2553	525	4	b.	b.	PROPN
ejpam-2553	525	5	brown	brown	PROPN
ejpam-2553	525	6	and	and	CCONJ
ejpam-2553	525	7	n.	n.	PROPN
ejpam-2553	525	8	h.	h.	PROPN
ejpam-2553	525	9	mccoy	mccoy	PROPN
ejpam-2553	525	10	.	.	PUNCT
ejpam-2553	526	1	the	the	DET
ejpam-2553	526	2	maximal	maximal	ADJ
ejpam-2553	526	3	regular	regular	ADJ
ejpam-2553	526	4	ideal	ideal	NOUN
ejpam-2553	526	5	of	of	ADP
ejpam-2553	526	6	a	a	DET
ejpam-2553	526	7	ring	ring	NOUN
ejpam-2553	526	8	,	,	PUNCT
ejpam-2553	526	9	proceedings	proceeding	NOUN
ejpam-2553	526	10	of	of	ADP
ejpam-2553	526	11	the	the	DET
ejpam-2553	526	12	american	american	PROPN
ejpam-2553	526	13	mathematical	mathematical	PROPN
ejpam-2553	526	14	society	society	NOUN
ejpam-2553	526	15	,	,	PUNCT
ejpam-2553	526	16	1	1	NUM
ejpam-2553	526	17	,	,	PUNCT
ejpam-2553	526	18	165	165	NUM
ejpam-2553	526	19	-	-	SYM
ejpam-2553	526	20	171	171	NUM
ejpam-2553	526	21	.	.	PUNCT
ejpam-2553	526	22	1950	1950	NUM
ejpam-2553	526	23	.	.	PUNCT
ejpam-2553	527	1	[	[	X
ejpam-2553	527	2	8	8	NUM
ejpam-2553	527	3	]	]	X
ejpam-2553	527	4	p.	p.	NOUN
ejpam-2553	527	5	corsini	corsini	PROPN
ejpam-2553	527	6	.	.	PUNCT
ejpam-2553	528	1	prolegomena	prolegomenon	NOUN
ejpam-2553	528	2	of	of	ADP
ejpam-2553	528	3	hypergroup	hypergroup	PROPN
ejpam-2553	528	4	theory	theory	NOUN
ejpam-2553	528	5	,	,	PUNCT
ejpam-2553	528	6	second	second	ADJ
ejpam-2553	528	7	edition	edition	NOUN
ejpam-2553	528	8	,	,	PUNCT
ejpam-2553	528	9	aviani	aviani	X
ejpam-2553	528	10	editore	editore	NOUN
ejpam-2553	528	11	,	,	PUNCT
ejpam-2553	528	12	1993	1993	NUM
ejpam-2553	528	13	.	.	PUNCT
ejpam-2553	529	1	[	[	X
ejpam-2553	529	2	9	9	NUM
ejpam-2553	529	3	]	]	PUNCT
ejpam-2553	529	4	p.	p.	NOUN
ejpam-2553	529	5	corsini	corsini	PROPN
ejpam-2553	529	6	and	and	CCONJ
ejpam-2553	529	7	v.	v.	ADP
ejpam-2553	529	8	leoreanu	leoreanu	PROPN
ejpam-2553	529	9	.	.	PUNCT
ejpam-2553	530	1	applications	application	NOUN
ejpam-2553	530	2	of	of	ADP
ejpam-2553	530	3	hyperstructures	hyperstructure	NOUN
ejpam-2553	530	4	theory	theory	NOUN
ejpam-2553	530	5	,	,	PUNCT
ejpam-2553	530	6	advances	advance	NOUN
ejpam-2553	530	7	in	in	ADP
ejpam-2553	530	8	mathematics	mathematic	NOUN
ejpam-2553	530	9	,	,	PUNCT
ejpam-2553	530	10	kluwer	kluwer	NOUN
ejpam-2553	530	11	academic	academic	ADJ
ejpam-2553	530	12	publishers	publisher	NOUN
ejpam-2553	530	13	,	,	PUNCT
ejpam-2553	530	14	2003	2003	NUM
ejpam-2553	530	15	.	.	PUNCT
ejpam-2553	531	1	[	[	X
ejpam-2553	531	2	10	10	NUM
ejpam-2553	531	3	]	]	X
ejpam-2553	531	4	u.	u.	PROPN
ejpam-2553	531	5	dasgupta	dasgupta	PROPN
ejpam-2553	531	6	.	.	PROPN
ejpam-2553	531	7	prime	prime	ADJ
ejpam-2553	531	8	and	and	CCONJ
ejpam-2553	531	9	primary	primary	ADJ
ejpam-2553	531	10	hyperideals	hyperideal	NOUN
ejpam-2553	531	11	of	of	ADP
ejpam-2553	531	12	a	a	DET
ejpam-2553	531	13	multiplicative	multiplicative	ADJ
ejpam-2553	531	14	hyperring	hyperring	NOUN
ejpam-2553	531	15	,	,	PUNCT
ejpam-2553	531	16	analele	analele	ADP
ejpam-2553	531	17	stiintifice	stiintifice	NOUN
ejpam-2553	531	18	ale	ale	NOUN
ejpam-2553	531	19	uniersitatii	uniersitatii	PROPN
ejpam-2553	531	20	al	al	PROPN
ejpam-2553	531	21	.	.	PUNCT
ejpam-2553	532	1	i	i	PRON
ejpam-2553	532	2	cuza	cuza	VERB
ejpam-2553	532	3	din	din	PROPN
ejpam-2553	532	4	iasi	iasi	PROPN
ejpam-2553	532	5	(	(	PUNCT
ejpam-2553	532	6	s.	s.	PROPN
ejpam-2553	532	7	n.	n.	PROPN
ejpam-2553	532	8	)	)	PUNCT
ejpam-2553	532	9	matematica	matematica	PROPN
ejpam-2553	532	10	,	,	PUNCT
ejpam-2553	532	11	lviii(1	lviii(1	PROPN
ejpam-2553	532	12	):	):	PUNCT
ejpam-2553	532	13	19	19	NUM
ejpam-2553	532	14	-	-	SYM
ejpam-2553	532	15	36	36	NUM
ejpam-2553	532	16	.	.	PUNCT
ejpam-2553	532	17	2012	2012	NUM
ejpam-2553	532	18	.	.	PUNCT
ejpam-2553	533	1	[	[	X
ejpam-2553	533	2	11	11	NUM
ejpam-2553	533	3	]	]	X
ejpam-2553	533	4	b.	b.	PROPN
ejpam-2553	533	5	davvaz	davvaz	PROPN
ejpam-2553	533	6	and	and	CCONJ
ejpam-2553	533	7	s.	s.	PROPN
ejpam-2553	533	8	mirvakili	mirvakili	PROPN
ejpam-2553	533	9	.	.	PUNCT
ejpam-2553	534	1	on	on	ADP
ejpam-2553	534	2	α	α	NOUN
ejpam-2553	534	3	-	-	PUNCT
ejpam-2553	534	4	relation	relation	NOUN
ejpam-2553	534	5	and	and	CCONJ
ejpam-2553	534	6	transitive	transitive	ADJ
ejpam-2553	534	7	condition	condition	NOUN
ejpam-2553	534	8	of	of	ADP
ejpam-2553	534	9	α	α	PRON
ejpam-2553	534	10	,	,	PUNCT
ejpam-2553	534	11	communications	communication	NOUN
ejpam-2553	534	12	in	in	ADP
ejpam-2553	534	13	algebra	algebra	NOUN
ejpam-2553	534	14	,	,	PUNCT
ejpam-2553	534	15	36(5	36(5	NUM
ejpam-2553	534	16	)	)	PUNCT
ejpam-2553	534	17	,	,	PUNCT
ejpam-2553	534	18	1695	1695	NUM
ejpam-2553	534	19	-	-	SYM
ejpam-2553	534	20	1703	1703	NUM
ejpam-2553	534	21	.	.	PUNCT
ejpam-2553	534	22	2008	2008	NUM
ejpam-2553	534	23	.	.	PUNCT
ejpam-2553	535	1	[	[	X
ejpam-2553	535	2	12	12	NUM
ejpam-2553	535	3	]	]	X
ejpam-2553	535	4	b.	b.	PROPN
ejpam-2553	535	5	davvaz	davvaz	PROPN
ejpam-2553	535	6	and	and	CCONJ
ejpam-2553	535	7	t.	t.	PROPN
ejpam-2553	535	8	vougiouklis	vougiouklis	PROPN
ejpam-2553	535	9	.	.	PUNCT
ejpam-2553	536	1	commutative	commutative	ADJ
ejpam-2553	536	2	rings	ring	NOUN
ejpam-2553	536	3	obtained	obtain	VERB
ejpam-2553	536	4	from	from	ADP
ejpam-2553	536	5	hyperrings	hyperring	NOUN
ejpam-2553	536	6	(	(	PUNCT
ejpam-2553	536	7	hv	hv	NOUN
ejpam-2553	536	8	-	-	PUNCT
ejpam-2553	536	9	rings	ring	NOUN
ejpam-2553	536	10	)	)	PUNCT
ejpam-2553	536	11	with	with	ADP
ejpam-2553	536	12	α∗-relations	α∗-relation	NOUN
ejpam-2553	536	13	,	,	PUNCT
ejpam-2553	536	14	communications	communication	NOUN
ejpam-2553	536	15	in	in	ADP
ejpam-2553	536	16	algebra	algebra	NOUN
ejpam-2553	536	17	,	,	PUNCT
ejpam-2553	536	18	35	35	NUM
ejpam-2553	536	19	,	,	PUNCT
ejpam-2553	536	20	3307	3307	NUM
ejpam-2553	536	21	-	-	SYM
ejpam-2553	536	22	3320	3320	NUM
ejpam-2553	536	23	.	.	PUNCT
ejpam-2553	537	1	2007	2007	NUM
ejpam-2553	537	2	.	.	PUNCT
ejpam-2553	538	1	[	[	X
ejpam-2553	538	2	13	13	NUM
ejpam-2553	538	3	]	]	X
ejpam-2553	538	4	b.	b.	PROPN
ejpam-2553	538	5	davvaz	davvaz	PROPN
ejpam-2553	538	6	and	and	CCONJ
ejpam-2553	538	7	v.	v.	ADP
ejpam-2553	538	8	leoreanu	leoreanu	NOUN
ejpam-2553	538	9	-	-	PUNCT
ejpam-2553	538	10	fotea	fotea	NOUN
ejpam-2553	538	11	.	.	PUNCT
ejpam-2553	539	1	hyperring	hyperre	VERB
ejpam-2553	539	2	theory	theory	NOUN
ejpam-2553	539	3	and	and	CCONJ
ejpam-2553	539	4	applications	application	NOUN
ejpam-2553	539	5	,	,	PUNCT
ejpam-2553	539	6	international	international	ADJ
ejpam-2553	539	7	academic	academic	ADJ
ejpam-2553	539	8	press	press	NOUN
ejpam-2553	539	9	,	,	PUNCT
ejpam-2553	539	10	usa	usa	PROPN
ejpam-2553	539	11	,	,	PUNCT
ejpam-2553	539	12	2007	2007	NUM
ejpam-2553	539	13	.	.	PUNCT
ejpam-2553	540	1	[	[	X
ejpam-2553	540	2	14	14	NUM
ejpam-2553	540	3	]	]	X
ejpam-2553	540	4	m.	m.	NOUN
ejpam-2553	540	5	de	de	NOUN
ejpam-2553	540	6	salvo	salvo	PROPN
ejpam-2553	540	7	and	and	CCONJ
ejpam-2553	540	8	g.	g.	PROPN
ejpam-2553	540	9	lo	lo	PROPN
ejpam-2553	540	10	faro	faro	PROPN
ejpam-2553	540	11	.	.	PUNCT
ejpam-2553	541	1	on	on	ADP
ejpam-2553	541	2	the	the	DET
ejpam-2553	541	3	n∗-complete	n∗-complete	ADJ
ejpam-2553	541	4	hypergroups	hypergroup	NOUN
ejpam-2553	541	5	,	,	PUNCT
ejpam-2553	541	6	discrete	discrete	ADJ
ejpam-2553	541	7	mathematics	mathematic	NOUN
ejpam-2553	541	8	,	,	PUNCT
ejpam-2553	541	9	208/209	208/209	PROPN
ejpam-2553	541	10	,	,	PUNCT
ejpam-2553	541	11	177	177	NUM
ejpam-2553	541	12	-	-	SYM
ejpam-2553	541	13	188	188	NUM
ejpam-2553	541	14	.	.	PUNCT
ejpam-2553	541	15	1990	1990	NUM
ejpam-2553	541	16	.	.	PUNCT
ejpam-2553	542	1	[	[	X
ejpam-2553	542	2	15	15	NUM
ejpam-2553	542	3	]	]	X
ejpam-2553	542	4	d.	d.	PROPN
ejpam-2553	542	5	freni	freni	PROPN
ejpam-2553	542	6	.	.	PUNCT
ejpam-2553	543	1	a	a	DET
ejpam-2553	543	2	new	new	ADJ
ejpam-2553	543	3	characterization	characterization	NOUN
ejpam-2553	543	4	of	of	ADP
ejpam-2553	543	5	the	the	DET
ejpam-2553	543	6	derived	derive	VERB
ejpam-2553	543	7	hypergroup	hypergroup	NOUN
ejpam-2553	543	8	via	via	ADP
ejpam-2553	543	9	strongly	strongly	ADV
ejpam-2553	543	10	regular	regular	ADJ
ejpam-2553	543	11	equivalences	equivalence	NOUN
ejpam-2553	543	12	,	,	PUNCT
ejpam-2553	543	13	communications	communication	NOUN
ejpam-2553	543	14	in	in	ADP
ejpam-2553	543	15	algebra	algebra	NOUN
ejpam-2553	543	16	,	,	PUNCT
ejpam-2553	543	17	30(8	30(8	NUM
ejpam-2553	543	18	)	)	PUNCT
ejpam-2553	543	19	,	,	PUNCT
ejpam-2553	543	20	3977	3977	NUM
ejpam-2553	543	21	-	-	SYM
ejpam-2553	543	22	3989	3989	NUM
ejpam-2553	543	23	.	.	PUNCT
ejpam-2553	544	1	2002	2002	NUM
ejpam-2553	544	2	.	.	PUNCT
ejpam-2553	545	1	[	[	X
ejpam-2553	545	2	16	16	NUM
ejpam-2553	545	3	]	]	X
ejpam-2553	545	4	d.	d.	PROPN
ejpam-2553	545	5	freni	freni	PROPN
ejpam-2553	545	6	.	.	PUNCT
ejpam-2553	546	1	strongly	strongly	ADV
ejpam-2553	546	2	transitive	transitive	ADJ
ejpam-2553	546	3	geometric	geometric	ADJ
ejpam-2553	546	4	spaces	space	NOUN
ejpam-2553	546	5	:	:	PUNCT
ejpam-2553	546	6	applications	application	NOUN
ejpam-2553	546	7	to	to	ADP
ejpam-2553	546	8	hypergroups	hypergroup	NOUN
ejpam-2553	546	9	and	and	CCONJ
ejpam-2553	546	10	semigroups	semigroup	NOUN
ejpam-2553	546	11	theory	theory	NOUN
ejpam-2553	546	12	,	,	PUNCT
ejpam-2553	546	13	communications	communication	NOUN
ejpam-2553	546	14	in	in	ADP
ejpam-2553	546	15	algebra	algebra	NOUN
ejpam-2553	546	16	,	,	PUNCT
ejpam-2553	546	17	32	32	NUM
ejpam-2553	546	18	,	,	PUNCT
ejpam-2553	546	19	969	969	NUM
ejpam-2553	546	20	-	-	SYM
ejpam-2553	546	21	988	988	NUM
ejpam-2553	546	22	.	.	PUNCT
ejpam-2553	546	23	2004	2004	NUM
ejpam-2553	546	24	.	.	PUNCT
ejpam-2553	547	1	[	[	X
ejpam-2553	547	2	17	17	NUM
ejpam-2553	547	3	]	]	PUNCT
ejpam-2553	547	4	m.	m.	NOUN
ejpam-2553	547	5	koskas	koskas	PROPN
ejpam-2553	547	6	.	.	PUNCT
ejpam-2553	548	1	groupoids	groupoids	PROPN
ejpam-2553	548	2	,	,	PUNCT
ejpam-2553	548	3	demi	demi	NOUN
ejpam-2553	548	4	-	-	PUNCT
ejpam-2553	548	5	groupes	groupe	NOUN
ejpam-2553	548	6	et	et	NOUN
ejpam-2553	548	7	hypergroupes	hypergroupe	NOUN
ejpam-2553	548	8	,	,	PUNCT
ejpam-2553	548	9	j.	j.	PROPN
ejpam-2553	548	10	math	math	PROPN
ejpam-2553	548	11	.	.	PUNCT
ejpam-2553	549	1	pures	pure	NOUN
ejpam-2553	549	2	appl	appl	PROPN
ejpam-2553	549	3	.	.	PROPN
ejpam-2553	549	4	,	,	PUNCT
ejpam-2553	549	5	49	49	NUM
ejpam-2553	549	6	,	,	PUNCT
ejpam-2553	549	7	155	155	NUM
ejpam-2553	549	8	-	-	SYM
ejpam-2553	549	9	192	192	NUM
ejpam-2553	549	10	.	.	PUNCT
ejpam-2553	549	11	1970	1970	NUM
ejpam-2553	549	12	.	.	PUNCT
ejpam-2553	550	1	[	[	X
ejpam-2553	550	2	18	18	NUM
ejpam-2553	550	3	]	]	PUNCT
ejpam-2553	550	4	m.	m.	NOUN
ejpam-2553	550	5	krasner	krasner	NOUN
ejpam-2553	550	6	.	.	PUNCT
ejpam-2553	551	1	a	a	DET
ejpam-2553	551	2	class	class	NOUN
ejpam-2553	551	3	of	of	ADP
ejpam-2553	551	4	hyperrings	hyperring	NOUN
ejpam-2553	551	5	and	and	CCONJ
ejpam-2553	551	6	hyperfields	hyperfield	NOUN
ejpam-2553	551	7	,	,	PUNCT
ejpam-2553	551	8	international	international	ADJ
ejpam-2553	551	9	journal	journal	NOUN
ejpam-2553	551	10	of	of	ADP
ejpam-2553	551	11	mathematics	mathematics	PROPN
ejpam-2553	551	12	and	and	CCONJ
ejpam-2553	551	13	mathematical	mathematical	ADJ
ejpam-2553	551	14	sciences	science	NOUN
ejpam-2553	551	15	,	,	PUNCT
ejpam-2553	551	16	2	2	NUM
ejpam-2553	551	17	,	,	PUNCT
ejpam-2553	551	18	307	307	NUM
ejpam-2553	551	19	-	-	SYM
ejpam-2553	551	20	312	312	NUM
ejpam-2553	551	21	.	.	NUM
ejpam-2553	551	22	1983	1983	NUM
ejpam-2553	551	23	.	.	PUNCT
ejpam-2553	552	1	[	[	X
ejpam-2553	552	2	19	19	NUM
ejpam-2553	552	3	]	]	X
ejpam-2553	552	4	c.	c.	PROPN
ejpam-2553	552	5	g.	g.	PROPN
ejpam-2553	552	6	massouros	massouros	PROPN
ejpam-2553	552	7	.	.	PUNCT
ejpam-2553	553	1	on	on	ADP
ejpam-2553	553	2	the	the	DET
ejpam-2553	553	3	theory	theory	NOUN
ejpam-2553	553	4	of	of	ADP
ejpam-2553	553	5	hyperrings	hyperring	NOUN
ejpam-2553	553	6	and	and	CCONJ
ejpam-2553	553	7	hyperfields	hyperfield	NOUN
ejpam-2553	553	8	,	,	PUNCT
ejpam-2553	553	9	algebra	algebra	NOUN
ejpam-2553	553	10	i	i	PRON
ejpam-2553	553	11	logika	logika	NOUN
ejpam-2553	553	12	,	,	PUNCT
ejpam-2553	553	13	24	24	NUM
ejpam-2553	553	14	,	,	PUNCT
ejpam-2553	553	15	728742	728742	NUM
ejpam-2553	553	16	.	.	PUNCT
ejpam-2553	554	1	1985	1985	NUM
ejpam-2553	554	2	.	.	PUNCT
ejpam-2553	555	1	[	[	X
ejpam-2553	555	2	20	20	NUM
ejpam-2553	555	3	]	]	PUNCT
ejpam-2553	555	4	j.	j.	PROPN
ejpam-2553	555	5	mittas	mittas	PROPN
ejpam-2553	555	6	.	.	PUNCT
ejpam-2553	556	1	hypergroups	hypergroup	NOUN
ejpam-2553	556	2	canoniques	canonique	NOUN
ejpam-2553	556	3	,	,	PUNCT
ejpam-2553	556	4	mathemaica	mathemaica	PROPN
ejpam-2553	556	5	balkanica	balkanica	PROPN
ejpam-2553	556	6	,	,	PUNCT
ejpam-2553	556	7	2	2	NUM
ejpam-2553	556	8	,	,	PUNCT
ejpam-2553	556	9	165	165	NUM
ejpam-2553	556	10	-	-	SYM
ejpam-2553	556	11	179	179	NUM
ejpam-2553	556	12	.	.	PUNCT
ejpam-2553	556	13	1972	1972	NUM
ejpam-2553	556	14	.	.	PUNCT
ejpam-2553	557	1	[	[	X
ejpam-2553	557	2	21	21	NUM
ejpam-2553	557	3	]	]	PUNCT
ejpam-2553	557	4	a.	a.	NOUN
ejpam-2553	557	5	nakassis	nakassis	NOUN
ejpam-2553	557	6	.	.	PUNCT
ejpam-2553	558	1	expository	expository	NOUN
ejpam-2553	558	2	and	and	CCONJ
ejpam-2553	558	3	survey	survey	NOUN
ejpam-2553	558	4	article	article	NOUN
ejpam-2553	558	5	recent	recent	ADJ
ejpam-2553	558	6	result	result	NOUN
ejpam-2553	558	7	in	in	ADP
ejpam-2553	558	8	hyperring	hyperring	NOUN
ejpam-2553	558	9	and	and	CCONJ
ejpam-2553	558	10	hyperfield	hyperfield	PROPN
ejpam-2553	558	11	theory	theory	NOUN
ejpam-2553	558	12	,	,	PUNCT
ejpam-2553	558	13	international	international	ADJ
ejpam-2553	558	14	journal	journal	NOUN
ejpam-2553	558	15	of	of	ADP
ejpam-2553	558	16	mathematics	mathematics	PROPN
ejpam-2553	558	17	and	and	CCONJ
ejpam-2553	558	18	mathematical	mathematical	ADJ
ejpam-2553	558	19	sciences	science	NOUN
ejpam-2553	558	20	,	,	PUNCT
ejpam-2553	558	21	11(2	11(2	NUM
ejpam-2553	558	22	)	)	PUNCT
ejpam-2553	558	23	,	,	PUNCT
ejpam-2553	558	24	209220	209220	NUM
ejpam-2553	558	25	.	.	PUNCT
ejpam-2553	559	1	1988	1988	NUM
ejpam-2553	559	2	.	.	PUNCT
ejpam-2553	560	1	references	reference	NOUN
ejpam-2553	560	2	418	418	NUM
ejpam-2553	561	1	[	[	X
ejpam-2553	561	2	22	22	NUM
ejpam-2553	561	3	]	]	X
ejpam-2553	561	4	d.	d.	PROPN
ejpam-2553	561	5	m.	m.	PROPN
ejpam-2553	561	6	olson	olson	PROPN
ejpam-2553	561	7	and	and	CCONJ
ejpam-2553	561	8	v.k	v.k	PROPN
ejpam-2553	561	9	.	.	PROPN
ejpam-2553	561	10	ward	ward	PROPN
ejpam-2553	561	11	.	.	PUNCT
ejpam-2553	562	1	a	a	DET
ejpam-2553	562	2	note	note	NOUN
ejpam-2553	562	3	on	on	ADP
ejpam-2553	562	4	multiplicative	multiplicative	ADJ
ejpam-2553	562	5	hyperrings	hyperring	NOUN
ejpam-2553	562	6	,	,	PUNCT
ejpam-2553	562	7	italian	italian	ADJ
ejpam-2553	562	8	journal	journal	NOUN
ejpam-2553	562	9	of	of	ADP
ejpam-2553	562	10	pure	pure	ADJ
ejpam-2553	562	11	and	and	CCONJ
ejpam-2553	562	12	applied	applied	ADJ
ejpam-2553	562	13	mathematics	mathematic	NOUN
ejpam-2553	562	14	,	,	PUNCT
ejpam-2553	562	15	1	1	NUM
ejpam-2553	562	16	,	,	PUNCT
ejpam-2553	562	17	77	77	NUM
ejpam-2553	562	18	-	-	SYM
ejpam-2553	562	19	84	84	NUM
ejpam-2553	562	20	.	.	PUNCT
ejpam-2553	563	1	1997	1997	NUM
ejpam-2553	563	2	.	.	PUNCT
ejpam-2553	564	1	[	[	X
ejpam-2553	564	2	23	23	NUM
ejpam-2553	564	3	]	]	X
ejpam-2553	564	4	r.	r.	PROPN
ejpam-2553	564	5	procesi	procesi	PROPN
ejpam-2553	564	6	-	-	PUNCT
ejpam-2553	564	7	ciampi	ciampi	NOUN
ejpam-2553	564	8	and	and	CCONJ
ejpam-2553	564	9	r.	r.	PROPN
ejpam-2553	564	10	rota	rota	PROPN
ejpam-2553	564	11	.	.	PUNCT
ejpam-2553	565	1	the	the	DET
ejpam-2553	565	2	hyperring	hyperring	NOUN
ejpam-2553	565	3	spectrum	spectrum	NOUN
ejpam-2553	565	4	,	,	PUNCT
ejpam-2553	565	5	riv	riv	PROPN
ejpam-2553	565	6	.	.	PROPN
ejpam-2553	565	7	mat	mat	PROPN
ejpam-2553	565	8	.	.	PUNCT
ejpam-2553	565	9	pura	pura	NOUN
ejpam-2553	565	10	appl	appl	PROPN
ejpam-2553	565	11	.	.	PROPN
ejpam-2553	565	12	,	,	PUNCT
ejpam-2553	565	13	1	1	NUM
ejpam-2553	565	14	,	,	PUNCT
ejpam-2553	565	15	71	71	NUM
ejpam-2553	565	16	-	-	SYM
ejpam-2553	565	17	80	80	NUM
ejpam-2553	565	18	.	.	PUNCT
ejpam-2553	565	19	1987	1987	NUM
ejpam-2553	565	20	.	.	PUNCT
ejpam-2553	566	1	[	[	X
ejpam-2553	566	2	24	24	NUM
ejpam-2553	566	3	]	]	X
ejpam-2553	566	4	r.	r.	PROPN
ejpam-2553	566	5	procesi	procesi	PROPN
ejpam-2553	566	6	and	and	CCONJ
ejpam-2553	566	7	r.	r.	PROPN
ejpam-2553	566	8	rota	rota	PROPN
ejpam-2553	566	9	.	.	PUNCT
ejpam-2553	567	1	on	on	ADP
ejpam-2553	567	2	some	some	DET
ejpam-2553	567	3	classes	class	NOUN
ejpam-2553	567	4	of	of	ADP
ejpam-2553	567	5	hyperstructures	hyperstructure	NOUN
ejpam-2553	567	6	,	,	PUNCT
ejpam-2553	567	7	discrete	discrete	ADJ
ejpam-2553	567	8	mathemaatics	mathemaatic	NOUN
ejpam-2553	567	9	,	,	PUNCT
ejpam-2553	567	10	208/209	208/209	NUM
ejpam-2553	567	11	,	,	PUNCT
ejpam-2553	567	12	485	485	NUM
ejpam-2553	567	13	-	-	SYM
ejpam-2553	567	14	497	497	NUM
ejpam-2553	567	15	.	.	PUNCT
ejpam-2553	567	16	1999	1999	NUM
ejpam-2553	567	17	.	.	PUNCT
ejpam-2553	568	1	[	[	X
ejpam-2553	568	2	25	25	NUM
ejpam-2553	568	3	]	]	PUNCT
ejpam-2553	568	4	a.	a.	PROPN
ejpam-2553	568	5	r.	r.	PROPN
ejpam-2553	568	6	barghi	barghi	PROPN
ejpam-2553	568	7	.	.	PUNCT
ejpam-2553	569	1	a	a	DET
ejpam-2553	569	2	class	class	NOUN
ejpam-2553	569	3	of	of	ADP
ejpam-2553	569	4	hyperrings	hyperring	NOUN
ejpam-2553	569	5	,	,	PUNCT
ejpam-2553	569	6	journal	journal	NOUN
ejpam-2553	569	7	of	of	ADP
ejpam-2553	569	8	discrete	discrete	ADJ
ejpam-2553	569	9	mathematical	mathematical	ADJ
ejpam-2553	569	10	sciences	science	NOUN
ejpam-2553	569	11	and	and	CCONJ
ejpam-2553	569	12	cryptography	cryptography	NOUN
ejpam-2553	569	13	,	,	PUNCT
ejpam-2553	569	14	6	6	NUM
ejpam-2553	569	15	,	,	PUNCT
ejpam-2553	569	16	227	227	NUM
ejpam-2553	569	17	-	-	SYM
ejpam-2553	569	18	233	233	NUM
ejpam-2553	569	19	.	.	PUNCT
ejpam-2553	569	20	2003	2003	NUM
ejpam-2553	569	21	.	.	PUNCT
ejpam-2553	570	1	[	[	X
ejpam-2553	570	2	26	26	NUM
ejpam-2553	570	3	]	]	X
ejpam-2553	570	4	r.	r.	PROPN
ejpam-2553	570	5	rota	rota	PROPN
ejpam-2553	570	6	.	.	PUNCT
ejpam-2553	571	1	strongly	strongly	ADV
ejpam-2553	571	2	distributive	distributive	ADJ
ejpam-2553	571	3	multiplicative	multiplicative	ADJ
ejpam-2553	571	4	hyperrings	hyperring	NOUN
ejpam-2553	571	5	,	,	PUNCT
ejpam-2553	571	6	journal	journal	NOUN
ejpam-2553	571	7	of	of	ADP
ejpam-2553	571	8	geometry	geometry	NOUN
ejpam-2553	571	9	,	,	PUNCT
ejpam-2553	571	10	39	39	NUM
ejpam-2553	571	11	,	,	PUNCT
ejpam-2553	571	12	130	130	NUM
ejpam-2553	571	13	-	-	SYM
ejpam-2553	571	14	138	138	NUM
ejpam-2553	571	15	.	.	PUNCT
ejpam-2553	571	16	1990	1990	NUM
ejpam-2553	571	17	.	.	PUNCT
ejpam-2553	572	1	[	[	X
ejpam-2553	572	2	27	27	NUM
ejpam-2553	572	3	]	]	X
ejpam-2553	572	4	r.	r.	PROPN
ejpam-2553	572	5	rota	rota	PROPN
ejpam-2553	572	6	.	.	PUNCT
ejpam-2553	573	1	sugli	sugli	PROPN
ejpam-2553	573	2	iperanelli	iperanelli	PROPN
ejpam-2553	573	3	moltiplicativi	moltiplicativi	PROPN
ejpam-2553	573	4	,	,	PUNCT
ejpam-2553	573	5	rend	rend	VERB
ejpam-2553	573	6	.	.	PUNCT
ejpam-2553	573	7	di	di	PROPN
ejpam-2553	573	8	mat	mat	PROPN
ejpam-2553	573	9	.	.	PROPN
ejpam-2553	573	10	,	,	PUNCT
ejpam-2553	573	11	series	series	PROPN
ejpam-2553	573	12	v	v	INTJ
ejpam-2553	574	1	i	i	PRON
ejpam-2553	574	2	i	i	PRON
ejpam-2553	574	3	,	,	PUNCT
ejpam-2553	574	4	4(2	4(2	NUM
ejpam-2553	574	5	)	)	PUNCT
ejpam-2553	574	6	,	,	PUNCT
ejpam-2553	574	7	711	711	NUM
ejpam-2553	574	8	-	-	SYM
ejpam-2553	574	9	724	724	NUM
ejpam-2553	574	10	.	.	PUNCT
ejpam-2553	574	11	1982	1982	NUM
ejpam-2553	574	12	.	.	PUNCT
ejpam-2553	575	1	[	[	X
ejpam-2553	575	2	28	28	NUM
ejpam-2553	575	3	]	]	X
ejpam-2553	575	4	r.	r.	PROPN
ejpam-2553	575	5	rota	rota	PROPN
ejpam-2553	575	6	.	.	PUNCT
ejpam-2553	576	1	congruenze	congruenze	PROPN
ejpam-2553	576	2	sugli	sugli	PROPN
ejpam-2553	576	3	iperanelli	iperanelli	PROPN
ejpam-2553	576	4	moltiplicativi	moltiplicativi	PROPN
ejpam-2553	576	5	,	,	PUNCT
ejpam-2553	576	6	rend	rend	VERB
ejpam-2553	576	7	.	.	PUNCT
ejpam-2553	576	8	di	di	PROPN
ejpam-2553	576	9	mat	mat	PROPN
ejpam-2553	576	10	.	.	PROPN
ejpam-2553	576	11	,	,	PUNCT
ejpam-2553	576	12	series	series	PROPN
ejpam-2553	576	13	v	v	INTJ
ejpam-2553	577	1	i	i	PRON
ejpam-2553	577	2	i	i	PRON
ejpam-2553	577	3	,	,	PUNCT
ejpam-2553	577	4	1(3	1(3	NUM
ejpam-2553	577	5	)	)	PUNCT
ejpam-2553	577	6	,	,	PUNCT
ejpam-2553	577	7	17	17	NUM
ejpam-2553	577	8	-	-	SYM
ejpam-2553	577	9	31	31	NUM
ejpam-2553	577	10	.	.	PUNCT
ejpam-2553	577	11	1983	1983	NUM
ejpam-2553	577	12	.	.	PUNCT
ejpam-2553	578	1	[	[	X
ejpam-2553	578	2	29	29	NUM
ejpam-2553	578	3	]	]	X
ejpam-2553	578	4	r.	r.	PROPN
ejpam-2553	578	5	rota	rota	PROPN
ejpam-2553	578	6	.	.	PUNCT
ejpam-2553	579	1	sulla	sulla	PROPN
ejpam-2553	579	2	categoria	categoria	PROPN
ejpam-2553	579	3	degli	degli	PROPN
ejpam-2553	579	4	iperanelli	iperanelli	PROPN
ejpam-2553	579	5	moltiplicativi	moltiplicativi	PROPN
ejpam-2553	579	6	,	,	PUNCT
ejpam-2553	579	7	rend	rend	VERB
ejpam-2553	579	8	.	.	PUNCT
ejpam-2553	579	9	di	di	PROPN
ejpam-2553	579	10	mat	mat	PROPN
ejpam-2553	579	11	.	.	PROPN
ejpam-2553	579	12	,	,	PUNCT
ejpam-2553	579	13	series	series	PROPN
ejpam-2553	579	14	v	v	INTJ
ejpam-2553	580	1	i	i	PRON
ejpam-2553	580	2	i	i	PRON
ejpam-2553	580	3	,	,	PUNCT
ejpam-2553	580	4	1(4	1(4	NUM
ejpam-2553	580	5	)	)	PUNCT
ejpam-2553	580	6	,	,	PUNCT
ejpam-2553	580	7	75	75	NUM
ejpam-2553	580	8	-	-	SYM
ejpam-2553	580	9	84	84	NUM
ejpam-2553	580	10	.	.	PUNCT
ejpam-2553	580	11	1984	1984	NUM
ejpam-2553	580	12	.	.	PUNCT
ejpam-2553	581	1	[	[	X
ejpam-2553	581	2	30	30	NUM
ejpam-2553	581	3	]	]	X
ejpam-2553	581	4	s.	s.	PROPN
ejpam-2553	581	5	spartalis	spartalis	PROPN
ejpam-2553	581	6	and	and	CCONJ
ejpam-2553	581	7	t.	t.	PROPN
ejpam-2553	581	8	vougiouklis	vougiouklis	PROPN
ejpam-2553	581	9	.	.	PUNCT
ejpam-2553	582	1	the	the	DET
ejpam-2553	582	2	fundamental	fundamental	ADJ
ejpam-2553	582	3	relations	relation	NOUN
ejpam-2553	582	4	on	on	ADP
ejpam-2553	582	5	hv	hv	NOUN
ejpam-2553	582	6	-	-	PUNCT
ejpam-2553	582	7	rings	ring	NOUN
ejpam-2553	582	8	,	,	PUNCT
ejpam-2553	582	9	rivista	rivista	PROPN
ejpam-2553	582	10	mat	mat	NOUN
ejpam-2553	582	11	.	.	PUNCT
ejpam-2553	582	12	pura	pura	NOUN
ejpam-2553	582	13	ed	ed	PROPN
ejpam-2553	582	14	appl	appl	PROPN
ejpam-2553	582	15	.	.	PROPN
ejpam-2553	582	16	,	,	PUNCT
ejpam-2553	582	17	13	13	NUM
ejpam-2553	582	18	,	,	PUNCT
ejpam-2553	582	19	7	7	NUM
ejpam-2553	582	20	-	-	SYM
ejpam-2553	582	21	20	20	NUM
ejpam-2553	582	22	.	.	PUNCT
ejpam-2553	582	23	1994	1994	NUM
ejpam-2553	582	24	.	.	PUNCT
ejpam-2553	583	1	[	[	X
ejpam-2553	583	2	31	31	NUM
ejpam-2553	583	3	]	]	PUNCT
ejpam-2553	583	4	t.	t.	PROPN
ejpam-2553	583	5	vougiouklis	vougiouklis	PROPN
ejpam-2553	583	6	.	.	PUNCT
ejpam-2553	584	1	hyperstructures	hyperstructure	NOUN
ejpam-2553	584	2	and	and	CCONJ
ejpam-2553	584	3	their	their	PRON
ejpam-2553	584	4	representations	representation	NOUN
ejpam-2553	584	5	,	,	PUNCT
ejpam-2553	584	6	115	115	NUM
ejpam-2553	584	7	hadronic	hadronic	ADJ
ejpam-2553	584	8	press	press	NOUN
ejpam-2553	584	9	,	,	PUNCT
ejpam-2553	584	10	inc	inc	PROPN
ejpam-2553	584	11	.	.	PROPN
ejpam-2553	584	12	,	,	PUNCT
ejpam-2553	584	13	palm	palm	PROPN
ejpam-2553	584	14	harber	harber	PROPN
ejpam-2553	584	15	,	,	PUNCT
ejpam-2553	584	16	usa	usa	PROPN
ejpam-2553	584	17	,	,	PUNCT
ejpam-2553	584	18	1994	1994	NUM
ejpam-2553	584	19	.	.	PUNCT
ejpam-2553	585	1	[	[	X
ejpam-2553	585	2	32	32	NUM
ejpam-2553	585	3	]	]	PUNCT
ejpam-2553	585	4	t.	t.	PROPN
ejpam-2553	585	5	vougiouklis	vougiouklis	PROPN
ejpam-2553	585	6	.	.	PUNCT
ejpam-2553	586	1	the	the	DET
ejpam-2553	586	2	fundamental	fundamental	ADJ
ejpam-2553	586	3	relation	relation	NOUN
ejpam-2553	586	4	in	in	ADP
ejpam-2553	586	5	hyperrings	hyperring	NOUN
ejpam-2553	586	6	,	,	PUNCT
ejpam-2553	586	7	the	the	DET
ejpam-2553	586	8	general	general	ADJ
ejpam-2553	586	9	hyperfield	hyperfield	NOUN
ejpam-2553	586	10	,	,	PUNCT
ejpam-2553	586	11	in	in	ADP
ejpam-2553	586	12	:	:	PUNCT
ejpam-2553	586	13	proceedings	proceeding	NOUN
ejpam-2553	586	14	of	of	ADP
ejpam-2553	586	15	the	the	DET
ejpam-2553	586	16	fourth	fourth	PROPN
ejpam-2553	586	17	international	international	ADJ
ejpam-2553	586	18	congress	congress	PROPN
ejpam-2553	586	19	on	on	ADP
ejpam-2553	586	20	algebraic	algebraic	PROPN
ejpam-2553	586	21	hyperstructures	hyperstructure	NOUN
ejpam-2553	586	22	and	and	CCONJ
ejpam-2553	586	23	applications	application	NOUN
ejpam-2553	586	24	,	,	PUNCT
ejpam-2553	586	25	aha	aha	INTJ
ejpam-2553	586	26	,	,	PUNCT
ejpam-2553	586	27	1990	1990	NUM
ejpam-2553	586	28	,	,	PUNCT
ejpam-2553	586	29	world	world	NOUN
ejpam-2553	586	30	scientific	scientific	ADJ
ejpam-2553	586	31	,	,	PUNCT
ejpam-2553	586	32	203	203	NUM
ejpam-2553	586	33	-	-	SYM
ejpam-2553	586	34	211	211	NUM
ejpam-2553	586	35	.	.	PUNCT
ejpam-2553	586	36	1991	1991	NUM
ejpam-2553	586	37	.	.	PUNCT
ejpam-2553	587	1	[	[	X
ejpam-2553	587	2	33	33	NUM
ejpam-2553	587	3	]	]	PUNCT
ejpam-2553	587	4	m.	m.	NOUN
ejpam-2553	587	5	m.	m.	PROPN
ejpam-2553	587	6	zahedi	zahedi	PROPN
ejpam-2553	587	7	and	and	CCONJ
ejpam-2553	587	8	r.	r.	PROPN
ejpam-2553	587	9	ameri	ameri	PROPN
ejpam-2553	587	10	.	.	PUNCT
ejpam-2553	588	1	on	on	ADP
ejpam-2553	588	2	the	the	DET
ejpam-2553	588	3	prime	prime	ADJ
ejpam-2553	588	4	,	,	PUNCT
ejpam-2553	588	5	primary	primary	ADJ
ejpam-2553	588	6	and	and	CCONJ
ejpam-2553	588	7	maximal	maximal	ADJ
ejpam-2553	588	8	subhypermodules	subhypermodule	NOUN
ejpam-2553	588	9	,	,	PUNCT
ejpam-2553	588	10	italian	italian	ADJ
ejpam-2553	588	11	journal	journal	NOUN
ejpam-2553	588	12	of	of	ADP
ejpam-2553	588	13	pure	pure	ADJ
ejpam-2553	588	14	and	and	CCONJ
ejpam-2553	588	15	applied	applied	ADJ
ejpam-2553	588	16	mathematics	mathematic	NOUN
ejpam-2553	588	17	,	,	PUNCT
ejpam-2553	588	18	5	5	NUM
ejpam-2553	588	19	,	,	PUNCT
ejpam-2553	588	20	61	61	NUM
ejpam-2553	588	21	-	-	SYM
ejpam-2553	588	22	80	80	NUM
ejpam-2553	588	23	.	.	PUNCT
ejpam-2553	588	24	1999	1999	NUM
ejpam-2553	588	25	.	.	PUNCT