id	sid	tid	token	lemma	pos
ejpam-4383	1	1	european	european	PROPN
ejpam-4383	1	2	journal	journal	PROPN
ejpam-4383	1	3	of	of	ADP
ejpam-4383	1	4	pure	pure	ADJ
ejpam-4383	1	5	and	and	CCONJ
ejpam-4383	1	6	applied	apply	VERB
ejpam-4383	1	7	mathematics	mathematic	NOUN
ejpam-4383	1	8	vol	vol	NOUN
ejpam-4383	1	9	.	.	PROPN
ejpam-4383	2	1	15	15	NUM
ejpam-4383	2	2	,	,	PUNCT
ejpam-4383	2	3	no	no	INTJ
ejpam-4383	2	4	.	.	NOUN
ejpam-4383	2	5	3	3	NUM
ejpam-4383	2	6	,	,	PUNCT
ejpam-4383	2	7	2022	2022	NUM
ejpam-4383	2	8	,	,	PUNCT
ejpam-4383	2	9	999	999	NUM
ejpam-4383	2	10	-	-	SYM
ejpam-4383	2	11	1014	1014	NUM
ejpam-4383	2	12	issn	issn	PROPN
ejpam-4383	2	13	1307	1307	NUM
ejpam-4383	2	14	-	-	SYM
ejpam-4383	2	15	5543	5543	NUM
ejpam-4383	2	16	–	–	PUNCT
ejpam-4383	2	17	ejpam.com	ejpam.com	X
ejpam-4383	2	18	published	publish	VERB
ejpam-4383	2	19	by	by	ADP
ejpam-4383	2	20	new	new	PROPN
ejpam-4383	2	21	york	york	PROPN
ejpam-4383	2	22	business	business	PROPN
ejpam-4383	2	23	global	global	ADJ
ejpam-4383	2	24	direct	direct	ADJ
ejpam-4383	2	25	product	product	NOUN
ejpam-4383	2	26	of	of	ADP
ejpam-4383	2	27	infinite	infinite	ADJ
ejpam-4383	2	28	family	family	NOUN
ejpam-4383	2	29	of	of	ADP
ejpam-4383	2	30	b	b	NOUN
ejpam-4383	2	31	-	-	PUNCT
ejpam-4383	2	32	algebras	algebras	ADJ
ejpam-4383	2	33	chatsuda	chatsuda	PROPN
ejpam-4383	2	34	chanmanee1	chanmanee1	PROPN
ejpam-4383	2	35	,	,	PUNCT
ejpam-4383	2	36	ronnason	ronnason	NOUN
ejpam-4383	2	37	chinram2	chinram2	PROPN
ejpam-4383	2	38	,	,	PUNCT
ejpam-4383	2	39	rukchart	rukchart	NOUN
ejpam-4383	2	40	prasertpong3	prasertpong3	NOUN
ejpam-4383	2	41	,	,	PUNCT
ejpam-4383	2	42	pongpun	pongpun	ADJ
ejpam-4383	2	43	julatha4	julatha4	PROPN
ejpam-4383	2	44	,	,	PUNCT
ejpam-4383	2	45	aiyared	aiyare	VERB
ejpam-4383	2	46	iampan1,∗	iampan1,∗	NOUN
ejpam-4383	2	47	1	1	NUM
ejpam-4383	2	48	fuzzy	fuzzy	ADJ
ejpam-4383	2	49	algebras	algebra	NOUN
ejpam-4383	2	50	and	and	CCONJ
ejpam-4383	2	51	decision	decision	NOUN
ejpam-4383	2	52	-	-	PUNCT
ejpam-4383	2	53	making	make	VERB
ejpam-4383	2	54	problems	problem	NOUN
ejpam-4383	2	55	research	research	NOUN
ejpam-4383	2	56	unit	unit	NOUN
ejpam-4383	2	57	,	,	PUNCT
ejpam-4383	2	58	department	department	NOUN
ejpam-4383	2	59	of	of	ADP
ejpam-4383	2	60	mathematics	mathematic	NOUN
ejpam-4383	2	61	,	,	PUNCT
ejpam-4383	2	62	school	school	NOUN
ejpam-4383	2	63	of	of	ADP
ejpam-4383	2	64	science	science	NOUN
ejpam-4383	2	65	,	,	PUNCT
ejpam-4383	2	66	university	university	NOUN
ejpam-4383	2	67	of	of	ADP
ejpam-4383	2	68	phayao	phayao	NOUN
ejpam-4383	2	69	,	,	PUNCT
ejpam-4383	2	70	mae	mae	PROPN
ejpam-4383	2	71	ka	ka	PROPN
ejpam-4383	2	72	,	,	PUNCT
ejpam-4383	2	73	mueang	mueang	PROPN
ejpam-4383	2	74	,	,	PUNCT
ejpam-4383	2	75	phayao	phayao	NOUN
ejpam-4383	2	76	56000	56000	NUM
ejpam-4383	2	77	,	,	PUNCT
ejpam-4383	2	78	thailand	thailand	PROPN
ejpam-4383	2	79	2	2	NUM
ejpam-4383	2	80	division	division	NOUN
ejpam-4383	2	81	of	of	ADP
ejpam-4383	2	82	computational	computational	ADJ
ejpam-4383	2	83	science	science	NOUN
ejpam-4383	2	84	,	,	PUNCT
ejpam-4383	2	85	faculty	faculty	NOUN
ejpam-4383	2	86	of	of	ADP
ejpam-4383	2	87	science	science	NOUN
ejpam-4383	2	88	,	,	PUNCT
ejpam-4383	2	89	prince	prince	NOUN
ejpam-4383	2	90	of	of	ADP
ejpam-4383	2	91	songkla	songkla	PROPN
ejpam-4383	2	92	university	university	PROPN
ejpam-4383	2	93	,	,	PUNCT
ejpam-4383	2	94	hat	hat	PROPN
ejpam-4383	2	95	yai	yai	PROPN
ejpam-4383	2	96	,	,	PUNCT
ejpam-4383	2	97	songkhla	songkhla	VERB
ejpam-4383	2	98	90110	90110	NUM
ejpam-4383	2	99	,	,	PUNCT
ejpam-4383	2	100	thailand	thailand	PROPN
ejpam-4383	2	101	3	3	NUM
ejpam-4383	2	102	division	division	NOUN
ejpam-4383	2	103	of	of	ADP
ejpam-4383	2	104	mathematics	mathematic	NOUN
ejpam-4383	2	105	and	and	CCONJ
ejpam-4383	2	106	statistics	statistic	NOUN
ejpam-4383	2	107	,	,	PUNCT
ejpam-4383	2	108	faculty	faculty	NOUN
ejpam-4383	2	109	of	of	ADP
ejpam-4383	2	110	science	science	NOUN
ejpam-4383	2	111	and	and	CCONJ
ejpam-4383	2	112	technology	technology	NOUN
ejpam-4383	2	113	,	,	PUNCT
ejpam-4383	2	114	nakhon	nakhon	PROPN
ejpam-4383	2	115	sawan	sawan	PROPN
ejpam-4383	2	116	rajabhat	rajabhat	PROPN
ejpam-4383	2	117	university	university	PROPN
ejpam-4383	2	118	,	,	PUNCT
ejpam-4383	2	119	nakhon	nakhon	PROPN
ejpam-4383	2	120	sawan	sawan	PROPN
ejpam-4383	2	121	60000	60000	NUM
ejpam-4383	2	122	,	,	PUNCT
ejpam-4383	2	123	thailand	thailand	PROPN
ejpam-4383	2	124	4	4	NUM
ejpam-4383	2	125	department	department	NOUN
ejpam-4383	2	126	of	of	ADP
ejpam-4383	2	127	mathematics	mathematic	NOUN
ejpam-4383	2	128	,	,	PUNCT
ejpam-4383	2	129	faculty	faculty	NOUN
ejpam-4383	2	130	of	of	ADP
ejpam-4383	2	131	science	science	NOUN
ejpam-4383	2	132	and	and	CCONJ
ejpam-4383	2	133	technology	technology	NOUN
ejpam-4383	2	134	,	,	PUNCT
ejpam-4383	2	135	pibulsongkram	pibulsongkram	PROPN
ejpam-4383	2	136	rajabhat	rajabhat	PROPN
ejpam-4383	2	137	university	university	NOUN
ejpam-4383	2	138	,	,	PUNCT
ejpam-4383	2	139	phitsanulok	phitsanulok	NOUN
ejpam-4383	2	140	65000	65000	NUM
ejpam-4383	2	141	,	,	PUNCT
ejpam-4383	2	142	thailand	thailand	PROPN
ejpam-4383	2	143	abstract	abstract	PROPN
ejpam-4383	2	144	.	.	PUNCT
ejpam-4383	3	1	the	the	DET
ejpam-4383	3	2	concept	concept	NOUN
ejpam-4383	3	3	of	of	ADP
ejpam-4383	3	4	the	the	DET
ejpam-4383	3	5	direct	direct	ADJ
ejpam-4383	3	6	product	product	NOUN
ejpam-4383	3	7	of	of	ADP
ejpam-4383	3	8	finite	finite	ADJ
ejpam-4383	3	9	family	family	NOUN
ejpam-4383	3	10	of	of	ADP
ejpam-4383	3	11	b	b	PROPN
ejpam-4383	3	12	-algebras	-algebras	PROPN
ejpam-4383	3	13	is	be	AUX
ejpam-4383	3	14	introduced	introduce	VERB
ejpam-4383	3	15	by	by	ADP
ejpam-4383	3	16	lingcong	lingcong	NOUN
ejpam-4383	3	17	and	and	CCONJ
ejpam-4383	3	18	endam	endam	NOUN
ejpam-4383	4	1	[	[	X
ejpam-4383	4	2	j.	j.	PROPN
ejpam-4383	4	3	a.	a.	PROPN
ejpam-4383	4	4	v.	v.	PROPN
ejpam-4383	4	5	lingcong	lingcong	PROPN
ejpam-4383	4	6	and	and	CCONJ
ejpam-4383	4	7	j.	j.	PROPN
ejpam-4383	4	8	c.	c.	PROPN
ejpam-4383	4	9	endam	endam	PROPN
ejpam-4383	4	10	,	,	PUNCT
ejpam-4383	4	11	direct	direct	ADJ
ejpam-4383	4	12	product	product	NOUN
ejpam-4383	4	13	of	of	ADP
ejpam-4383	4	14	b	b	NOUN
ejpam-4383	4	15	-algebras	-algebras	PROPN
ejpam-4383	4	16	,	,	PUNCT
ejpam-4383	4	17	int	int	NOUN
ejpam-4383	4	18	.	.	PUNCT
ejpam-4383	5	1	j.	j.	PROPN
ejpam-4383	5	2	algebra	algebra	PROPN
ejpam-4383	5	3	,	,	PUNCT
ejpam-4383	5	4	10(1):33	10(1):33	NUM
ejpam-4383	5	5	-	-	SYM
ejpam-4383	5	6	40	40	NUM
ejpam-4383	5	7	,	,	PUNCT
ejpam-4383	5	8	2016	2016	NUM
ejpam-4383	5	9	.	.	PUNCT
ejpam-4383	5	10	]	]	PUNCT
ejpam-4383	5	11	.	.	PUNCT
ejpam-4383	6	1	in	in	ADP
ejpam-4383	6	2	this	this	DET
ejpam-4383	6	3	paper	paper	NOUN
ejpam-4383	6	4	,	,	PUNCT
ejpam-4383	6	5	we	we	PRON
ejpam-4383	6	6	introduce	introduce	VERB
ejpam-4383	6	7	the	the	DET
ejpam-4383	6	8	concept	concept	NOUN
ejpam-4383	6	9	of	of	ADP
ejpam-4383	6	10	the	the	DET
ejpam-4383	6	11	direct	direct	ADJ
ejpam-4383	6	12	product	product	NOUN
ejpam-4383	6	13	of	of	ADP
ejpam-4383	6	14	infinite	infinite	ADJ
ejpam-4383	6	15	family	family	NOUN
ejpam-4383	6	16	of	of	ADP
ejpam-4383	6	17	b	b	PROPN
ejpam-4383	6	18	-algebras	-algebras	PROPN
ejpam-4383	6	19	,	,	PUNCT
ejpam-4383	6	20	we	we	PRON
ejpam-4383	6	21	call	call	VERB
ejpam-4383	6	22	the	the	DET
ejpam-4383	6	23	external	external	ADJ
ejpam-4383	6	24	direct	direct	ADJ
ejpam-4383	6	25	product	product	NOUN
ejpam-4383	6	26	,	,	PUNCT
ejpam-4383	6	27	which	which	PRON
ejpam-4383	6	28	is	be	AUX
ejpam-4383	6	29	a	a	DET
ejpam-4383	6	30	generalization	generalization	NOUN
ejpam-4383	6	31	of	of	ADP
ejpam-4383	6	32	the	the	DET
ejpam-4383	6	33	direct	direct	ADJ
ejpam-4383	6	34	product	product	NOUN
ejpam-4383	6	35	in	in	ADP
ejpam-4383	6	36	the	the	DET
ejpam-4383	6	37	sense	sense	NOUN
ejpam-4383	6	38	of	of	ADP
ejpam-4383	6	39	lingcong	lingcong	NOUN
ejpam-4383	6	40	and	and	CCONJ
ejpam-4383	6	41	endam	endam	NOUN
ejpam-4383	6	42	.	.	PUNCT
ejpam-4383	7	1	also	also	ADV
ejpam-4383	7	2	,	,	PUNCT
ejpam-4383	7	3	we	we	PRON
ejpam-4383	7	4	introduce	introduce	VERB
ejpam-4383	7	5	the	the	DET
ejpam-4383	7	6	concept	concept	NOUN
ejpam-4383	7	7	of	of	ADP
ejpam-4383	7	8	the	the	DET
ejpam-4383	7	9	weak	weak	ADJ
ejpam-4383	7	10	direct	direct	ADJ
ejpam-4383	7	11	product	product	NOUN
ejpam-4383	7	12	of	of	ADP
ejpam-4383	7	13	b	b	NOUN
ejpam-4383	7	14	-algebras	-algebras	PROPN
ejpam-4383	7	15	.	.	PUNCT
ejpam-4383	8	1	finally	finally	ADV
ejpam-4383	8	2	,	,	PUNCT
ejpam-4383	8	3	we	we	PRON
ejpam-4383	8	4	provide	provide	VERB
ejpam-4383	8	5	several	several	ADJ
ejpam-4383	8	6	fundamental	fundamental	ADJ
ejpam-4383	8	7	theorems	theorem	NOUN
ejpam-4383	8	8	of	of	ADP
ejpam-4383	8	9	(	(	PUNCT
ejpam-4383	8	10	anti-)b	anti-)b	INTJ
ejpam-4383	8	11	-homomorphisms	-homomorphism	NOUN
ejpam-4383	8	12	in	in	ADP
ejpam-4383	8	13	view	view	NOUN
ejpam-4383	8	14	of	of	ADP
ejpam-4383	8	15	the	the	DET
ejpam-4383	8	16	external	external	ADJ
ejpam-4383	8	17	direct	direct	ADJ
ejpam-4383	8	18	product	product	NOUN
ejpam-4383	8	19	b	b	NOUN
ejpam-4383	8	20	-algebras	-algebra	NOUN
ejpam-4383	8	21	.	.	PUNCT
ejpam-4383	8	22	2020	2020	NUM
ejpam-4383	8	23	mathematics	mathematic	NOUN
ejpam-4383	8	24	subject	subject	NOUN
ejpam-4383	8	25	classifications	classification	NOUN
ejpam-4383	8	26	:	:	PUNCT
ejpam-4383	8	27	03g25	03g25	NUM
ejpam-4383	8	28	,	,	PUNCT
ejpam-4383	8	29	20k25	20k25	NUM
ejpam-4383	8	30	key	key	ADJ
ejpam-4383	8	31	words	word	NOUN
ejpam-4383	8	32	and	and	CCONJ
ejpam-4383	8	33	phrases	phrase	NOUN
ejpam-4383	8	34	:	:	PUNCT
ejpam-4383	8	35	b	b	X
ejpam-4383	8	36	-algebra	-algebra	NOUN
ejpam-4383	8	37	,	,	PUNCT
ejpam-4383	8	38	external	external	ADJ
ejpam-4383	8	39	direct	direct	ADJ
ejpam-4383	8	40	product	product	NOUN
ejpam-4383	8	41	,	,	PUNCT
ejpam-4383	8	42	weak	weak	ADJ
ejpam-4383	8	43	direct	direct	ADJ
ejpam-4383	8	44	product	product	NOUN
ejpam-4383	8	45	,	,	PUNCT
ejpam-4383	8	46	b	b	PROPN
ejpam-4383	8	47	-homomorphism	-homomorphism	PROPN
ejpam-4383	8	48	,	,	PUNCT
ejpam-4383	8	49	anti	anti	ADJ
ejpam-4383	8	50	-	-	ADJ
ejpam-4383	8	51	b	b	ADJ
ejpam-4383	8	52	-homomorphism	-homomorphism	NOUN
ejpam-4383	8	53	1	1	NUM
ejpam-4383	8	54	.	.	PUNCT
ejpam-4383	8	55	introduction	introduction	NOUN
ejpam-4383	8	56	and	and	CCONJ
ejpam-4383	8	57	preliminaries	preliminary	NOUN
ejpam-4383	8	58	imai	imai	PROPN
ejpam-4383	8	59	and	and	CCONJ
ejpam-4383	8	60	iséki	iséki	NUM
ejpam-4383	8	61	introduced	introduce	VERB
ejpam-4383	8	62	two	two	NUM
ejpam-4383	8	63	classes	class	NOUN
ejpam-4383	8	64	of	of	ADP
ejpam-4383	8	65	abstract	abstract	ADJ
ejpam-4383	8	66	algebras	algebra	NOUN
ejpam-4383	8	67	called	call	VERB
ejpam-4383	8	68	bck	bck	PROPN
ejpam-4383	8	69	-algebras	-algebras	PROPN
ejpam-4383	8	70	and	and	CCONJ
ejpam-4383	8	71	bci	bci	PROPN
ejpam-4383	8	72	-algebras	-algebras	PROPN
ejpam-4383	8	73	.	.	PUNCT
ejpam-4383	9	1	it	it	PRON
ejpam-4383	9	2	is	be	AUX
ejpam-4383	9	3	known	know	VERB
ejpam-4383	9	4	that	that	SCONJ
ejpam-4383	9	5	the	the	DET
ejpam-4383	9	6	class	class	NOUN
ejpam-4383	9	7	of	of	ADP
ejpam-4383	9	8	bck	bck	PROPN
ejpam-4383	9	9	-algebras	-algebras	PROPN
ejpam-4383	9	10	is	be	AUX
ejpam-4383	9	11	a	a	DET
ejpam-4383	9	12	proper	proper	ADJ
ejpam-4383	9	13	subclass	subclass	NOUN
ejpam-4383	9	14	of	of	ADP
ejpam-4383	9	15	the	the	DET
ejpam-4383	9	16	class	class	NOUN
ejpam-4383	9	17	of	of	ADP
ejpam-4383	9	18	bci	bci	PROPN
ejpam-4383	9	19	-algebras	-algebras	PROPN
ejpam-4383	9	20	[	[	X
ejpam-4383	9	21	8	8	NUM
ejpam-4383	9	22	,	,	PUNCT
ejpam-4383	9	23	9	9	NUM
ejpam-4383	9	24	]	]	PUNCT
ejpam-4383	9	25	.	.	PUNCT
ejpam-4383	10	1	in	in	ADP
ejpam-4383	10	2	2002	2002	NUM
ejpam-4383	10	3	,	,	PUNCT
ejpam-4383	10	4	neggers	negger	NOUN
ejpam-4383	10	5	and	and	CCONJ
ejpam-4383	10	6	kim	kim	PROPN
ejpam-4383	11	1	[	[	X
ejpam-4383	11	2	17	17	NUM
ejpam-4383	11	3	]	]	PUNCT
ejpam-4383	11	4	constructed	construct	VERB
ejpam-4383	11	5	a	a	DET
ejpam-4383	11	6	new	new	ADJ
ejpam-4383	11	7	algebraic	algebraic	ADJ
ejpam-4383	11	8	structure	structure	NOUN
ejpam-4383	11	9	.	.	PUNCT
ejpam-4383	12	1	they	they	PRON
ejpam-4383	12	2	took	take	VERB
ejpam-4383	12	3	some	some	DET
ejpam-4383	12	4	properties	property	NOUN
ejpam-4383	12	5	from	from	ADP
ejpam-4383	12	6	bci	bci	PROPN
ejpam-4383	12	7	and	and	CCONJ
ejpam-4383	12	8	bck	bck	PROPN
ejpam-4383	12	9	-algebras	-algebras	PROPN
ejpam-4383	12	10	be	be	AUX
ejpam-4383	12	11	called	call	VERB
ejpam-4383	12	12	a	a	DET
ejpam-4383	12	13	b	b	NOUN
ejpam-4383	12	14	-algebra	-algebra	NOUN
ejpam-4383	12	15	.	.	PUNCT
ejpam-4383	13	1	a	a	DET
ejpam-4383	13	2	b	b	NOUN
ejpam-4383	13	3	-algebra	-algebra	NOUN
ejpam-4383	13	4	x	x	X
ejpam-4383	13	5	=	=	SYM
ejpam-4383	13	6	(	(	PUNCT
ejpam-4383	13	7	x	x	NOUN
ejpam-4383	13	8	;	;	PUNCT
ejpam-4383	13	9	∗	∗	NOUN
ejpam-4383	13	10	,	,	PUNCT
ejpam-4383	13	11	0	0	NUM
ejpam-4383	13	12	)	)	PUNCT
ejpam-4383	13	13	is	be	AUX
ejpam-4383	13	14	an	an	DET
ejpam-4383	13	15	algebra	algebra	NOUN
ejpam-4383	13	16	of	of	ADP
ejpam-4383	13	17	type	type	NOUN
ejpam-4383	13	18	(	(	PUNCT
ejpam-4383	13	19	2	2	NUM
ejpam-4383	13	20	,	,	PUNCT
ejpam-4383	13	21	0	0	NUM
ejpam-4383	13	22	)	)	PUNCT
ejpam-4383	13	23	,	,	PUNCT
ejpam-4383	13	24	that	that	ADV
ejpam-4383	13	25	is	is	ADV
ejpam-4383	13	26	,	,	PUNCT
ejpam-4383	13	27	a	a	DET
ejpam-4383	13	28	nonempty	nonempty	ADV
ejpam-4383	13	29	set	set	VERB
ejpam-4383	13	30	x	x	PUNCT
ejpam-4383	13	31	together	together	ADV
ejpam-4383	13	32	with	with	ADP
ejpam-4383	13	33	a	a	DET
ejpam-4383	13	34	binary	binary	ADJ
ejpam-4383	13	35	operation	operation	NOUN
ejpam-4383	13	36	∗	∗	NOUN
ejpam-4383	13	37	and	and	CCONJ
ejpam-4383	13	38	a	a	DET
ejpam-4383	13	39	constant	constant	ADJ
ejpam-4383	13	40	0	0	NUM
ejpam-4383	13	41	satisfying	satisfy	VERB
ejpam-4383	13	42	some	some	DET
ejpam-4383	13	43	axioms	axiom	NOUN
ejpam-4383	13	44	.	.	PUNCT
ejpam-4383	14	1	∗corresponding	∗corresponde	VERB
ejpam-4383	14	2	author	author	NOUN
ejpam-4383	14	3	.	.	PUNCT
ejpam-4383	15	1	doi	doi	NOUN
ejpam-4383	15	2	:	:	PUNCT
ejpam-4383	15	3	https://doi.org/10.29020/nybg.ejpam.v15i3.4383	https://doi.org/10.29020/nybg.ejpam.v15i3.4383	VERB
ejpam-4383	15	4	email	email	NOUN
ejpam-4383	15	5	addresses	address	NOUN
ejpam-4383	15	6	:	:	PUNCT
ejpam-4383	15	7	chatsuda.chanmanee@gmail.com	chatsuda.chanmanee@gmail.com	X
ejpam-4383	15	8	(	(	PUNCT
ejpam-4383	15	9	c.	c.	PROPN
ejpam-4383	15	10	chanmanee	chanmanee	PROPN
ejpam-4383	15	11	)	)	PUNCT
ejpam-4383	15	12	,	,	PUNCT
ejpam-4383	15	13	ronnason.c@psu.ac.th	ronnason.c@psu.ac.th	PROPN
ejpam-4383	15	14	(	(	PUNCT
ejpam-4383	15	15	r.	r.	PROPN
ejpam-4383	15	16	chinram	chinram	PROPN
ejpam-4383	15	17	)	)	PUNCT
ejpam-4383	15	18	,	,	PUNCT
ejpam-4383	15	19	rukchart.p@nsru.ac.th	rukchart.p@nsru.ac.th	PROPN
ejpam-4383	15	20	(	(	PUNCT
ejpam-4383	15	21	r.	r.	PROPN
ejpam-4383	15	22	prasertpong	prasertpong	PROPN
ejpam-4383	15	23	)	)	PUNCT
ejpam-4383	15	24	,	,	PUNCT
ejpam-4383	15	25	pongpun.j@psru.ac.th	pongpun.j@psru.ac.th	PROPN
ejpam-4383	15	26	(	(	PUNCT
ejpam-4383	15	27	p.	p.	NOUN
ejpam-4383	15	28	julatha	julatha	NOUN
ejpam-4383	15	29	)	)	PUNCT
ejpam-4383	15	30	,	,	PUNCT
ejpam-4383	15	31	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-4383	15	32	(	(	PUNCT
ejpam-4383	15	33	a.	a.	NOUN
ejpam-4383	15	34	iampan	iampan	PROPN
ejpam-4383	15	35	)	)	PUNCT
ejpam-4383	15	36	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4383	15	37	999	999	NUM
ejpam-4383	16	1	©	©	ADP
ejpam-4383	16	2	2022	2022	NUM
ejpam-4383	16	3	ejpam	ejpam	VERB
ejpam-4383	16	4	all	all	DET
ejpam-4383	16	5	rights	right	NOUN
ejpam-4383	16	6	reserved	reserve	VERB
ejpam-4383	16	7	.	.	PUNCT
ejpam-4383	17	1	a.	a.	PROPN
ejpam-4383	17	2	iampan	iampan	PROPN
ejpam-4383	17	3	et	et	PROPN
ejpam-4383	17	4	al	al	PROPN
ejpam-4383	17	5	.	.	PUNCT
ejpam-4383	17	6	/	/	SYM
ejpam-4383	17	7	eur	eur	PROPN
ejpam-4383	17	8	.	.	PUNCT
ejpam-4383	18	1	j.	j.	PROPN
ejpam-4383	18	2	pure	pure	PROPN
ejpam-4383	18	3	appl	appl	PROPN
ejpam-4383	18	4	.	.	PROPN
ejpam-4383	18	5	math	math	PROPN
ejpam-4383	18	6	,	,	PUNCT
ejpam-4383	18	7	15	15	NUM
ejpam-4383	18	8	(	(	PUNCT
ejpam-4383	18	9	3	3	NUM
ejpam-4383	18	10	)	)	PUNCT
ejpam-4383	18	11	(	(	PUNCT
ejpam-4383	18	12	2022	2022	NUM
ejpam-4383	18	13	)	)	PUNCT
ejpam-4383	18	14	,	,	PUNCT
ejpam-4383	18	15	999	999	NUM
ejpam-4383	18	16	-	-	SYM
ejpam-4383	18	17	1014	1014	NUM
ejpam-4383	18	18	1000	1000	NUM
ejpam-4383	18	19	b	b	NOUN
ejpam-4383	18	20	-algebras	-algebra	NOUN
ejpam-4383	18	21	and	and	CCONJ
ejpam-4383	18	22	some	some	PRON
ejpam-4383	18	23	of	of	ADP
ejpam-4383	18	24	their	their	PRON
ejpam-4383	18	25	properties	property	NOUN
ejpam-4383	18	26	have	have	AUX
ejpam-4383	18	27	been	be	AUX
ejpam-4383	18	28	discussed	discuss	VERB
ejpam-4383	18	29	,	,	PUNCT
ejpam-4383	18	30	e.g.	e.g.	ADV
ejpam-4383	18	31	,	,	PUNCT
ejpam-4383	18	32	some	some	DET
ejpam-4383	18	33	axiomatizations	axiomatization	NOUN
ejpam-4383	18	34	of	of	ADP
ejpam-4383	18	35	b	b	NOUN
ejpam-4383	18	36	-algebras	-algebra	NOUN
ejpam-4383	18	37	by	by	ADP
ejpam-4383	18	38	walendziak	walendziak	NOUN
ejpam-4383	18	39	in	in	ADP
ejpam-4383	18	40	2006	2006	NUM
ejpam-4383	18	41	[	[	X
ejpam-4383	18	42	22	22	NUM
ejpam-4383	18	43	]	]	PUNCT
ejpam-4383	18	44	,	,	PUNCT
ejpam-4383	18	45	medial	medial	ADJ
ejpam-4383	18	46	b	b	NOUN
ejpam-4383	18	47	-algebras	-algebra	NOUN
ejpam-4383	18	48	by	by	ADP
ejpam-4383	18	49	kim	kim	PROPN
ejpam-4383	18	50	in	in	ADP
ejpam-4383	18	51	2014	2014	NUM
ejpam-4383	19	1	[	[	X
ejpam-4383	19	2	11	11	NUM
ejpam-4383	19	3	]	]	PUNCT
ejpam-4383	19	4	,	,	PUNCT
ejpam-4383	19	5	fuzzy	fuzzy	ADJ
ejpam-4383	19	6	order	order	NOUN
ejpam-4383	19	7	relative	relative	ADJ
ejpam-4383	19	8	to	to	ADP
ejpam-4383	19	9	fuzzy	fuzzy	ADJ
ejpam-4383	19	10	b	b	NOUN
ejpam-4383	19	11	-algebras	-algebra	NOUN
ejpam-4383	19	12	by	by	ADP
ejpam-4383	19	13	gonzaga	gonzaga	NOUN
ejpam-4383	19	14	,	,	PUNCT
ejpam-4383	19	15	jr	jr	PROPN
ejpam-4383	19	16	.	.	PROPN
ejpam-4383	19	17	and	and	CCONJ
ejpam-4383	19	18	vilela	vilela	NOUN
ejpam-4383	19	19	in	in	ADP
ejpam-4383	19	20	2019	2019	NUM
ejpam-4383	20	1	[	[	X
ejpam-4383	20	2	7	7	NUM
ejpam-4383	20	3	]	]	PUNCT
ejpam-4383	20	4	,	,	PUNCT
ejpam-4383	20	5	b	b	X
ejpam-4383	20	6	-ideals	-ideal	NOUN
ejpam-4383	20	7	in	in	ADP
ejpam-4383	20	8	a	a	DET
ejpam-4383	20	9	topological	topological	ADJ
ejpam-4383	20	10	b	b	PROPN
ejpam-4383	20	11	-algebra	-algebra	PROPN
ejpam-4383	20	12	and	and	CCONJ
ejpam-4383	20	13	the	the	DET
ejpam-4383	20	14	uniform	uniform	ADJ
ejpam-4383	20	15	b	b	ADP
ejpam-4383	20	16	-topological	-topological	ADJ
ejpam-4383	20	17	space	space	NOUN
ejpam-4383	20	18	by	by	ADP
ejpam-4383	20	19	belleza	belleza	NOUN
ejpam-4383	20	20	and	and	CCONJ
ejpam-4383	20	21	vilela	vilela	NOUN
ejpam-4383	20	22	in	in	ADP
ejpam-4383	20	23	2020	2020	NUM
ejpam-4383	21	1	[	[	X
ejpam-4383	21	2	3	3	NUM
ejpam-4383	21	3	]	]	PUNCT
ejpam-4383	21	4	.	.	PUNCT
ejpam-4383	22	1	in	in	ADP
ejpam-4383	22	2	2021	2021	NUM
ejpam-4383	22	3	,	,	PUNCT
ejpam-4383	22	4	gan	gin	VERB
ejpam-4383	22	5	et	et	PROPN
ejpam-4383	22	6	al	al	PROPN
ejpam-4383	22	7	.	.	PUNCT
ejpam-4383	23	1	[	[	X
ejpam-4383	23	2	6	6	NUM
ejpam-4383	23	3	]	]	PUNCT
ejpam-4383	23	4	guaranteed	guarantee	VERB
ejpam-4383	23	5	the	the	DET
ejpam-4383	23	6	existences	existence	NOUN
ejpam-4383	23	7	of	of	ADP
ejpam-4383	23	8	both	both	CCONJ
ejpam-4383	23	9	direct	direct	ADJ
ejpam-4383	23	10	limits	limit	NOUN
ejpam-4383	23	11	and	and	CCONJ
ejpam-4383	23	12	inverse	inverse	NOUN
ejpam-4383	23	13	limits	limit	NOUN
ejpam-4383	23	14	in	in	ADP
ejpam-4383	23	15	the	the	DET
ejpam-4383	23	16	categories	category	NOUN
ejpam-4383	23	17	of	of	ADP
ejpam-4383	23	18	quantum	quantum	PROPN
ejpam-4383	23	19	b	b	PROPN
ejpam-4383	23	20	-algebras	-algebras	NOUN
ejpam-4383	23	21	with	with	ADP
ejpam-4383	23	22	morphisms	morphism	NOUN
ejpam-4383	23	23	of	of	ADP
ejpam-4383	23	24	exact	exact	ADJ
ejpam-4383	23	25	ones	one	NOUN
ejpam-4383	23	26	or	or	CCONJ
ejpam-4383	23	27	spectral	spectral	ADJ
ejpam-4383	23	28	ones	one	NOUN
ejpam-4383	23	29	,	,	PUNCT
ejpam-4383	23	30	etc	etc	X
ejpam-4383	23	31	.	.	X
ejpam-4383	24	1	the	the	DET
ejpam-4383	24	2	concept	concept	NOUN
ejpam-4383	24	3	of	of	ADP
ejpam-4383	24	4	the	the	DET
ejpam-4383	24	5	direct	direct	ADJ
ejpam-4383	24	6	product	product	NOUN
ejpam-4383	24	7	[	[	X
ejpam-4383	24	8	19	19	NUM
ejpam-4383	24	9	]	]	PUNCT
ejpam-4383	24	10	was	be	AUX
ejpam-4383	24	11	first	first	ADV
ejpam-4383	24	12	defined	define	VERB
ejpam-4383	24	13	in	in	ADP
ejpam-4383	24	14	the	the	DET
ejpam-4383	24	15	group	group	NOUN
ejpam-4383	24	16	and	and	CCONJ
ejpam-4383	24	17	obtained	obtain	VERB
ejpam-4383	24	18	some	some	DET
ejpam-4383	24	19	properties	property	NOUN
ejpam-4383	24	20	.	.	PUNCT
ejpam-4383	25	1	for	for	ADP
ejpam-4383	25	2	example	example	NOUN
ejpam-4383	25	3	,	,	PUNCT
ejpam-4383	25	4	a	a	DET
ejpam-4383	25	5	direct	direct	ADJ
ejpam-4383	25	6	product	product	NOUN
ejpam-4383	25	7	of	of	ADP
ejpam-4383	25	8	the	the	DET
ejpam-4383	25	9	group	group	NOUN
ejpam-4383	25	10	is	be	AUX
ejpam-4383	25	11	also	also	ADV
ejpam-4383	25	12	a	a	DET
ejpam-4383	25	13	group	group	NOUN
ejpam-4383	25	14	,	,	PUNCT
ejpam-4383	25	15	and	and	CCONJ
ejpam-4383	25	16	a	a	DET
ejpam-4383	25	17	direct	direct	ADJ
ejpam-4383	25	18	product	product	NOUN
ejpam-4383	25	19	of	of	ADP
ejpam-4383	25	20	the	the	DET
ejpam-4383	25	21	abelian	abelian	ADJ
ejpam-4383	25	22	group	group	NOUN
ejpam-4383	25	23	is	be	AUX
ejpam-4383	25	24	also	also	ADV
ejpam-4383	25	25	an	an	DET
ejpam-4383	25	26	abelian	abelian	ADJ
ejpam-4383	25	27	group	group	NOUN
ejpam-4383	25	28	.	.	PUNCT
ejpam-4383	26	1	then	then	ADV
ejpam-4383	26	2	,	,	PUNCT
ejpam-4383	26	3	direct	direct	ADJ
ejpam-4383	26	4	product	product	NOUN
ejpam-4383	26	5	groups	group	NOUN
ejpam-4383	26	6	are	be	AUX
ejpam-4383	26	7	applied	apply	VERB
ejpam-4383	26	8	to	to	ADP
ejpam-4383	26	9	other	other	ADJ
ejpam-4383	26	10	algebraic	algebraic	ADJ
ejpam-4383	26	11	structures	structure	NOUN
ejpam-4383	26	12	.	.	PUNCT
ejpam-4383	27	1	in	in	ADP
ejpam-4383	27	2	2016	2016	NUM
ejpam-4383	27	3	,	,	PUNCT
ejpam-4383	27	4	lingcong	lingcong	NOUN
ejpam-4383	27	5	and	and	CCONJ
ejpam-4383	27	6	endam	endam	ADJ
ejpam-4383	27	7	[	[	X
ejpam-4383	27	8	12	12	NUM
ejpam-4383	27	9	]	]	PUNCT
ejpam-4383	27	10	discussed	discuss	VERB
ejpam-4383	27	11	the	the	DET
ejpam-4383	27	12	notion	notion	NOUN
ejpam-4383	27	13	of	of	ADP
ejpam-4383	27	14	the	the	DET
ejpam-4383	27	15	direct	direct	ADJ
ejpam-4383	27	16	product	product	NOUN
ejpam-4383	27	17	of	of	ADP
ejpam-4383	27	18	b	b	NOUN
ejpam-4383	27	19	-algebras	-algebras	PROPN
ejpam-4383	27	20	,	,	PUNCT
ejpam-4383	27	21	0	0	NUM
ejpam-4383	27	22	-	-	PUNCT
ejpam-4383	27	23	commutative	commutative	ADJ
ejpam-4383	27	24	b	b	PROPN
ejpam-4383	27	25	-algebras	-algebra	NOUN
ejpam-4383	27	26	,	,	PUNCT
ejpam-4383	27	27	and	and	CCONJ
ejpam-4383	27	28	b	b	X
ejpam-4383	27	29	-homomorphisms	-homomorphism	NOUN
ejpam-4383	27	30	and	and	CCONJ
ejpam-4383	27	31	obtained	obtain	VERB
ejpam-4383	27	32	related	related	ADJ
ejpam-4383	27	33	properties	property	NOUN
ejpam-4383	27	34	,	,	PUNCT
ejpam-4383	27	35	one	one	NUM
ejpam-4383	27	36	of	of	ADP
ejpam-4383	27	37	which	which	PRON
ejpam-4383	27	38	is	be	AUX
ejpam-4383	27	39	a	a	DET
ejpam-4383	27	40	direct	direct	ADJ
ejpam-4383	27	41	product	product	NOUN
ejpam-4383	27	42	of	of	ADP
ejpam-4383	27	43	two	two	NUM
ejpam-4383	27	44	b	b	NOUN
ejpam-4383	27	45	-algebras	-algebra	NOUN
ejpam-4383	27	46	,	,	PUNCT
ejpam-4383	27	47	which	which	PRON
ejpam-4383	27	48	is	be	AUX
ejpam-4383	27	49	also	also	ADV
ejpam-4383	27	50	a	a	DET
ejpam-4383	27	51	b	b	NOUN
ejpam-4383	27	52	-algebra	-algebra	NOUN
ejpam-4383	27	53	.	.	PUNCT
ejpam-4383	28	1	then	then	ADV
ejpam-4383	28	2	,	,	PUNCT
ejpam-4383	28	3	they	they	PRON
ejpam-4383	28	4	extended	extend	VERB
ejpam-4383	28	5	the	the	DET
ejpam-4383	28	6	concept	concept	NOUN
ejpam-4383	28	7	of	of	ADP
ejpam-4383	28	8	the	the	DET
ejpam-4383	28	9	direct	direct	ADJ
ejpam-4383	28	10	product	product	NOUN
ejpam-4383	28	11	of	of	ADP
ejpam-4383	28	12	b	b	PROPN
ejpam-4383	28	13	-algebra	-algebra	NOUN
ejpam-4383	28	14	to	to	PART
ejpam-4383	28	15	finite	finite	VERB
ejpam-4383	28	16	family	family	PROPN
ejpam-4383	28	17	b	b	PROPN
ejpam-4383	28	18	-algebra	-algebra	PROPN
ejpam-4383	28	19	,	,	PUNCT
ejpam-4383	28	20	and	and	CCONJ
ejpam-4383	28	21	some	some	PRON
ejpam-4383	28	22	of	of	ADP
ejpam-4383	28	23	the	the	DET
ejpam-4383	28	24	related	relate	VERB
ejpam-4383	28	25	properties	property	NOUN
ejpam-4383	28	26	were	be	AUX
ejpam-4383	28	27	investigated	investigate	VERB
ejpam-4383	28	28	.	.	PUNCT
ejpam-4383	29	1	also	also	ADV
ejpam-4383	29	2	,	,	PUNCT
ejpam-4383	29	3	they	they	PRON
ejpam-4383	29	4	introduced	introduce	VERB
ejpam-4383	29	5	two	two	NUM
ejpam-4383	29	6	canonical	canonical	ADJ
ejpam-4383	29	7	mappings	mapping	NOUN
ejpam-4383	29	8	of	of	ADP
ejpam-4383	29	9	the	the	DET
ejpam-4383	29	10	direct	direct	ADJ
ejpam-4383	29	11	product	product	NOUN
ejpam-4383	29	12	of	of	ADP
ejpam-4383	29	13	b	b	NOUN
ejpam-4383	29	14	-algebras	-algebra	NOUN
ejpam-4383	29	15	and	and	CCONJ
ejpam-4383	29	16	we	we	PRON
ejpam-4383	29	17	obtained	obtain	VERB
ejpam-4383	29	18	some	some	PRON
ejpam-4383	29	19	of	of	ADP
ejpam-4383	29	20	their	their	PRON
ejpam-4383	29	21	properties	property	NOUN
ejpam-4383	29	22	[	[	X
ejpam-4383	29	23	13	13	NUM
ejpam-4383	29	24	]	]	PUNCT
ejpam-4383	29	25	.	.	PUNCT
ejpam-4383	30	1	in	in	ADP
ejpam-4383	30	2	the	the	DET
ejpam-4383	30	3	same	same	ADJ
ejpam-4383	30	4	year	year	NOUN
ejpam-4383	30	5	,	,	PUNCT
ejpam-4383	30	6	endam	endam	NOUN
ejpam-4383	30	7	and	and	CCONJ
ejpam-4383	30	8	teves	teve	NOUN
ejpam-4383	30	9	[	[	X
ejpam-4383	30	10	5	5	NUM
ejpam-4383	30	11	]	]	PUNCT
ejpam-4383	30	12	defined	define	VERB
ejpam-4383	30	13	the	the	DET
ejpam-4383	30	14	direct	direct	ADJ
ejpam-4383	30	15	product	product	NOUN
ejpam-4383	30	16	of	of	ADP
ejpam-4383	30	17	bf	bf	NOUN
ejpam-4383	30	18	-algebras	-algebra	NOUN
ejpam-4383	30	19	,	,	PUNCT
ejpam-4383	30	20	0	0	NUM
ejpam-4383	30	21	-	-	PUNCT
ejpam-4383	30	22	commutative	commutative	ADJ
ejpam-4383	30	23	bf	bf	NOUN
ejpam-4383	30	24	-algebras	-algebra	NOUN
ejpam-4383	30	25	,	,	PUNCT
ejpam-4383	30	26	and	and	CCONJ
ejpam-4383	30	27	bf	bf	NOUN
ejpam-4383	30	28	-homomorphism	-homomorphism	PROPN
ejpam-4383	30	29	and	and	CCONJ
ejpam-4383	30	30	obtained	obtain	VERB
ejpam-4383	30	31	related	related	ADJ
ejpam-4383	30	32	properties	property	NOUN
ejpam-4383	30	33	.	.	PUNCT
ejpam-4383	31	1	in	in	ADP
ejpam-4383	31	2	2018	2018	NUM
ejpam-4383	31	3	,	,	PUNCT
ejpam-4383	31	4	abebe	abebe	NOUN
ejpam-4383	31	5	[	[	X
ejpam-4383	31	6	1	1	NUM
ejpam-4383	31	7	]	]	PUNCT
ejpam-4383	31	8	introduced	introduce	VERB
ejpam-4383	31	9	the	the	DET
ejpam-4383	31	10	concept	concept	NOUN
ejpam-4383	31	11	of	of	ADP
ejpam-4383	31	12	the	the	DET
ejpam-4383	31	13	finite	finite	ADJ
ejpam-4383	31	14	direct	direct	ADJ
ejpam-4383	31	15	product	product	NOUN
ejpam-4383	31	16	of	of	ADP
ejpam-4383	31	17	brk	brk	PROPN
ejpam-4383	31	18	algebras	algebras	PROPN
ejpam-4383	31	19	and	and	CCONJ
ejpam-4383	31	20	proved	prove	VERB
ejpam-4383	31	21	that	that	SCONJ
ejpam-4383	31	22	the	the	DET
ejpam-4383	31	23	finite	finite	ADJ
ejpam-4383	31	24	direct	direct	ADJ
ejpam-4383	31	25	product	product	NOUN
ejpam-4383	31	26	of	of	ADP
ejpam-4383	31	27	brk	brk	PROPN
ejpam-4383	31	28	-algebras	-algebras	PROPN
ejpam-4383	31	29	is	be	AUX
ejpam-4383	31	30	a	a	DET
ejpam-4383	31	31	brk	brk	PROPN
ejpam-4383	31	32	-algebra	-algebra	PROPN
ejpam-4383	31	33	.	.	PUNCT
ejpam-4383	32	1	in	in	ADP
ejpam-4383	32	2	2019	2019	NUM
ejpam-4383	32	3	,	,	PUNCT
ejpam-4383	32	4	widianto	widianto	PROPN
ejpam-4383	32	5	et	et	PROPN
ejpam-4383	32	6	al	al	PROPN
ejpam-4383	32	7	.	.	PUNCT
ejpam-4383	33	1	[	[	X
ejpam-4383	33	2	23	23	NUM
ejpam-4383	33	3	]	]	PUNCT
ejpam-4383	33	4	defined	define	VERB
ejpam-4383	33	5	the	the	DET
ejpam-4383	33	6	direct	direct	ADJ
ejpam-4383	33	7	product	product	NOUN
ejpam-4383	33	8	of	of	ADP
ejpam-4383	33	9	bg	bg	PROPN
ejpam-4383	33	10	-	-	PUNCT
ejpam-4383	33	11	algebras	algebras	PROPN
ejpam-4383	33	12	,	,	PUNCT
ejpam-4383	33	13	0	0	NUM
ejpam-4383	33	14	-	-	PUNCT
ejpam-4383	33	15	commutative	commutative	ADJ
ejpam-4383	33	16	bg	bg	NOUN
ejpam-4383	33	17	-	-	PUNCT
ejpam-4383	33	18	algebras	algebras	PROPN
ejpam-4383	33	19	,	,	PUNCT
ejpam-4383	33	20	and	and	CCONJ
ejpam-4383	33	21	bg	bg	PROPN
ejpam-4383	33	22	-	-	PUNCT
ejpam-4383	33	23	homomorphism	homomorphism	PROPN
ejpam-4383	33	24	,	,	PUNCT
ejpam-4383	33	25	including	include	VERB
ejpam-4383	33	26	related	related	ADJ
ejpam-4383	33	27	properties	property	NOUN
ejpam-4383	33	28	of	of	ADP
ejpam-4383	33	29	bg	bg	PROPN
ejpam-4383	33	30	-	-	PUNCT
ejpam-4383	33	31	algebras	algebras	PROPN
ejpam-4383	33	32	.	.	PUNCT
ejpam-4383	34	1	in	in	ADP
ejpam-4383	34	2	2020	2020	NUM
ejpam-4383	34	3	,	,	PUNCT
ejpam-4383	34	4	setiani	setiani	PROPN
ejpam-4383	34	5	et	et	PROPN
ejpam-4383	34	6	al	al	PROPN
ejpam-4383	34	7	.	.	PUNCT
ejpam-4383	35	1	[	[	X
ejpam-4383	35	2	19	19	NUM
ejpam-4383	35	3	]	]	PUNCT
ejpam-4383	35	4	defined	define	VERB
ejpam-4383	35	5	the	the	DET
ejpam-4383	35	6	direct	direct	ADJ
ejpam-4383	35	7	product	product	NOUN
ejpam-4383	35	8	of	of	ADP
ejpam-4383	35	9	bp	bp	PROPN
ejpam-4383	35	10	-algebras	-algebras	PROPN
ejpam-4383	35	11	,	,	PUNCT
ejpam-4383	35	12	which	which	PRON
ejpam-4383	35	13	is	be	AUX
ejpam-4383	35	14	equivalent	equivalent	ADJ
ejpam-4383	35	15	to	to	ADP
ejpam-4383	35	16	b	b	NOUN
ejpam-4383	35	17	-algebras	-algebras	PROPN
ejpam-4383	35	18	.	.	PUNCT
ejpam-4383	36	1	they	they	PRON
ejpam-4383	36	2	obtained	obtain	VERB
ejpam-4383	36	3	the	the	DET
ejpam-4383	36	4	relevant	relevant	ADJ
ejpam-4383	36	5	property	property	NOUN
ejpam-4383	36	6	of	of	ADP
ejpam-4383	36	7	the	the	DET
ejpam-4383	36	8	direct	direct	ADJ
ejpam-4383	36	9	product	product	NOUN
ejpam-4383	36	10	of	of	ADP
ejpam-4383	36	11	bp	bp	PROPN
ejpam-4383	36	12	-algebras	-algebras	PROPN
ejpam-4383	36	13	and	and	CCONJ
ejpam-4383	36	14	then	then	ADV
ejpam-4383	36	15	defined	define	VERB
ejpam-4383	36	16	the	the	DET
ejpam-4383	36	17	direct	direct	ADJ
ejpam-4383	36	18	product	product	NOUN
ejpam-4383	36	19	of	of	ADP
ejpam-4383	36	20	bp	bp	PROPN
ejpam-4383	36	21	-algebras	-algebras	PROPN
ejpam-4383	36	22	as	as	ADP
ejpam-4383	36	23	applied	apply	VERB
ejpam-4383	36	24	to	to	ADP
ejpam-4383	36	25	finite	finite	ADJ
ejpam-4383	36	26	sets	set	NOUN
ejpam-4383	36	27	of	of	ADP
ejpam-4383	36	28	bp	bp	PROPN
ejpam-4383	36	29	-algebras	-algebras	PROPN
ejpam-4383	36	30	,	,	PUNCT
ejpam-4383	36	31	finite	finite	ADJ
ejpam-4383	36	32	family	family	NOUN
ejpam-4383	36	33	0	0	NUM
ejpam-4383	36	34	-	-	PUNCT
ejpam-4383	36	35	commutative	commutative	ADJ
ejpam-4383	36	36	bp	bp	PROPN
ejpam-4383	36	37	-algebras	-algebras	PROPN
ejpam-4383	36	38	,	,	PUNCT
ejpam-4383	36	39	and	and	CCONJ
ejpam-4383	36	40	finite	finite	PROPN
ejpam-4383	36	41	family	family	NOUN
ejpam-4383	36	42	bp	bp	PROPN
ejpam-4383	36	43	-homomorphisms	-homomorphisms	PROPN
ejpam-4383	36	44	.	.	PUNCT
ejpam-4383	37	1	in	in	ADP
ejpam-4383	37	2	2021	2021	NUM
ejpam-4383	37	3	,	,	PUNCT
ejpam-4383	37	4	kavitha	kavitha	PROPN
ejpam-4383	37	5	and	and	CCONJ
ejpam-4383	37	6	gowri	gowri	PROPN
ejpam-4383	38	1	[	[	X
ejpam-4383	38	2	10	10	NUM
ejpam-4383	38	3	]	]	PUNCT
ejpam-4383	38	4	defined	define	VERB
ejpam-4383	38	5	the	the	DET
ejpam-4383	38	6	direct	direct	ADJ
ejpam-4383	38	7	product	product	NOUN
ejpam-4383	38	8	of	of	ADP
ejpam-4383	38	9	gk	gk	PROPN
ejpam-4383	38	10	algebra	algebra	PROPN
ejpam-4383	38	11	.	.	PUNCT
ejpam-4383	39	1	they	they	PRON
ejpam-4383	39	2	derived	derive	VERB
ejpam-4383	39	3	the	the	DET
ejpam-4383	39	4	finite	finite	ADJ
ejpam-4383	39	5	form	form	NOUN
ejpam-4383	39	6	of	of	ADP
ejpam-4383	39	7	the	the	DET
ejpam-4383	39	8	direct	direct	ADJ
ejpam-4383	39	9	product	product	NOUN
ejpam-4383	39	10	of	of	ADP
ejpam-4383	39	11	gk	gk	PROPN
ejpam-4383	39	12	algebra	algebra	NOUN
ejpam-4383	39	13	and	and	CCONJ
ejpam-4383	39	14	function	function	NOUN
ejpam-4383	39	15	as	as	ADV
ejpam-4383	39	16	well	well	ADV
ejpam-4383	39	17	.	.	PUNCT
ejpam-4383	40	1	they	they	PRON
ejpam-4383	40	2	investigated	investigate	VERB
ejpam-4383	40	3	and	and	CCONJ
ejpam-4383	40	4	applied	apply	VERB
ejpam-4383	40	5	the	the	DET
ejpam-4383	40	6	concept	concept	NOUN
ejpam-4383	40	7	of	of	ADP
ejpam-4383	40	8	the	the	DET
ejpam-4383	40	9	direct	direct	ADJ
ejpam-4383	40	10	product	product	NOUN
ejpam-4383	40	11	of	of	ADP
ejpam-4383	40	12	gk	gk	PROPN
ejpam-4383	40	13	algebra	algebra	PROPN
ejpam-4383	40	14	in	in	ADP
ejpam-4383	40	15	gk	gk	PROPN
ejpam-4383	40	16	function	function	NOUN
ejpam-4383	40	17	and	and	CCONJ
ejpam-4383	40	18	gk	gk	PROPN
ejpam-4383	40	19	kernel	kernel	PROPN
ejpam-4383	40	20	and	and	CCONJ
ejpam-4383	40	21	obtained	obtain	VERB
ejpam-4383	40	22	interesting	interesting	ADJ
ejpam-4383	40	23	results	result	NOUN
ejpam-4383	40	24	.	.	PUNCT
ejpam-4383	41	1	in	in	ADP
ejpam-4383	41	2	this	this	DET
ejpam-4383	41	3	paper	paper	NOUN
ejpam-4383	41	4	,	,	PUNCT
ejpam-4383	41	5	we	we	PRON
ejpam-4383	41	6	introduce	introduce	VERB
ejpam-4383	41	7	the	the	DET
ejpam-4383	41	8	concept	concept	NOUN
ejpam-4383	41	9	of	of	ADP
ejpam-4383	41	10	the	the	DET
ejpam-4383	41	11	direct	direct	ADJ
ejpam-4383	41	12	product	product	NOUN
ejpam-4383	41	13	of	of	ADP
ejpam-4383	41	14	infinite	infinite	ADJ
ejpam-4383	41	15	family	family	NOUN
ejpam-4383	41	16	of	of	ADP
ejpam-4383	41	17	b	b	PROPN
ejpam-4383	41	18	-algebras	-algebras	PROPN
ejpam-4383	41	19	,	,	PUNCT
ejpam-4383	41	20	we	we	PRON
ejpam-4383	41	21	call	call	VERB
ejpam-4383	41	22	the	the	DET
ejpam-4383	41	23	external	external	ADJ
ejpam-4383	41	24	direct	direct	ADJ
ejpam-4383	41	25	product	product	NOUN
ejpam-4383	41	26	,	,	PUNCT
ejpam-4383	41	27	which	which	PRON
ejpam-4383	41	28	is	be	AUX
ejpam-4383	41	29	a	a	DET
ejpam-4383	41	30	generalization	generalization	NOUN
ejpam-4383	41	31	of	of	ADP
ejpam-4383	41	32	the	the	DET
ejpam-4383	41	33	direct	direct	ADJ
ejpam-4383	41	34	product	product	NOUN
ejpam-4383	41	35	in	in	ADP
ejpam-4383	41	36	the	the	DET
ejpam-4383	41	37	sense	sense	NOUN
ejpam-4383	41	38	of	of	ADP
ejpam-4383	41	39	lingcong	lingcong	NOUN
ejpam-4383	41	40	and	and	CCONJ
ejpam-4383	41	41	endam	endam	NOUN
ejpam-4383	42	1	[	[	X
ejpam-4383	42	2	12	12	NUM
ejpam-4383	42	3	]	]	PUNCT
ejpam-4383	42	4	.	.	PUNCT
ejpam-4383	43	1	moreover	moreover	ADV
ejpam-4383	43	2	,	,	PUNCT
ejpam-4383	43	3	we	we	PRON
ejpam-4383	43	4	introduce	introduce	VERB
ejpam-4383	43	5	the	the	DET
ejpam-4383	43	6	concept	concept	NOUN
ejpam-4383	43	7	of	of	ADP
ejpam-4383	43	8	the	the	DET
ejpam-4383	43	9	weak	weak	ADJ
ejpam-4383	43	10	direct	direct	ADJ
ejpam-4383	43	11	product	product	NOUN
ejpam-4383	43	12	of	of	ADP
ejpam-4383	43	13	b	b	NOUN
ejpam-4383	43	14	-algebras	-algebras	PROPN
ejpam-4383	43	15	.	.	PUNCT
ejpam-4383	44	1	finally	finally	ADV
ejpam-4383	44	2	,	,	PUNCT
ejpam-4383	44	3	we	we	PRON
ejpam-4383	44	4	discuss	discuss	VERB
ejpam-4383	44	5	several	several	ADJ
ejpam-4383	44	6	(	(	PUNCT
ejpam-4383	44	7	anti-)b	anti-)b	NUM
ejpam-4383	44	8	-homomorphism	-homomorphism	NOUN
ejpam-4383	44	9	theorems	theorem	VERB
ejpam-4383	44	10	in	in	ADP
ejpam-4383	44	11	view	view	NOUN
ejpam-4383	44	12	of	of	ADP
ejpam-4383	44	13	the	the	DET
ejpam-4383	44	14	external	external	ADJ
ejpam-4383	44	15	direct	direct	ADJ
ejpam-4383	44	16	product	product	NOUN
ejpam-4383	44	17	b	b	NOUN
ejpam-4383	44	18	-algebras	-algebra	NOUN
ejpam-4383	44	19	.	.	PUNCT
ejpam-4383	45	1	first	first	ADV
ejpam-4383	45	2	of	of	ADP
ejpam-4383	45	3	all	all	PRON
ejpam-4383	45	4	,	,	PUNCT
ejpam-4383	45	5	we	we	PRON
ejpam-4383	45	6	start	start	VERB
ejpam-4383	45	7	with	with	ADP
ejpam-4383	45	8	the	the	DET
ejpam-4383	45	9	definitions	definition	NOUN
ejpam-4383	45	10	and	and	CCONJ
ejpam-4383	45	11	examples	example	NOUN
ejpam-4383	45	12	of	of	ADP
ejpam-4383	45	13	b	b	NOUN
ejpam-4383	45	14	-algebras	-algebra	NOUN
ejpam-4383	45	15	as	as	ADV
ejpam-4383	45	16	well	well	ADV
ejpam-4383	45	17	as	as	ADP
ejpam-4383	45	18	other	other	ADJ
ejpam-4383	45	19	relevant	relevant	ADJ
ejpam-4383	45	20	definitions	definition	NOUN
ejpam-4383	45	21	for	for	ADP
ejpam-4383	45	22	the	the	DET
ejpam-4383	45	23	study	study	NOUN
ejpam-4383	45	24	in	in	ADP
ejpam-4383	45	25	this	this	DET
ejpam-4383	45	26	paper	paper	NOUN
ejpam-4383	45	27	as	as	SCONJ
ejpam-4383	45	28	follows	follow	VERB
ejpam-4383	45	29	:	:	PUNCT
ejpam-4383	45	30	definition	definition	NOUN
ejpam-4383	45	31	1	1	NUM
ejpam-4383	45	32	.	.	PUNCT
ejpam-4383	46	1	[	[	X
ejpam-4383	46	2	17	17	NUM
ejpam-4383	46	3	]	]	PUNCT
ejpam-4383	46	4	a	a	DET
ejpam-4383	46	5	b	b	X
ejpam-4383	46	6	-	-	PUNCT
ejpam-4383	46	7	algebra	algebra	NOUN
ejpam-4383	46	8	p	p	NOUN
ejpam-4383	46	9	=	=	X
ejpam-4383	46	10	(	(	PUNCT
ejpam-4383	46	11	p	p	NOUN
ejpam-4383	46	12	;	;	PUNCT
ejpam-4383	46	13	∗	∗	NOUN
ejpam-4383	46	14	,	,	PUNCT
ejpam-4383	46	15	0	0	NUM
ejpam-4383	46	16	)	)	PUNCT
ejpam-4383	46	17	is	be	AUX
ejpam-4383	46	18	an	an	DET
ejpam-4383	46	19	algebra	algebra	NOUN
ejpam-4383	46	20	of	of	ADP
ejpam-4383	46	21	type	type	NOUN
ejpam-4383	46	22	(	(	PUNCT
ejpam-4383	46	23	2	2	NUM
ejpam-4383	46	24	,	,	PUNCT
ejpam-4383	46	25	0	0	NUM
ejpam-4383	46	26	)	)	PUNCT
ejpam-4383	46	27	,	,	PUNCT
ejpam-4383	46	28	that	that	ADV
ejpam-4383	46	29	is	is	ADV
ejpam-4383	46	30	,	,	PUNCT
ejpam-4383	46	31	a	a	DET
ejpam-4383	46	32	nonempty	nonempty	NOUN
ejpam-4383	46	33	set	set	VERB
ejpam-4383	46	34	p	p	NOUN
ejpam-4383	46	35	together	together	ADV
ejpam-4383	46	36	with	with	ADP
ejpam-4383	46	37	a	a	DET
ejpam-4383	46	38	binary	binary	ADJ
ejpam-4383	46	39	operation	operation	NOUN
ejpam-4383	46	40	∗	∗	NOUN
ejpam-4383	46	41	and	and	CCONJ
ejpam-4383	46	42	a	a	DET
ejpam-4383	46	43	constant	constant	ADJ
ejpam-4383	46	44	0	0	NUM
ejpam-4383	46	45	satisfying	satisfy	VERB
ejpam-4383	46	46	the	the	DET
ejpam-4383	46	47	following	follow	VERB
ejpam-4383	46	48	axioms	axiom	NOUN
ejpam-4383	46	49	:	:	PUNCT
ejpam-4383	46	50	(	(	PUNCT
ejpam-4383	46	51	∀x	∀x	X
ejpam-4383	46	52	∈	∈	PROPN
ejpam-4383	46	53	p	p	NOUN
ejpam-4383	46	54	)	)	PUNCT
ejpam-4383	46	55	(	(	PUNCT
ejpam-4383	46	56	x	x	X
ejpam-4383	46	57	∗	∗	NOUN
ejpam-4383	46	58	x	x	SYM
ejpam-4383	46	59	=	=	NOUN
ejpam-4383	46	60	0	0	NUM
ejpam-4383	46	61	)	)	PUNCT
ejpam-4383	46	62	,	,	PUNCT
ejpam-4383	46	63	(	(	PUNCT
ejpam-4383	46	64	b-1	b-1	PROPN
ejpam-4383	46	65	)	)	PUNCT
ejpam-4383	46	66	(	(	PUNCT
ejpam-4383	46	67	∀x	∀x	X
ejpam-4383	46	68	∈	∈	PROPN
ejpam-4383	46	69	p	p	NOUN
ejpam-4383	46	70	)	)	PUNCT
ejpam-4383	46	71	(	(	PUNCT
ejpam-4383	46	72	x	x	X
ejpam-4383	46	73	∗	∗	NOUN
ejpam-4383	46	74	0	0	NUM
ejpam-4383	47	1	=	=	SYM
ejpam-4383	47	2	x	x	NOUN
ejpam-4383	47	3	)	)	PUNCT
ejpam-4383	47	4	,	,	PUNCT
ejpam-4383	47	5	(	(	PUNCT
ejpam-4383	47	6	b-2	b-2	NOUN
ejpam-4383	47	7	)	)	PUNCT
ejpam-4383	47	8	a.	a.	NOUN
ejpam-4383	47	9	iampan	iampan	PROPN
ejpam-4383	47	10	et	et	PROPN
ejpam-4383	47	11	al	al	PROPN
ejpam-4383	47	12	.	.	PUNCT
ejpam-4383	47	13	/	/	SYM
ejpam-4383	47	14	eur	eur	PROPN
ejpam-4383	47	15	.	.	PUNCT
ejpam-4383	48	1	j.	j.	PROPN
ejpam-4383	48	2	pure	pure	PROPN
ejpam-4383	48	3	appl	appl	PROPN
ejpam-4383	48	4	.	.	PROPN
ejpam-4383	48	5	math	math	PROPN
ejpam-4383	48	6	,	,	PUNCT
ejpam-4383	48	7	15	15	NUM
ejpam-4383	48	8	(	(	PUNCT
ejpam-4383	48	9	3	3	NUM
ejpam-4383	48	10	)	)	PUNCT
ejpam-4383	48	11	(	(	PUNCT
ejpam-4383	48	12	2022	2022	NUM
ejpam-4383	48	13	)	)	PUNCT
ejpam-4383	48	14	,	,	PUNCT
ejpam-4383	48	15	999	999	NUM
ejpam-4383	48	16	-	-	SYM
ejpam-4383	48	17	1014	1014	NUM
ejpam-4383	48	18	1001	1001	NUM
ejpam-4383	48	19	(	(	PUNCT
ejpam-4383	48	20	∀x	∀x	NUM
ejpam-4383	48	21	,	,	PUNCT
ejpam-4383	48	22	y	y	PROPN
ejpam-4383	48	23	,	,	PUNCT
ejpam-4383	48	24	z	z	PROPN
ejpam-4383	48	25	∈	∈	PROPN
ejpam-4383	48	26	p	p	NOUN
ejpam-4383	48	27	)	)	PUNCT
ejpam-4383	48	28	(	(	PUNCT
ejpam-4383	48	29	(	(	PUNCT
ejpam-4383	48	30	x	x	SYM
ejpam-4383	48	31	∗	∗	PROPN
ejpam-4383	48	32	y	y	NOUN
ejpam-4383	48	33	)	)	PUNCT
ejpam-4383	48	34	∗	∗	NOUN
ejpam-4383	48	35	z	z	NOUN
ejpam-4383	49	1	=	=	SYM
ejpam-4383	49	2	x	x	X
ejpam-4383	49	3	∗	∗	NOUN
ejpam-4383	49	4	(	(	PUNCT
ejpam-4383	49	5	z	z	NOUN
ejpam-4383	49	6	∗	∗	NOUN
ejpam-4383	49	7	(	(	PUNCT
ejpam-4383	49	8	0	0	NUM
ejpam-4383	49	9	∗	∗	PROPN
ejpam-4383	49	10	y	y	PROPN
ejpam-4383	49	11	)	)	PUNCT
ejpam-4383	49	12	)	)	PUNCT
ejpam-4383	49	13	)	)	PUNCT
ejpam-4383	49	14	.	.	PUNCT
ejpam-4383	50	1	(	(	PUNCT
ejpam-4383	50	2	b-3	b-3	PROPN
ejpam-4383	50	3	)	)	PUNCT
ejpam-4383	50	4	example	example	NOUN
ejpam-4383	51	1	1	1	X
ejpam-4383	51	2	.	.	PUNCT
ejpam-4383	51	3	let	let	VERB
ejpam-4383	51	4	p	p	NOUN
ejpam-4383	51	5	=	=	X
ejpam-4383	51	6	{	{	PUNCT
ejpam-4383	51	7	0	0	NUM
ejpam-4383	51	8	,	,	PUNCT
ejpam-4383	51	9	1	1	NUM
ejpam-4383	51	10	,	,	PUNCT
ejpam-4383	51	11	2	2	NUM
ejpam-4383	51	12	,	,	PUNCT
ejpam-4383	51	13	3	3	NUM
ejpam-4383	51	14	,	,	PUNCT
ejpam-4383	51	15	4	4	NUM
ejpam-4383	51	16	,	,	PUNCT
ejpam-4383	51	17	5	5	NUM
ejpam-4383	51	18	}	}	PUNCT
ejpam-4383	51	19	be	be	AUX
ejpam-4383	51	20	a	a	DET
ejpam-4383	51	21	set	set	NOUN
ejpam-4383	51	22	with	with	ADP
ejpam-4383	51	23	the	the	DET
ejpam-4383	51	24	cayley	cayley	ADJ
ejpam-4383	51	25	table	table	NOUN
ejpam-4383	51	26	as	as	SCONJ
ejpam-4383	51	27	follows	follow	VERB
ejpam-4383	51	28	:	:	PUNCT
ejpam-4383	51	29	∗	∗	NOUN
ejpam-4383	51	30	0	0	NUM
ejpam-4383	51	31	1	1	NUM
ejpam-4383	51	32	2	2	NUM
ejpam-4383	51	33	3	3	NUM
ejpam-4383	51	34	4	4	NUM
ejpam-4383	51	35	5	5	NUM
ejpam-4383	51	36	0	0	NUM
ejpam-4383	51	37	0	0	NUM
ejpam-4383	51	38	1	1	NUM
ejpam-4383	51	39	4	4	NUM
ejpam-4383	51	40	5	5	NUM
ejpam-4383	51	41	2	2	NUM
ejpam-4383	51	42	3	3	NUM
ejpam-4383	51	43	1	1	NUM
ejpam-4383	51	44	1	1	NUM
ejpam-4383	51	45	0	0	NUM
ejpam-4383	51	46	5	5	NUM
ejpam-4383	51	47	4	4	NUM
ejpam-4383	51	48	3	3	NUM
ejpam-4383	51	49	2	2	NUM
ejpam-4383	51	50	2	2	NUM
ejpam-4383	51	51	2	2	NUM
ejpam-4383	51	52	3	3	NUM
ejpam-4383	51	53	0	0	NUM
ejpam-4383	51	54	1	1	NUM
ejpam-4383	51	55	4	4	NUM
ejpam-4383	51	56	5	5	NUM
ejpam-4383	51	57	3	3	NUM
ejpam-4383	51	58	3	3	NUM
ejpam-4383	51	59	2	2	NUM
ejpam-4383	51	60	1	1	NUM
ejpam-4383	51	61	0	0	NUM
ejpam-4383	51	62	5	5	NUM
ejpam-4383	51	63	4	4	NUM
ejpam-4383	51	64	4	4	NUM
ejpam-4383	51	65	4	4	NUM
ejpam-4383	51	66	5	5	NUM
ejpam-4383	51	67	2	2	NUM
ejpam-4383	51	68	3	3	NUM
ejpam-4383	51	69	0	0	NUM
ejpam-4383	51	70	1	1	NUM
ejpam-4383	51	71	5	5	NUM
ejpam-4383	51	72	5	5	NUM
ejpam-4383	51	73	4	4	NUM
ejpam-4383	51	74	3	3	NUM
ejpam-4383	51	75	2	2	NUM
ejpam-4383	51	76	1	1	NUM
ejpam-4383	51	77	0	0	NUM
ejpam-4383	51	78	then	then	ADV
ejpam-4383	51	79	p	p	NOUN
ejpam-4383	51	80	=	=	PUNCT
ejpam-4383	51	81	(	(	PUNCT
ejpam-4383	51	82	p	p	NOUN
ejpam-4383	51	83	;	;	PUNCT
ejpam-4383	51	84	∗	∗	NOUN
ejpam-4383	51	85	,	,	PUNCT
ejpam-4383	51	86	0	0	NUM
ejpam-4383	51	87	)	)	PUNCT
ejpam-4383	51	88	is	be	AUX
ejpam-4383	51	89	a	a	DET
ejpam-4383	51	90	b	b	NOUN
ejpam-4383	51	91	-	-	PUNCT
ejpam-4383	51	92	algebra	algebra	NOUN
ejpam-4383	51	93	.	.	PUNCT
ejpam-4383	52	1	definition	definition	NOUN
ejpam-4383	52	2	2	2	NUM
ejpam-4383	52	3	.	.	PUNCT
ejpam-4383	53	1	[	[	X
ejpam-4383	53	2	17	17	NUM
ejpam-4383	53	3	]	]	PUNCT
ejpam-4383	53	4	a	a	DET
ejpam-4383	53	5	b	b	X
ejpam-4383	53	6	-	-	PUNCT
ejpam-4383	53	7	algebra	algebra	NOUN
ejpam-4383	53	8	(	(	PUNCT
ejpam-4383	53	9	p	p	NOUN
ejpam-4383	53	10	;	;	PUNCT
ejpam-4383	53	11	∗	∗	NOUN
ejpam-4383	53	12	,	,	PUNCT
ejpam-4383	53	13	0	0	NUM
ejpam-4383	53	14	)	)	PUNCT
ejpam-4383	53	15	is	be	AUX
ejpam-4383	53	16	said	say	VERB
ejpam-4383	53	17	to	to	PART
ejpam-4383	53	18	be	be	AUX
ejpam-4383	53	19	commutative	commutative	ADJ
ejpam-4383	53	20	if	if	SCONJ
ejpam-4383	53	21	(	(	PUNCT
ejpam-4383	53	22	∀x	∀x	X
ejpam-4383	53	23	,	,	PUNCT
ejpam-4383	53	24	y	y	PROPN
ejpam-4383	53	25	∈	∈	PROPN
ejpam-4383	53	26	p	p	NOUN
ejpam-4383	53	27	)	)	PUNCT
ejpam-4383	53	28	(	(	PUNCT
ejpam-4383	53	29	x	x	SYM
ejpam-4383	53	30	∗	∗	NOUN
ejpam-4383	53	31	(	(	PUNCT
ejpam-4383	53	32	0	0	NUM
ejpam-4383	53	33	∗	∗	NUM
ejpam-4383	53	34	y	y	NOUN
ejpam-4383	53	35	)	)	PUNCT
ejpam-4383	54	1	=	=	SYM
ejpam-4383	54	2	y	y	PROPN
ejpam-4383	54	3	∗	∗	NOUN
ejpam-4383	54	4	(	(	PUNCT
ejpam-4383	54	5	0	0	NUM
ejpam-4383	54	6	∗	∗	NOUN
ejpam-4383	54	7	x	x	NOUN
ejpam-4383	54	8	)	)	PUNCT
ejpam-4383	54	9	)	)	PUNCT
ejpam-4383	54	10	.	.	PUNCT
ejpam-4383	54	11	example	example	NOUN
ejpam-4383	55	1	2	2	NUM
ejpam-4383	55	2	.	.	X
ejpam-4383	55	3	from	from	ADP
ejpam-4383	55	4	example	example	NOUN
ejpam-4383	55	5	1	1	NUM
ejpam-4383	55	6	,	,	PUNCT
ejpam-4383	55	7	we	we	PRON
ejpam-4383	55	8	have	have	VERB
ejpam-4383	55	9	p	p	NOUN
ejpam-4383	55	10	=	=	X
ejpam-4383	55	11	(	(	PUNCT
ejpam-4383	55	12	p	p	NOUN
ejpam-4383	55	13	;	;	PUNCT
ejpam-4383	55	14	∗	∗	NOUN
ejpam-4383	55	15	,	,	PUNCT
ejpam-4383	55	16	0	0	NUM
ejpam-4383	55	17	)	)	PUNCT
ejpam-4383	55	18	is	be	AUX
ejpam-4383	55	19	commutative	commutative	ADJ
ejpam-4383	55	20	.	.	PUNCT
ejpam-4383	56	1	definition	definition	NOUN
ejpam-4383	56	2	3	3	NUM
ejpam-4383	56	3	.	.	PUNCT
ejpam-4383	57	1	[	[	X
ejpam-4383	57	2	16	16	NUM
ejpam-4383	57	3	]	]	PUNCT
ejpam-4383	57	4	a	a	DET
ejpam-4383	57	5	nonempty	nonempty	NOUN
ejpam-4383	57	6	subset	subset	VERB
ejpam-4383	57	7	n	n	PROPN
ejpam-4383	57	8	of	of	ADP
ejpam-4383	57	9	a	a	DET
ejpam-4383	57	10	b	b	NOUN
ejpam-4383	57	11	-	-	PUNCT
ejpam-4383	57	12	algebra	algebra	NOUN
ejpam-4383	57	13	p	p	NOUN
ejpam-4383	57	14	=	=	X
ejpam-4383	57	15	(	(	PUNCT
ejpam-4383	57	16	p	p	NOUN
ejpam-4383	57	17	;	;	PUNCT
ejpam-4383	57	18	∗	∗	NOUN
ejpam-4383	57	19	,	,	PUNCT
ejpam-4383	57	20	0	0	NUM
ejpam-4383	57	21	)	)	PUNCT
ejpam-4383	57	22	is	be	AUX
ejpam-4383	57	23	said	say	VERB
ejpam-4383	57	24	to	to	PART
ejpam-4383	57	25	be	be	AUX
ejpam-4383	57	26	a	a	DET
ejpam-4383	57	27	b	b	NOUN
ejpam-4383	57	28	-	-	PUNCT
ejpam-4383	57	29	subalgebra	subalgebra	NOUN
ejpam-4383	57	30	of	of	ADP
ejpam-4383	57	31	p	p	PRON
ejpam-4383	57	32	if	if	SCONJ
ejpam-4383	57	33	(	(	PUNCT
ejpam-4383	57	34	∀x	∀x	X
ejpam-4383	57	35	,	,	PUNCT
ejpam-4383	57	36	y	y	PROPN
ejpam-4383	57	37	∈	∈	PROPN
ejpam-4383	57	38	n)(x	n)(x	PROPN
ejpam-4383	57	39	∗	∗	VERB
ejpam-4383	57	40	y	y	PROPN
ejpam-4383	57	41	∈	∈	PROPN
ejpam-4383	57	42	n	n	CCONJ
ejpam-4383	57	43	)	)	PUNCT
ejpam-4383	57	44	.	.	PUNCT
ejpam-4383	58	1	definition	definition	NOUN
ejpam-4383	58	2	4	4	NUM
ejpam-4383	58	3	.	.	PUNCT
ejpam-4383	59	1	[	[	X
ejpam-4383	59	2	2	2	X
ejpam-4383	59	3	]	]	PUNCT
ejpam-4383	59	4	a	a	DET
ejpam-4383	59	5	nonempty	nonempty	NOUN
ejpam-4383	59	6	subset	subset	VERB
ejpam-4383	59	7	i	i	PRON
ejpam-4383	59	8	of	of	ADP
ejpam-4383	59	9	a	a	DET
ejpam-4383	59	10	b	b	NOUN
ejpam-4383	59	11	-	-	PUNCT
ejpam-4383	59	12	algebra	algebra	NOUN
ejpam-4383	59	13	p	p	NOUN
ejpam-4383	59	14	=	=	X
ejpam-4383	59	15	(	(	PUNCT
ejpam-4383	59	16	p	p	NOUN
ejpam-4383	59	17	;	;	PUNCT
ejpam-4383	59	18	∗	∗	NOUN
ejpam-4383	59	19	,	,	PUNCT
ejpam-4383	59	20	0	0	NUM
ejpam-4383	59	21	)	)	PUNCT
ejpam-4383	59	22	is	be	AUX
ejpam-4383	59	23	called	call	VERB
ejpam-4383	59	24	a	a	DET
ejpam-4383	59	25	b	b	NOUN
ejpam-4383	59	26	-	-	PUNCT
ejpam-4383	59	27	ideal	ideal	NOUN
ejpam-4383	59	28	of	of	ADP
ejpam-4383	59	29	p	p	NOUN
ejpam-4383	59	30	if	if	SCONJ
ejpam-4383	59	31	it	it	PRON
ejpam-4383	59	32	satisfies	satisfy	VERB
ejpam-4383	59	33	following	follow	VERB
ejpam-4383	59	34	conditions	condition	NOUN
ejpam-4383	59	35	:	:	PUNCT
ejpam-4383	59	36	0	0	NUM
ejpam-4383	60	1	∈	∈	PROPN
ejpam-4383	60	2	i	i	PRON
ejpam-4383	60	3	,	,	PUNCT
ejpam-4383	60	4	(	(	PUNCT
ejpam-4383	60	5	bi-1	bi-1	X
ejpam-4383	60	6	)	)	PUNCT
ejpam-4383	60	7	(	(	PUNCT
ejpam-4383	60	8	∀x	∀x	X
ejpam-4383	60	9	,	,	PUNCT
ejpam-4383	60	10	y	y	PROPN
ejpam-4383	60	11	∈	∈	PROPN
ejpam-4383	60	12	p	p	PROPN
ejpam-4383	60	13	)	)	PUNCT
ejpam-4383	60	14	(	(	PUNCT
ejpam-4383	60	15	(	(	PUNCT
ejpam-4383	60	16	x	x	SYM
ejpam-4383	60	17	∗	∗	VERB
ejpam-4383	60	18	y	y	PROPN
ejpam-4383	60	19	∈	∈	PROPN
ejpam-4383	61	1	i	i	PRON
ejpam-4383	61	2	,	,	PUNCT
ejpam-4383	61	3	y	y	PROPN
ejpam-4383	61	4	∈	∈	PROPN
ejpam-4383	61	5	i	i	NOUN
ejpam-4383	61	6	)	)	PUNCT
ejpam-4383	61	7	⇒	⇒	VERB
ejpam-4383	61	8	x	x	X
ejpam-4383	61	9	∈	∈	PROPN
ejpam-4383	61	10	i	i	PROPN
ejpam-4383	61	11	)	)	PUNCT
ejpam-4383	61	12	.	.	PUNCT
ejpam-4383	62	1	(	(	PUNCT
ejpam-4383	62	2	bi-2	bi-2	NUM
ejpam-4383	62	3	)	)	PUNCT
ejpam-4383	62	4	by	by	ADP
ejpam-4383	62	5	(	(	PUNCT
ejpam-4383	62	6	b-1	b-1	PROPN
ejpam-4383	62	7	)	)	PUNCT
ejpam-4383	62	8	,	,	PUNCT
ejpam-4383	62	9	we	we	PRON
ejpam-4383	62	10	have	have	VERB
ejpam-4383	62	11	every	every	DET
ejpam-4383	62	12	b	b	NOUN
ejpam-4383	62	13	-subalgebra	-subalgebra	NOUN
ejpam-4383	62	14	of	of	ADP
ejpam-4383	62	15	a	a	DET
ejpam-4383	62	16	b	b	NOUN
ejpam-4383	62	17	-algebra	-algebra	NOUN
ejpam-4383	62	18	satisfies	satisfie	NOUN
ejpam-4383	62	19	(	(	PUNCT
ejpam-4383	62	20	bi-1	bi-1	NUM
ejpam-4383	62	21	)	)	PUNCT
ejpam-4383	62	22	.	.	PUNCT
ejpam-4383	63	1	definition	definition	NOUN
ejpam-4383	63	2	5	5	NUM
ejpam-4383	63	3	.	.	PUNCT
ejpam-4383	64	1	[	[	X
ejpam-4383	64	2	16	16	NUM
ejpam-4383	64	3	]	]	PUNCT
ejpam-4383	64	4	a	a	DET
ejpam-4383	64	5	nonempty	nonempty	NOUN
ejpam-4383	64	6	subset	subset	VERB
ejpam-4383	64	7	n	n	PROPN
ejpam-4383	64	8	of	of	ADP
ejpam-4383	64	9	a	a	DET
ejpam-4383	64	10	b	b	NOUN
ejpam-4383	64	11	-	-	PUNCT
ejpam-4383	64	12	algebra	algebra	NOUN
ejpam-4383	64	13	p	p	NOUN
ejpam-4383	64	14	=	=	X
ejpam-4383	64	15	(	(	PUNCT
ejpam-4383	64	16	p	p	NOUN
ejpam-4383	64	17	;	;	PUNCT
ejpam-4383	64	18	∗	∗	NOUN
ejpam-4383	64	19	,	,	PUNCT
ejpam-4383	64	20	0	0	NUM
ejpam-4383	64	21	)	)	PUNCT
ejpam-4383	64	22	is	be	AUX
ejpam-4383	64	23	said	say	VERB
ejpam-4383	64	24	to	to	PART
ejpam-4383	64	25	be	be	AUX
ejpam-4383	64	26	normal	normal	ADJ
ejpam-4383	64	27	of	of	ADP
ejpam-4383	64	28	p	p	PRON
ejpam-4383	64	29	if	if	SCONJ
ejpam-4383	64	30	(	(	PUNCT
ejpam-4383	64	31	∀x	∀x	NUM
ejpam-4383	64	32	,	,	PUNCT
ejpam-4383	64	33	y	y	PROPN
ejpam-4383	64	34	,	,	PUNCT
ejpam-4383	64	35	a	a	PRON
ejpam-4383	64	36	,	,	PUNCT
ejpam-4383	64	37	b	b	PROPN
ejpam-4383	64	38	∈	∈	PROPN
ejpam-4383	64	39	p	p	NOUN
ejpam-4383	64	40	)	)	PUNCT
ejpam-4383	64	41	(	(	PUNCT
ejpam-4383	64	42	x	x	SYM
ejpam-4383	64	43	∗	∗	PROPN
ejpam-4383	64	44	y	y	PROPN
ejpam-4383	64	45	,	,	PUNCT
ejpam-4383	64	46	a	a	DET
ejpam-4383	64	47	∗	∗	NOUN
ejpam-4383	64	48	b	b	NOUN
ejpam-4383	64	49	∈	∈	PROPN
ejpam-4383	64	50	n	n	PART
ejpam-4383	64	51	⇒	⇒	NOUN
ejpam-4383	64	52	(	(	PUNCT
ejpam-4383	64	53	x	x	X
ejpam-4383	64	54	∗	∗	X
ejpam-4383	64	55	a	a	NOUN
ejpam-4383	64	56	)	)	PUNCT
ejpam-4383	64	57	∗	∗	NOUN
ejpam-4383	64	58	(	(	PUNCT
ejpam-4383	64	59	y	y	PROPN
ejpam-4383	64	60	∗	∗	PUNCT
ejpam-4383	64	61	b	b	NOUN
ejpam-4383	64	62	)	)	PUNCT
ejpam-4383	64	63	∈	∈	PROPN
ejpam-4383	64	64	n	n	CCONJ
ejpam-4383	64	65	)	)	PUNCT
ejpam-4383	64	66	.	.	PUNCT
ejpam-4383	65	1	theorem	theorem	NOUN
ejpam-4383	65	2	1	1	NUM
ejpam-4383	65	3	.	.	PUNCT
ejpam-4383	66	1	[	[	X
ejpam-4383	66	2	16	16	NUM
ejpam-4383	66	3	]	]	PUNCT
ejpam-4383	66	4	every	every	DET
ejpam-4383	66	5	normal	normal	ADJ
ejpam-4383	66	6	subset	subset	NOUN
ejpam-4383	66	7	of	of	ADP
ejpam-4383	66	8	a	a	PRON
ejpam-4383	66	9	b	b	X
ejpam-4383	66	10	-	-	PUNCT
ejpam-4383	66	11	algebra	algebra	NOUN
ejpam-4383	66	12	is	be	AUX
ejpam-4383	66	13	a	a	DET
ejpam-4383	66	14	b	b	NOUN
ejpam-4383	66	15	-	-	PUNCT
ejpam-4383	66	16	subalgebra	subalgebra	NOUN
ejpam-4383	66	17	and	and	CCONJ
ejpam-4383	66	18	hence	hence	ADV
ejpam-4383	66	19	,	,	PUNCT
ejpam-4383	66	20	it	it	PRON
ejpam-4383	66	21	satisfies	satisfy	VERB
ejpam-4383	66	22	(	(	PUNCT
ejpam-4383	66	23	bi-1	bi-1	NUM
ejpam-4383	66	24	)	)	PUNCT
ejpam-4383	66	25	.	.	PUNCT
ejpam-4383	67	1	the	the	DET
ejpam-4383	67	2	concept	concept	NOUN
ejpam-4383	67	3	of	of	ADP
ejpam-4383	67	4	b	b	NOUN
ejpam-4383	67	5	-homomorphisms	-homomorphism	NOUN
ejpam-4383	67	6	was	be	AUX
ejpam-4383	67	7	also	also	ADV
ejpam-4383	67	8	introduced	introduce	VERB
ejpam-4383	67	9	by	by	ADP
ejpam-4383	67	10	neggers	negger	NOUN
ejpam-4383	67	11	and	and	CCONJ
ejpam-4383	67	12	kim	kim	PROPN
ejpam-4383	68	1	[	[	X
ejpam-4383	68	2	16	16	NUM
ejpam-4383	68	3	]	]	PUNCT
ejpam-4383	68	4	.	.	PUNCT
ejpam-4383	69	1	let	let	VERB
ejpam-4383	69	2	a	a	PRON
ejpam-4383	69	3	=	=	X
ejpam-4383	69	4	(	(	PUNCT
ejpam-4383	69	5	a	a	X
ejpam-4383	69	6	;	;	PUNCT
ejpam-4383	69	7	∗a	∗a	PROPN
ejpam-4383	69	8	,	,	PUNCT
ejpam-4383	69	9	0a	0a	NUM
ejpam-4383	69	10	)	)	PUNCT
ejpam-4383	69	11	and	and	CCONJ
ejpam-4383	69	12	b	b	X
ejpam-4383	69	13	=	=	SYM
ejpam-4383	69	14	(	(	PUNCT
ejpam-4383	69	15	b	b	NOUN
ejpam-4383	69	16	;	;	PUNCT
ejpam-4383	69	17	∗b	∗b	PROPN
ejpam-4383	69	18	,	,	PUNCT
ejpam-4383	69	19	0b	0b	NUM
ejpam-4383	69	20	)	)	PUNCT
ejpam-4383	69	21	be	be	AUX
ejpam-4383	69	22	b	b	NOUN
ejpam-4383	69	23	-algebras	-algebra	NOUN
ejpam-4383	69	24	.	.	PUNCT
ejpam-4383	70	1	a	a	DET
ejpam-4383	70	2	map	map	NOUN
ejpam-4383	70	3	φ	φ	X
ejpam-4383	70	4	:	:	PUNCT
ejpam-4383	70	5	a	a	DET
ejpam-4383	70	6	→	→	SYM
ejpam-4383	70	7	b	b	PROPN
ejpam-4383	70	8	is	be	AUX
ejpam-4383	70	9	called	call	VERB
ejpam-4383	70	10	a	a	DET
ejpam-4383	70	11	b	b	NOUN
ejpam-4383	70	12	-	-	PUNCT
ejpam-4383	70	13	homomorphism	homomorphism	NOUN
ejpam-4383	70	14	if	if	SCONJ
ejpam-4383	70	15	(	(	PUNCT
ejpam-4383	70	16	∀x	∀x	X
ejpam-4383	70	17	,	,	PUNCT
ejpam-4383	70	18	y	y	PROPN
ejpam-4383	70	19	∈	∈	PROPN
ejpam-4383	70	20	a)(φ(x	a)(φ(x	PUNCT
ejpam-4383	70	21	∗a	∗a	PROPN
ejpam-4383	70	22	y	y	PROPN
ejpam-4383	70	23	)	)	PUNCT
ejpam-4383	70	24	=	=	SYM
ejpam-4383	70	25	φ(x	φ(x	X
ejpam-4383	70	26	)	)	PUNCT
ejpam-4383	70	27	∗b	∗b	PROPN
ejpam-4383	70	28	φ(y	φ(y	NOUN
ejpam-4383	70	29	)	)	PUNCT
ejpam-4383	70	30	)	)	PUNCT
ejpam-4383	70	31	,	,	PUNCT
ejpam-4383	70	32	an	an	DET
ejpam-4383	70	33	anti	anti	ADJ
ejpam-4383	70	34	-	-	ADJ
ejpam-4383	70	35	b	b	NOUN
ejpam-4383	70	36	-	-	PUNCT
ejpam-4383	70	37	homomorphism	homomorphism	NOUN
ejpam-4383	70	38	if	if	SCONJ
ejpam-4383	70	39	(	(	PUNCT
ejpam-4383	70	40	∀x	∀x	X
ejpam-4383	70	41	,	,	PUNCT
ejpam-4383	70	42	y	y	PROPN
ejpam-4383	70	43	∈	∈	PROPN
ejpam-4383	70	44	a)(φ(x	a)(φ(x	PUNCT
ejpam-4383	70	45	∗a	∗a	PROPN
ejpam-4383	70	46	y	y	PROPN
ejpam-4383	70	47	)	)	PUNCT
ejpam-4383	70	48	=	=	SYM
ejpam-4383	70	49	φ(y	φ(y	ADJ
ejpam-4383	70	50	)	)	PUNCT
ejpam-4383	70	51	∗b	∗b	PROPN
ejpam-4383	70	52	φ(x	φ(x	NOUN
ejpam-4383	70	53	)	)	PUNCT
ejpam-4383	70	54	)	)	PUNCT
ejpam-4383	70	55	.	.	PUNCT
ejpam-4383	71	1	a.	a.	PROPN
ejpam-4383	71	2	iampan	iampan	PROPN
ejpam-4383	71	3	et	et	PROPN
ejpam-4383	71	4	al	al	PROPN
ejpam-4383	71	5	.	.	PUNCT
ejpam-4383	71	6	/	/	SYM
ejpam-4383	71	7	eur	eur	PROPN
ejpam-4383	71	8	.	.	PUNCT
ejpam-4383	72	1	j.	j.	PROPN
ejpam-4383	72	2	pure	pure	PROPN
ejpam-4383	72	3	appl	appl	PROPN
ejpam-4383	72	4	.	.	PROPN
ejpam-4383	72	5	math	math	PROPN
ejpam-4383	72	6	,	,	PUNCT
ejpam-4383	72	7	15	15	NUM
ejpam-4383	72	8	(	(	PUNCT
ejpam-4383	72	9	3	3	NUM
ejpam-4383	72	10	)	)	PUNCT
ejpam-4383	72	11	(	(	PUNCT
ejpam-4383	72	12	2022	2022	NUM
ejpam-4383	72	13	)	)	PUNCT
ejpam-4383	72	14	,	,	PUNCT
ejpam-4383	72	15	999	999	NUM
ejpam-4383	72	16	-	-	SYM
ejpam-4383	72	17	1014	1014	NUM
ejpam-4383	72	18	1002	1002	NUM
ejpam-4383	72	19	the	the	DET
ejpam-4383	72	20	kernel	kernel	NOUN
ejpam-4383	72	21	of	of	ADP
ejpam-4383	72	22	φ	φ	PROPN
ejpam-4383	72	23	,	,	PUNCT
ejpam-4383	72	24	denoted	denote	VERB
ejpam-4383	72	25	by	by	ADP
ejpam-4383	72	26	kerφ	kerφ	PROPN
ejpam-4383	72	27	,	,	PUNCT
ejpam-4383	72	28	is	be	AUX
ejpam-4383	72	29	defined	define	VERB
ejpam-4383	72	30	to	to	PART
ejpam-4383	72	31	be	be	AUX
ejpam-4383	72	32	the	the	DET
ejpam-4383	72	33	{	{	PUNCT
ejpam-4383	72	34	x	x	SYM
ejpam-4383	72	35	∈	∈	PROPN
ejpam-4383	72	36	a	a	DET
ejpam-4383	72	37	|	|	NOUN
ejpam-4383	72	38	φ(x	φ(x	NOUN
ejpam-4383	72	39	)	)	PUNCT
ejpam-4383	72	40	=	=	SYM
ejpam-4383	72	41	0b	0b	NOUN
ejpam-4383	72	42	}	}	PUNCT
ejpam-4383	72	43	.	.	PUNCT
ejpam-4383	73	1	the	the	DET
ejpam-4383	73	2	kerφ	kerφ	PROPN
ejpam-4383	73	3	is	be	AUX
ejpam-4383	73	4	a	a	DET
ejpam-4383	73	5	normal	normal	ADJ
ejpam-4383	73	6	b	b	NOUN
ejpam-4383	73	7	-subalgebra	-subalgebra	NOUN
ejpam-4383	73	8	of	of	ADP
ejpam-4383	73	9	a	a	PRON
ejpam-4383	73	10	,	,	PUNCT
ejpam-4383	73	11	and	and	CCONJ
ejpam-4383	73	12	kerφ	kerφ	PROPN
ejpam-4383	73	13	=	=	PUNCT
ejpam-4383	73	14	{	{	PUNCT
ejpam-4383	73	15	0a	0a	PROPN
ejpam-4383	73	16	}	}	PUNCT
ejpam-4383	73	17	if	if	SCONJ
ejpam-4383	73	18	and	and	CCONJ
ejpam-4383	73	19	only	only	ADV
ejpam-4383	73	20	if	if	SCONJ
ejpam-4383	73	21	φ	φ	PROPN
ejpam-4383	73	22	is	be	AUX
ejpam-4383	73	23	injective	injective	ADJ
ejpam-4383	73	24	.	.	PUNCT
ejpam-4383	74	1	a	a	DET
ejpam-4383	74	2	(	(	PUNCT
ejpam-4383	74	3	anti-)b	anti-)b	PRON
ejpam-4383	74	4	-homomorphism	-homomorphism	PROPN
ejpam-4383	74	5	φ	φ	PROPN
ejpam-4383	74	6	is	be	AUX
ejpam-4383	74	7	called	call	VERB
ejpam-4383	74	8	a	a	DET
ejpam-4383	74	9	(	(	PUNCT
ejpam-4383	74	10	anti-)b	anti-)b	X
ejpam-4383	74	11	-monomorphism	-monomorphism	NOUN
ejpam-4383	74	12	,	,	PUNCT
ejpam-4383	74	13	(	(	PUNCT
ejpam-4383	74	14	anti-)b	anti-)b	X
ejpam-4383	74	15	-epimorphism	-epimorphism	PROPN
ejpam-4383	74	16	,	,	PUNCT
ejpam-4383	74	17	or	or	CCONJ
ejpam-4383	74	18	(	(	PUNCT
ejpam-4383	74	19	anti-)b	anti-)b	PRON
ejpam-4383	74	20	-isomorphism	-isomorphism	PROPN
ejpam-4383	74	21	if	if	SCONJ
ejpam-4383	74	22	φ	φ	PROPN
ejpam-4383	74	23	is	be	AUX
ejpam-4383	74	24	injective	injective	ADJ
ejpam-4383	74	25	,	,	PUNCT
ejpam-4383	74	26	surjective	surjective	ADJ
ejpam-4383	74	27	,	,	PUNCT
ejpam-4383	74	28	or	or	CCONJ
ejpam-4383	74	29	bijective	bijective	ADJ
ejpam-4383	74	30	,	,	PUNCT
ejpam-4383	74	31	respectively	respectively	ADV
ejpam-4383	74	32	.	.	PUNCT
ejpam-4383	75	1	theorem	theorem	NOUN
ejpam-4383	75	2	2	2	NUM
ejpam-4383	75	3	.	.	PUNCT
ejpam-4383	76	1	[	[	X
ejpam-4383	76	2	16	16	NUM
ejpam-4383	76	3	]	]	PUNCT
ejpam-4383	76	4	let	let	VERB
ejpam-4383	76	5	n	n	PRON
ejpam-4383	76	6	be	be	AUX
ejpam-4383	76	7	a	a	DET
ejpam-4383	76	8	nonempty	nonempty	ADJ
ejpam-4383	76	9	subset	subset	NOUN
ejpam-4383	76	10	of	of	ADP
ejpam-4383	76	11	a	a	DET
ejpam-4383	76	12	b	b	NOUN
ejpam-4383	76	13	-	-	PUNCT
ejpam-4383	76	14	algebra	algebra	NOUN
ejpam-4383	76	15	p	p	NOUN
ejpam-4383	76	16	=	=	X
ejpam-4383	76	17	(	(	PUNCT
ejpam-4383	76	18	p	p	NOUN
ejpam-4383	76	19	;	;	PUNCT
ejpam-4383	76	20	∗	∗	NOUN
ejpam-4383	76	21	,	,	PUNCT
ejpam-4383	76	22	0	0	NUM
ejpam-4383	76	23	)	)	PUNCT
ejpam-4383	76	24	.	.	PUNCT
ejpam-4383	77	1	then	then	ADV
ejpam-4383	77	2	the	the	DET
ejpam-4383	77	3	following	follow	VERB
ejpam-4383	77	4	statements	statement	NOUN
ejpam-4383	77	5	are	be	AUX
ejpam-4383	77	6	equivalent	equivalent	ADJ
ejpam-4383	77	7	:	:	PUNCT
ejpam-4383	77	8	(	(	PUNCT
ejpam-4383	77	9	i	i	NOUN
ejpam-4383	77	10	)	)	PUNCT
ejpam-4383	77	11	n	n	PRON
ejpam-4383	77	12	is	be	AUX
ejpam-4383	77	13	a	a	DET
ejpam-4383	77	14	b	b	NOUN
ejpam-4383	77	15	-	-	PUNCT
ejpam-4383	77	16	subalgebra	subalgebra	NOUN
ejpam-4383	77	17	of	of	ADP
ejpam-4383	77	18	p	p	PROPN
ejpam-4383	77	19	.	.	PUNCT
ejpam-4383	78	1	(	(	PUNCT
ejpam-4383	78	2	ii	ii	NOUN
ejpam-4383	78	3	)	)	PUNCT
ejpam-4383	78	4	x	x	SYM
ejpam-4383	79	1	∗	∗	NOUN
ejpam-4383	79	2	(	(	PUNCT
ejpam-4383	79	3	0	0	NUM
ejpam-4383	79	4	∗	∗	PROPN
ejpam-4383	79	5	y	y	PROPN
ejpam-4383	79	6	)	)	PUNCT
ejpam-4383	79	7	,	,	PUNCT
ejpam-4383	79	8	0	0	NUM
ejpam-4383	79	9	∗	∗	NOUN
ejpam-4383	79	10	y	y	PROPN
ejpam-4383	79	11	∈	∈	PROPN
ejpam-4383	79	12	n	n	PROPN
ejpam-4383	79	13	for	for	ADP
ejpam-4383	79	14	all	all	DET
ejpam-4383	79	15	x	x	NOUN
ejpam-4383	79	16	,	,	PUNCT
ejpam-4383	79	17	y	y	PROPN
ejpam-4383	79	18	∈	∈	PROPN
ejpam-4383	79	19	n	n	ADV
ejpam-4383	79	20	.	.	PUNCT
ejpam-4383	80	1	2	2	X
ejpam-4383	80	2	.	.	X
ejpam-4383	80	3	external	external	ADJ
ejpam-4383	80	4	direct	direct	ADJ
ejpam-4383	80	5	product	product	NOUN
ejpam-4383	80	6	of	of	ADP
ejpam-4383	80	7	b	b	NOUN
ejpam-4383	80	8	-	-	PUNCT
ejpam-4383	80	9	algebras	algebras	PROPN
ejpam-4383	80	10	lingcong	lingcong	PROPN
ejpam-4383	80	11	and	and	CCONJ
ejpam-4383	80	12	endam	endam	NOUN
ejpam-4383	81	1	[	[	X
ejpam-4383	81	2	12	12	NUM
ejpam-4383	81	3	]	]	PUNCT
ejpam-4383	81	4	discussed	discuss	VERB
ejpam-4383	81	5	the	the	DET
ejpam-4383	81	6	notion	notion	NOUN
ejpam-4383	81	7	of	of	ADP
ejpam-4383	81	8	the	the	DET
ejpam-4383	81	9	direct	direct	ADJ
ejpam-4383	81	10	product	product	NOUN
ejpam-4383	81	11	of	of	ADP
ejpam-4383	81	12	b	b	NOUN
ejpam-4383	81	13	-algebras	-algebras	PROPN
ejpam-4383	81	14	,	,	PUNCT
ejpam-4383	81	15	0	0	NUM
ejpam-4383	81	16	-	-	PUNCT
ejpam-4383	81	17	commutative	commutative	ADJ
ejpam-4383	81	18	b	b	PROPN
ejpam-4383	81	19	-algebras	-algebra	NOUN
ejpam-4383	81	20	,	,	PUNCT
ejpam-4383	81	21	and	and	CCONJ
ejpam-4383	81	22	b	b	X
ejpam-4383	81	23	-homomorphisms	-homomorphism	NOUN
ejpam-4383	81	24	and	and	CCONJ
ejpam-4383	81	25	obtained	obtain	VERB
ejpam-4383	81	26	related	related	ADJ
ejpam-4383	81	27	properties	property	NOUN
ejpam-4383	81	28	,	,	PUNCT
ejpam-4383	81	29	one	one	NUM
ejpam-4383	81	30	of	of	ADP
ejpam-4383	81	31	which	which	PRON
ejpam-4383	81	32	is	be	AUX
ejpam-4383	81	33	a	a	DET
ejpam-4383	81	34	direct	direct	ADJ
ejpam-4383	81	35	product	product	NOUN
ejpam-4383	81	36	of	of	ADP
ejpam-4383	81	37	two	two	NUM
ejpam-4383	81	38	b	b	NOUN
ejpam-4383	81	39	-algebras	-algebra	NOUN
ejpam-4383	81	40	,	,	PUNCT
ejpam-4383	81	41	which	which	PRON
ejpam-4383	81	42	is	be	AUX
ejpam-4383	81	43	also	also	ADV
ejpam-4383	81	44	a	a	DET
ejpam-4383	81	45	b	b	NOUN
ejpam-4383	81	46	-algebra	-algebra	NOUN
ejpam-4383	81	47	.	.	PUNCT
ejpam-4383	82	1	then	then	ADV
ejpam-4383	82	2	,	,	PUNCT
ejpam-4383	82	3	they	they	PRON
ejpam-4383	82	4	extended	extend	VERB
ejpam-4383	82	5	the	the	DET
ejpam-4383	82	6	concept	concept	NOUN
ejpam-4383	82	7	of	of	ADP
ejpam-4383	82	8	the	the	DET
ejpam-4383	82	9	direct	direct	ADJ
ejpam-4383	82	10	product	product	NOUN
ejpam-4383	82	11	of	of	ADP
ejpam-4383	82	12	b	b	PROPN
ejpam-4383	82	13	-algebra	-algebra	NOUN
ejpam-4383	82	14	to	to	PART
ejpam-4383	82	15	finite	finite	VERB
ejpam-4383	82	16	family	family	PROPN
ejpam-4383	82	17	b	b	PROPN
ejpam-4383	82	18	-algebra	-algebra	PROPN
ejpam-4383	82	19	,	,	PUNCT
ejpam-4383	82	20	and	and	CCONJ
ejpam-4383	82	21	some	some	PRON
ejpam-4383	82	22	of	of	ADP
ejpam-4383	82	23	the	the	DET
ejpam-4383	82	24	related	relate	VERB
ejpam-4383	82	25	properties	property	NOUN
ejpam-4383	82	26	were	be	AUX
ejpam-4383	82	27	investigated	investigate	VERB
ejpam-4383	82	28	as	as	SCONJ
ejpam-4383	82	29	follows	follow	VERB
ejpam-4383	82	30	:	:	PUNCT
ejpam-4383	82	31	definition	definition	NOUN
ejpam-4383	82	32	6	6	NUM
ejpam-4383	82	33	.	.	PUNCT
ejpam-4383	83	1	[	[	X
ejpam-4383	83	2	12	12	NUM
ejpam-4383	83	3	]	]	X
ejpam-4383	83	4	let	let	ADJ
ejpam-4383	83	5	(	(	PUNCT
ejpam-4383	83	6	pi	pi	NOUN
ejpam-4383	83	7	;	;	PUNCT
ejpam-4383	83	8	∗i	∗i	X
ejpam-4383	83	9	)	)	PUNCT
ejpam-4383	83	10	be	be	VERB
ejpam-4383	83	11	an	an	DET
ejpam-4383	83	12	algebra	algebra	NOUN
ejpam-4383	83	13	for	for	ADP
ejpam-4383	83	14	each	each	DET
ejpam-4383	83	15	i	i	PRON
ejpam-4383	83	16	∈	∈	PROPN
ejpam-4383	83	17	{	{	PUNCT
ejpam-4383	83	18	1	1	NUM
ejpam-4383	83	19	,	,	PUNCT
ejpam-4383	83	20	2	2	NUM
ejpam-4383	83	21	,	,	PUNCT
ejpam-4383	83	22	...	...	PUNCT
ejpam-4383	83	23	,	,	PUNCT
ejpam-4383	83	24	k	k	NOUN
ejpam-4383	83	25	}	}	PUNCT
ejpam-4383	83	26	.	.	PUNCT
ejpam-4383	84	1	define	define	VERB
ejpam-4383	84	2	the	the	DET
ejpam-4383	84	3	direct	direct	ADJ
ejpam-4383	84	4	product	product	NOUN
ejpam-4383	84	5	of	of	ADP
ejpam-4383	84	6	algebras	algebras	PROPN
ejpam-4383	84	7	p1	p1	PROPN
ejpam-4383	84	8	,	,	PUNCT
ejpam-4383	84	9	p2	p2	NOUN
ejpam-4383	84	10	,	,	PUNCT
ejpam-4383	84	11	...	...	PUNCT
ejpam-4383	84	12	,	,	PUNCT
ejpam-4383	84	13	pk	pk	NOUN
ejpam-4383	84	14	to	to	PART
ejpam-4383	84	15	be	be	AUX
ejpam-4383	84	16	the	the	DET
ejpam-4383	84	17	structure	structure	NOUN
ejpam-4383	84	18	(	(	PUNCT
ejpam-4383	84	19	∏k	∏k	X
ejpam-4383	84	20	i=1	i=1	PROPN
ejpam-4383	84	21	pi;⊗	pi;⊗	PROPN
ejpam-4383	84	22	)	)	PUNCT
ejpam-4383	84	23	,	,	PUNCT
ejpam-4383	84	24	where	where	SCONJ
ejpam-4383	84	25	k∏	k∏	PROPN
ejpam-4383	84	26	i=1	i=1	PROPN
ejpam-4383	84	27	pi	pi	NOUN
ejpam-4383	84	28	=	=	SYM
ejpam-4383	84	29	p1	p1	PROPN
ejpam-4383	84	30	×	×	NOUN
ejpam-4383	84	31	p2	p2	PROPN
ejpam-4383	84	32	×	×	NOUN
ejpam-4383	84	33	...	...	PUNCT
ejpam-4383	84	34	×	×	PROPN
ejpam-4383	84	35	pk	pk	NOUN
ejpam-4383	84	36	=	=	SYM
ejpam-4383	84	37	{	{	PUNCT
ejpam-4383	84	38	(	(	PUNCT
ejpam-4383	84	39	p1	p1	NOUN
ejpam-4383	84	40	,	,	PUNCT
ejpam-4383	84	41	p2	p2	NOUN
ejpam-4383	84	42	,	,	PUNCT
ejpam-4383	84	43	...	...	PUNCT
ejpam-4383	84	44	,	,	PUNCT
ejpam-4383	84	45	pk	pk	NOUN
ejpam-4383	84	46	)	)	PUNCT
ejpam-4383	85	1	|	|	ADV
ejpam-4383	85	2	pi	pi	NOUN
ejpam-4383	85	3	∈	∈	PROPN
ejpam-4383	85	4	pi	pi	NOUN
ejpam-4383	85	5	∀i	∀i	NOUN
ejpam-4383	85	6	=	=	SYM
ejpam-4383	85	7	1	1	NUM
ejpam-4383	85	8	,	,	PUNCT
ejpam-4383	85	9	2	2	NUM
ejpam-4383	85	10	,	,	PUNCT
ejpam-4383	85	11	...	...	PUNCT
ejpam-4383	85	12	,	,	PUNCT
ejpam-4383	85	13	k	k	NOUN
ejpam-4383	85	14	}	}	PUNCT
ejpam-4383	85	15	and	and	CCONJ
ejpam-4383	85	16	whose	whose	DET
ejpam-4383	85	17	operation	operation	NOUN
ejpam-4383	85	18	⊗	⊗	PROPN
ejpam-4383	85	19	is	be	AUX
ejpam-4383	85	20	given	give	VERB
ejpam-4383	85	21	by	by	ADP
ejpam-4383	85	22	(	(	PUNCT
ejpam-4383	85	23	p1	p1	PROPN
ejpam-4383	85	24	,	,	PUNCT
ejpam-4383	85	25	p2	p2	NOUN
ejpam-4383	85	26	,	,	PUNCT
ejpam-4383	85	27	...	...	PUNCT
ejpam-4383	85	28	,	,	PUNCT
ejpam-4383	85	29	pk)⊗	pk)⊗	PROPN
ejpam-4383	85	30	(	(	PUNCT
ejpam-4383	85	31	q1	q1	PROPN
ejpam-4383	85	32	,	,	PUNCT
ejpam-4383	85	33	q2	q2	NOUN
ejpam-4383	85	34	,	,	PUNCT
ejpam-4383	85	35	...	...	PUNCT
ejpam-4383	85	36	,	,	PUNCT
ejpam-4383	85	37	qk	qk	NOUN
ejpam-4383	85	38	)	)	PUNCT
ejpam-4383	85	39	=	=	SYM
ejpam-4383	86	1	(	(	PUNCT
ejpam-4383	86	2	p1	p1	PROPN
ejpam-4383	86	3	∗1	∗1	PROPN
ejpam-4383	86	4	q1	q1	PROPN
ejpam-4383	86	5	,	,	PUNCT
ejpam-4383	86	6	p2	p2	PROPN
ejpam-4383	86	7	∗2	∗2	PROPN
ejpam-4383	86	8	q2	q2	NOUN
ejpam-4383	86	9	,	,	PUNCT
ejpam-4383	86	10	...	...	PUNCT
ejpam-4383	86	11	,	,	PUNCT
ejpam-4383	86	12	pk	pk	PROPN
ejpam-4383	86	13	∗k	∗k	PROPN
ejpam-4383	86	14	qk	qk	NOUN
ejpam-4383	86	15	)	)	PUNCT
ejpam-4383	86	16	for	for	ADP
ejpam-4383	86	17	all	all	DET
ejpam-4383	86	18	(	(	PUNCT
ejpam-4383	86	19	p1	p1	PROPN
ejpam-4383	86	20	,	,	PUNCT
ejpam-4383	86	21	p2	p2	NOUN
ejpam-4383	86	22	,	,	PUNCT
ejpam-4383	86	23	...	...	PUNCT
ejpam-4383	86	24	,	,	PUNCT
ejpam-4383	86	25	pk	pk	NOUN
ejpam-4383	86	26	)	)	PUNCT
ejpam-4383	86	27	,	,	PUNCT
ejpam-4383	86	28	(	(	PUNCT
ejpam-4383	86	29	q1	q1	PROPN
ejpam-4383	86	30	,	,	PUNCT
ejpam-4383	86	31	q2	q2	NOUN
ejpam-4383	86	32	,	,	PUNCT
ejpam-4383	86	33	...	...	PUNCT
ejpam-4383	86	34	,	,	PUNCT
ejpam-4383	86	35	qk	qk	INTJ
ejpam-4383	86	36	)	)	PUNCT
ejpam-4383	86	37	∈	∈	PROPN
ejpam-4383	87	1	∏k	∏k	X
ejpam-4383	87	2	i=1	i=1	PROPN
ejpam-4383	87	3	pi	pi	NOUN
ejpam-4383	87	4	.	.	PUNCT
ejpam-4383	88	1	theorem	theorem	NOUN
ejpam-4383	88	2	3	3	NUM
ejpam-4383	88	3	.	.	PUNCT
ejpam-4383	89	1	[	[	X
ejpam-4383	89	2	12	12	NUM
ejpam-4383	89	3	]	]	PUNCT
ejpam-4383	89	4	(	(	PUNCT
ejpam-4383	89	5	pi	pi	NOUN
ejpam-4383	89	6	;	;	PUNCT
ejpam-4383	89	7	∗i	∗i	PROPN
ejpam-4383	89	8	,	,	PUNCT
ejpam-4383	89	9	0i	0i	NOUN
ejpam-4383	89	10	)	)	PUNCT
ejpam-4383	89	11	is	be	AUX
ejpam-4383	89	12	a	a	DET
ejpam-4383	89	13	b	b	NOUN
ejpam-4383	89	14	-	-	PUNCT
ejpam-4383	89	15	algebra	algebra	NOUN
ejpam-4383	89	16	for	for	ADP
ejpam-4383	89	17	all	all	DET
ejpam-4383	89	18	i	i	PRON
ejpam-4383	89	19	=	=	NOUN
ejpam-4383	89	20	1	1	NUM
ejpam-4383	89	21	,	,	PUNCT
ejpam-4383	89	22	2	2	NUM
ejpam-4383	89	23	,	,	PUNCT
ejpam-4383	89	24	...	...	PUNCT
ejpam-4383	89	25	,	,	PUNCT
ejpam-4383	89	26	k	k	PROPN
ejpam-4383	90	1	if	if	SCONJ
ejpam-4383	90	2	and	and	CCONJ
ejpam-4383	90	3	only	only	ADV
ejpam-4383	90	4	if	if	SCONJ
ejpam-4383	90	5	(	(	PUNCT
ejpam-4383	90	6	k∏	k∏	PROPN
ejpam-4383	90	7	i=1	i=1	PROPN
ejpam-4383	90	8	pi;⊗	pi;⊗	PROPN
ejpam-4383	90	9	,	,	PUNCT
ejpam-4383	90	10	(	(	PUNCT
ejpam-4383	90	11	01	01	NUM
ejpam-4383	90	12	,	,	PUNCT
ejpam-4383	90	13	02	02	NUM
ejpam-4383	90	14	,	,	PUNCT
ejpam-4383	90	15	...	...	PUNCT
ejpam-4383	90	16	,	,	PUNCT
ejpam-4383	90	17	0k	0k	NOUN
ejpam-4383	90	18	)	)	PUNCT
ejpam-4383	90	19	)	)	PUNCT
ejpam-4383	90	20	is	be	AUX
ejpam-4383	90	21	a	a	DET
ejpam-4383	90	22	b	b	NOUN
ejpam-4383	90	23	-	-	PUNCT
ejpam-4383	90	24	algebra	algebra	NOUN
ejpam-4383	90	25	,	,	PUNCT
ejpam-4383	90	26	where	where	SCONJ
ejpam-4383	90	27	the	the	DET
ejpam-4383	90	28	binary	binary	PROPN
ejpam-4383	90	29	operation	operation	PROPN
ejpam-4383	90	30	⊗	⊗	PROPN
ejpam-4383	90	31	is	be	AUX
ejpam-4383	90	32	defined	define	VERB
ejpam-4383	90	33	in	in	ADP
ejpam-4383	90	34	definition	definition	NOUN
ejpam-4383	90	35	6	6	NUM
ejpam-4383	90	36	.	.	PUNCT
ejpam-4383	91	1	now	now	ADV
ejpam-4383	91	2	,	,	PUNCT
ejpam-4383	91	3	we	we	PRON
ejpam-4383	91	4	extend	extend	VERB
ejpam-4383	91	5	the	the	DET
ejpam-4383	91	6	concept	concept	NOUN
ejpam-4383	91	7	of	of	ADP
ejpam-4383	91	8	the	the	DET
ejpam-4383	91	9	direct	direct	ADJ
ejpam-4383	91	10	product	product	NOUN
ejpam-4383	91	11	to	to	PART
ejpam-4383	91	12	infinite	infinite	VERB
ejpam-4383	91	13	family	family	NOUN
ejpam-4383	91	14	of	of	ADP
ejpam-4383	91	15	b	b	NOUN
ejpam-4383	91	16	-algebras	-algebras	NOUN
ejpam-4383	91	17	and	and	CCONJ
ejpam-4383	91	18	provide	provide	VERB
ejpam-4383	91	19	some	some	PRON
ejpam-4383	91	20	of	of	ADP
ejpam-4383	91	21	its	its	PRON
ejpam-4383	91	22	properties	property	NOUN
ejpam-4383	91	23	.	.	PUNCT
ejpam-4383	92	1	definition	definition	NOUN
ejpam-4383	92	2	7	7	NUM
ejpam-4383	92	3	.	.	PUNCT
ejpam-4383	93	1	let	let	VERB
ejpam-4383	93	2	pi	pi	PRON
ejpam-4383	93	3	be	be	AUX
ejpam-4383	93	4	a	a	DET
ejpam-4383	93	5	nonempty	nonempty	NOUN
ejpam-4383	93	6	set	set	VERB
ejpam-4383	93	7	for	for	ADP
ejpam-4383	93	8	each	each	DET
ejpam-4383	93	9	i	i	PRON
ejpam-4383	93	10	∈	∈	PROPN
ejpam-4383	93	11	i.	i.	NOUN
ejpam-4383	93	12	define	define	VERB
ejpam-4383	93	13	the	the	DET
ejpam-4383	93	14	external	external	ADJ
ejpam-4383	93	15	direct	direct	ADJ
ejpam-4383	93	16	product	product	NOUN
ejpam-4383	93	17	of	of	ADP
ejpam-4383	93	18	sets	set	NOUN
ejpam-4383	93	19	pi	pi	VERB
ejpam-4383	93	20	for	for	ADP
ejpam-4383	93	21	all	all	PRON
ejpam-4383	93	22	i	i	PRON
ejpam-4383	93	23	∈	∈	VERB
ejpam-4383	94	1	i	i	PRON
ejpam-4383	94	2	to	to	PART
ejpam-4383	94	3	be	be	AUX
ejpam-4383	94	4	the	the	DET
ejpam-4383	94	5	set	set	ADJ
ejpam-4383	94	6	∏	∏	PROPN
ejpam-4383	94	7	i∈i	i∈i	ADJ
ejpam-4383	94	8	pi	pi	NOUN
ejpam-4383	94	9	,	,	PUNCT
ejpam-4383	94	10	where∏	where∏	PROPN
ejpam-4383	94	11	i∈i	i∈i	ADJ
ejpam-4383	94	12	pi	pi	NOUN
ejpam-4383	95	1	=	=	PUNCT
ejpam-4383	95	2	{	{	PUNCT
ejpam-4383	95	3	f	f	X
ejpam-4383	95	4	:	:	PUNCT
ejpam-4383	95	5	i	i	PRON
ejpam-4383	95	6	→	→	SYM
ejpam-4383	95	7	⋃	⋃	ADP
ejpam-4383	95	8	i∈i	i∈i	ADJ
ejpam-4383	95	9	pi	pi	NOUN
ejpam-4383	95	10	|	|	ADV
ejpam-4383	95	11	f(i	f(i	NOUN
ejpam-4383	95	12	)	)	PUNCT
ejpam-4383	95	13	∈	∈	PROPN
ejpam-4383	95	14	pi	pi	NOUN
ejpam-4383	95	15	∀i	∀i	NOUN
ejpam-4383	95	16	∈	∈	PROPN
ejpam-4383	95	17	i	i	NOUN
ejpam-4383	95	18	}	}	PUNCT
ejpam-4383	95	19	.	.	PUNCT
ejpam-4383	96	1	for	for	ADP
ejpam-4383	96	2	convenience	convenience	NOUN
ejpam-4383	96	3	,	,	PUNCT
ejpam-4383	96	4	we	we	PRON
ejpam-4383	96	5	define	define	VERB
ejpam-4383	96	6	an	an	DET
ejpam-4383	96	7	element	element	NOUN
ejpam-4383	96	8	of	of	ADP
ejpam-4383	96	9	∏	∏	NUM
ejpam-4383	96	10	i∈i	i∈i	ADJ
ejpam-4383	96	11	pi	pi	NOUN
ejpam-4383	96	12	with	with	ADP
ejpam-4383	96	13	a	a	DET
ejpam-4383	96	14	function	function	NOUN
ejpam-4383	96	15	(	(	PUNCT
ejpam-4383	96	16	pi)i∈i	pi)i∈i	NUM
ejpam-4383	96	17	:	:	PUNCT
ejpam-4383	96	18	i	i	PRON
ejpam-4383	96	19	→	→	SYM
ejpam-4383	96	20	⋃	⋃	ADP
ejpam-4383	96	21	i∈i	i∈i	ADJ
ejpam-4383	96	22	pi	pi	NOUN
ejpam-4383	96	23	,	,	PUNCT
ejpam-4383	96	24	where	where	SCONJ
ejpam-4383	96	25	i	i	PRON
ejpam-4383	96	26	7→	7→	NUM
ejpam-4383	96	27	pi	pi	NOUN
ejpam-4383	96	28	∈	∈	NOUN
ejpam-4383	96	29	pi	pi	NOUN
ejpam-4383	96	30	for	for	ADP
ejpam-4383	96	31	all	all	PRON
ejpam-4383	96	32	i	i	PRON
ejpam-4383	96	33	∈	∈	PROPN
ejpam-4383	96	34	i.	i.	PROPN
ejpam-4383	96	35	a.	a.	PROPN
ejpam-4383	96	36	iampan	iampan	PROPN
ejpam-4383	96	37	et	et	PROPN
ejpam-4383	96	38	al	al	PROPN
ejpam-4383	96	39	.	.	PUNCT
ejpam-4383	96	40	/	/	SYM
ejpam-4383	96	41	eur	eur	PROPN
ejpam-4383	96	42	.	.	PUNCT
ejpam-4383	97	1	j.	j.	PROPN
ejpam-4383	97	2	pure	pure	PROPN
ejpam-4383	97	3	appl	appl	PROPN
ejpam-4383	97	4	.	.	PROPN
ejpam-4383	97	5	math	math	PROPN
ejpam-4383	97	6	,	,	PUNCT
ejpam-4383	97	7	15	15	NUM
ejpam-4383	97	8	(	(	PUNCT
ejpam-4383	97	9	3	3	NUM
ejpam-4383	97	10	)	)	PUNCT
ejpam-4383	97	11	(	(	PUNCT
ejpam-4383	97	12	2022	2022	NUM
ejpam-4383	97	13	)	)	PUNCT
ejpam-4383	97	14	,	,	PUNCT
ejpam-4383	97	15	999	999	NUM
ejpam-4383	97	16	-	-	SYM
ejpam-4383	97	17	1014	1014	NUM
ejpam-4383	97	18	1003	1003	NUM
ejpam-4383	97	19	definition	definition	NOUN
ejpam-4383	97	20	8	8	NUM
ejpam-4383	97	21	.	.	PUNCT
ejpam-4383	98	1	let	let	VERB
ejpam-4383	98	2	pi	pi	NOUN
ejpam-4383	98	3	=	=	PUNCT
ejpam-4383	98	4	(	(	PUNCT
ejpam-4383	98	5	pi	pi	NOUN
ejpam-4383	98	6	;	;	PUNCT
ejpam-4383	98	7	∗i	∗i	X
ejpam-4383	98	8	)	)	PUNCT
ejpam-4383	98	9	be	be	VERB
ejpam-4383	98	10	an	an	DET
ejpam-4383	98	11	algebra	algebra	NOUN
ejpam-4383	98	12	for	for	ADP
ejpam-4383	98	13	all	all	PRON
ejpam-4383	98	14	i	i	PRON
ejpam-4383	98	15	∈	∈	PROPN
ejpam-4383	98	16	i.	i.	NOUN
ejpam-4383	98	17	define	define	VERB
ejpam-4383	98	18	the	the	DET
ejpam-4383	98	19	binary	binary	PROPN
ejpam-4383	98	20	operation	operation	NOUN
ejpam-4383	98	21	⊗	⊗	PROPN
ejpam-4383	98	22	on	on	ADP
ejpam-4383	98	23	the	the	DET
ejpam-4383	98	24	external	external	ADJ
ejpam-4383	98	25	direct	direct	ADJ
ejpam-4383	98	26	product	product	NOUN
ejpam-4383	98	27	∏	∏	NUM
ejpam-4383	98	28	i∈i	i∈i	ADJ
ejpam-4383	98	29	pi	pi	NOUN
ejpam-4383	99	1	=	=	PUNCT
ejpam-4383	99	2	(	(	PUNCT
ejpam-4383	99	3	∏	∏	PROPN
ejpam-4383	99	4	i∈i	i∈i	NOUN
ejpam-4383	99	5	pi;⊗	pi;⊗	PROPN
ejpam-4383	99	6	)	)	PUNCT
ejpam-4383	99	7	as	as	SCONJ
ejpam-4383	99	8	follows	follow	VERB
ejpam-4383	99	9	:	:	PUNCT
ejpam-4383	99	10	(	(	PUNCT
ejpam-4383	99	11	∀(pi)i∈i	∀(pi)i∈i	X
ejpam-4383	99	12	,	,	PUNCT
ejpam-4383	99	13	(	(	PUNCT
ejpam-4383	99	14	qi)i∈i	qi)i∈i	NUM
ejpam-4383	99	15	∈	∈	PROPN
ejpam-4383	99	16	∏	∏	PROPN
ejpam-4383	99	17	i∈i	i∈i	NOUN
ejpam-4383	99	18	pi)((pi)i∈i	pi)((pi)i∈i	PROPN
ejpam-4383	99	19	⊗	⊗	PROPN
ejpam-4383	99	20	(	(	PUNCT
ejpam-4383	99	21	qi)i∈i	qi)i∈i	NUM
ejpam-4383	99	22	=	=	SYM
ejpam-4383	99	23	(	(	PUNCT
ejpam-4383	99	24	pi	pi	NOUN
ejpam-4383	99	25	∗i	∗i	PROPN
ejpam-4383	99	26	qi)i∈i	qi)i∈i	NUM
ejpam-4383	99	27	)	)	PUNCT
ejpam-4383	99	28	.	.	PUNCT
ejpam-4383	100	1	(	(	PUNCT
ejpam-4383	100	2	2.1	2.1	NUM
ejpam-4383	100	3	)	)	PUNCT
ejpam-4383	100	4	we	we	PRON
ejpam-4383	100	5	shall	shall	AUX
ejpam-4383	100	6	show	show	VERB
ejpam-4383	100	7	that	that	SCONJ
ejpam-4383	100	8	⊗	⊗	PROPN
ejpam-4383	100	9	is	be	AUX
ejpam-4383	100	10	a	a	DET
ejpam-4383	100	11	binary	binary	ADJ
ejpam-4383	100	12	operation	operation	NOUN
ejpam-4383	100	13	on	on	ADP
ejpam-4383	100	14	∏	∏	PROPN
ejpam-4383	100	15	i∈i	i∈i	ADJ
ejpam-4383	100	16	pi	pi	NOUN
ejpam-4383	100	17	.	.	PUNCT
ejpam-4383	101	1	let	let	VERB
ejpam-4383	101	2	(	(	PUNCT
ejpam-4383	101	3	pi)i∈i	pi)i∈i	NUM
ejpam-4383	101	4	,	,	PUNCT
ejpam-4383	101	5	(	(	PUNCT
ejpam-4383	101	6	qi)i∈i	qi)i∈i	NUM
ejpam-4383	101	7	∈	∈	PROPN
ejpam-4383	101	8	∏	∏	PROPN
ejpam-4383	101	9	i∈i	i∈i	ADJ
ejpam-4383	101	10	pi	pi	NOUN
ejpam-4383	101	11	.	.	PUNCT
ejpam-4383	102	1	since	since	SCONJ
ejpam-4383	102	2	∗i	∗i	PROPN
ejpam-4383	102	3	is	be	AUX
ejpam-4383	102	4	a	a	DET
ejpam-4383	102	5	binary	binary	ADJ
ejpam-4383	102	6	operation	operation	NOUN
ejpam-4383	102	7	on	on	ADP
ejpam-4383	102	8	pi	pi	NOUN
ejpam-4383	102	9	for	for	ADP
ejpam-4383	102	10	all	all	PRON
ejpam-4383	102	11	i	i	PRON
ejpam-4383	102	12	∈	∈	PROPN
ejpam-4383	103	1	i	i	PRON
ejpam-4383	103	2	,	,	PUNCT
ejpam-4383	103	3	we	we	PRON
ejpam-4383	103	4	have	have	VERB
ejpam-4383	103	5	pi	pi	NOUN
ejpam-4383	103	6	∗i	∗i	PROPN
ejpam-4383	103	7	qi	qi	PROPN
ejpam-4383	103	8	∈	∈	PROPN
ejpam-4383	103	9	pi	pi	NOUN
ejpam-4383	103	10	for	for	ADP
ejpam-4383	103	11	all	all	PRON
ejpam-4383	103	12	i	i	PRON
ejpam-4383	103	13	∈	∈	PROPN
ejpam-4383	103	14	i.	i.	NOUN
ejpam-4383	103	15	then	then	ADV
ejpam-4383	103	16	(	(	PUNCT
ejpam-4383	103	17	pi	pi	NOUN
ejpam-4383	103	18	∗i	∗i	PROPN
ejpam-4383	103	19	qi)i∈i	qi)i∈i	NUM
ejpam-4383	103	20	∈	∈	PROPN
ejpam-4383	103	21	∏	∏	PROPN
ejpam-4383	103	22	i∈i	i∈i	ADJ
ejpam-4383	103	23	pi	pi	NOUN
ejpam-4383	103	24	such	such	ADJ
ejpam-4383	103	25	that	that	SCONJ
ejpam-4383	103	26	(	(	PUNCT
ejpam-4383	103	27	pi)i∈i	pi)i∈i	NUM
ejpam-4383	103	28	⊗	⊗	PROPN
ejpam-4383	103	29	(	(	PUNCT
ejpam-4383	103	30	qi)i∈i	qi)i∈i	NUM
ejpam-4383	103	31	=	=	SYM
ejpam-4383	103	32	(	(	PUNCT
ejpam-4383	103	33	pi	pi	NOUN
ejpam-4383	103	34	∗i	∗i	PROPN
ejpam-4383	103	35	qi)i∈i	qi)i∈i	NUM
ejpam-4383	103	36	.	.	PUNCT
ejpam-4383	104	1	let	let	VERB
ejpam-4383	104	2	(	(	PUNCT
ejpam-4383	104	3	pi)i∈i	pi)i∈i	NUM
ejpam-4383	104	4	,	,	PUNCT
ejpam-4383	104	5	(	(	PUNCT
ejpam-4383	104	6	qi)i∈i	qi)i∈i	NUM
ejpam-4383	104	7	,	,	PUNCT
ejpam-4383	104	8	(	(	PUNCT
ejpam-4383	104	9	p	p	NOUN
ejpam-4383	104	10	′	′	NUM
ejpam-4383	104	11	i)i∈i	i)i∈i	PROPN
ejpam-4383	104	12	,	,	PUNCT
ejpam-4383	104	13	(	(	PUNCT
ejpam-4383	104	14	q	q	NOUN
ejpam-4383	104	15	′	′	NUM
ejpam-4383	104	16	i)i∈i	i)i∈i	PROPN
ejpam-4383	104	17	∈	∈	PROPN
ejpam-4383	104	18	∏	∏	PROPN
ejpam-4383	104	19	i∈i	i∈i	ADJ
ejpam-4383	104	20	pi	pi	NOUN
ejpam-4383	104	21	be	be	AUX
ejpam-4383	104	22	such	such	ADJ
ejpam-4383	104	23	that	that	SCONJ
ejpam-4383	104	24	(	(	PUNCT
ejpam-4383	104	25	pi)i∈i	pi)i∈i	NUM
ejpam-4383	104	26	=	=	NOUN
ejpam-4383	104	27	(	(	PUNCT
ejpam-4383	104	28	qi)i∈i	qi)i∈i	NUM
ejpam-4383	104	29	and	and	CCONJ
ejpam-4383	104	30	(	(	PUNCT
ejpam-4383	104	31	p′i)i∈i	p′i)i∈i	NOUN
ejpam-4383	104	32	=	=	PUNCT
ejpam-4383	104	33	(	(	PUNCT
ejpam-4383	104	34	q′i)i∈i	q′i)i∈i	ADV
ejpam-4383	104	35	.	.	PUNCT
ejpam-4383	105	1	we	we	PRON
ejpam-4383	105	2	shall	shall	AUX
ejpam-4383	105	3	show	show	VERB
ejpam-4383	105	4	that	that	SCONJ
ejpam-4383	105	5	(	(	PUNCT
ejpam-4383	105	6	pi)i∈i	pi)i∈i	NUM
ejpam-4383	105	7	⊗	⊗	PROPN
ejpam-4383	105	8	(	(	PUNCT
ejpam-4383	105	9	p′i)i∈i	p′i)i∈i	X
ejpam-4383	105	10	=	=	PUNCT
ejpam-4383	105	11	(	(	PUNCT
ejpam-4383	105	12	qi)i∈i	qi)i∈i	NUM
ejpam-4383	105	13	⊗	⊗	PROPN
ejpam-4383	105	14	(	(	PUNCT
ejpam-4383	105	15	q′i)i∈i	q′i)i∈i	ADV
ejpam-4383	105	16	.	.	PUNCT
ejpam-4383	106	1	then	then	ADV
ejpam-4383	106	2	pi	pi	NOUN
ejpam-4383	106	3	=	=	PUNCT
ejpam-4383	106	4	qi	qi	PROPN
ejpam-4383	106	5	for	for	ADP
ejpam-4383	106	6	all	all	PRON
ejpam-4383	106	7	i	i	PRON
ejpam-4383	106	8	∈	∈	VERB
ejpam-4383	107	1	i	i	PRON
ejpam-4383	107	2	and	and	CCONJ
ejpam-4383	107	3	p′i	p′i	NOUN
ejpam-4383	107	4	=	=	PUNCT
ejpam-4383	107	5	q′i	q′i	ADJ
ejpam-4383	107	6	for	for	ADP
ejpam-4383	107	7	all	all	PRON
ejpam-4383	107	8	i	i	PRON
ejpam-4383	107	9	∈	∈	PROPN
ejpam-4383	107	10	i.	i.	NOUN
ejpam-4383	107	11	since	since	SCONJ
ejpam-4383	107	12	∗i	∗i	PROPN
ejpam-4383	107	13	is	be	AUX
ejpam-4383	107	14	a	a	DET
ejpam-4383	107	15	binary	binary	ADJ
ejpam-4383	107	16	operation	operation	NOUN
ejpam-4383	107	17	on	on	ADP
ejpam-4383	107	18	pi	pi	NOUN
ejpam-4383	107	19	for	for	ADP
ejpam-4383	107	20	all	all	PRON
ejpam-4383	107	21	i	i	PRON
ejpam-4383	107	22	∈	∈	PROPN
ejpam-4383	108	1	i	i	PRON
ejpam-4383	108	2	,	,	PUNCT
ejpam-4383	108	3	we	we	PRON
ejpam-4383	108	4	have	have	VERB
ejpam-4383	108	5	pi	pi	NOUN
ejpam-4383	108	6	∗i	∗i	PROPN
ejpam-4383	108	7	p′i	p′i	NOUN
ejpam-4383	108	8	=	=	SYM
ejpam-4383	108	9	qi	qi	PROPN
ejpam-4383	108	10	∗i	∗i	PROPN
ejpam-4383	108	11	q′i	q′i	VERB
ejpam-4383	108	12	for	for	ADP
ejpam-4383	108	13	all	all	PRON
ejpam-4383	108	14	i	i	PRON
ejpam-4383	108	15	∈	∈	PROPN
ejpam-4383	108	16	i.	i.	NOUN
ejpam-4383	108	17	thus	thus	ADV
ejpam-4383	108	18	(	(	PUNCT
ejpam-4383	108	19	pi)i∈i	pi)i∈i	NUM
ejpam-4383	108	20	⊗	⊗	PROPN
ejpam-4383	108	21	(	(	PUNCT
ejpam-4383	108	22	p′i)i∈i	p′i)i∈i	X
ejpam-4383	108	23	=	=	SYM
ejpam-4383	108	24	(	(	PUNCT
ejpam-4383	108	25	pi	pi	NOUN
ejpam-4383	108	26	∗i	∗i	PROPN
ejpam-4383	108	27	p′i)i∈i	p′i)i∈i	NOUN
ejpam-4383	108	28	=	=	PUNCT
ejpam-4383	108	29	(	(	PUNCT
ejpam-4383	108	30	qi	qi	X
ejpam-4383	108	31	∗i	∗i	PROPN
ejpam-4383	108	32	q′i)i∈i	q′i)i∈i	ADV
ejpam-4383	108	33	=	=	PUNCT
ejpam-4383	108	34	(	(	PUNCT
ejpam-4383	108	35	qi)i∈i	qi)i∈i	NUM
ejpam-4383	108	36	⊗	⊗	PROPN
ejpam-4383	108	37	(	(	PUNCT
ejpam-4383	108	38	q′i)i∈i	q′i)i∈i	ADV
ejpam-4383	108	39	.	.	PUNCT
ejpam-4383	109	1	hence	hence	ADV
ejpam-4383	109	2	,	,	PUNCT
ejpam-4383	109	3	⊗	⊗	PROPN
ejpam-4383	109	4	is	be	AUX
ejpam-4383	109	5	a	a	DET
ejpam-4383	109	6	binary	binary	ADJ
ejpam-4383	109	7	operation	operation	NOUN
ejpam-4383	109	8	on	on	ADP
ejpam-4383	109	9	∏	∏	PROPN
ejpam-4383	109	10	i∈i	i∈i	ADJ
ejpam-4383	109	11	pi	pi	NOUN
ejpam-4383	109	12	.	.	PUNCT
ejpam-4383	110	1	let	let	AUX
ejpam-4383	110	2	pi	pi	NOUN
ejpam-4383	110	3	=	=	PUNCT
ejpam-4383	110	4	(	(	PUNCT
ejpam-4383	110	5	pi	pi	NOUN
ejpam-4383	110	6	;	;	PUNCT
ejpam-4383	110	7	∗i	∗i	PROPN
ejpam-4383	110	8	,	,	PUNCT
ejpam-4383	110	9	0i	0i	NOUN
ejpam-4383	110	10	)	)	PUNCT
ejpam-4383	110	11	be	be	VERB
ejpam-4383	110	12	a	a	DET
ejpam-4383	110	13	b	b	NOUN
ejpam-4383	110	14	-algebra	-algebra	NOUN
ejpam-4383	110	15	for	for	ADP
ejpam-4383	110	16	all	all	PRON
ejpam-4383	110	17	i	i	PRON
ejpam-4383	110	18	∈	∈	PROPN
ejpam-4383	110	19	i.	i.	NOUN
ejpam-4383	110	20	for	for	ADP
ejpam-4383	110	21	i	i	PROPN
ejpam-4383	110	22	∈	∈	PROPN
ejpam-4383	111	1	i	i	PRON
ejpam-4383	111	2	,	,	PUNCT
ejpam-4383	111	3	let	let	VERB
ejpam-4383	111	4	pi	pi	NOUN
ejpam-4383	111	5	∈	∈	PROPN
ejpam-4383	111	6	pi	pi	NOUN
ejpam-4383	111	7	.	.	PUNCT
ejpam-4383	112	1	we	we	PRON
ejpam-4383	112	2	define	define	VERB
ejpam-4383	112	3	the	the	DET
ejpam-4383	112	4	function	function	NOUN
ejpam-4383	112	5	fpi	fpi	NOUN
ejpam-4383	112	6	:	:	PUNCT
ejpam-4383	112	7	i	i	PRON
ejpam-4383	112	8	→	→	SYM
ejpam-4383	112	9	⋃	⋃	NOUN
ejpam-4383	112	10	i∈i	i∈i	ADJ
ejpam-4383	112	11	pi	pi	NOUN
ejpam-4383	112	12	as	as	SCONJ
ejpam-4383	112	13	follows	follow	VERB
ejpam-4383	112	14	:	:	PUNCT
ejpam-4383	112	15	(	(	PUNCT
ejpam-4383	112	16	∀j	∀j	PROPN
ejpam-4383	112	17	∈	∈	PROPN
ejpam-4383	112	18	i	i	NOUN
ejpam-4383	112	19	)	)	PUNCT
ejpam-4383	112	20	(	(	PUNCT
ejpam-4383	112	21	fpi(j	fpi(j	ADV
ejpam-4383	112	22	)	)	PUNCT
ejpam-4383	112	23	=	=	PRON
ejpam-4383	112	24	{	{	PUNCT
ejpam-4383	112	25	pi	pi	NOUN
ejpam-4383	112	26	if	if	SCONJ
ejpam-4383	112	27	j	j	PROPN
ejpam-4383	112	28	=	=	PUNCT
ejpam-4383	112	29	i	i	PRON
ejpam-4383	112	30	0j	0j	VERB
ejpam-4383	112	31	otherwise	otherwise	ADV
ejpam-4383	112	32	)	)	PUNCT
ejpam-4383	112	33	.	.	PUNCT
ejpam-4383	113	1	(	(	PUNCT
ejpam-4383	113	2	2.2	2.2	NUM
ejpam-4383	113	3	)	)	PUNCT
ejpam-4383	113	4	then	then	ADV
ejpam-4383	113	5	fpi	fpi	PROPN
ejpam-4383	113	6	∈	∈	PROPN
ejpam-4383	113	7	∏	∏	PROPN
ejpam-4383	113	8	i∈i	i∈i	ADJ
ejpam-4383	113	9	pi	pi	NOUN
ejpam-4383	113	10	.	.	PUNCT
ejpam-4383	114	1	lemma	lemma	PROPN
ejpam-4383	114	2	1	1	X
ejpam-4383	114	3	.	.	PUNCT
ejpam-4383	115	1	let	let	VERB
ejpam-4383	115	2	pi	pi	NOUN
ejpam-4383	115	3	=	=	PUNCT
ejpam-4383	115	4	(	(	PUNCT
ejpam-4383	115	5	pi	pi	NOUN
ejpam-4383	115	6	;	;	PUNCT
ejpam-4383	115	7	∗i	∗i	PROPN
ejpam-4383	115	8	,	,	PUNCT
ejpam-4383	115	9	0i	0i	NOUN
ejpam-4383	115	10	)	)	PUNCT
ejpam-4383	115	11	be	be	VERB
ejpam-4383	115	12	a	a	DET
ejpam-4383	115	13	b	b	NOUN
ejpam-4383	115	14	-	-	PUNCT
ejpam-4383	115	15	algebra	algebra	NOUN
ejpam-4383	115	16	for	for	ADP
ejpam-4383	115	17	all	all	PRON
ejpam-4383	115	18	i	i	PRON
ejpam-4383	115	19	∈	∈	PROPN
ejpam-4383	115	20	i.	i.	NOUN
ejpam-4383	115	21	for	for	ADP
ejpam-4383	115	22	i	i	PROPN
ejpam-4383	115	23	∈	∈	PROPN
ejpam-4383	116	1	i	i	PRON
ejpam-4383	116	2	,	,	PUNCT
ejpam-4383	116	3	let	let	VERB
ejpam-4383	116	4	pi	pi	ADV
ejpam-4383	116	5	,	,	PUNCT
ejpam-4383	116	6	qi	qi	PROPN
ejpam-4383	116	7	∈	∈	PROPN
ejpam-4383	116	8	pi	pi	NOUN
ejpam-4383	116	9	.	.	PUNCT
ejpam-4383	117	1	then	then	ADV
ejpam-4383	117	2	fpi	fpi	PROPN
ejpam-4383	117	3	⊗	⊗	PROPN
ejpam-4383	117	4	fqi	fqi	NOUN
ejpam-4383	117	5	=	=	SYM
ejpam-4383	117	6	fpi∗iqi	fpi∗iqi	NOUN
ejpam-4383	117	7	.	.	PUNCT
ejpam-4383	118	1	proof	proof	NOUN
ejpam-4383	118	2	.	.	PUNCT
ejpam-4383	119	1	now	now	ADV
ejpam-4383	119	2	,	,	PUNCT
ejpam-4383	119	3	(	(	PUNCT
ejpam-4383	119	4	∀j	∀j	PROPN
ejpam-4383	119	5	∈	∈	PROPN
ejpam-4383	119	6	i	i	NOUN
ejpam-4383	119	7	)	)	PUNCT
ejpam-4383	119	8	(	(	PUNCT
ejpam-4383	119	9	(	(	PUNCT
ejpam-4383	119	10	fpi	fpi	PROPN
ejpam-4383	119	11	⊗	⊗	PROPN
ejpam-4383	119	12	fqi)(j	fqi)(j	PROPN
ejpam-4383	119	13	)	)	PUNCT
ejpam-4383	119	14	=	=	PRON
ejpam-4383	119	15	{	{	PUNCT
ejpam-4383	119	16	pi	pi	NOUN
ejpam-4383	119	17	∗i	∗i	PROPN
ejpam-4383	119	18	qi	qi	PROPN
ejpam-4383	119	19	if	if	SCONJ
ejpam-4383	119	20	j	j	PROPN
ejpam-4383	119	21	=	=	PRON
ejpam-4383	120	1	i	i	PRON
ejpam-4383	120	2	0j	0j	VERB
ejpam-4383	120	3	∗j	∗j	PROPN
ejpam-4383	120	4	0j	0j	NOUN
ejpam-4383	120	5	otherwise	otherwise	ADV
ejpam-4383	120	6	)	)	PUNCT
ejpam-4383	120	7	.	.	PUNCT
ejpam-4383	121	1	by	by	ADP
ejpam-4383	121	2	(	(	PUNCT
ejpam-4383	121	3	b-1	b-1	PROPN
ejpam-4383	121	4	)	)	PUNCT
ejpam-4383	121	5	,	,	PUNCT
ejpam-4383	121	6	we	we	PRON
ejpam-4383	121	7	have	have	VERB
ejpam-4383	121	8	(	(	PUNCT
ejpam-4383	121	9	∀j	∀j	PROPN
ejpam-4383	121	10	∈	∈	PROPN
ejpam-4383	121	11	i	i	NOUN
ejpam-4383	121	12	)	)	PUNCT
ejpam-4383	121	13	(	(	PUNCT
ejpam-4383	121	14	(	(	PUNCT
ejpam-4383	121	15	fpi	fpi	PROPN
ejpam-4383	121	16	⊗	⊗	PROPN
ejpam-4383	121	17	fqi)(j	fqi)(j	PROPN
ejpam-4383	121	18	)	)	PUNCT
ejpam-4383	121	19	=	=	PRON
ejpam-4383	121	20	{	{	PUNCT
ejpam-4383	122	1	pi	pi	NOUN
ejpam-4383	122	2	∗i	∗i	PROPN
ejpam-4383	122	3	qi	qi	PROPN
ejpam-4383	122	4	if	if	SCONJ
ejpam-4383	122	5	j	j	PROPN
ejpam-4383	122	6	=	=	PUNCT
ejpam-4383	123	1	i	i	PRON
ejpam-4383	123	2	0j	0j	VERB
ejpam-4383	123	3	otherwise	otherwise	ADV
ejpam-4383	123	4	)	)	PUNCT
ejpam-4383	123	5	.	.	PUNCT
ejpam-4383	124	1	by	by	ADP
ejpam-4383	124	2	(	(	PUNCT
ejpam-4383	124	3	2.2	2.2	NUM
ejpam-4383	124	4	)	)	PUNCT
ejpam-4383	124	5	,	,	PUNCT
ejpam-4383	124	6	we	we	PRON
ejpam-4383	124	7	have	have	VERB
ejpam-4383	124	8	fpi	fpi	PROPN
ejpam-4383	124	9	⊗	⊗	PROPN
ejpam-4383	124	10	fqi	fqi	NOUN
ejpam-4383	124	11	=	=	SYM
ejpam-4383	124	12	fpi∗iqi	fpi∗iqi	PROPN
ejpam-4383	124	13	.	.	PUNCT
ejpam-4383	125	1	the	the	DET
ejpam-4383	125	2	following	follow	VERB
ejpam-4383	125	3	theorem	theorem	NOUN
ejpam-4383	125	4	shows	show	VERB
ejpam-4383	125	5	that	that	SCONJ
ejpam-4383	125	6	the	the	DET
ejpam-4383	125	7	direct	direct	ADJ
ejpam-4383	125	8	product	product	NOUN
ejpam-4383	125	9	of	of	ADP
ejpam-4383	125	10	b	b	NOUN
ejpam-4383	125	11	-algebras	-algebra	NOUN
ejpam-4383	125	12	in	in	ADP
ejpam-4383	125	13	term	term	NOUN
ejpam-4383	125	14	of	of	ADP
ejpam-4383	125	15	infinite	infinite	ADJ
ejpam-4383	125	16	family	family	NOUN
ejpam-4383	125	17	of	of	ADP
ejpam-4383	125	18	b	b	PROPN
ejpam-4383	125	19	-algebras	-algebras	PROPN
ejpam-4383	125	20	is	be	AUX
ejpam-4383	125	21	also	also	ADV
ejpam-4383	125	22	a	a	DET
ejpam-4383	125	23	b	b	NOUN
ejpam-4383	125	24	-algebra	-algebra	NOUN
ejpam-4383	125	25	which	which	PRON
ejpam-4383	125	26	is	be	AUX
ejpam-4383	125	27	more	more	ADV
ejpam-4383	125	28	generalized	generalized	ADJ
ejpam-4383	125	29	than	than	ADP
ejpam-4383	125	30	theorem	theorem	ADJ
ejpam-4383	125	31	3	3	NUM
ejpam-4383	125	32	.	.	PUNCT
ejpam-4383	125	33	a.	a.	NOUN
ejpam-4383	125	34	iampan	iampan	PROPN
ejpam-4383	125	35	et	et	PROPN
ejpam-4383	125	36	al	al	PROPN
ejpam-4383	125	37	.	.	PUNCT
ejpam-4383	125	38	/	/	SYM
ejpam-4383	125	39	eur	eur	PROPN
ejpam-4383	125	40	.	.	PUNCT
ejpam-4383	126	1	j.	j.	PROPN
ejpam-4383	126	2	pure	pure	PROPN
ejpam-4383	126	3	appl	appl	PROPN
ejpam-4383	126	4	.	.	PROPN
ejpam-4383	126	5	math	math	PROPN
ejpam-4383	126	6	,	,	PUNCT
ejpam-4383	126	7	15	15	NUM
ejpam-4383	126	8	(	(	PUNCT
ejpam-4383	126	9	3	3	NUM
ejpam-4383	126	10	)	)	PUNCT
ejpam-4383	126	11	(	(	PUNCT
ejpam-4383	126	12	2022	2022	NUM
ejpam-4383	126	13	)	)	PUNCT
ejpam-4383	126	14	,	,	PUNCT
ejpam-4383	126	15	999	999	NUM
ejpam-4383	126	16	-	-	SYM
ejpam-4383	126	17	1014	1014	NUM
ejpam-4383	126	18	1004	1004	NUM
ejpam-4383	126	19	theorem	theorem	VERB
ejpam-4383	126	20	4	4	NUM
ejpam-4383	126	21	.	.	PUNCT
ejpam-4383	126	22	pi	pi	NOUN
ejpam-4383	127	1	=	=	PUNCT
ejpam-4383	127	2	(	(	PUNCT
ejpam-4383	127	3	pi	pi	NOUN
ejpam-4383	127	4	;	;	PUNCT
ejpam-4383	127	5	∗i	∗i	PROPN
ejpam-4383	127	6	,	,	PUNCT
ejpam-4383	127	7	0i	0i	NOUN
ejpam-4383	127	8	)	)	PUNCT
ejpam-4383	127	9	is	be	AUX
ejpam-4383	127	10	a	a	DET
ejpam-4383	127	11	b	b	NOUN
ejpam-4383	127	12	-	-	PUNCT
ejpam-4383	127	13	algebra	algebra	NOUN
ejpam-4383	127	14	for	for	ADP
ejpam-4383	127	15	all	all	PRON
ejpam-4383	127	16	i	i	PRON
ejpam-4383	127	17	∈	∈	VERB
ejpam-4383	128	1	i	i	PRON
ejpam-4383	128	2	if	if	SCONJ
ejpam-4383	128	3	and	and	CCONJ
ejpam-4383	128	4	only	only	ADV
ejpam-4383	129	1	if	if	SCONJ
ejpam-4383	129	2	∏	∏	NUM
ejpam-4383	129	3	i∈i	i∈i	ADJ
ejpam-4383	129	4	pi	pi	NOUN
ejpam-4383	129	5	=	=	PUNCT
ejpam-4383	129	6	(	(	PUNCT
ejpam-4383	129	7	∏	∏	X
ejpam-4383	129	8	i∈i	i∈i	ADJ
ejpam-4383	129	9	pi;⊗	pi;⊗	PROPN
ejpam-4383	129	10	,	,	PUNCT
ejpam-4383	129	11	(	(	PUNCT
ejpam-4383	129	12	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	129	13	)	)	PUNCT
ejpam-4383	129	14	is	be	AUX
ejpam-4383	129	15	a	a	DET
ejpam-4383	129	16	b	b	NOUN
ejpam-4383	129	17	-	-	PUNCT
ejpam-4383	129	18	algebra	algebra	NOUN
ejpam-4383	129	19	,	,	PUNCT
ejpam-4383	129	20	where	where	SCONJ
ejpam-4383	129	21	the	the	DET
ejpam-4383	129	22	binary	binary	PROPN
ejpam-4383	129	23	operation	operation	PROPN
ejpam-4383	129	24	⊗	⊗	PROPN
ejpam-4383	129	25	is	be	AUX
ejpam-4383	129	26	defined	define	VERB
ejpam-4383	129	27	in	in	ADP
ejpam-4383	129	28	definition	definition	NOUN
ejpam-4383	129	29	8	8	NUM
ejpam-4383	129	30	.	.	PUNCT
ejpam-4383	130	1	proof	proof	NOUN
ejpam-4383	130	2	.	.	PUNCT
ejpam-4383	131	1	assume	assume	VERB
ejpam-4383	131	2	that	that	SCONJ
ejpam-4383	131	3	pi	pi	NOUN
ejpam-4383	131	4	=	=	PUNCT
ejpam-4383	131	5	(	(	PUNCT
ejpam-4383	131	6	pi	pi	NOUN
ejpam-4383	131	7	;	;	PUNCT
ejpam-4383	131	8	∗i	∗i	PROPN
ejpam-4383	131	9	,	,	PUNCT
ejpam-4383	131	10	0i	0i	NOUN
ejpam-4383	131	11	)	)	PUNCT
ejpam-4383	131	12	is	be	AUX
ejpam-4383	131	13	a	a	DET
ejpam-4383	131	14	b	b	NOUN
ejpam-4383	131	15	-algebra	-algebra	NOUN
ejpam-4383	131	16	for	for	ADP
ejpam-4383	131	17	all	all	PRON
ejpam-4383	131	18	i	i	PRON
ejpam-4383	131	19	∈	∈	PROPN
ejpam-4383	131	20	i.	i.	NOUN
ejpam-4383	131	21	(	(	PUNCT
ejpam-4383	131	22	b-1	b-1	PROPN
ejpam-4383	131	23	)	)	PUNCT
ejpam-4383	132	1	let	let	VERB
ejpam-4383	132	2	(	(	PUNCT
ejpam-4383	132	3	pi)i∈i	pi)i∈i	NUM
ejpam-4383	132	4	∈	∈	PROPN
ejpam-4383	132	5	∏	∏	PROPN
ejpam-4383	132	6	i∈i	i∈i	ADJ
ejpam-4383	132	7	pi	pi	NOUN
ejpam-4383	132	8	.	.	PUNCT
ejpam-4383	133	1	since	since	SCONJ
ejpam-4383	133	2	pi	pi	NOUN
ejpam-4383	133	3	satisfies	satisfie	NOUN
ejpam-4383	133	4	(	(	PUNCT
ejpam-4383	133	5	b-1	b-1	PROPN
ejpam-4383	133	6	)	)	PUNCT
ejpam-4383	133	7	,	,	PUNCT
ejpam-4383	133	8	we	we	PRON
ejpam-4383	133	9	have	have	VERB
ejpam-4383	133	10	pi	pi	NOUN
ejpam-4383	133	11	∗i	∗i	PROPN
ejpam-4383	133	12	pi	pi	NOUN
ejpam-4383	133	13	=	=	PUNCT
ejpam-4383	133	14	0i	0i	NOUN
ejpam-4383	133	15	for	for	ADP
ejpam-4383	133	16	all	all	PRON
ejpam-4383	133	17	i	i	PRON
ejpam-4383	133	18	∈	∈	PROPN
ejpam-4383	133	19	i.	i.	NOUN
ejpam-4383	133	20	thus	thus	ADV
ejpam-4383	133	21	(	(	PUNCT
ejpam-4383	133	22	pi)i∈i	pi)i∈i	NUM
ejpam-4383	133	23	⊗	⊗	PROPN
ejpam-4383	133	24	(	(	PUNCT
ejpam-4383	133	25	pi)i∈i	pi)i∈i	NUM
ejpam-4383	133	26	=	=	SYM
ejpam-4383	133	27	(	(	PUNCT
ejpam-4383	133	28	pi	pi	NOUN
ejpam-4383	133	29	∗i	∗i	PROPN
ejpam-4383	133	30	pi)i∈i	pi)i∈i	NUM
ejpam-4383	133	31	=	=	SYM
ejpam-4383	133	32	(	(	PUNCT
ejpam-4383	133	33	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	133	34	.	.	PUNCT
ejpam-4383	134	1	(	(	PUNCT
ejpam-4383	134	2	b-2	b-2	NOUN
ejpam-4383	134	3	)	)	PUNCT
ejpam-4383	134	4	let	let	VERB
ejpam-4383	134	5	(	(	PUNCT
ejpam-4383	134	6	pi)i∈i	pi)i∈i	NUM
ejpam-4383	134	7	∈	∈	PROPN
ejpam-4383	134	8	∏	∏	PROPN
ejpam-4383	134	9	i∈i	i∈i	ADJ
ejpam-4383	134	10	pi	pi	NOUN
ejpam-4383	134	11	.	.	PUNCT
ejpam-4383	135	1	since	since	SCONJ
ejpam-4383	135	2	pi	pi	NOUN
ejpam-4383	135	3	satisfies	satisfie	NOUN
ejpam-4383	135	4	(	(	PUNCT
ejpam-4383	135	5	b-2	b-2	NOUN
ejpam-4383	135	6	)	)	PUNCT
ejpam-4383	135	7	,	,	PUNCT
ejpam-4383	135	8	we	we	PRON
ejpam-4383	135	9	have	have	VERB
ejpam-4383	135	10	pi	pi	NOUN
ejpam-4383	135	11	∗i	∗i	PROPN
ejpam-4383	135	12	0i	0i	NOUN
ejpam-4383	136	1	=	=	PUNCT
ejpam-4383	136	2	pi	pi	NOUN
ejpam-4383	136	3	for	for	ADP
ejpam-4383	136	4	all	all	DET
ejpam-4383	136	5	i	i	PRON
ejpam-4383	136	6	∈	∈	PROPN
ejpam-4383	136	7	i.	i.	NOUN
ejpam-4383	136	8	thus	thus	ADV
ejpam-4383	136	9	(	(	PUNCT
ejpam-4383	136	10	pi)i∈i	pi)i∈i	NUM
ejpam-4383	136	11	⊗	⊗	PROPN
ejpam-4383	136	12	(	(	PUNCT
ejpam-4383	136	13	0i)i∈i	0i)i∈i	NUM
ejpam-4383	136	14	=	=	SYM
ejpam-4383	136	15	(	(	PUNCT
ejpam-4383	136	16	pi	pi	NOUN
ejpam-4383	136	17	∗i	∗i	PROPN
ejpam-4383	136	18	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	137	1	=	=	SYM
ejpam-4383	137	2	(	(	PUNCT
ejpam-4383	137	3	pi)i∈i	pi)i∈i	NUM
ejpam-4383	137	4	.	.	PUNCT
ejpam-4383	138	1	(	(	PUNCT
ejpam-4383	138	2	b-3	b-3	PROPN
ejpam-4383	138	3	)	)	PUNCT
ejpam-4383	138	4	let	let	VERB
ejpam-4383	138	5	(	(	PUNCT
ejpam-4383	138	6	pi)i∈i	pi)i∈i	NUM
ejpam-4383	138	7	,	,	PUNCT
ejpam-4383	138	8	(	(	PUNCT
ejpam-4383	138	9	qi)i∈i	qi)i∈i	NUM
ejpam-4383	138	10	,	,	PUNCT
ejpam-4383	138	11	(	(	PUNCT
ejpam-4383	138	12	ri)i∈i	ri)i∈i	NUM
ejpam-4383	138	13	∈	∈	PROPN
ejpam-4383	138	14	∏	∏	PROPN
ejpam-4383	138	15	i∈i	i∈i	ADJ
ejpam-4383	138	16	pi	pi	NOUN
ejpam-4383	138	17	.	.	PUNCT
ejpam-4383	139	1	since	since	SCONJ
ejpam-4383	139	2	pi	pi	NOUN
ejpam-4383	139	3	satisfies	satisfie	NOUN
ejpam-4383	139	4	(	(	PUNCT
ejpam-4383	139	5	b-3	b-3	PROPN
ejpam-4383	139	6	)	)	PUNCT
ejpam-4383	139	7	,	,	PUNCT
ejpam-4383	139	8	we	we	PRON
ejpam-4383	139	9	have	have	VERB
ejpam-4383	139	10	(	(	PUNCT
ejpam-4383	139	11	pi	pi	NOUN
ejpam-4383	139	12	∗i	∗i	PROPN
ejpam-4383	139	13	qi	qi	PROPN
ejpam-4383	139	14	)	)	PUNCT
ejpam-4383	139	15	∗i	∗i	PROPN
ejpam-4383	139	16	ri	ri	PROPN
ejpam-4383	139	17	=	=	PUNCT
ejpam-4383	139	18	pi	pi	NOUN
ejpam-4383	139	19	∗i	∗i	PROPN
ejpam-4383	139	20	(	(	PUNCT
ejpam-4383	139	21	ri	ri	PROPN
ejpam-4383	139	22	∗i	∗i	PROPN
ejpam-4383	139	23	(	(	PUNCT
ejpam-4383	139	24	0i	0i	X
ejpam-4383	139	25	∗i	∗i	PROPN
ejpam-4383	139	26	qi	qi	PROPN
ejpam-4383	139	27	)	)	PUNCT
ejpam-4383	139	28	)	)	PUNCT
ejpam-4383	139	29	for	for	ADP
ejpam-4383	139	30	all	all	PRON
ejpam-4383	139	31	i	i	PRON
ejpam-4383	139	32	∈	∈	PROPN
ejpam-4383	139	33	i.	i.	NOUN
ejpam-4383	139	34	thus	thus	ADV
ejpam-4383	139	35	(	(	PUNCT
ejpam-4383	139	36	(	(	PUNCT
ejpam-4383	139	37	pi)i∈i	pi)i∈i	NUM
ejpam-4383	139	38	⊗	⊗	PROPN
ejpam-4383	139	39	(	(	PUNCT
ejpam-4383	139	40	qi)i∈i)⊗	qi)i∈i)⊗	NOUN
ejpam-4383	139	41	(	(	PUNCT
ejpam-4383	139	42	ri)i∈i	ri)i∈i	NUM
ejpam-4383	139	43	=	=	NOUN
ejpam-4383	139	44	(	(	PUNCT
ejpam-4383	139	45	pi	pi	NOUN
ejpam-4383	139	46	∗i	∗i	PROPN
ejpam-4383	139	47	qi)i∈i	qi)i∈i	NUM
ejpam-4383	139	48	⊗	⊗	NUM
ejpam-4383	139	49	(	(	PUNCT
ejpam-4383	139	50	ri)i∈i	ri)i∈i	NUM
ejpam-4383	139	51	=	=	SYM
ejpam-4383	139	52	(	(	PUNCT
ejpam-4383	139	53	(	(	PUNCT
ejpam-4383	139	54	pi	pi	NOUN
ejpam-4383	139	55	∗i	∗i	PROPN
ejpam-4383	139	56	qi	qi	PROPN
ejpam-4383	139	57	)	)	PUNCT
ejpam-4383	139	58	∗i	∗i	PROPN
ejpam-4383	139	59	ri)i∈i	ri)i∈i	ADJ
ejpam-4383	139	60	=	=	SYM
ejpam-4383	139	61	(	(	PUNCT
ejpam-4383	139	62	pi	pi	NOUN
ejpam-4383	139	63	∗i	∗i	PROPN
ejpam-4383	139	64	(	(	PUNCT
ejpam-4383	139	65	ri	ri	PROPN
ejpam-4383	139	66	∗i	∗i	PROPN
ejpam-4383	139	67	(	(	PUNCT
ejpam-4383	139	68	0i	0i	X
ejpam-4383	139	69	∗i	∗i	PROPN
ejpam-4383	139	70	qi)))i∈i	qi)))i∈i	PROPN
ejpam-4383	139	71	=	=	PUNCT
ejpam-4383	139	72	(	(	PUNCT
ejpam-4383	139	73	pi)i∈i	pi)i∈i	NUM
ejpam-4383	139	74	⊗	⊗	PROPN
ejpam-4383	139	75	(	(	PUNCT
ejpam-4383	139	76	ri	ri	PROPN
ejpam-4383	139	77	∗i	∗i	PROPN
ejpam-4383	139	78	(	(	PUNCT
ejpam-4383	139	79	0i	0i	X
ejpam-4383	139	80	∗i	∗i	PROPN
ejpam-4383	139	81	qi))i∈i	qi))i∈i	PROPN
ejpam-4383	139	82	=	=	SYM
ejpam-4383	139	83	(	(	PUNCT
ejpam-4383	139	84	pi)i∈i	pi)i∈i	NUM
ejpam-4383	139	85	⊗	⊗	PROPN
ejpam-4383	139	86	(	(	PUNCT
ejpam-4383	139	87	(	(	PUNCT
ejpam-4383	139	88	ri)i∈i	ri)i∈i	ADJ
ejpam-4383	139	89	⊗	⊗	PROPN
ejpam-4383	139	90	(	(	PUNCT
ejpam-4383	139	91	0i	0i	X
ejpam-4383	139	92	∗i	∗i	PROPN
ejpam-4383	139	93	qi)i∈i	qi)i∈i	NUM
ejpam-4383	139	94	)	)	PUNCT
ejpam-4383	140	1	=	=	SYM
ejpam-4383	140	2	(	(	PUNCT
ejpam-4383	140	3	pi)i∈i	pi)i∈i	NUM
ejpam-4383	140	4	⊗	⊗	PROPN
ejpam-4383	140	5	(	(	PUNCT
ejpam-4383	140	6	(	(	PUNCT
ejpam-4383	140	7	ri)i∈i	ri)i∈i	ADJ
ejpam-4383	140	8	⊗	⊗	PROPN
ejpam-4383	140	9	(	(	PUNCT
ejpam-4383	140	10	(	(	PUNCT
ejpam-4383	140	11	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	140	12	⊗	⊗	PROPN
ejpam-4383	140	13	(	(	PUNCT
ejpam-4383	140	14	qi)i∈i	qi)i∈i	NUM
ejpam-4383	140	15	)	)	PUNCT
ejpam-4383	140	16	)	)	PUNCT
ejpam-4383	140	17	.	.	PUNCT
ejpam-4383	141	1	hence	hence	ADV
ejpam-4383	141	2	,	,	PUNCT
ejpam-4383	141	3	∏	∏	PROPN
ejpam-4383	141	4	i∈i	i∈i	ADJ
ejpam-4383	141	5	pi	pi	NOUN
ejpam-4383	141	6	=	=	PUNCT
ejpam-4383	141	7	(	(	PUNCT
ejpam-4383	141	8	∏	∏	X
ejpam-4383	141	9	i∈i	i∈i	ADJ
ejpam-4383	141	10	pi;⊗	pi;⊗	PROPN
ejpam-4383	141	11	,	,	PUNCT
ejpam-4383	141	12	(	(	PUNCT
ejpam-4383	141	13	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	141	14	)	)	PUNCT
ejpam-4383	141	15	is	be	AUX
ejpam-4383	141	16	a	a	DET
ejpam-4383	141	17	b	b	NOUN
ejpam-4383	141	18	-algebra	-algebra	NOUN
ejpam-4383	141	19	.	.	PUNCT
ejpam-4383	142	1	conversely	conversely	ADV
ejpam-4383	142	2	,	,	PUNCT
ejpam-4383	142	3	assume	assume	VERB
ejpam-4383	142	4	that	that	SCONJ
ejpam-4383	142	5	∏	∏	NUM
ejpam-4383	142	6	i∈i	i∈i	ADJ
ejpam-4383	142	7	pi	pi	NOUN
ejpam-4383	142	8	=	=	PUNCT
ejpam-4383	142	9	(	(	PUNCT
ejpam-4383	142	10	∏	∏	X
ejpam-4383	142	11	i∈i	i∈i	ADJ
ejpam-4383	142	12	pi;⊗	pi;⊗	PROPN
ejpam-4383	142	13	,	,	PUNCT
ejpam-4383	142	14	(	(	PUNCT
ejpam-4383	142	15	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	142	16	)	)	PUNCT
ejpam-4383	142	17	is	be	AUX
ejpam-4383	142	18	a	a	DET
ejpam-4383	142	19	b	b	NOUN
ejpam-4383	142	20	-algebra	-algebra	NOUN
ejpam-4383	142	21	,	,	PUNCT
ejpam-4383	142	22	where	where	SCONJ
ejpam-4383	142	23	the	the	DET
ejpam-4383	142	24	binary	binary	PROPN
ejpam-4383	142	25	operation	operation	PROPN
ejpam-4383	142	26	⊗	⊗	PROPN
ejpam-4383	142	27	is	be	AUX
ejpam-4383	142	28	defined	define	VERB
ejpam-4383	142	29	in	in	ADP
ejpam-4383	142	30	definition	definition	NOUN
ejpam-4383	142	31	8	8	NUM
ejpam-4383	142	32	.	.	PUNCT
ejpam-4383	143	1	let	let	VERB
ejpam-4383	143	2	i	i	PRON
ejpam-4383	143	3	∈	∈	PROPN
ejpam-4383	143	4	i.	i.	NOUN
ejpam-4383	143	5	(	(	PUNCT
ejpam-4383	143	6	b-1	b-1	PROPN
ejpam-4383	143	7	)	)	PUNCT
ejpam-4383	143	8	let	let	VERB
ejpam-4383	143	9	pi	pi	NOUN
ejpam-4383	143	10	∈	∈	PROPN
ejpam-4383	143	11	pi	pi	NOUN
ejpam-4383	143	12	.	.	PUNCT
ejpam-4383	144	1	then	then	ADV
ejpam-4383	144	2	fpi	fpi	PROPN
ejpam-4383	144	3	∈	∈	PROPN
ejpam-4383	144	4	∏	∏	PROPN
ejpam-4383	144	5	i∈i	i∈i	ADJ
ejpam-4383	144	6	pi	pi	NOUN
ejpam-4383	144	7	,	,	PUNCT
ejpam-4383	144	8	which	which	PRON
ejpam-4383	144	9	is	be	AUX
ejpam-4383	144	10	defined	define	VERB
ejpam-4383	144	11	by	by	ADP
ejpam-4383	144	12	(	(	PUNCT
ejpam-4383	144	13	2.2	2.2	NUM
ejpam-4383	144	14	)	)	PUNCT
ejpam-4383	144	15	.	.	PUNCT
ejpam-4383	145	1	since	since	SCONJ
ejpam-4383	145	2	∏	∏	NUM
ejpam-4383	145	3	i∈i	i∈i	ADJ
ejpam-4383	145	4	pi	pi	NOUN
ejpam-4383	145	5	satisfies	satisfie	NOUN
ejpam-4383	145	6	(	(	PUNCT
ejpam-4383	145	7	b-1	b-1	PROPN
ejpam-4383	145	8	)	)	PUNCT
ejpam-4383	145	9	,	,	PUNCT
ejpam-4383	145	10	we	we	PRON
ejpam-4383	145	11	have	have	VERB
ejpam-4383	145	12	fpi	fpi	PROPN
ejpam-4383	145	13	⊗	⊗	PROPN
ejpam-4383	145	14	fpi	fpi	PROPN
ejpam-4383	145	15	=	=	PROPN
ejpam-4383	145	16	(	(	PUNCT
ejpam-4383	145	17	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	145	18	.	.	PUNCT
ejpam-4383	146	1	now	now	ADV
ejpam-4383	146	2	,	,	PUNCT
ejpam-4383	146	3	(	(	PUNCT
ejpam-4383	146	4	∀j	∀j	PROPN
ejpam-4383	146	5	∈	∈	PROPN
ejpam-4383	146	6	i	i	NOUN
ejpam-4383	146	7	)	)	PUNCT
ejpam-4383	146	8	(	(	PUNCT
ejpam-4383	146	9	(	(	PUNCT
ejpam-4383	146	10	fpi	fpi	PROPN
ejpam-4383	146	11	⊗	⊗	PROPN
ejpam-4383	146	12	fpi)(j	fpi)(j	NOUN
ejpam-4383	146	13	)	)	PUNCT
ejpam-4383	146	14	=	=	PRON
ejpam-4383	146	15	{	{	PUNCT
ejpam-4383	146	16	pi	pi	NOUN
ejpam-4383	146	17	∗i	∗i	PROPN
ejpam-4383	146	18	pi	pi	NOUN
ejpam-4383	146	19	if	if	SCONJ
ejpam-4383	146	20	j	j	PROPN
ejpam-4383	146	21	=	=	PRON
ejpam-4383	147	1	i	i	PRON
ejpam-4383	147	2	0j	0j	VERB
ejpam-4383	147	3	∗j	∗j	PROPN
ejpam-4383	147	4	0j	0j	NOUN
ejpam-4383	147	5	otherwise	otherwise	ADV
ejpam-4383	147	6	)	)	PUNCT
ejpam-4383	147	7	,	,	PUNCT
ejpam-4383	147	8	this	this	PRON
ejpam-4383	147	9	implies	imply	VERB
ejpam-4383	147	10	that	that	SCONJ
ejpam-4383	147	11	pi	pi	NOUN
ejpam-4383	147	12	∗i	∗i	PROPN
ejpam-4383	147	13	pi	pi	NOUN
ejpam-4383	147	14	=	=	PUNCT
ejpam-4383	147	15	0i	0i	NOUN
ejpam-4383	147	16	.	.	PUNCT
ejpam-4383	148	1	(	(	PUNCT
ejpam-4383	148	2	b-2	b-2	NOUN
ejpam-4383	148	3	)	)	PUNCT
ejpam-4383	148	4	let	let	VERB
ejpam-4383	148	5	pi	pi	NOUN
ejpam-4383	148	6	∈	∈	PROPN
ejpam-4383	148	7	pi	pi	NOUN
ejpam-4383	148	8	.	.	PUNCT
ejpam-4383	149	1	then	then	ADV
ejpam-4383	149	2	fpi	fpi	PROPN
ejpam-4383	149	3	∈	∈	PROPN
ejpam-4383	149	4	∏	∏	PROPN
ejpam-4383	149	5	i∈i	i∈i	ADJ
ejpam-4383	149	6	pi	pi	NOUN
ejpam-4383	149	7	,	,	PUNCT
ejpam-4383	149	8	which	which	PRON
ejpam-4383	149	9	is	be	AUX
ejpam-4383	149	10	defined	define	VERB
ejpam-4383	149	11	by	by	ADP
ejpam-4383	149	12	(	(	PUNCT
ejpam-4383	149	13	2.2	2.2	NUM
ejpam-4383	149	14	)	)	PUNCT
ejpam-4383	149	15	.	.	PUNCT
ejpam-4383	150	1	since	since	SCONJ
ejpam-4383	150	2	∏	∏	NUM
ejpam-4383	150	3	i∈i	i∈i	ADJ
ejpam-4383	150	4	pi	pi	NOUN
ejpam-4383	150	5	satisfies	satisfie	NOUN
ejpam-4383	150	6	(	(	PUNCT
ejpam-4383	150	7	b-2	b-2	NOUN
ejpam-4383	150	8	)	)	PUNCT
ejpam-4383	150	9	,	,	PUNCT
ejpam-4383	150	10	we	we	PRON
ejpam-4383	150	11	have	have	VERB
ejpam-4383	150	12	fpi	fpi	PROPN
ejpam-4383	150	13	⊗	⊗	PROPN
ejpam-4383	150	14	(	(	PUNCT
ejpam-4383	150	15	0i)i∈i	0i)i∈i	NUM
ejpam-4383	150	16	=	=	SYM
ejpam-4383	150	17	fpi	fpi	PROPN
ejpam-4383	150	18	.	.	PUNCT
ejpam-4383	151	1	now	now	ADV
ejpam-4383	151	2	,	,	PUNCT
ejpam-4383	151	3	(	(	PUNCT
ejpam-4383	151	4	∀j	∀j	PROPN
ejpam-4383	151	5	∈	∈	PROPN
ejpam-4383	151	6	i	i	NOUN
ejpam-4383	151	7	)	)	PUNCT
ejpam-4383	151	8	(	(	PUNCT
ejpam-4383	151	9	(	(	PUNCT
ejpam-4383	151	10	fpi	fpi	PROPN
ejpam-4383	151	11	⊗	⊗	PROPN
ejpam-4383	151	12	(	(	PUNCT
ejpam-4383	151	13	0i)i∈i)(j	0i)i∈i)(j	NOUN
ejpam-4383	151	14	)	)	PUNCT
ejpam-4383	151	15	=	=	PRON
ejpam-4383	151	16	{	{	PUNCT
ejpam-4383	151	17	pi	pi	NOUN
ejpam-4383	151	18	∗i	∗i	PROPN
ejpam-4383	151	19	0i	0i	X
ejpam-4383	151	20	if	if	SCONJ
ejpam-4383	151	21	j	j	PROPN
ejpam-4383	151	22	=	=	PUNCT
ejpam-4383	152	1	i	i	PRON
ejpam-4383	152	2	0j	0j	VERB
ejpam-4383	152	3	∗j	∗j	PROPN
ejpam-4383	152	4	0j	0j	NOUN
ejpam-4383	152	5	otherwise	otherwise	ADV
ejpam-4383	152	6	)	)	PUNCT
ejpam-4383	152	7	,	,	PUNCT
ejpam-4383	152	8	this	this	PRON
ejpam-4383	152	9	implies	imply	VERB
ejpam-4383	152	10	that	that	SCONJ
ejpam-4383	152	11	pi	pi	NOUN
ejpam-4383	152	12	∗i	∗i	PROPN
ejpam-4383	152	13	0i	0i	X
ejpam-4383	153	1	=	=	SYM
ejpam-4383	154	1	pi	pi	NOUN
ejpam-4383	154	2	.	.	PUNCT
ejpam-4383	155	1	a.	a.	PROPN
ejpam-4383	155	2	iampan	iampan	PROPN
ejpam-4383	155	3	et	et	PROPN
ejpam-4383	155	4	al	al	PROPN
ejpam-4383	155	5	.	.	PUNCT
ejpam-4383	155	6	/	/	SYM
ejpam-4383	155	7	eur	eur	PROPN
ejpam-4383	155	8	.	.	PUNCT
ejpam-4383	156	1	j.	j.	PROPN
ejpam-4383	156	2	pure	pure	PROPN
ejpam-4383	156	3	appl	appl	PROPN
ejpam-4383	156	4	.	.	PROPN
ejpam-4383	156	5	math	math	PROPN
ejpam-4383	156	6	,	,	PUNCT
ejpam-4383	156	7	15	15	NUM
ejpam-4383	156	8	(	(	PUNCT
ejpam-4383	156	9	3	3	NUM
ejpam-4383	156	10	)	)	PUNCT
ejpam-4383	156	11	(	(	PUNCT
ejpam-4383	156	12	2022	2022	NUM
ejpam-4383	156	13	)	)	PUNCT
ejpam-4383	156	14	,	,	PUNCT
ejpam-4383	156	15	999	999	NUM
ejpam-4383	156	16	-	-	SYM
ejpam-4383	156	17	1014	1014	NUM
ejpam-4383	156	18	1005	1005	NUM
ejpam-4383	156	19	(	(	PUNCT
ejpam-4383	156	20	b-3	b-3	PROPN
ejpam-4383	156	21	)	)	PUNCT
ejpam-4383	156	22	let	let	VERB
ejpam-4383	156	23	pi	pi	NOUN
ejpam-4383	156	24	,	,	PUNCT
ejpam-4383	156	25	qi	qi	PROPN
ejpam-4383	156	26	,	,	PUNCT
ejpam-4383	156	27	ri	ri	PROPN
ejpam-4383	156	28	∈	∈	PROPN
ejpam-4383	156	29	pi	pi	NOUN
ejpam-4383	156	30	.	.	PUNCT
ejpam-4383	157	1	then	then	ADV
ejpam-4383	157	2	fpi	fpi	PROPN
ejpam-4383	157	3	,	,	PUNCT
ejpam-4383	157	4	fqi	fqi	VERB
ejpam-4383	157	5	,	,	PUNCT
ejpam-4383	157	6	fri	fri	PROPN
ejpam-4383	157	7	∈	∈	PROPN
ejpam-4383	157	8	∏	∏	PROPN
ejpam-4383	157	9	i∈i	i∈i	ADJ
ejpam-4383	157	10	pi	pi	NOUN
ejpam-4383	157	11	,	,	PUNCT
ejpam-4383	157	12	which	which	PRON
ejpam-4383	157	13	is	be	AUX
ejpam-4383	157	14	defined	define	VERB
ejpam-4383	157	15	by	by	ADP
ejpam-4383	157	16	(	(	PUNCT
ejpam-4383	157	17	2.2	2.2	NUM
ejpam-4383	157	18	)	)	PUNCT
ejpam-4383	157	19	.	.	PUNCT
ejpam-4383	158	1	since∏	since∏	PROPN
ejpam-4383	159	1	i∈i	i∈i	ADJ
ejpam-4383	159	2	pi	pi	NOUN
ejpam-4383	159	3	satisfies	satisfie	NOUN
ejpam-4383	159	4	(	(	PUNCT
ejpam-4383	159	5	b-3	b-3	PROPN
ejpam-4383	159	6	)	)	PUNCT
ejpam-4383	159	7	,	,	PUNCT
ejpam-4383	159	8	we	we	PRON
ejpam-4383	159	9	have	have	VERB
ejpam-4383	159	10	(	(	PUNCT
ejpam-4383	159	11	fpi	fpi	PROPN
ejpam-4383	159	12	⊗	⊗	PROPN
ejpam-4383	159	13	fqi)⊗	fqi)⊗	PROPN
ejpam-4383	159	14	fri	fri	NOUN
ejpam-4383	159	15	=	=	PROPN
ejpam-4383	159	16	fpi	fpi	PROPN
ejpam-4383	159	17	⊗	⊗	PROPN
ejpam-4383	159	18	(	(	PUNCT
ejpam-4383	159	19	fri	fri	PROPN
ejpam-4383	159	20	⊗	⊗	PROPN
ejpam-4383	159	21	(	(	PUNCT
ejpam-4383	159	22	(	(	PUNCT
ejpam-4383	159	23	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	159	24	⊗	⊗	PROPN
ejpam-4383	159	25	fqi	fqi	NOUN
ejpam-4383	159	26	)	)	PUNCT
ejpam-4383	159	27	)	)	PUNCT
ejpam-4383	159	28	.	.	PUNCT
ejpam-4383	160	1	now	now	ADV
ejpam-4383	160	2	,	,	PUNCT
ejpam-4383	160	3	(	(	PUNCT
ejpam-4383	160	4	∀j	∀j	PROPN
ejpam-4383	160	5	∈	∈	PROPN
ejpam-4383	160	6	i	i	NOUN
ejpam-4383	160	7	)	)	PUNCT
ejpam-4383	160	8	(	(	PUNCT
ejpam-4383	160	9	(	(	PUNCT
ejpam-4383	160	10	fpi	fpi	PROPN
ejpam-4383	160	11	⊗	⊗	PROPN
ejpam-4383	160	12	fqi)⊗	fqi)⊗	PROPN
ejpam-4383	160	13	fri)(j	fri)(j	X
ejpam-4383	160	14	)	)	PUNCT
ejpam-4383	161	1	=	=	PRON
ejpam-4383	161	2	{	{	PUNCT
ejpam-4383	161	3	(	(	PUNCT
ejpam-4383	161	4	pi	pi	NOUN
ejpam-4383	161	5	∗i	∗i	PROPN
ejpam-4383	161	6	qi	qi	PROPN
ejpam-4383	161	7	)	)	PUNCT
ejpam-4383	161	8	∗i	∗i	PROPN
ejpam-4383	161	9	ri	ri	NOUN
ejpam-4383	161	10	if	if	SCONJ
ejpam-4383	161	11	j	j	PROPN
ejpam-4383	161	12	=	=	VERB
ejpam-4383	161	13	i	i	PROPN
ejpam-4383	161	14	(	(	PUNCT
ejpam-4383	161	15	0j	0j	NOUN
ejpam-4383	161	16	∗j	∗j	PROPN
ejpam-4383	161	17	0j	0j	NOUN
ejpam-4383	161	18	)	)	PUNCT
ejpam-4383	162	1	∗j	∗j	NOUN
ejpam-4383	162	2	0j	0j	NOUN
ejpam-4383	162	3	otherwise	otherwise	ADV
ejpam-4383	162	4	)	)	PUNCT
ejpam-4383	162	5	and	and	CCONJ
ejpam-4383	162	6	(	(	PUNCT
ejpam-4383	162	7	∀j	∀j	PROPN
ejpam-4383	162	8	∈	∈	PROPN
ejpam-4383	162	9	i	i	PROPN
ejpam-4383	162	10	)	)	PUNCT
ejpam-4383	162	11	(	(	PUNCT
ejpam-4383	162	12	fpi	fpi	PROPN
ejpam-4383	162	13	⊗	⊗	PROPN
ejpam-4383	162	14	(	(	PUNCT
ejpam-4383	162	15	fri	fri	PROPN
ejpam-4383	162	16	⊗	⊗	PROPN
ejpam-4383	162	17	(	(	PUNCT
ejpam-4383	162	18	(	(	PUNCT
ejpam-4383	162	19	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	162	20	⊗	⊗	PROPN
ejpam-4383	162	21	fqi)))(j	fqi)))(j	PROPN
ejpam-4383	162	22	)	)	PUNCT
ejpam-4383	163	1	=	=	PRON
ejpam-4383	163	2	{	{	PUNCT
ejpam-4383	163	3	pi	pi	NOUN
ejpam-4383	163	4	∗i	∗i	PROPN
ejpam-4383	163	5	(	(	PUNCT
ejpam-4383	163	6	ri	ri	PROPN
ejpam-4383	163	7	∗i	∗i	PROPN
ejpam-4383	163	8	(	(	PUNCT
ejpam-4383	163	9	0i	0i	X
ejpam-4383	163	10	∗i	∗i	PROPN
ejpam-4383	163	11	qi	qi	PROPN
ejpam-4383	163	12	)	)	PUNCT
ejpam-4383	163	13	)	)	PUNCT
ejpam-4383	164	1	if	if	SCONJ
ejpam-4383	164	2	j	j	PROPN
ejpam-4383	164	3	=	=	PUNCT
ejpam-4383	165	1	i	i	PRON
ejpam-4383	165	2	0j	0j	VERB
ejpam-4383	165	3	∗j	∗j	PROPN
ejpam-4383	165	4	(	(	PUNCT
ejpam-4383	165	5	0j	0j	PROPN
ejpam-4383	165	6	∗j	∗j	PROPN
ejpam-4383	165	7	(	(	PUNCT
ejpam-4383	165	8	0j	0j	NOUN
ejpam-4383	165	9	∗j	∗j	PROPN
ejpam-4383	165	10	0j	0j	NOUN
ejpam-4383	165	11	)	)	PUNCT
ejpam-4383	165	12	)	)	PUNCT
ejpam-4383	165	13	otherwise	otherwise	ADV
ejpam-4383	165	14	)	)	PUNCT
ejpam-4383	165	15	,	,	PUNCT
ejpam-4383	165	16	this	this	PRON
ejpam-4383	165	17	implies	imply	VERB
ejpam-4383	165	18	that	that	SCONJ
ejpam-4383	165	19	(	(	PUNCT
ejpam-4383	165	20	pi	pi	NOUN
ejpam-4383	165	21	∗i	∗i	PROPN
ejpam-4383	165	22	qi	qi	PROPN
ejpam-4383	165	23	)	)	PUNCT
ejpam-4383	165	24	∗i	∗i	PROPN
ejpam-4383	165	25	ri	ri	PROPN
ejpam-4383	165	26	=	=	PUNCT
ejpam-4383	165	27	pi	pi	NOUN
ejpam-4383	165	28	∗i	∗i	PROPN
ejpam-4383	165	29	(	(	PUNCT
ejpam-4383	165	30	ri	ri	PROPN
ejpam-4383	165	31	∗i	∗i	PROPN
ejpam-4383	165	32	(	(	PUNCT
ejpam-4383	165	33	0i	0i	X
ejpam-4383	165	34	∗i	∗i	PROPN
ejpam-4383	165	35	qi	qi	PROPN
ejpam-4383	165	36	)	)	PUNCT
ejpam-4383	165	37	)	)	PUNCT
ejpam-4383	165	38	.	.	PUNCT
ejpam-4383	166	1	hence	hence	ADV
ejpam-4383	166	2	,	,	PUNCT
ejpam-4383	166	3	pi	pi	NOUN
ejpam-4383	166	4	=	=	PUNCT
ejpam-4383	166	5	(	(	PUNCT
ejpam-4383	166	6	pi	pi	NOUN
ejpam-4383	166	7	;	;	PUNCT
ejpam-4383	166	8	∗i	∗i	PROPN
ejpam-4383	166	9	,	,	PUNCT
ejpam-4383	166	10	0i	0i	NOUN
ejpam-4383	166	11	)	)	PUNCT
ejpam-4383	166	12	is	be	AUX
ejpam-4383	166	13	a	a	DET
ejpam-4383	166	14	b	b	NOUN
ejpam-4383	166	15	-algebra	-algebra	NOUN
ejpam-4383	166	16	for	for	ADP
ejpam-4383	166	17	all	all	PRON
ejpam-4383	166	18	i	i	PRON
ejpam-4383	166	19	∈	∈	PROPN
ejpam-4383	166	20	i.	i.	NOUN
ejpam-4383	166	21	we	we	PRON
ejpam-4383	166	22	call	call	VERB
ejpam-4383	166	23	the	the	DET
ejpam-4383	166	24	b	b	NOUN
ejpam-4383	166	25	-algebra	-algebra	PROPN
ejpam-4383	166	26	∏	∏	PROPN
ejpam-4383	166	27	i∈i	i∈i	ADJ
ejpam-4383	166	28	pi	pi	NOUN
ejpam-4383	166	29	=	=	PUNCT
ejpam-4383	166	30	(	(	PUNCT
ejpam-4383	166	31	∏	∏	X
ejpam-4383	166	32	i∈i	i∈i	ADJ
ejpam-4383	166	33	pi;⊗	pi;⊗	PROPN
ejpam-4383	166	34	,	,	PUNCT
ejpam-4383	166	35	(	(	PUNCT
ejpam-4383	166	36	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	166	37	)	)	PUNCT
ejpam-4383	166	38	in	in	ADP
ejpam-4383	166	39	theorem	theorem	NOUN
ejpam-4383	166	40	4	4	NUM
ejpam-4383	166	41	the	the	DET
ejpam-4383	166	42	external	external	ADJ
ejpam-4383	166	43	direct	direct	ADJ
ejpam-4383	166	44	product	product	NOUN
ejpam-4383	166	45	b	b	PROPN
ejpam-4383	166	46	-algebra	-algebra	NOUN
ejpam-4383	166	47	induced	induce	VERB
ejpam-4383	166	48	by	by	ADP
ejpam-4383	166	49	a	a	DET
ejpam-4383	166	50	b	b	PROPN
ejpam-4383	166	51	-algebra	-algebra	NOUN
ejpam-4383	166	52	pi	pi	NOUN
ejpam-4383	166	53	=	=	PUNCT
ejpam-4383	166	54	(	(	PUNCT
ejpam-4383	166	55	pi	pi	NOUN
ejpam-4383	166	56	;	;	PUNCT
ejpam-4383	166	57	∗i	∗i	PROPN
ejpam-4383	166	58	,	,	PUNCT
ejpam-4383	166	59	0i	0i	NOUN
ejpam-4383	166	60	)	)	PUNCT
ejpam-4383	166	61	for	for	ADP
ejpam-4383	166	62	all	all	PRON
ejpam-4383	166	63	i	i	PRON
ejpam-4383	166	64	∈	∈	PROPN
ejpam-4383	166	65	i.	i.	NOUN
ejpam-4383	166	66	theorem	theorem	VERB
ejpam-4383	166	67	5	5	NUM
ejpam-4383	166	68	.	.	PUNCT
ejpam-4383	167	1	let	let	VERB
ejpam-4383	167	2	pi	pi	NOUN
ejpam-4383	167	3	=	=	PUNCT
ejpam-4383	167	4	(	(	PUNCT
ejpam-4383	167	5	pi	pi	NOUN
ejpam-4383	167	6	;	;	PUNCT
ejpam-4383	167	7	∗i	∗i	PROPN
ejpam-4383	167	8	,	,	PUNCT
ejpam-4383	167	9	0i	0i	NOUN
ejpam-4383	167	10	)	)	PUNCT
ejpam-4383	167	11	be	be	VERB
ejpam-4383	167	12	a	a	DET
ejpam-4383	167	13	b	b	NOUN
ejpam-4383	167	14	-	-	PUNCT
ejpam-4383	167	15	algebra	algebra	NOUN
ejpam-4383	167	16	for	for	ADP
ejpam-4383	167	17	all	all	PRON
ejpam-4383	167	18	i	i	PRON
ejpam-4383	167	19	∈	∈	PROPN
ejpam-4383	167	20	i.	i.	NOUN
ejpam-4383	167	21	then	then	ADV
ejpam-4383	167	22	pi	pi	PROPN
ejpam-4383	167	23	is	be	AUX
ejpam-4383	167	24	commutative	commutative	ADJ
ejpam-4383	167	25	for	for	ADP
ejpam-4383	167	26	all	all	PRON
ejpam-4383	167	27	i	i	PRON
ejpam-4383	167	28	∈	∈	VERB
ejpam-4383	168	1	i	i	PRON
ejpam-4383	168	2	if	if	SCONJ
ejpam-4383	168	3	and	and	CCONJ
ejpam-4383	168	4	only	only	ADV
ejpam-4383	169	1	if	if	SCONJ
ejpam-4383	169	2	∏	∏	NUM
ejpam-4383	169	3	i∈i	i∈i	ADJ
ejpam-4383	169	4	pi	pi	NOUN
ejpam-4383	169	5	=	=	PUNCT
ejpam-4383	169	6	(	(	PUNCT
ejpam-4383	169	7	∏	∏	X
ejpam-4383	169	8	i∈i	i∈i	ADJ
ejpam-4383	169	9	pi;⊗	pi;⊗	PROPN
ejpam-4383	169	10	,	,	PUNCT
ejpam-4383	169	11	(	(	PUNCT
ejpam-4383	169	12	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	169	13	)	)	PUNCT
ejpam-4383	169	14	is	be	AUX
ejpam-4383	169	15	commutative	commutative	ADJ
ejpam-4383	169	16	,	,	PUNCT
ejpam-4383	169	17	where	where	SCONJ
ejpam-4383	169	18	the	the	DET
ejpam-4383	169	19	binary	binary	PROPN
ejpam-4383	169	20	operation	operation	PROPN
ejpam-4383	169	21	⊗	⊗	PROPN
ejpam-4383	169	22	is	be	AUX
ejpam-4383	169	23	defined	define	VERB
ejpam-4383	169	24	in	in	ADP
ejpam-4383	169	25	definition	definition	NOUN
ejpam-4383	169	26	8	8	NUM
ejpam-4383	169	27	.	.	PUNCT
ejpam-4383	170	1	proof	proof	NOUN
ejpam-4383	170	2	.	.	PUNCT
ejpam-4383	171	1	by	by	ADP
ejpam-4383	171	2	theorem	theorem	NOUN
ejpam-4383	171	3	4	4	NUM
ejpam-4383	171	4	,	,	PUNCT
ejpam-4383	171	5	we	we	PRON
ejpam-4383	171	6	have	have	VERB
ejpam-4383	171	7	pi	pi	NOUN
ejpam-4383	171	8	=	=	SYM
ejpam-4383	171	9	(	(	PUNCT
ejpam-4383	171	10	pi	pi	NOUN
ejpam-4383	171	11	;	;	PUNCT
ejpam-4383	171	12	∗i	∗i	PROPN
ejpam-4383	171	13	,	,	PUNCT
ejpam-4383	171	14	0i	0i	NOUN
ejpam-4383	171	15	)	)	PUNCT
ejpam-4383	171	16	is	be	AUX
ejpam-4383	171	17	a	a	DET
ejpam-4383	171	18	b	b	NOUN
ejpam-4383	171	19	-algebra	-algebra	NOUN
ejpam-4383	171	20	for	for	ADP
ejpam-4383	171	21	all	all	PRON
ejpam-4383	172	1	i	i	PRON
ejpam-4383	172	2	∈	∈	VERB
ejpam-4383	173	1	i	i	PRON
ejpam-4383	173	2	if	if	SCONJ
ejpam-4383	173	3	and	and	CCONJ
ejpam-4383	173	4	only	only	ADV
ejpam-4383	174	1	if	if	SCONJ
ejpam-4383	174	2	∏	∏	NUM
ejpam-4383	174	3	i∈i	i∈i	ADJ
ejpam-4383	174	4	pi	pi	NOUN
ejpam-4383	174	5	=	=	PUNCT
ejpam-4383	174	6	(	(	PUNCT
ejpam-4383	174	7	∏	∏	X
ejpam-4383	174	8	i∈i	i∈i	ADJ
ejpam-4383	174	9	pi;⊗	pi;⊗	PROPN
ejpam-4383	174	10	,	,	PUNCT
ejpam-4383	174	11	(	(	PUNCT
ejpam-4383	174	12	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	174	13	)	)	PUNCT
ejpam-4383	174	14	is	be	AUX
ejpam-4383	174	15	a	a	DET
ejpam-4383	174	16	b	b	NOUN
ejpam-4383	174	17	-algebra	-algebra	NOUN
ejpam-4383	174	18	,	,	PUNCT
ejpam-4383	174	19	where	where	SCONJ
ejpam-4383	174	20	the	the	DET
ejpam-4383	174	21	binary	binary	PROPN
ejpam-4383	174	22	operation	operation	PROPN
ejpam-4383	174	23	⊗	⊗	PROPN
ejpam-4383	174	24	is	be	AUX
ejpam-4383	174	25	defined	define	VERB
ejpam-4383	174	26	in	in	ADP
ejpam-4383	174	27	definition	definition	NOUN
ejpam-4383	174	28	8	8	NUM
ejpam-4383	174	29	.	.	PUNCT
ejpam-4383	175	1	we	we	PRON
ejpam-4383	175	2	are	be	AUX
ejpam-4383	175	3	left	leave	VERB
ejpam-4383	175	4	to	to	PART
ejpam-4383	175	5	prove	prove	VERB
ejpam-4383	175	6	that	that	PRON
ejpam-4383	175	7	pi	pi	NOUN
ejpam-4383	175	8	is	be	AUX
ejpam-4383	175	9	commutative	commutative	ADJ
ejpam-4383	175	10	for	for	ADP
ejpam-4383	175	11	all	all	DET
ejpam-4383	175	12	i	i	PRON
ejpam-4383	175	13	∈	∈	VERB
ejpam-4383	176	1	i	i	PRON
ejpam-4383	176	2	if	if	SCONJ
ejpam-4383	176	3	and	and	CCONJ
ejpam-4383	176	4	only	only	ADV
ejpam-4383	176	5	if∏	if∏	NOUN
ejpam-4383	176	6	i∈i	i∈i	ADJ
ejpam-4383	176	7	pi	pi	NOUN
ejpam-4383	176	8	is	be	AUX
ejpam-4383	176	9	commutative	commutative	ADJ
ejpam-4383	176	10	.	.	PUNCT
ejpam-4383	177	1	assume	assume	VERB
ejpam-4383	177	2	that	that	SCONJ
ejpam-4383	177	3	pi	pi	NOUN
ejpam-4383	177	4	is	be	AUX
ejpam-4383	177	5	commutative	commutative	ADJ
ejpam-4383	177	6	for	for	SCONJ
ejpam-4383	177	7	all	all	PRON
ejpam-4383	177	8	i	i	PRON
ejpam-4383	177	9	∈	∈	PROPN
ejpam-4383	177	10	i.	i.	NOUN
ejpam-4383	177	11	let	let	VERB
ejpam-4383	177	12	(	(	PUNCT
ejpam-4383	177	13	pi)i∈i	pi)i∈i	NUM
ejpam-4383	177	14	,	,	PUNCT
ejpam-4383	177	15	(	(	PUNCT
ejpam-4383	177	16	qi)i∈i	qi)i∈i	NUM
ejpam-4383	177	17	∈	∈	PROPN
ejpam-4383	177	18	∏	∏	PROPN
ejpam-4383	177	19	i∈i	i∈i	ADJ
ejpam-4383	177	20	pi	pi	NOUN
ejpam-4383	177	21	.	.	PUNCT
ejpam-4383	178	1	since	since	SCONJ
ejpam-4383	178	2	pi	pi	NOUN
ejpam-4383	178	3	is	be	AUX
ejpam-4383	178	4	commutative	commutative	ADJ
ejpam-4383	178	5	for	for	ADP
ejpam-4383	178	6	all	all	PRON
ejpam-4383	178	7	i	i	PRON
ejpam-4383	178	8	∈	∈	PROPN
ejpam-4383	179	1	i	i	PRON
ejpam-4383	179	2	,	,	PUNCT
ejpam-4383	179	3	we	we	PRON
ejpam-4383	179	4	have	have	VERB
ejpam-4383	179	5	pi	pi	NOUN
ejpam-4383	179	6	∗i	∗i	PROPN
ejpam-4383	179	7	(	(	PUNCT
ejpam-4383	179	8	0i	0i	X
ejpam-4383	179	9	∗i	∗i	PROPN
ejpam-4383	179	10	qi	qi	PROPN
ejpam-4383	179	11	)	)	PUNCT
ejpam-4383	179	12	=	=	SYM
ejpam-4383	179	13	qi	qi	PROPN
ejpam-4383	179	14	∗i	∗i	PROPN
ejpam-4383	179	15	(	(	PUNCT
ejpam-4383	179	16	0i	0i	X
ejpam-4383	179	17	∗i	∗i	PROPN
ejpam-4383	179	18	pi	pi	NOUN
ejpam-4383	179	19	)	)	PUNCT
ejpam-4383	179	20	for	for	ADP
ejpam-4383	179	21	all	all	PRON
ejpam-4383	179	22	i	i	PRON
ejpam-4383	179	23	∈	∈	PROPN
ejpam-4383	179	24	i.	i.	NOUN
ejpam-4383	179	25	thus	thus	ADV
ejpam-4383	179	26	(	(	PUNCT
ejpam-4383	179	27	pi)i∈i	pi)i∈i	NUM
ejpam-4383	179	28	⊗	⊗	PROPN
ejpam-4383	179	29	(	(	PUNCT
ejpam-4383	179	30	(	(	PUNCT
ejpam-4383	179	31	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	179	32	⊗	⊗	PROPN
ejpam-4383	179	33	(	(	PUNCT
ejpam-4383	179	34	qi)i∈i	qi)i∈i	NUM
ejpam-4383	179	35	)	)	PUNCT
ejpam-4383	179	36	=	=	NOUN
ejpam-4383	179	37	(	(	PUNCT
ejpam-4383	179	38	pi)i∈i	pi)i∈i	NUM
ejpam-4383	179	39	⊗	⊗	PROPN
ejpam-4383	179	40	(	(	PUNCT
ejpam-4383	179	41	0i	0i	X
ejpam-4383	179	42	∗i	∗i	PROPN
ejpam-4383	179	43	qi)i∈i	qi)i∈i	NUM
ejpam-4383	179	44	=	=	SYM
ejpam-4383	179	45	(	(	PUNCT
ejpam-4383	179	46	pi	pi	NOUN
ejpam-4383	179	47	∗i	∗i	PROPN
ejpam-4383	179	48	(	(	PUNCT
ejpam-4383	179	49	0i	0i	X
ejpam-4383	179	50	∗i	∗i	PROPN
ejpam-4383	179	51	qi))i∈i	qi))i∈i	NOUN
ejpam-4383	179	52	=	=	SYM
ejpam-4383	179	53	(	(	PUNCT
ejpam-4383	179	54	qi	qi	PROPN
ejpam-4383	179	55	∗i	∗i	PROPN
ejpam-4383	179	56	(	(	PUNCT
ejpam-4383	179	57	0i	0i	X
ejpam-4383	179	58	∗i	∗i	PROPN
ejpam-4383	179	59	pi))i∈i	pi))i∈i	PROPN
ejpam-4383	179	60	=	=	SYM
ejpam-4383	179	61	(	(	PUNCT
ejpam-4383	179	62	qi)i∈i	qi)i∈i	NUM
ejpam-4383	179	63	⊗	⊗	PROPN
ejpam-4383	179	64	(	(	PUNCT
ejpam-4383	179	65	0i	0i	X
ejpam-4383	179	66	∗i	∗i	PROPN
ejpam-4383	179	67	pi)i∈i	pi)i∈i	NUM
ejpam-4383	179	68	=	=	PUNCT
ejpam-4383	179	69	(	(	PUNCT
ejpam-4383	179	70	qi)i∈i	qi)i∈i	NUM
ejpam-4383	179	71	⊗	⊗	PROPN
ejpam-4383	179	72	(	(	PUNCT
ejpam-4383	179	73	(	(	PUNCT
ejpam-4383	179	74	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	179	75	⊗	⊗	PROPN
ejpam-4383	179	76	(	(	PUNCT
ejpam-4383	179	77	pi)i∈i	pi)i∈i	NUM
ejpam-4383	179	78	)	)	PUNCT
ejpam-4383	179	79	.	.	PUNCT
ejpam-4383	180	1	hence	hence	ADV
ejpam-4383	180	2	,	,	PUNCT
ejpam-4383	180	3	∏	∏	PROPN
ejpam-4383	180	4	i∈i	i∈i	ADJ
ejpam-4383	180	5	pi	pi	NOUN
ejpam-4383	180	6	is	be	AUX
ejpam-4383	180	7	commutative	commutative	ADJ
ejpam-4383	180	8	.	.	PUNCT
ejpam-4383	181	1	conversely	conversely	ADV
ejpam-4383	181	2	,	,	PUNCT
ejpam-4383	181	3	assume	assume	VERB
ejpam-4383	181	4	that	that	SCONJ
ejpam-4383	181	5	∏	∏	PROPN
ejpam-4383	181	6	i∈i	i∈i	ADJ
ejpam-4383	181	7	pi	pi	NOUN
ejpam-4383	181	8	is	be	AUX
ejpam-4383	181	9	commutative	commutative	ADJ
ejpam-4383	181	10	.	.	PUNCT
ejpam-4383	182	1	let	let	VERB
ejpam-4383	182	2	i	i	PRON
ejpam-4383	182	3	∈	∈	PROPN
ejpam-4383	182	4	i.	i.	NOUN
ejpam-4383	182	5	let	let	VERB
ejpam-4383	182	6	pi	pi	NOUN
ejpam-4383	182	7	,	,	PUNCT
ejpam-4383	182	8	qi	qi	PROPN
ejpam-4383	182	9	∈	∈	PROPN
ejpam-4383	182	10	pi	pi	NOUN
ejpam-4383	182	11	.	.	PUNCT
ejpam-4383	183	1	then	then	ADV
ejpam-4383	183	2	fpi	fpi	PROPN
ejpam-4383	183	3	,	,	PUNCT
ejpam-4383	183	4	fqi	fqi	VERB
ejpam-4383	183	5	∈	∈	PROPN
ejpam-4383	183	6	∏	∏	PROPN
ejpam-4383	183	7	i∈i	i∈i	ADJ
ejpam-4383	183	8	pi	pi	NOUN
ejpam-4383	183	9	,	,	PUNCT
ejpam-4383	183	10	which	which	PRON
ejpam-4383	183	11	is	be	AUX
ejpam-4383	183	12	defined	define	VERB
ejpam-4383	183	13	by	by	ADP
ejpam-4383	183	14	(	(	PUNCT
ejpam-4383	183	15	2.2	2.2	NUM
ejpam-4383	183	16	)	)	PUNCT
ejpam-4383	183	17	.	.	PUNCT
ejpam-4383	184	1	since	since	SCONJ
ejpam-4383	184	2	∏	∏	PROPN
ejpam-4383	184	3	i∈i	i∈i	ADJ
ejpam-4383	184	4	pi	pi	NOUN
ejpam-4383	184	5	is	be	AUX
ejpam-4383	184	6	commutative	commutative	ADJ
ejpam-4383	184	7	,	,	PUNCT
ejpam-4383	184	8	we	we	PRON
ejpam-4383	184	9	have	have	VERB
ejpam-4383	184	10	fpi	fpi	PROPN
ejpam-4383	184	11	⊗	⊗	PROPN
ejpam-4383	184	12	(	(	PUNCT
ejpam-4383	184	13	(	(	PUNCT
ejpam-4383	184	14	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	184	15	⊗	⊗	PROPN
ejpam-4383	184	16	fqi	fqi	NOUN
ejpam-4383	184	17	)	)	PUNCT
ejpam-4383	185	1	=	=	VERB
ejpam-4383	185	2	fqi	fqi	VERB
ejpam-4383	185	3	⊗	⊗	X
ejpam-4383	185	4	(	(	PUNCT
ejpam-4383	185	5	(	(	PUNCT
ejpam-4383	185	6	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	185	7	⊗	⊗	PROPN
ejpam-4383	185	8	fpi	fpi	PROPN
ejpam-4383	185	9	)	)	PUNCT
ejpam-4383	185	10	.	.	PUNCT
ejpam-4383	186	1	now	now	ADV
ejpam-4383	186	2	,	,	PUNCT
ejpam-4383	186	3	(	(	PUNCT
ejpam-4383	186	4	∀j	∀j	PROPN
ejpam-4383	186	5	∈	∈	PROPN
ejpam-4383	186	6	i	i	NOUN
ejpam-4383	186	7	)	)	PUNCT
ejpam-4383	186	8	(	(	PUNCT
ejpam-4383	186	9	(	(	PUNCT
ejpam-4383	186	10	fpi	fpi	PROPN
ejpam-4383	186	11	⊗	⊗	PROPN
ejpam-4383	186	12	(	(	PUNCT
ejpam-4383	186	13	(	(	PUNCT
ejpam-4383	186	14	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	186	15	⊗	⊗	PROPN
ejpam-4383	186	16	fqi))(j	fqi))(j	PROPN
ejpam-4383	186	17	)	)	PUNCT
ejpam-4383	186	18	=	=	PRON
ejpam-4383	186	19	{	{	PUNCT
ejpam-4383	186	20	pi	pi	NOUN
ejpam-4383	186	21	∗i	∗i	PROPN
ejpam-4383	186	22	(	(	PUNCT
ejpam-4383	186	23	0i	0i	X
ejpam-4383	186	24	∗i	∗i	PROPN
ejpam-4383	186	25	qi	qi	PROPN
ejpam-4383	186	26	)	)	PUNCT
ejpam-4383	186	27	if	if	SCONJ
ejpam-4383	186	28	j	j	PROPN
ejpam-4383	186	29	=	=	PUNCT
ejpam-4383	187	1	i	i	PRON
ejpam-4383	187	2	0j	0j	VERB
ejpam-4383	187	3	∗j	∗j	PROPN
ejpam-4383	187	4	(	(	PUNCT
ejpam-4383	187	5	0j	0j	NOUN
ejpam-4383	187	6	∗j	∗j	PROPN
ejpam-4383	187	7	0j	0j	NOUN
ejpam-4383	187	8	)	)	PUNCT
ejpam-4383	187	9	otherwise	otherwise	ADV
ejpam-4383	187	10	)	)	PUNCT
ejpam-4383	187	11	a.	a.	NOUN
ejpam-4383	187	12	iampan	iampan	NOUN
ejpam-4383	187	13	et	et	PROPN
ejpam-4383	188	1	al	al	PROPN
ejpam-4383	188	2	.	.	PUNCT
ejpam-4383	188	3	/	/	SYM
ejpam-4383	188	4	eur	eur	PROPN
ejpam-4383	188	5	.	.	PUNCT
ejpam-4383	189	1	j.	j.	PROPN
ejpam-4383	189	2	pure	pure	PROPN
ejpam-4383	189	3	appl	appl	PROPN
ejpam-4383	189	4	.	.	PROPN
ejpam-4383	189	5	math	math	PROPN
ejpam-4383	189	6	,	,	PUNCT
ejpam-4383	189	7	15	15	NUM
ejpam-4383	189	8	(	(	PUNCT
ejpam-4383	189	9	3	3	NUM
ejpam-4383	189	10	)	)	PUNCT
ejpam-4383	189	11	(	(	PUNCT
ejpam-4383	189	12	2022	2022	NUM
ejpam-4383	189	13	)	)	PUNCT
ejpam-4383	189	14	,	,	PUNCT
ejpam-4383	189	15	999	999	NUM
ejpam-4383	189	16	-	-	SYM
ejpam-4383	189	17	1014	1014	NUM
ejpam-4383	189	18	1006	1006	NUM
ejpam-4383	189	19	and	and	CCONJ
ejpam-4383	189	20	(	(	PUNCT
ejpam-4383	189	21	∀j	∀j	PROPN
ejpam-4383	189	22	∈	∈	PROPN
ejpam-4383	189	23	i	i	NOUN
ejpam-4383	189	24	)	)	PUNCT
ejpam-4383	189	25	(	(	PUNCT
ejpam-4383	189	26	(	(	PUNCT
ejpam-4383	189	27	fqi	fqi	VERB
ejpam-4383	189	28	⊗	⊗	PROPN
ejpam-4383	189	29	(	(	PUNCT
ejpam-4383	189	30	(	(	PUNCT
ejpam-4383	189	31	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	189	32	⊗	⊗	PROPN
ejpam-4383	189	33	fpi))(j	fpi))(j	PROPN
ejpam-4383	189	34	)	)	PUNCT
ejpam-4383	189	35	=	=	PRON
ejpam-4383	189	36	{	{	PUNCT
ejpam-4383	189	37	qi	qi	PROPN
ejpam-4383	189	38	∗i	∗i	PROPN
ejpam-4383	189	39	(	(	PUNCT
ejpam-4383	189	40	0i	0i	X
ejpam-4383	189	41	∗i	∗i	PROPN
ejpam-4383	189	42	pi	pi	NOUN
ejpam-4383	189	43	)	)	PUNCT
ejpam-4383	189	44	if	if	SCONJ
ejpam-4383	189	45	j	j	PROPN
ejpam-4383	189	46	=	=	PUNCT
ejpam-4383	190	1	i	i	PRON
ejpam-4383	190	2	0j	0j	VERB
ejpam-4383	190	3	∗j	∗j	PROPN
ejpam-4383	190	4	(	(	PUNCT
ejpam-4383	190	5	0j	0j	NOUN
ejpam-4383	190	6	∗j	∗j	PROPN
ejpam-4383	190	7	0j	0j	NOUN
ejpam-4383	190	8	)	)	PUNCT
ejpam-4383	190	9	otherwise	otherwise	ADV
ejpam-4383	190	10	)	)	PUNCT
ejpam-4383	190	11	,	,	PUNCT
ejpam-4383	190	12	this	this	PRON
ejpam-4383	190	13	implies	imply	VERB
ejpam-4383	190	14	that	that	SCONJ
ejpam-4383	190	15	pi	pi	NOUN
ejpam-4383	190	16	∗i	∗i	PROPN
ejpam-4383	190	17	(	(	PUNCT
ejpam-4383	190	18	0i	0i	X
ejpam-4383	190	19	∗i	∗i	PROPN
ejpam-4383	190	20	qi	qi	PROPN
ejpam-4383	190	21	)	)	PUNCT
ejpam-4383	190	22	=	=	SYM
ejpam-4383	190	23	qi	qi	PROPN
ejpam-4383	190	24	∗i	∗i	PROPN
ejpam-4383	190	25	(	(	PUNCT
ejpam-4383	190	26	0i	0i	X
ejpam-4383	190	27	∗i	∗i	PROPN
ejpam-4383	190	28	pi	pi	NOUN
ejpam-4383	190	29	)	)	PUNCT
ejpam-4383	190	30	.	.	PUNCT
ejpam-4383	191	1	hence	hence	ADV
ejpam-4383	191	2	,	,	PUNCT
ejpam-4383	191	3	pi	pi	PROPN
ejpam-4383	191	4	is	be	AUX
ejpam-4383	191	5	commutative	commutative	ADJ
ejpam-4383	191	6	for	for	ADP
ejpam-4383	191	7	all	all	DET
ejpam-4383	191	8	i	i	PRON
ejpam-4383	191	9	∈	∈	PROPN
ejpam-4383	191	10	i.	i.	NOUN
ejpam-4383	191	11	next	next	ADV
ejpam-4383	191	12	,	,	PUNCT
ejpam-4383	191	13	we	we	PRON
ejpam-4383	191	14	introduce	introduce	VERB
ejpam-4383	191	15	the	the	DET
ejpam-4383	191	16	concept	concept	NOUN
ejpam-4383	191	17	of	of	ADP
ejpam-4383	191	18	the	the	DET
ejpam-4383	191	19	weak	weak	ADJ
ejpam-4383	191	20	direct	direct	ADJ
ejpam-4383	191	21	product	product	NOUN
ejpam-4383	191	22	of	of	ADP
ejpam-4383	191	23	infinite	infinite	ADJ
ejpam-4383	191	24	family	family	NOUN
ejpam-4383	191	25	of	of	ADP
ejpam-4383	191	26	b	b	PROPN
ejpam-4383	191	27	algebras	algebra	NOUN
ejpam-4383	191	28	and	and	CCONJ
ejpam-4383	191	29	obtain	obtain	VERB
ejpam-4383	191	30	some	some	PRON
ejpam-4383	191	31	of	of	ADP
ejpam-4383	191	32	its	its	PRON
ejpam-4383	191	33	properties	property	NOUN
ejpam-4383	191	34	as	as	SCONJ
ejpam-4383	191	35	follows	follow	VERB
ejpam-4383	191	36	:	:	PUNCT
ejpam-4383	191	37	definition	definition	NOUN
ejpam-4383	191	38	9	9	NUM
ejpam-4383	191	39	.	.	PUNCT
ejpam-4383	192	1	let	let	AUX
ejpam-4383	192	2	pi	pi	NOUN
ejpam-4383	192	3	=	=	PUNCT
ejpam-4383	192	4	(	(	PUNCT
ejpam-4383	192	5	pi	pi	NOUN
ejpam-4383	192	6	;	;	PUNCT
ejpam-4383	192	7	∗i	∗i	PROPN
ejpam-4383	192	8	,	,	PUNCT
ejpam-4383	192	9	0i	0i	NOUN
ejpam-4383	192	10	)	)	PUNCT
ejpam-4383	192	11	be	be	VERB
ejpam-4383	192	12	a	a	DET
ejpam-4383	192	13	b	b	NOUN
ejpam-4383	192	14	-	-	PUNCT
ejpam-4383	192	15	algebra	algebra	NOUN
ejpam-4383	192	16	for	for	ADP
ejpam-4383	192	17	all	all	PRON
ejpam-4383	192	18	i	i	PRON
ejpam-4383	192	19	∈	∈	PROPN
ejpam-4383	192	20	i.	i.	NOUN
ejpam-4383	192	21	define	define	VERB
ejpam-4383	192	22	the	the	DET
ejpam-4383	192	23	weak	weak	ADJ
ejpam-4383	192	24	direct	direct	ADJ
ejpam-4383	192	25	product	product	NOUN
ejpam-4383	192	26	of	of	ADP
ejpam-4383	192	27	a	a	DET
ejpam-4383	192	28	b	b	NOUN
ejpam-4383	192	29	-	-	PUNCT
ejpam-4383	192	30	algebra	algebra	NOUN
ejpam-4383	192	31	pi	pi	NOUN
ejpam-4383	192	32	for	for	ADP
ejpam-4383	192	33	all	all	PRON
ejpam-4383	192	34	i	i	PRON
ejpam-4383	192	35	∈	∈	VERB
ejpam-4383	193	1	i	i	PRON
ejpam-4383	193	2	to	to	PART
ejpam-4383	193	3	be	be	AUX
ejpam-4383	193	4	the	the	DET
ejpam-4383	193	5	structure	structure	NOUN
ejpam-4383	193	6	∏w	∏w	NOUN
ejpam-4383	193	7	i∈i	i∈i	ADJ
ejpam-4383	193	8	pi	pi	NOUN
ejpam-4383	193	9	=	=	PUNCT
ejpam-4383	193	10	(	(	PUNCT
ejpam-4383	193	11	∏w	∏w	X
ejpam-4383	193	12	i∈i	i∈i	ADJ
ejpam-4383	193	13	pi;⊗	pi;⊗	PROPN
ejpam-4383	193	14	)	)	PUNCT
ejpam-4383	193	15	,	,	PUNCT
ejpam-4383	194	1	where	where	SCONJ
ejpam-4383	194	2	w∏	w∏	PROPN
ejpam-4383	194	3	i∈i	i∈i	ADJ
ejpam-4383	194	4	pi	pi	NOUN
ejpam-4383	194	5	=	=	PUNCT
ejpam-4383	194	6	{	{	PUNCT
ejpam-4383	194	7	(	(	PUNCT
ejpam-4383	194	8	pi)i∈i	pi)i∈i	NUM
ejpam-4383	194	9	∈	∈	PROPN
ejpam-4383	194	10	∏	∏	PROPN
ejpam-4383	194	11	i∈i	i∈i	ADJ
ejpam-4383	194	12	pi	pi	NOUN
ejpam-4383	195	1	|	|	ADV
ejpam-4383	195	2	pi	pi	NOUN
ejpam-4383	195	3	̸=	̸=	PROPN
ejpam-4383	195	4	0i	0i	NOUN
ejpam-4383	195	5	,	,	PUNCT
ejpam-4383	195	6	where	where	SCONJ
ejpam-4383	195	7	the	the	DET
ejpam-4383	195	8	number	number	NOUN
ejpam-4383	195	9	of	of	ADP
ejpam-4383	195	10	such	such	ADJ
ejpam-4383	195	11	i	i	PRON
ejpam-4383	195	12	is	be	AUX
ejpam-4383	195	13	finite	finite	ADJ
ejpam-4383	195	14	}	}	PUNCT
ejpam-4383	195	15	.	.	PUNCT
ejpam-4383	196	1	then	then	ADV
ejpam-4383	196	2	(	(	PUNCT
ejpam-4383	196	3	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	196	4	∈	∈	PROPN
ejpam-4383	196	5	∏w	∏w	X
ejpam-4383	196	6	i∈i	i∈i	ADJ
ejpam-4383	196	7	pi	pi	NOUN
ejpam-4383	196	8	⊆	⊆	NUM
ejpam-4383	196	9	∏	∏	PROPN
ejpam-4383	196	10	i∈i	i∈i	ADJ
ejpam-4383	196	11	pi	pi	NOUN
ejpam-4383	196	12	.	.	PUNCT
ejpam-4383	197	1	theorem	theorem	VERB
ejpam-4383	197	2	6	6	NUM
ejpam-4383	197	3	.	.	PUNCT
ejpam-4383	198	1	let	let	VERB
ejpam-4383	198	2	pi	pi	NOUN
ejpam-4383	198	3	=	=	PUNCT
ejpam-4383	198	4	(	(	PUNCT
ejpam-4383	198	5	pi	pi	NOUN
ejpam-4383	198	6	;	;	PUNCT
ejpam-4383	198	7	∗i	∗i	PROPN
ejpam-4383	198	8	,	,	PUNCT
ejpam-4383	198	9	0i	0i	NOUN
ejpam-4383	198	10	)	)	PUNCT
ejpam-4383	198	11	be	be	VERB
ejpam-4383	198	12	a	a	DET
ejpam-4383	198	13	b	b	NOUN
ejpam-4383	198	14	-	-	PUNCT
ejpam-4383	198	15	algebra	algebra	NOUN
ejpam-4383	198	16	for	for	ADP
ejpam-4383	198	17	all	all	PRON
ejpam-4383	198	18	i	i	PRON
ejpam-4383	198	19	∈	∈	PROPN
ejpam-4383	198	20	i.	i.	NOUN
ejpam-4383	198	21	then	then	ADV
ejpam-4383	198	22	∏w	∏w	X
ejpam-4383	198	23	i∈i	i∈i	ADJ
ejpam-4383	198	24	pi	pi	NOUN
ejpam-4383	198	25	is	be	AUX
ejpam-4383	198	26	a	a	DET
ejpam-4383	198	27	bsubalgebra	bsubalgebra	NOUN
ejpam-4383	198	28	of	of	ADP
ejpam-4383	198	29	the	the	DET
ejpam-4383	198	30	external	external	ADJ
ejpam-4383	198	31	direct	direct	ADJ
ejpam-4383	198	32	product	product	NOUN
ejpam-4383	198	33	b	b	X
ejpam-4383	198	34	-	-	PUNCT
ejpam-4383	198	35	algebra	algebra	NOUN
ejpam-4383	198	36	∏	∏	PROPN
ejpam-4383	198	37	i∈i	i∈i	ADJ
ejpam-4383	198	38	pi	pi	NOUN
ejpam-4383	199	1	=	=	PUNCT
ejpam-4383	199	2	(	(	PUNCT
ejpam-4383	199	3	∏	∏	X
ejpam-4383	199	4	i∈i	i∈i	ADJ
ejpam-4383	199	5	pi;⊗	pi;⊗	PROPN
ejpam-4383	199	6	,	,	PUNCT
ejpam-4383	199	7	(	(	PUNCT
ejpam-4383	199	8	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	199	9	)	)	PUNCT
ejpam-4383	199	10	.	.	PUNCT
ejpam-4383	200	1	proof	proof	NOUN
ejpam-4383	200	2	.	.	PUNCT
ejpam-4383	201	1	we	we	PRON
ejpam-4383	201	2	see	see	VERB
ejpam-4383	201	3	that	that	SCONJ
ejpam-4383	201	4	(	(	PUNCT
ejpam-4383	201	5	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	201	6	∈	∈	PROPN
ejpam-4383	201	7	∏w	∏w	X
ejpam-4383	201	8	i∈i	i∈i	ADJ
ejpam-4383	201	9	pi	pi	NOUN
ejpam-4383	201	10	̸=	̸=	PROPN
ejpam-4383	201	11	∅.	∅.	ADV
ejpam-4383	201	12	let	let	VERB
ejpam-4383	201	13	(	(	PUNCT
ejpam-4383	201	14	pi)i∈i	pi)i∈i	NUM
ejpam-4383	201	15	,	,	PUNCT
ejpam-4383	201	16	(	(	PUNCT
ejpam-4383	201	17	qi)i∈i	qi)i∈i	NUM
ejpam-4383	201	18	∈	∈	PROPN
ejpam-4383	201	19	∏w	∏w	NOUN
ejpam-4383	201	20	i∈i	i∈i	ADJ
ejpam-4383	201	21	pi	pi	NOUN
ejpam-4383	201	22	,	,	PUNCT
ejpam-4383	201	23	where	where	SCONJ
ejpam-4383	201	24	i1	i1	PROPN
ejpam-4383	201	25	=	=	PUNCT
ejpam-4383	201	26	{	{	PUNCT
ejpam-4383	202	1	i	i	NOUN
ejpam-4383	202	2	∈	∈	PROPN
ejpam-4383	203	1	i	i	PRON
ejpam-4383	203	2	|	|	ADV
ejpam-4383	203	3	pi	pi	NOUN
ejpam-4383	203	4	̸=	̸=	PROPN
ejpam-4383	203	5	0i	0i	PROPN
ejpam-4383	203	6	}	}	PUNCT
ejpam-4383	203	7	and	and	CCONJ
ejpam-4383	203	8	i2	i2	PROPN
ejpam-4383	203	9	=	=	PUNCT
ejpam-4383	204	1	{	{	PUNCT
ejpam-4383	204	2	i	i	NOUN
ejpam-4383	204	3	∈	∈	PROPN
ejpam-4383	205	1	i	i	PRON
ejpam-4383	205	2	|	|	ADV
ejpam-4383	205	3	qi	qi	PROPN
ejpam-4383	205	4	̸=	̸=	PROPN
ejpam-4383	205	5	0i	0i	PROPN
ejpam-4383	205	6	}	}	PUNCT
ejpam-4383	205	7	are	be	AUX
ejpam-4383	205	8	finite	finite	ADJ
ejpam-4383	205	9	.	.	PUNCT
ejpam-4383	206	1	then	then	ADV
ejpam-4383	206	2	|i1	|i1	PROPN
ejpam-4383	206	3	∪	∪	ADP
ejpam-4383	206	4	i2|	i2|	PROPN
ejpam-4383	206	5	is	be	AUX
ejpam-4383	206	6	finite	finite	ADJ
ejpam-4383	206	7	.	.	PUNCT
ejpam-4383	207	1	thus	thus	ADV
ejpam-4383	207	2	(	(	PUNCT
ejpam-4383	207	3	∀j	∀j	PROPN
ejpam-4383	207	4	∈	∈	PROPN
ejpam-4383	207	5	i	i	NOUN
ejpam-4383	207	6	)	)	PUNCT
ejpam-4383	207	7	((pi)i∈i	((pi)i∈i	ADJ
ejpam-4383	207	8	⊗	⊗	PROPN
ejpam-4383	207	9	(	(	PUNCT
ejpam-4383	207	10	qi)i∈i)(j	qi)i∈i)(j	NOUN
ejpam-4383	207	11	)	)	PUNCT
ejpam-4383	207	12	=	=	PUNCT
ejpam-4383	207	13			VERB
ejpam-4383	207	14	pj	pj	PROPN
ejpam-4383	207	15	∗j	∗j	PROPN
ejpam-4383	207	16	0j	0j	VERB
ejpam-4383	207	17	if	if	SCONJ
ejpam-4383	207	18	j	j	PROPN
ejpam-4383	207	19	∈	∈	PROPN
ejpam-4383	207	20	i1	i1	PROPN
ejpam-4383	207	21	−	−	PROPN
ejpam-4383	207	22	i2	i2	PROPN
ejpam-4383	207	23	pj	pj	PROPN
ejpam-4383	207	24	∗j	∗j	PROPN
ejpam-4383	207	25	qj	qj	PROPN
ejpam-4383	207	26	if	if	SCONJ
ejpam-4383	207	27	j	j	PROPN
ejpam-4383	207	28	∈	∈	PROPN
ejpam-4383	207	29	i1	i1	PROPN
ejpam-4383	207	30	∩	∩	PROPN
ejpam-4383	207	31	i2	i2	PROPN
ejpam-4383	207	32	0j	0j	PROPN
ejpam-4383	207	33	∗j	∗j	PROPN
ejpam-4383	207	34	qj	qj	PROPN
ejpam-4383	207	35	if	if	SCONJ
ejpam-4383	207	36	j	j	PROPN
ejpam-4383	207	37	∈	∈	PROPN
ejpam-4383	207	38	i2	i2	PROPN
ejpam-4383	207	39	−	−	PROPN
ejpam-4383	207	40	i1	i1	PROPN
ejpam-4383	207	41	0j	0j	VERB
ejpam-4383	207	42	∗j	∗j	PROPN
ejpam-4383	207	43	0j	0j	NOUN
ejpam-4383	207	44	otherwise	otherwise	ADV
ejpam-4383	207	45			NOUN
ejpam-4383	207	46	.	.	PUNCT
ejpam-4383	208	1	by	by	ADP
ejpam-4383	208	2	(	(	PUNCT
ejpam-4383	208	3	b-1	b-1	PROPN
ejpam-4383	208	4	)	)	PUNCT
ejpam-4383	208	5	and	and	CCONJ
ejpam-4383	208	6	(	(	PUNCT
ejpam-4383	208	7	b-2	b-2	NOUN
ejpam-4383	208	8	)	)	PUNCT
ejpam-4383	208	9	,	,	PUNCT
ejpam-4383	208	10	we	we	PRON
ejpam-4383	208	11	have	have	VERB
ejpam-4383	208	12	(	(	PUNCT
ejpam-4383	208	13	∀j	∀j	PROPN
ejpam-4383	208	14	∈	∈	PROPN
ejpam-4383	208	15	i	i	NOUN
ejpam-4383	208	16	)	)	PUNCT
ejpam-4383	209	1	((pi)i∈i	((pi)i∈i	ADJ
ejpam-4383	209	2	⊗	⊗	PROPN
ejpam-4383	209	3	(	(	PUNCT
ejpam-4383	209	4	qi)i∈i)(j	qi)i∈i)(j	NOUN
ejpam-4383	209	5	)	)	PUNCT
ejpam-4383	209	6	=	=	PUNCT
ejpam-4383	210	1			VERB
ejpam-4383	210	2	pj	pj	PROPN
ejpam-4383	211	1	if	if	SCONJ
ejpam-4383	211	2	j	j	PROPN
ejpam-4383	211	3	∈	∈	PROPN
ejpam-4383	211	4	i1	i1	PROPN
ejpam-4383	211	5	−	−	PROPN
ejpam-4383	211	6	i2	i2	PROPN
ejpam-4383	211	7	pj	pj	PROPN
ejpam-4383	211	8	∗j	∗j	PROPN
ejpam-4383	211	9	qj	qj	PROPN
ejpam-4383	211	10	if	if	SCONJ
ejpam-4383	211	11	j	j	PROPN
ejpam-4383	211	12	∈	∈	PROPN
ejpam-4383	211	13	i1	i1	PROPN
ejpam-4383	211	14	∩	∩	PROPN
ejpam-4383	211	15	i2	i2	PROPN
ejpam-4383	211	16	0j	0j	PROPN
ejpam-4383	211	17	∗j	∗j	PROPN
ejpam-4383	211	18	qj	qj	PROPN
ejpam-4383	211	19	if	if	SCONJ
ejpam-4383	211	20	j	j	PROPN
ejpam-4383	211	21	∈	∈	PROPN
ejpam-4383	211	22	i2	i2	PROPN
ejpam-4383	211	23	−	−	PROPN
ejpam-4383	211	24	i1	i1	PROPN
ejpam-4383	211	25	0j	0j	NOUN
ejpam-4383	211	26	otherwise	otherwise	ADV
ejpam-4383	211	27			NOUN
ejpam-4383	211	28	.	.	PUNCT
ejpam-4383	212	1	this	this	PRON
ejpam-4383	212	2	implies	imply	VERB
ejpam-4383	212	3	that	that	SCONJ
ejpam-4383	212	4	the	the	DET
ejpam-4383	212	5	number	number	NOUN
ejpam-4383	212	6	of	of	ADP
ejpam-4383	212	7	such	such	ADJ
ejpam-4383	212	8	(	(	PUNCT
ejpam-4383	212	9	(	(	PUNCT
ejpam-4383	212	10	pi)i∈i⊗(qi)i∈i)(j	pi)i∈i⊗(qi)i∈i)(j	ADJ
ejpam-4383	212	11	)	)	PUNCT
ejpam-4383	212	12	is	be	AUX
ejpam-4383	212	13	not	not	PART
ejpam-4383	212	14	more	more	ADJ
ejpam-4383	212	15	than	than	ADP
ejpam-4383	212	16	|i1∪i2|	|i1∪i2|	NUM
ejpam-4383	212	17	,	,	PUNCT
ejpam-4383	212	18	that	that	ADV
ejpam-4383	212	19	is	is	ADV
ejpam-4383	212	20	,	,	PUNCT
ejpam-4383	212	21	it	it	PRON
ejpam-4383	212	22	is	be	AUX
ejpam-4383	212	23	finite	finite	ADJ
ejpam-4383	212	24	.	.	PUNCT
ejpam-4383	213	1	thus	thus	ADV
ejpam-4383	213	2	(	(	PUNCT
ejpam-4383	213	3	pi)i∈i	pi)i∈i	NUM
ejpam-4383	213	4	⊗	⊗	PROPN
ejpam-4383	213	5	(	(	PUNCT
ejpam-4383	213	6	qi)i∈i	qi)i∈i	NUM
ejpam-4383	213	7	∈	∈	PROPN
ejpam-4383	213	8	∏w	∏w	X
ejpam-4383	213	9	i∈i	i∈i	ADJ
ejpam-4383	213	10	pi	pi	NOUN
ejpam-4383	213	11	.	.	PUNCT
ejpam-4383	214	1	hence	hence	ADV
ejpam-4383	214	2	,	,	PUNCT
ejpam-4383	214	3	∏w	∏w	X
ejpam-4383	214	4	i∈i	i∈i	ADJ
ejpam-4383	214	5	pi	pi	NOUN
ejpam-4383	214	6	is	be	AUX
ejpam-4383	214	7	a	a	DET
ejpam-4383	214	8	b	b	NOUN
ejpam-4383	214	9	-subalgebra	-subalgebra	NOUN
ejpam-4383	214	10	of	of	ADP
ejpam-4383	214	11	∏	∏	PROPN
ejpam-4383	214	12	i∈i	i∈i	ADJ
ejpam-4383	214	13	pi	pi	NOUN
ejpam-4383	214	14	.	.	PUNCT
ejpam-4383	215	1	theorem	theorem	ADJ
ejpam-4383	215	2	7	7	NUM
ejpam-4383	215	3	.	.	PUNCT
ejpam-4383	216	1	let	let	VERB
ejpam-4383	216	2	pi	pi	NOUN
ejpam-4383	216	3	=	=	PUNCT
ejpam-4383	216	4	(	(	PUNCT
ejpam-4383	216	5	pi	pi	NOUN
ejpam-4383	216	6	;	;	PUNCT
ejpam-4383	216	7	∗i	∗i	PROPN
ejpam-4383	216	8	,	,	PUNCT
ejpam-4383	216	9	0i	0i	NOUN
ejpam-4383	216	10	)	)	PUNCT
ejpam-4383	216	11	be	be	VERB
ejpam-4383	216	12	a	a	DET
ejpam-4383	216	13	b	b	NOUN
ejpam-4383	216	14	-	-	PUNCT
ejpam-4383	216	15	algebra	algebra	NOUN
ejpam-4383	216	16	and	and	CCONJ
ejpam-4383	216	17	qi	qi	PRON
ejpam-4383	216	18	a	a	DET
ejpam-4383	216	19	subset	subset	NOUN
ejpam-4383	216	20	of	of	ADP
ejpam-4383	216	21	pi	pi	NOUN
ejpam-4383	216	22	for	for	ADP
ejpam-4383	216	23	all	all	PRON
ejpam-4383	216	24	i	i	PRON
ejpam-4383	216	25	∈	∈	PROPN
ejpam-4383	216	26	i.	i.	NOUN
ejpam-4383	216	27	then	then	ADV
ejpam-4383	216	28	qi	qi	PROPN
ejpam-4383	216	29	is	be	AUX
ejpam-4383	216	30	a	a	DET
ejpam-4383	216	31	b	b	NOUN
ejpam-4383	216	32	-	-	PUNCT
ejpam-4383	216	33	subalgebra	subalgebra	NOUN
ejpam-4383	216	34	of	of	ADP
ejpam-4383	216	35	pi	pi	NOUN
ejpam-4383	216	36	for	for	ADP
ejpam-4383	216	37	all	all	PRON
ejpam-4383	216	38	i	i	PRON
ejpam-4383	216	39	∈	∈	VERB
ejpam-4383	217	1	i	i	PRON
ejpam-4383	217	2	if	if	SCONJ
ejpam-4383	217	3	and	and	CCONJ
ejpam-4383	217	4	only	only	ADV
ejpam-4383	217	5	if	if	SCONJ
ejpam-4383	217	6	∏	∏	PROPN
ejpam-4383	217	7	i∈i	i∈i	NOUN
ejpam-4383	217	8	qi	qi	PROPN
ejpam-4383	217	9	is	be	AUX
ejpam-4383	217	10	a	a	DET
ejpam-4383	217	11	b	b	NOUN
ejpam-4383	217	12	-	-	PUNCT
ejpam-4383	217	13	subalgebra	subalgebra	NOUN
ejpam-4383	217	14	of	of	ADP
ejpam-4383	217	15	the	the	DET
ejpam-4383	217	16	external	external	ADJ
ejpam-4383	217	17	direct	direct	ADJ
ejpam-4383	217	18	product	product	NOUN
ejpam-4383	217	19	b	b	X
ejpam-4383	217	20	-	-	PUNCT
ejpam-4383	217	21	algebra	algebra	NOUN
ejpam-4383	217	22	∏	∏	PROPN
ejpam-4383	217	23	i∈i	i∈i	ADJ
ejpam-4383	217	24	pi	pi	NOUN
ejpam-4383	217	25	=	=	PUNCT
ejpam-4383	217	26	(	(	PUNCT
ejpam-4383	217	27	∏	∏	X
ejpam-4383	217	28	i∈i	i∈i	ADJ
ejpam-4383	217	29	pi;⊗	pi;⊗	PROPN
ejpam-4383	217	30	,	,	PUNCT
ejpam-4383	217	31	(	(	PUNCT
ejpam-4383	217	32	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	217	33	)	)	PUNCT
ejpam-4383	217	34	.	.	PUNCT
ejpam-4383	218	1	a.	a.	PROPN
ejpam-4383	218	2	iampan	iampan	PROPN
ejpam-4383	218	3	et	et	PROPN
ejpam-4383	218	4	al	al	PROPN
ejpam-4383	218	5	.	.	PUNCT
ejpam-4383	218	6	/	/	SYM
ejpam-4383	218	7	eur	eur	PROPN
ejpam-4383	218	8	.	.	PUNCT
ejpam-4383	219	1	j.	j.	PROPN
ejpam-4383	219	2	pure	pure	PROPN
ejpam-4383	219	3	appl	appl	PROPN
ejpam-4383	219	4	.	.	PROPN
ejpam-4383	219	5	math	math	PROPN
ejpam-4383	219	6	,	,	PUNCT
ejpam-4383	219	7	15	15	NUM
ejpam-4383	219	8	(	(	PUNCT
ejpam-4383	219	9	3	3	NUM
ejpam-4383	219	10	)	)	PUNCT
ejpam-4383	219	11	(	(	PUNCT
ejpam-4383	219	12	2022	2022	NUM
ejpam-4383	219	13	)	)	PUNCT
ejpam-4383	219	14	,	,	PUNCT
ejpam-4383	219	15	999	999	NUM
ejpam-4383	219	16	-	-	SYM
ejpam-4383	219	17	1014	1014	NUM
ejpam-4383	219	18	1007	1007	NUM
ejpam-4383	219	19	proof	proof	NOUN
ejpam-4383	219	20	.	.	PUNCT
ejpam-4383	220	1	assume	assume	VERB
ejpam-4383	220	2	that	that	SCONJ
ejpam-4383	220	3	qi	qi	PROPN
ejpam-4383	220	4	is	be	AUX
ejpam-4383	220	5	a	a	DET
ejpam-4383	220	6	b	b	NOUN
ejpam-4383	220	7	-subalgebra	-subalgebra	NOUN
ejpam-4383	220	8	of	of	ADP
ejpam-4383	220	9	pi	pi	NOUN
ejpam-4383	220	10	for	for	ADP
ejpam-4383	220	11	all	all	DET
ejpam-4383	220	12	i	i	PRON
ejpam-4383	220	13	∈	∈	PROPN
ejpam-4383	220	14	i.	i.	NOUN
ejpam-4383	220	15	then	then	ADV
ejpam-4383	220	16	0i	0i	PROPN
ejpam-4383	220	17	∈	∈	PROPN
ejpam-4383	220	18	qi	qi	PROPN
ejpam-4383	220	19	for	for	ADP
ejpam-4383	220	20	all	all	PRON
ejpam-4383	220	21	i	i	PRON
ejpam-4383	220	22	∈	∈	PROPN
ejpam-4383	221	1	i	i	PRON
ejpam-4383	221	2	,	,	PUNCT
ejpam-4383	221	3	so	so	CCONJ
ejpam-4383	221	4	(	(	PUNCT
ejpam-4383	221	5	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	221	6	∈	∈	PROPN
ejpam-4383	221	7	∏	∏	PROPN
ejpam-4383	221	8	i∈i	i∈i	ADJ
ejpam-4383	221	9	qi	qi	PROPN
ejpam-4383	221	10	̸=	̸=	PROPN
ejpam-4383	221	11	∅.	∅.	ADV
ejpam-4383	221	12	let	let	VERB
ejpam-4383	221	13	(	(	PUNCT
ejpam-4383	221	14	pi)i∈i	pi)i∈i	NUM
ejpam-4383	221	15	,	,	PUNCT
ejpam-4383	221	16	(	(	PUNCT
ejpam-4383	221	17	qi)i∈i	qi)i∈i	NUM
ejpam-4383	221	18	∈	∈	PROPN
ejpam-4383	221	19	∏	∏	PROPN
ejpam-4383	221	20	i∈i	i∈i	PROPN
ejpam-4383	221	21	qi	qi	PROPN
ejpam-4383	221	22	.	.	PUNCT
ejpam-4383	222	1	then	then	ADV
ejpam-4383	222	2	pi	pi	PROPN
ejpam-4383	222	3	,	,	PUNCT
ejpam-4383	222	4	qi	qi	PROPN
ejpam-4383	222	5	∈	∈	PROPN
ejpam-4383	222	6	qi	qi	PROPN
ejpam-4383	222	7	for	for	ADP
ejpam-4383	222	8	all	all	PRON
ejpam-4383	222	9	i	i	PRON
ejpam-4383	222	10	∈	∈	PROPN
ejpam-4383	222	11	i.	i.	NOUN
ejpam-4383	222	12	thus	thus	ADV
ejpam-4383	222	13	pi	pi	VERB
ejpam-4383	222	14	∗i	∗i	PROPN
ejpam-4383	222	15	qi	qi	PROPN
ejpam-4383	222	16	∈	∈	PROPN
ejpam-4383	222	17	qi	qi	PROPN
ejpam-4383	222	18	for	for	ADP
ejpam-4383	222	19	all	all	PRON
ejpam-4383	222	20	i	i	PRON
ejpam-4383	222	21	∈	∈	PROPN
ejpam-4383	223	1	i	i	PRON
ejpam-4383	223	2	,	,	PUNCT
ejpam-4383	223	3	so	so	CCONJ
ejpam-4383	223	4	(	(	PUNCT
ejpam-4383	223	5	pi)i∈i	pi)i∈i	NUM
ejpam-4383	223	6	⊗	⊗	PROPN
ejpam-4383	223	7	(	(	PUNCT
ejpam-4383	223	8	qi)i∈i	qi)i∈i	NUM
ejpam-4383	223	9	=	=	SYM
ejpam-4383	223	10	(	(	PUNCT
ejpam-4383	223	11	pi	pi	NOUN
ejpam-4383	223	12	∗i	∗i	PROPN
ejpam-4383	223	13	qi)i∈i	qi)i∈i	NUM
ejpam-4383	223	14	∈	∈	PROPN
ejpam-4383	223	15	∏	∏	PROPN
ejpam-4383	223	16	i∈i	i∈i	PROPN
ejpam-4383	223	17	qi	qi	PROPN
ejpam-4383	223	18	.	.	PUNCT
ejpam-4383	224	1	hence	hence	ADV
ejpam-4383	224	2	,	,	PUNCT
ejpam-4383	224	3	∏	∏	PROPN
ejpam-4383	224	4	i∈i	i∈i	ADJ
ejpam-4383	224	5	qi	qi	PROPN
ejpam-4383	224	6	is	be	AUX
ejpam-4383	224	7	a	a	DET
ejpam-4383	224	8	b	b	NOUN
ejpam-4383	224	9	-subalgebra	-subalgebra	NOUN
ejpam-4383	224	10	of	of	ADP
ejpam-4383	224	11	∏	∏	NUM
ejpam-4383	224	12	i∈i	i∈i	ADJ
ejpam-4383	224	13	pi	pi	NOUN
ejpam-4383	224	14	.	.	PUNCT
ejpam-4383	225	1	conversely	conversely	ADV
ejpam-4383	225	2	,	,	PUNCT
ejpam-4383	225	3	assume	assume	VERB
ejpam-4383	225	4	that	that	SCONJ
ejpam-4383	225	5	∏	∏	PROPN
ejpam-4383	225	6	i∈i	i∈i	NOUN
ejpam-4383	225	7	qi	qi	PROPN
ejpam-4383	225	8	is	be	AUX
ejpam-4383	225	9	a	a	DET
ejpam-4383	225	10	b	b	NOUN
ejpam-4383	225	11	-subalgebra	-subalgebra	NOUN
ejpam-4383	225	12	of	of	ADP
ejpam-4383	225	13	∏	∏	NUM
ejpam-4383	225	14	i∈i	i∈i	ADJ
ejpam-4383	225	15	pi	pi	NOUN
ejpam-4383	225	16	.	.	PUNCT
ejpam-4383	226	1	then	then	ADV
ejpam-4383	226	2	(	(	PUNCT
ejpam-4383	226	3	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	226	4	∈	∈	PROPN
ejpam-4383	226	5	∏	∏	PROPN
ejpam-4383	226	6	i∈i	i∈i	PROPN
ejpam-4383	226	7	qi	qi	PROPN
ejpam-4383	226	8	,	,	PUNCT
ejpam-4383	226	9	so	so	SCONJ
ejpam-4383	226	10	0i	0i	PROPN
ejpam-4383	226	11	∈	∈	PROPN
ejpam-4383	226	12	qi	qi	PROPN
ejpam-4383	226	13	̸=	̸=	PROPN
ejpam-4383	226	14	∅	∅	NOUN
ejpam-4383	226	15	for	for	ADP
ejpam-4383	226	16	all	all	PRON
ejpam-4383	226	17	i	i	PRON
ejpam-4383	226	18	∈	∈	PROPN
ejpam-4383	226	19	i.	i.	NOUN
ejpam-4383	226	20	let	let	VERB
ejpam-4383	226	21	i	i	PRON
ejpam-4383	226	22	∈	∈	VERB
ejpam-4383	226	23	i	i	PRON
ejpam-4383	226	24	and	and	CCONJ
ejpam-4383	226	25	let	let	VERB
ejpam-4383	226	26	pi	pi	NOUN
ejpam-4383	226	27	,	,	PUNCT
ejpam-4383	226	28	qi	qi	PROPN
ejpam-4383	226	29	∈	∈	PROPN
ejpam-4383	226	30	qi	qi	PROPN
ejpam-4383	226	31	.	.	PUNCT
ejpam-4383	227	1	then	then	ADV
ejpam-4383	227	2	fpi	fpi	PROPN
ejpam-4383	227	3	,	,	PUNCT
ejpam-4383	227	4	fqi	fqi	VERB
ejpam-4383	227	5	∈	∈	PROPN
ejpam-4383	227	6	∏	∏	PROPN
ejpam-4383	227	7	i∈i	i∈i	PROPN
ejpam-4383	227	8	qi	qi	PROPN
ejpam-4383	227	9	,	,	PUNCT
ejpam-4383	227	10	which	which	PRON
ejpam-4383	227	11	is	be	AUX
ejpam-4383	227	12	defined	define	VERB
ejpam-4383	227	13	by	by	ADP
ejpam-4383	227	14	(	(	PUNCT
ejpam-4383	227	15	2.2	2.2	NUM
ejpam-4383	227	16	)	)	PUNCT
ejpam-4383	227	17	.	.	PUNCT
ejpam-4383	228	1	since	since	SCONJ
ejpam-4383	228	2	∏	∏	PROPN
ejpam-4383	228	3	i∈i	i∈i	NOUN
ejpam-4383	228	4	qi	qi	PROPN
ejpam-4383	228	5	is	be	AUX
ejpam-4383	228	6	a	a	DET
ejpam-4383	228	7	b	b	NOUN
ejpam-4383	228	8	-subalgebra	-subalgebra	NOUN
ejpam-4383	228	9	of	of	ADP
ejpam-4383	228	10	∏	∏	PROPN
ejpam-4383	228	11	i∈i	i∈i	ADJ
ejpam-4383	228	12	pi	pi	NOUN
ejpam-4383	228	13	,	,	PUNCT
ejpam-4383	228	14	we	we	PRON
ejpam-4383	228	15	have	have	VERB
ejpam-4383	228	16	fpi	fpi	PROPN
ejpam-4383	228	17	⊗	⊗	PROPN
ejpam-4383	228	18	fqi	fqi	VERB
ejpam-4383	228	19	∈	∈	PROPN
ejpam-4383	228	20	∏	∏	PROPN
ejpam-4383	228	21	i∈i	i∈i	PROPN
ejpam-4383	228	22	qi	qi	PROPN
ejpam-4383	228	23	.	.	PUNCT
ejpam-4383	229	1	now	now	ADV
ejpam-4383	229	2	,	,	PUNCT
ejpam-4383	229	3	(	(	PUNCT
ejpam-4383	229	4	∀j	∀j	PROPN
ejpam-4383	229	5	∈	∈	PROPN
ejpam-4383	229	6	i	i	NOUN
ejpam-4383	229	7	)	)	PUNCT
ejpam-4383	229	8	(	(	PUNCT
ejpam-4383	229	9	(	(	PUNCT
ejpam-4383	229	10	fpi	fpi	PROPN
ejpam-4383	229	11	⊗	⊗	PROPN
ejpam-4383	229	12	fqi)(j	fqi)(j	PROPN
ejpam-4383	229	13	)	)	PUNCT
ejpam-4383	229	14	=	=	PRON
ejpam-4383	229	15	{	{	PUNCT
ejpam-4383	229	16	pi	pi	NOUN
ejpam-4383	229	17	∗i	∗i	PROPN
ejpam-4383	229	18	qi	qi	PROPN
ejpam-4383	229	19	if	if	SCONJ
ejpam-4383	229	20	j	j	PROPN
ejpam-4383	229	21	=	=	PRON
ejpam-4383	230	1	i	i	PRON
ejpam-4383	230	2	0j	0j	VERB
ejpam-4383	230	3	∗j	∗j	PROPN
ejpam-4383	230	4	0j	0j	NOUN
ejpam-4383	230	5	otherwise	otherwise	ADV
ejpam-4383	230	6	)	)	PUNCT
ejpam-4383	230	7	,	,	PUNCT
ejpam-4383	230	8	this	this	PRON
ejpam-4383	230	9	implies	imply	VERB
ejpam-4383	230	10	that	that	SCONJ
ejpam-4383	230	11	pi	pi	NOUN
ejpam-4383	230	12	∗i	∗i	PROPN
ejpam-4383	230	13	qi	qi	PROPN
ejpam-4383	230	14	∈	∈	PROPN
ejpam-4383	230	15	qi	qi	PROPN
ejpam-4383	230	16	.	.	PUNCT
ejpam-4383	231	1	hence	hence	ADV
ejpam-4383	231	2	,	,	PUNCT
ejpam-4383	231	3	qi	qi	PROPN
ejpam-4383	231	4	is	be	AUX
ejpam-4383	231	5	a	a	DET
ejpam-4383	231	6	b	b	NOUN
ejpam-4383	231	7	-subalgebra	-subalgebra	NOUN
ejpam-4383	231	8	of	of	ADP
ejpam-4383	231	9	pi	pi	NOUN
ejpam-4383	231	10	for	for	ADP
ejpam-4383	231	11	all	all	PRON
ejpam-4383	231	12	i	i	PRON
ejpam-4383	231	13	∈	∈	PROPN
ejpam-4383	231	14	i.	i.	NOUN
ejpam-4383	231	15	theorem	theorem	VERB
ejpam-4383	231	16	8	8	NUM
ejpam-4383	231	17	.	.	PUNCT
ejpam-4383	232	1	let	let	VERB
ejpam-4383	232	2	pi	pi	NOUN
ejpam-4383	232	3	=	=	PUNCT
ejpam-4383	232	4	(	(	PUNCT
ejpam-4383	232	5	pi	pi	NOUN
ejpam-4383	232	6	;	;	PUNCT
ejpam-4383	232	7	∗i	∗i	PROPN
ejpam-4383	232	8	,	,	PUNCT
ejpam-4383	232	9	0i	0i	NOUN
ejpam-4383	232	10	)	)	PUNCT
ejpam-4383	232	11	be	be	VERB
ejpam-4383	232	12	a	a	DET
ejpam-4383	232	13	b	b	NOUN
ejpam-4383	232	14	-	-	PUNCT
ejpam-4383	232	15	algebra	algebra	NOUN
ejpam-4383	232	16	and	and	CCONJ
ejpam-4383	232	17	qi	qi	PRON
ejpam-4383	232	18	a	a	DET
ejpam-4383	232	19	subset	subset	NOUN
ejpam-4383	232	20	of	of	ADP
ejpam-4383	232	21	pi	pi	NOUN
ejpam-4383	232	22	for	for	ADP
ejpam-4383	232	23	all	all	PRON
ejpam-4383	232	24	i	i	PRON
ejpam-4383	232	25	∈	∈	PROPN
ejpam-4383	232	26	i.	i.	NOUN
ejpam-4383	232	27	then	then	ADV
ejpam-4383	232	28	qi	qi	PROPN
ejpam-4383	232	29	is	be	AUX
ejpam-4383	232	30	normal	normal	ADJ
ejpam-4383	232	31	of	of	ADP
ejpam-4383	232	32	pi	pi	NOUN
ejpam-4383	232	33	for	for	ADP
ejpam-4383	232	34	all	all	PRON
ejpam-4383	232	35	i	i	PRON
ejpam-4383	232	36	∈	∈	VERB
ejpam-4383	233	1	i	i	PRON
ejpam-4383	233	2	if	if	SCONJ
ejpam-4383	233	3	and	and	CCONJ
ejpam-4383	233	4	only	only	ADV
ejpam-4383	233	5	if	if	SCONJ
ejpam-4383	233	6	∏	∏	PROPN
ejpam-4383	233	7	i∈i	i∈i	NOUN
ejpam-4383	233	8	qi	qi	PROPN
ejpam-4383	233	9	is	be	AUX
ejpam-4383	233	10	normal	normal	ADJ
ejpam-4383	233	11	of	of	ADP
ejpam-4383	233	12	the	the	DET
ejpam-4383	233	13	external	external	ADJ
ejpam-4383	233	14	direct	direct	ADJ
ejpam-4383	233	15	product	product	NOUN
ejpam-4383	233	16	b	b	X
ejpam-4383	233	17	-	-	PUNCT
ejpam-4383	233	18	algebra	algebra	NOUN
ejpam-4383	233	19	∏	∏	PROPN
ejpam-4383	233	20	i∈i	i∈i	ADJ
ejpam-4383	233	21	pi	pi	NOUN
ejpam-4383	233	22	=	=	PUNCT
ejpam-4383	233	23	(	(	PUNCT
ejpam-4383	233	24	∏	∏	X
ejpam-4383	233	25	i∈i	i∈i	ADJ
ejpam-4383	233	26	pi;⊗	pi;⊗	PROPN
ejpam-4383	233	27	,	,	PUNCT
ejpam-4383	233	28	(	(	PUNCT
ejpam-4383	233	29	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	233	30	)	)	PUNCT
ejpam-4383	233	31	.	.	PUNCT
ejpam-4383	234	1	proof	proof	NOUN
ejpam-4383	234	2	.	.	PUNCT
ejpam-4383	235	1	assume	assume	VERB
ejpam-4383	235	2	that	that	SCONJ
ejpam-4383	235	3	qi	qi	PROPN
ejpam-4383	235	4	is	be	AUX
ejpam-4383	235	5	normal	normal	ADJ
ejpam-4383	235	6	of	of	ADP
ejpam-4383	235	7	pi	pi	NOUN
ejpam-4383	235	8	for	for	ADP
ejpam-4383	235	9	all	all	PRON
ejpam-4383	235	10	i	i	PRON
ejpam-4383	235	11	∈	∈	PROPN
ejpam-4383	235	12	i.	i.	NOUN
ejpam-4383	235	13	then	then	ADV
ejpam-4383	235	14	qi	qi	PROPN
ejpam-4383	235	15	̸=	̸=	PROPN
ejpam-4383	235	16	∅	∅	NOUN
ejpam-4383	235	17	for	for	ADP
ejpam-4383	235	18	all	all	PRON
ejpam-4383	235	19	i	i	PRON
ejpam-4383	235	20	∈	∈	PROPN
ejpam-4383	236	1	i	i	PRON
ejpam-4383	236	2	,	,	PUNCT
ejpam-4383	236	3	so	so	ADV
ejpam-4383	236	4	∏	∏	PROPN
ejpam-4383	236	5	i∈i	i∈i	ADJ
ejpam-4383	236	6	qi	qi	PROPN
ejpam-4383	236	7	̸=	̸=	PROPN
ejpam-4383	236	8	∅.	∅.	ADV
ejpam-4383	236	9	let	let	VERB
ejpam-4383	236	10	(	(	PUNCT
ejpam-4383	236	11	pi)i∈i	pi)i∈i	NUM
ejpam-4383	236	12	,	,	PUNCT
ejpam-4383	236	13	(	(	PUNCT
ejpam-4383	236	14	p	p	NOUN
ejpam-4383	236	15	′	′	NUM
ejpam-4383	237	1	i)i∈i	i)i∈i	PROPN
ejpam-4383	237	2	,	,	PUNCT
ejpam-4383	237	3	(	(	PUNCT
ejpam-4383	237	4	qi)i∈i	qi)i∈i	NUM
ejpam-4383	237	5	,	,	PUNCT
ejpam-4383	237	6	(	(	PUNCT
ejpam-4383	237	7	q	q	NOUN
ejpam-4383	237	8	′	′	NUM
ejpam-4383	237	9	i)i∈i	i)i∈i	PROPN
ejpam-4383	237	10	∈	∈	PROPN
ejpam-4383	237	11	∏	∏	PROPN
ejpam-4383	237	12	i∈i	i∈i	ADJ
ejpam-4383	237	13	pi	pi	NOUN
ejpam-4383	237	14	be	be	AUX
ejpam-4383	237	15	such	such	ADJ
ejpam-4383	237	16	that	that	SCONJ
ejpam-4383	237	17	(	(	PUNCT
ejpam-4383	237	18	pi)i∈i	pi)i∈i	NUM
ejpam-4383	237	19	⊗	⊗	PROPN
ejpam-4383	237	20	(	(	PUNCT
ejpam-4383	237	21	qi)i∈i	qi)i∈i	NUM
ejpam-4383	237	22	,	,	PUNCT
ejpam-4383	237	23	(	(	PUNCT
ejpam-4383	237	24	p	p	NOUN
ejpam-4383	237	25	′	′	NUM
ejpam-4383	237	26	i)i∈i	i)i∈i	PROPN
ejpam-4383	237	27	⊗	⊗	PROPN
ejpam-4383	237	28	(	(	PUNCT
ejpam-4383	237	29	q′i)i∈i	q′i)i∈i	ADP
ejpam-4383	237	30	∈	∈	PROPN
ejpam-4383	237	31	∏	∏	PROPN
ejpam-4383	237	32	i∈i	i∈i	PROPN
ejpam-4383	237	33	qi	qi	PROPN
ejpam-4383	237	34	.	.	PUNCT
ejpam-4383	238	1	then	then	ADV
ejpam-4383	238	2	pi	pi	VERB
ejpam-4383	238	3	∗i	∗i	PROPN
ejpam-4383	238	4	qi	qi	PROPN
ejpam-4383	238	5	,	,	PUNCT
ejpam-4383	238	6	p′i	p′i	NOUN
ejpam-4383	238	7	∗i	∗i	PROPN
ejpam-4383	238	8	q′i	q′i	PROPN
ejpam-4383	238	9	∈	∈	PROPN
ejpam-4383	238	10	qi	qi	NOUN
ejpam-4383	238	11	for	for	ADP
ejpam-4383	238	12	all	all	PRON
ejpam-4383	238	13	i	i	PRON
ejpam-4383	238	14	∈	∈	PROPN
ejpam-4383	238	15	i.	i.	NOUN
ejpam-4383	238	16	since	since	SCONJ
ejpam-4383	238	17	qi	qi	PROPN
ejpam-4383	238	18	is	be	AUX
ejpam-4383	238	19	normal	normal	ADJ
ejpam-4383	238	20	of	of	ADP
ejpam-4383	238	21	pi	pi	NOUN
ejpam-4383	238	22	,	,	PUNCT
ejpam-4383	238	23	we	we	PRON
ejpam-4383	238	24	have	have	VERB
ejpam-4383	238	25	(	(	PUNCT
ejpam-4383	238	26	pi	pi	NOUN
ejpam-4383	238	27	∗i	∗i	PROPN
ejpam-4383	238	28	p′i	p′i	NOUN
ejpam-4383	238	29	)	)	PUNCT
ejpam-4383	239	1	∗i	∗i	PROPN
ejpam-4383	239	2	(	(	PUNCT
ejpam-4383	239	3	qi	qi	PROPN
ejpam-4383	239	4	∗i	∗i	PROPN
ejpam-4383	239	5	q′i	q′i	PROPN
ejpam-4383	239	6	)	)	PUNCT
ejpam-4383	239	7	∈	∈	PROPN
ejpam-4383	239	8	qi	qi	PROPN
ejpam-4383	239	9	for	for	ADP
ejpam-4383	239	10	all	all	DET
ejpam-4383	239	11	i	i	PRON
ejpam-4383	239	12	∈	∈	PROPN
ejpam-4383	239	13	i.	i.	NOUN
ejpam-4383	239	14	thus	thus	ADV
ejpam-4383	239	15	(	(	PUNCT
ejpam-4383	239	16	(	(	PUNCT
ejpam-4383	239	17	pi)i∈i⊗(p′i)i∈i)⊗((qi)i∈i⊗(q′i)i∈i	pi)i∈i⊗(p′i)i∈i)⊗((qi)i∈i⊗(q′i)i∈i	ADJ
ejpam-4383	239	18	)	)	PUNCT
ejpam-4383	239	19	=	=	SYM
ejpam-4383	239	20	(	(	PUNCT
ejpam-4383	239	21	pi∗ip′i)i∈i⊗(qi∗iq′i)i∈i	pi∗ip′i)i∈i⊗(qi∗iq′i)i∈i	NOUN
ejpam-4383	239	22	=	=	SYM
ejpam-4383	239	23	(	(	PUNCT
ejpam-4383	239	24	(	(	PUNCT
ejpam-4383	239	25	pi∗ip′i)∗i(qi∗iq′i))i∈i	pi∗ip′i)∗i(qi∗iq′i))i∈i	PROPN
ejpam-4383	239	26	∈	∈	PROPN
ejpam-4383	239	27	∏	∏	PROPN
ejpam-4383	239	28	i∈i	i∈i	PROPN
ejpam-4383	239	29	qi	qi	PROPN
ejpam-4383	239	30	.	.	PUNCT
ejpam-4383	240	1	hence	hence	ADV
ejpam-4383	240	2	,	,	PUNCT
ejpam-4383	240	3	∏	∏	PROPN
ejpam-4383	240	4	i∈i	i∈i	ADJ
ejpam-4383	240	5	qi	qi	PROPN
ejpam-4383	240	6	is	be	AUX
ejpam-4383	240	7	normal	normal	ADJ
ejpam-4383	240	8	of	of	ADP
ejpam-4383	240	9	∏	∏	NUM
ejpam-4383	240	10	i∈i	i∈i	ADJ
ejpam-4383	240	11	pi	pi	NOUN
ejpam-4383	240	12	.	.	PUNCT
ejpam-4383	241	1	conversely	conversely	ADV
ejpam-4383	241	2	,	,	PUNCT
ejpam-4383	241	3	assume	assume	VERB
ejpam-4383	241	4	that	that	SCONJ
ejpam-4383	241	5	∏	∏	PROPN
ejpam-4383	241	6	i∈i	i∈i	NOUN
ejpam-4383	241	7	qi	qi	PROPN
ejpam-4383	241	8	is	be	AUX
ejpam-4383	241	9	normal	normal	ADJ
ejpam-4383	241	10	of	of	ADP
ejpam-4383	241	11	∏	∏	NUM
ejpam-4383	241	12	i∈i	i∈i	ADJ
ejpam-4383	241	13	pi	pi	NOUN
ejpam-4383	241	14	.	.	PUNCT
ejpam-4383	242	1	then	then	ADV
ejpam-4383	242	2	∏	∏	PROPN
ejpam-4383	242	3	i∈i	i∈i	ADJ
ejpam-4383	242	4	qi	qi	PROPN
ejpam-4383	242	5	̸=	̸=	PROPN
ejpam-4383	242	6	∅	∅	NOUN
ejpam-4383	242	7	,	,	PUNCT
ejpam-4383	242	8	so	so	ADV
ejpam-4383	242	9	qi	qi	PROPN
ejpam-4383	242	10	̸=	̸=	PROPN
ejpam-4383	242	11	∅	∅	NOUN
ejpam-4383	242	12	for	for	ADP
ejpam-4383	242	13	all	all	PRON
ejpam-4383	242	14	i	i	PRON
ejpam-4383	242	15	∈	∈	PROPN
ejpam-4383	242	16	i.	i.	NOUN
ejpam-4383	242	17	by	by	ADP
ejpam-4383	242	18	theorem	theorem	NOUN
ejpam-4383	242	19	1	1	NUM
ejpam-4383	242	20	,	,	PUNCT
ejpam-4383	242	21	we	we	PRON
ejpam-4383	242	22	have	have	VERB
ejpam-4383	242	23	(	(	PUNCT
ejpam-4383	242	24	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	242	25	∈	∈	PROPN
ejpam-4383	242	26	∏	∏	PROPN
ejpam-4383	242	27	i∈i	i∈i	PROPN
ejpam-4383	242	28	qi	qi	PROPN
ejpam-4383	242	29	.	.	PUNCT
ejpam-4383	243	1	thus	thus	ADV
ejpam-4383	243	2	0i	0i	X
ejpam-4383	243	3	∈	∈	PROPN
ejpam-4383	243	4	qi	qi	PROPN
ejpam-4383	243	5	for	for	ADP
ejpam-4383	243	6	all	all	PRON
ejpam-4383	243	7	i	i	PRON
ejpam-4383	243	8	∈	∈	PROPN
ejpam-4383	243	9	i.	i.	NOUN
ejpam-4383	243	10	let	let	VERB
ejpam-4383	243	11	i	i	PRON
ejpam-4383	243	12	∈	∈	VERB
ejpam-4383	243	13	i	i	PRON
ejpam-4383	243	14	and	and	CCONJ
ejpam-4383	243	15	let	let	VERB
ejpam-4383	243	16	pi	pi	NOUN
ejpam-4383	243	17	,	,	PUNCT
ejpam-4383	243	18	qi	qi	PROPN
ejpam-4383	243	19	,	,	PUNCT
ejpam-4383	243	20	p	p	NOUN
ejpam-4383	243	21	′	′	NUM
ejpam-4383	244	1	i	i	PRON
ejpam-4383	244	2	,	,	PUNCT
ejpam-4383	244	3	q	q	PROPN
ejpam-4383	244	4	′	′	NUM
ejpam-4383	245	1	i	i	PRON
ejpam-4383	245	2	∈	∈	INTJ
ejpam-4383	245	3	pi	pi	NOUN
ejpam-4383	245	4	be	be	AUX
ejpam-4383	245	5	such	such	ADJ
ejpam-4383	245	6	that	that	DET
ejpam-4383	245	7	pi	pi	NOUN
ejpam-4383	245	8	∗i	∗i	PROPN
ejpam-4383	245	9	qi	qi	PROPN
ejpam-4383	245	10	,	,	PUNCT
ejpam-4383	245	11	p′i	p′i	NOUN
ejpam-4383	245	12	∗i	∗i	PROPN
ejpam-4383	245	13	q′i	q′i	PROPN
ejpam-4383	245	14	∈	∈	PROPN
ejpam-4383	245	15	qi	qi	PROPN
ejpam-4383	245	16	.	.	PUNCT
ejpam-4383	246	1	then	then	ADV
ejpam-4383	246	2	fpi	fpi	PROPN
ejpam-4383	246	3	,	,	PUNCT
ejpam-4383	246	4	fqi	fqi	VERB
ejpam-4383	246	5	,	,	PUNCT
ejpam-4383	246	6	fp′i	fp′i	PROPN
ejpam-4383	246	7	,	,	PUNCT
ejpam-4383	246	8	fq′i	fq′i	PROPN
ejpam-4383	246	9	∈	∈	PROPN
ejpam-4383	246	10	∏	∏	PROPN
ejpam-4383	246	11	i∈i	i∈i	ADJ
ejpam-4383	246	12	pi	pi	NOUN
ejpam-4383	246	13	,	,	PUNCT
ejpam-4383	246	14	which	which	PRON
ejpam-4383	246	15	is	be	AUX
ejpam-4383	246	16	defined	define	VERB
ejpam-4383	246	17	by	by	ADP
ejpam-4383	246	18	(	(	PUNCT
ejpam-4383	246	19	2.2	2.2	NUM
ejpam-4383	246	20	)	)	PUNCT
ejpam-4383	246	21	.	.	PUNCT
ejpam-4383	247	1	now	now	ADV
ejpam-4383	247	2	,	,	PUNCT
ejpam-4383	247	3	(	(	PUNCT
ejpam-4383	247	4	∀j	∀j	PROPN
ejpam-4383	247	5	∈	∈	PROPN
ejpam-4383	247	6	i	i	NOUN
ejpam-4383	247	7	)	)	PUNCT
ejpam-4383	247	8	(	(	PUNCT
ejpam-4383	247	9	(	(	PUNCT
ejpam-4383	247	10	fpi	fpi	PROPN
ejpam-4383	247	11	⊗	⊗	PROPN
ejpam-4383	247	12	fqi)(j	fqi)(j	PROPN
ejpam-4383	247	13	)	)	PUNCT
ejpam-4383	247	14	=	=	PRON
ejpam-4383	247	15	{	{	PUNCT
ejpam-4383	247	16	pi	pi	NOUN
ejpam-4383	247	17	∗i	∗i	PROPN
ejpam-4383	247	18	qi	qi	PROPN
ejpam-4383	247	19	if	if	SCONJ
ejpam-4383	247	20	j	j	PROPN
ejpam-4383	247	21	=	=	PRON
ejpam-4383	248	1	i	i	PRON
ejpam-4383	248	2	0j	0j	VERB
ejpam-4383	248	3	∗j	∗j	PROPN
ejpam-4383	248	4	0j	0j	NOUN
ejpam-4383	248	5	otherwise	otherwise	ADV
ejpam-4383	248	6	)	)	PUNCT
ejpam-4383	248	7	.	.	PUNCT
ejpam-4383	249	1	by	by	ADP
ejpam-4383	249	2	(	(	PUNCT
ejpam-4383	249	3	b-1	b-1	PROPN
ejpam-4383	249	4	)	)	PUNCT
ejpam-4383	249	5	,	,	PUNCT
ejpam-4383	249	6	we	we	PRON
ejpam-4383	249	7	have	have	VERB
ejpam-4383	249	8	(	(	PUNCT
ejpam-4383	249	9	∀j	∀j	PROPN
ejpam-4383	249	10	∈	∈	PROPN
ejpam-4383	249	11	i	i	NOUN
ejpam-4383	249	12	)	)	PUNCT
ejpam-4383	249	13	(	(	PUNCT
ejpam-4383	249	14	(	(	PUNCT
ejpam-4383	249	15	fpi	fpi	PROPN
ejpam-4383	249	16	⊗	⊗	PROPN
ejpam-4383	249	17	fqi)(j	fqi)(j	PROPN
ejpam-4383	249	18	)	)	PUNCT
ejpam-4383	249	19	=	=	PRON
ejpam-4383	249	20	{	{	PUNCT
ejpam-4383	250	1	pi	pi	NOUN
ejpam-4383	250	2	∗i	∗i	PROPN
ejpam-4383	250	3	qi	qi	PROPN
ejpam-4383	250	4	if	if	SCONJ
ejpam-4383	250	5	j	j	PROPN
ejpam-4383	250	6	=	=	PUNCT
ejpam-4383	251	1	i	i	PRON
ejpam-4383	251	2	0j	0j	VERB
ejpam-4383	251	3	otherwise	otherwise	ADV
ejpam-4383	251	4	)	)	PUNCT
ejpam-4383	251	5	,	,	PUNCT
ejpam-4383	251	6	this	this	PRON
ejpam-4383	251	7	implies	imply	VERB
ejpam-4383	251	8	that	that	SCONJ
ejpam-4383	251	9	fpi	fpi	PROPN
ejpam-4383	251	10	⊗	⊗	PROPN
ejpam-4383	251	11	fqi	fqi	VERB
ejpam-4383	251	12	∈	∈	PROPN
ejpam-4383	251	13	∏	∏	PROPN
ejpam-4383	251	14	i∈i	i∈i	PROPN
ejpam-4383	251	15	qi	qi	PROPN
ejpam-4383	251	16	.	.	PUNCT
ejpam-4383	252	1	similarly	similarly	ADV
ejpam-4383	252	2	,	,	PUNCT
ejpam-4383	252	3	fp′i	fp′i	PROPN
ejpam-4383	252	4	⊗	⊗	PROPN
ejpam-4383	252	5	fq′i	fq′i	PROPN
ejpam-4383	252	6	∈	∈	PROPN
ejpam-4383	252	7	∏	∏	PROPN
ejpam-4383	252	8	i∈i	i∈i	PROPN
ejpam-4383	252	9	qi	qi	PROPN
ejpam-4383	252	10	.	.	PUNCT
ejpam-4383	253	1	since	since	SCONJ
ejpam-4383	253	2	∏	∏	PROPN
ejpam-4383	253	3	i∈i	i∈i	NOUN
ejpam-4383	253	4	qi	qi	PROPN
ejpam-4383	253	5	is	be	AUX
ejpam-4383	253	6	normal	normal	ADJ
ejpam-4383	253	7	of	of	ADP
ejpam-4383	253	8	∏	∏	NUM
ejpam-4383	253	9	i∈i	i∈i	ADJ
ejpam-4383	253	10	pi	pi	NOUN
ejpam-4383	253	11	,	,	PUNCT
ejpam-4383	253	12	we	we	PRON
ejpam-4383	253	13	have	have	VERB
ejpam-4383	253	14	(	(	PUNCT
ejpam-4383	253	15	fpi	fpi	PROPN
ejpam-4383	253	16	⊗	⊗	PROPN
ejpam-4383	253	17	fp′i)⊗	fp′i)⊗	NOUN
ejpam-4383	253	18	(	(	PUNCT
ejpam-4383	253	19	fqi	fqi	VERB
ejpam-4383	253	20	⊗	⊗	PROPN
ejpam-4383	253	21	fq′i	fq′i	PROPN
ejpam-4383	253	22	)	)	PUNCT
ejpam-4383	253	23	∈	∈	PROPN
ejpam-4383	253	24	∏	∏	PROPN
ejpam-4383	253	25	i∈i	i∈i	PROPN
ejpam-4383	253	26	qi	qi	PROPN
ejpam-4383	253	27	.	.	PUNCT
ejpam-4383	254	1	now	now	ADV
ejpam-4383	254	2	,	,	PUNCT
ejpam-4383	254	3	(	(	PUNCT
ejpam-4383	254	4	∀j	∀j	PROPN
ejpam-4383	254	5	∈	∈	PROPN
ejpam-4383	254	6	i	i	NOUN
ejpam-4383	254	7	)	)	PUNCT
ejpam-4383	254	8	(	(	PUNCT
ejpam-4383	254	9	(	(	PUNCT
ejpam-4383	254	10	(	(	PUNCT
ejpam-4383	254	11	fpi	fpi	PROPN
ejpam-4383	254	12	⊗	⊗	PROPN
ejpam-4383	254	13	fp′i)⊗	fp′i)⊗	NOUN
ejpam-4383	254	14	(	(	PUNCT
ejpam-4383	254	15	fqi	fqi	VERB
ejpam-4383	254	16	⊗	⊗	PROPN
ejpam-4383	254	17	fq′i))(j	fq′i))(j	ADJ
ejpam-4383	254	18	)	)	PUNCT
ejpam-4383	254	19	=	=	PRON
ejpam-4383	254	20	{	{	PUNCT
ejpam-4383	254	21	(	(	PUNCT
ejpam-4383	254	22	pi	pi	NOUN
ejpam-4383	254	23	∗i	∗i	PROPN
ejpam-4383	254	24	p′i	p′i	NOUN
ejpam-4383	254	25	)	)	PUNCT
ejpam-4383	254	26	∗i	∗i	PROPN
ejpam-4383	254	27	(	(	PUNCT
ejpam-4383	254	28	qi	qi	PROPN
ejpam-4383	254	29	∗i	∗i	PROPN
ejpam-4383	254	30	q′i	q′i	PROPN
ejpam-4383	254	31	)	)	PUNCT
ejpam-4383	254	32	if	if	SCONJ
ejpam-4383	254	33	j	j	PROPN
ejpam-4383	254	34	=	=	VERB
ejpam-4383	254	35	i	i	PROPN
ejpam-4383	254	36	(	(	PUNCT
ejpam-4383	254	37	0j	0j	NOUN
ejpam-4383	254	38	∗j	∗j	PROPN
ejpam-4383	254	39	0j	0j	NOUN
ejpam-4383	254	40	)	)	PUNCT
ejpam-4383	254	41	∗j	∗j	NOUN
ejpam-4383	254	42	(	(	PUNCT
ejpam-4383	254	43	0j	0j	NOUN
ejpam-4383	254	44	∗j	∗j	PROPN
ejpam-4383	254	45	0j	0j	NOUN
ejpam-4383	254	46	)	)	PUNCT
ejpam-4383	254	47	otherwise	otherwise	ADV
ejpam-4383	254	48	)	)	PUNCT
ejpam-4383	254	49	,	,	PUNCT
ejpam-4383	254	50	a.	a.	NOUN
ejpam-4383	254	51	iampan	iampan	NOUN
ejpam-4383	254	52	et	et	PROPN
ejpam-4383	254	53	al	al	PROPN
ejpam-4383	254	54	.	.	PUNCT
ejpam-4383	254	55	/	/	SYM
ejpam-4383	254	56	eur	eur	PROPN
ejpam-4383	254	57	.	.	PUNCT
ejpam-4383	255	1	j.	j.	PROPN
ejpam-4383	255	2	pure	pure	PROPN
ejpam-4383	255	3	appl	appl	PROPN
ejpam-4383	255	4	.	.	PROPN
ejpam-4383	255	5	math	math	PROPN
ejpam-4383	255	6	,	,	PUNCT
ejpam-4383	255	7	15	15	NUM
ejpam-4383	255	8	(	(	PUNCT
ejpam-4383	255	9	3	3	NUM
ejpam-4383	255	10	)	)	PUNCT
ejpam-4383	255	11	(	(	PUNCT
ejpam-4383	255	12	2022	2022	NUM
ejpam-4383	255	13	)	)	PUNCT
ejpam-4383	255	14	,	,	PUNCT
ejpam-4383	255	15	999	999	NUM
ejpam-4383	255	16	-	-	SYM
ejpam-4383	255	17	1014	1014	NUM
ejpam-4383	255	18	1008	1008	NUM
ejpam-4383	255	19	this	this	PRON
ejpam-4383	255	20	implies	imply	VERB
ejpam-4383	255	21	that	that	SCONJ
ejpam-4383	255	22	(	(	PUNCT
ejpam-4383	255	23	pi	pi	NOUN
ejpam-4383	255	24	∗i	∗i	PROPN
ejpam-4383	255	25	p′i	p′i	NOUN
ejpam-4383	255	26	)	)	PUNCT
ejpam-4383	256	1	∗i	∗i	PROPN
ejpam-4383	256	2	(	(	PUNCT
ejpam-4383	256	3	qi	qi	PROPN
ejpam-4383	256	4	∗i	∗i	PROPN
ejpam-4383	256	5	q′i	q′i	PROPN
ejpam-4383	256	6	)	)	PUNCT
ejpam-4383	256	7	∈	∈	PROPN
ejpam-4383	256	8	qi	qi	PROPN
ejpam-4383	256	9	.	.	PUNCT
ejpam-4383	257	1	hence	hence	ADV
ejpam-4383	257	2	,	,	PUNCT
ejpam-4383	257	3	qi	qi	PROPN
ejpam-4383	257	4	is	be	AUX
ejpam-4383	257	5	normal	normal	ADJ
ejpam-4383	257	6	of	of	ADP
ejpam-4383	257	7	pi	pi	NOUN
ejpam-4383	257	8	for	for	ADP
ejpam-4383	257	9	all	all	PRON
ejpam-4383	257	10	i	i	PRON
ejpam-4383	257	11	∈	∈	PROPN
ejpam-4383	257	12	i.	i.	NOUN
ejpam-4383	257	13	theorem	theorem	VERB
ejpam-4383	257	14	9	9	NUM
ejpam-4383	257	15	.	.	PUNCT
ejpam-4383	258	1	let	let	VERB
ejpam-4383	258	2	pi	pi	NOUN
ejpam-4383	258	3	=	=	PUNCT
ejpam-4383	258	4	(	(	PUNCT
ejpam-4383	258	5	pi	pi	NOUN
ejpam-4383	258	6	;	;	PUNCT
ejpam-4383	258	7	∗i	∗i	PROPN
ejpam-4383	258	8	,	,	PUNCT
ejpam-4383	258	9	0i	0i	NOUN
ejpam-4383	258	10	)	)	PUNCT
ejpam-4383	258	11	be	be	VERB
ejpam-4383	258	12	a	a	DET
ejpam-4383	258	13	b	b	NOUN
ejpam-4383	258	14	-	-	PUNCT
ejpam-4383	258	15	algebra	algebra	NOUN
ejpam-4383	258	16	and	and	CCONJ
ejpam-4383	258	17	qi	qi	PRON
ejpam-4383	258	18	a	a	DET
ejpam-4383	258	19	subset	subset	NOUN
ejpam-4383	258	20	of	of	ADP
ejpam-4383	258	21	pi	pi	NOUN
ejpam-4383	258	22	for	for	ADP
ejpam-4383	258	23	all	all	PRON
ejpam-4383	258	24	i	i	PRON
ejpam-4383	258	25	∈	∈	PROPN
ejpam-4383	258	26	i.	i.	NOUN
ejpam-4383	258	27	then	then	ADV
ejpam-4383	258	28	qi	qi	PROPN
ejpam-4383	258	29	is	be	AUX
ejpam-4383	258	30	a	a	DET
ejpam-4383	258	31	b	b	NOUN
ejpam-4383	258	32	-	-	PUNCT
ejpam-4383	258	33	ideal	ideal	NOUN
ejpam-4383	258	34	of	of	ADP
ejpam-4383	258	35	pi	pi	NOUN
ejpam-4383	258	36	for	for	ADP
ejpam-4383	258	37	all	all	PRON
ejpam-4383	258	38	i	i	PRON
ejpam-4383	258	39	∈	∈	VERB
ejpam-4383	259	1	i	i	PRON
ejpam-4383	259	2	if	if	SCONJ
ejpam-4383	259	3	and	and	CCONJ
ejpam-4383	259	4	only	only	ADV
ejpam-4383	259	5	if	if	SCONJ
ejpam-4383	259	6	∏	∏	PROPN
ejpam-4383	259	7	i∈i	i∈i	NOUN
ejpam-4383	259	8	qi	qi	PROPN
ejpam-4383	259	9	is	be	AUX
ejpam-4383	259	10	a	a	DET
ejpam-4383	259	11	b	b	NOUN
ejpam-4383	259	12	-	-	PUNCT
ejpam-4383	259	13	ideal	ideal	NOUN
ejpam-4383	259	14	of	of	ADP
ejpam-4383	259	15	the	the	DET
ejpam-4383	259	16	external	external	ADJ
ejpam-4383	259	17	direct	direct	ADJ
ejpam-4383	259	18	product	product	NOUN
ejpam-4383	259	19	b	b	X
ejpam-4383	259	20	-	-	PUNCT
ejpam-4383	259	21	algebra	algebra	NOUN
ejpam-4383	259	22	∏	∏	PROPN
ejpam-4383	259	23	i∈i	i∈i	ADJ
ejpam-4383	259	24	pi	pi	NOUN
ejpam-4383	259	25	=	=	PUNCT
ejpam-4383	259	26	(	(	PUNCT
ejpam-4383	259	27	∏	∏	X
ejpam-4383	259	28	i∈i	i∈i	ADJ
ejpam-4383	259	29	pi;⊗	pi;⊗	PROPN
ejpam-4383	259	30	,	,	PUNCT
ejpam-4383	259	31	(	(	PUNCT
ejpam-4383	259	32	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	259	33	)	)	PUNCT
ejpam-4383	259	34	.	.	PUNCT
ejpam-4383	260	1	proof	proof	NOUN
ejpam-4383	260	2	.	.	PUNCT
ejpam-4383	261	1	assume	assume	VERB
ejpam-4383	261	2	that	that	SCONJ
ejpam-4383	261	3	qi	qi	PROPN
ejpam-4383	261	4	is	be	AUX
ejpam-4383	261	5	a	a	DET
ejpam-4383	261	6	b	b	NOUN
ejpam-4383	261	7	-ideal	-ideal	NOUN
ejpam-4383	261	8	of	of	ADP
ejpam-4383	261	9	pi	pi	NOUN
ejpam-4383	261	10	for	for	ADP
ejpam-4383	261	11	all	all	PRON
ejpam-4383	261	12	i	i	PRON
ejpam-4383	261	13	∈	∈	PROPN
ejpam-4383	261	14	i.	i.	NOUN
ejpam-4383	261	15	(	(	PUNCT
ejpam-4383	261	16	bi-1	bi-1	NOUN
ejpam-4383	261	17	)	)	PUNCT
ejpam-4383	261	18	by	by	ADP
ejpam-4383	261	19	(	(	PUNCT
ejpam-4383	261	20	bi-1	bi-1	NOUN
ejpam-4383	261	21	)	)	PUNCT
ejpam-4383	261	22	,	,	PUNCT
ejpam-4383	261	23	we	we	PRON
ejpam-4383	261	24	have	have	VERB
ejpam-4383	261	25	0i	0i	NOUN
ejpam-4383	261	26	∈	∈	PROPN
ejpam-4383	261	27	qi	qi	PROPN
ejpam-4383	261	28	for	for	ADP
ejpam-4383	261	29	all	all	PRON
ejpam-4383	261	30	i	i	PRON
ejpam-4383	261	31	∈	∈	PROPN
ejpam-4383	261	32	i.	i.	NOUN
ejpam-4383	261	33	then	then	ADV
ejpam-4383	261	34	(	(	PUNCT
ejpam-4383	261	35	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	261	36	∈	∈	PROPN
ejpam-4383	261	37	∏	∏	PROPN
ejpam-4383	261	38	i∈i	i∈i	PROPN
ejpam-4383	261	39	qi	qi	PROPN
ejpam-4383	261	40	.	.	PUNCT
ejpam-4383	262	1	(	(	PUNCT
ejpam-4383	262	2	bi-2	bi-2	NUM
ejpam-4383	262	3	)	)	PUNCT
ejpam-4383	262	4	let	let	VERB
ejpam-4383	262	5	(	(	PUNCT
ejpam-4383	262	6	pi)i∈i	pi)i∈i	NUM
ejpam-4383	262	7	,	,	PUNCT
ejpam-4383	262	8	(	(	PUNCT
ejpam-4383	262	9	qi)i∈i	qi)i∈i	NUM
ejpam-4383	262	10	∈	∈	PROPN
ejpam-4383	262	11	∏	∏	PROPN
ejpam-4383	262	12	i∈i	i∈i	ADJ
ejpam-4383	262	13	pi	pi	NOUN
ejpam-4383	262	14	be	be	AUX
ejpam-4383	262	15	such	such	ADJ
ejpam-4383	262	16	that	that	SCONJ
ejpam-4383	262	17	(	(	PUNCT
ejpam-4383	262	18	pi)i∈i	pi)i∈i	NUM
ejpam-4383	262	19	⊗	⊗	PROPN
ejpam-4383	262	20	(	(	PUNCT
ejpam-4383	262	21	qi)i∈i	qi)i∈i	NUM
ejpam-4383	262	22	∈	∈	PROPN
ejpam-4383	262	23	∏	∏	PROPN
ejpam-4383	262	24	i∈i	i∈i	ADJ
ejpam-4383	262	25	qi	qi	PROPN
ejpam-4383	262	26	and	and	CCONJ
ejpam-4383	262	27	(	(	PUNCT
ejpam-4383	262	28	qi)i∈i	qi)i∈i	NUM
ejpam-4383	262	29	∈	∈	PROPN
ejpam-4383	262	30	∏	∏	PROPN
ejpam-4383	262	31	i∈i	i∈i	PROPN
ejpam-4383	262	32	qi	qi	PROPN
ejpam-4383	262	33	.	.	PUNCT
ejpam-4383	263	1	then	then	ADV
ejpam-4383	263	2	(	(	PUNCT
ejpam-4383	263	3	pi	pi	NOUN
ejpam-4383	263	4	∗i	∗i	PROPN
ejpam-4383	263	5	qi)i∈i	qi)i∈i	NUM
ejpam-4383	263	6	∈	∈	PROPN
ejpam-4383	263	7	∏	∏	PROPN
ejpam-4383	263	8	i∈i	i∈i	PROPN
ejpam-4383	263	9	qi	qi	PROPN
ejpam-4383	263	10	.	.	PUNCT
ejpam-4383	264	1	thus	thus	ADV
ejpam-4383	264	2	pi	pi	VERB
ejpam-4383	264	3	∗i	∗i	PROPN
ejpam-4383	264	4	qi	qi	PROPN
ejpam-4383	264	5	∈	∈	PROPN
ejpam-4383	264	6	qi	qi	PROPN
ejpam-4383	264	7	and	and	CCONJ
ejpam-4383	264	8	qi	qi	PROPN
ejpam-4383	264	9	∈	∈	PROPN
ejpam-4383	264	10	qi	qi	PROPN
ejpam-4383	264	11	,	,	PUNCT
ejpam-4383	264	12	it	it	PRON
ejpam-4383	264	13	follows	follow	VERB
ejpam-4383	264	14	from	from	ADP
ejpam-4383	264	15	(	(	PUNCT
ejpam-4383	264	16	bi-2	bi-2	NUM
ejpam-4383	264	17	)	)	PUNCT
ejpam-4383	264	18	that	that	PRON
ejpam-4383	264	19	pi	pi	NOUN
ejpam-4383	264	20	∈	∈	PROPN
ejpam-4383	264	21	qi	qi	PROPN
ejpam-4383	264	22	for	for	ADP
ejpam-4383	264	23	all	all	PRON
ejpam-4383	264	24	i	i	PRON
ejpam-4383	264	25	∈	∈	PROPN
ejpam-4383	264	26	i.	i.	NOUN
ejpam-4383	264	27	thus	thus	ADV
ejpam-4383	264	28	(	(	PUNCT
ejpam-4383	264	29	pi)i∈i	pi)i∈i	NUM
ejpam-4383	264	30	∈	∈	PROPN
ejpam-4383	264	31	∏	∏	PROPN
ejpam-4383	264	32	i∈i	i∈i	PROPN
ejpam-4383	264	33	qi	qi	PROPN
ejpam-4383	264	34	.	.	PUNCT
ejpam-4383	265	1	hence	hence	ADV
ejpam-4383	265	2	,	,	PUNCT
ejpam-4383	265	3	∏	∏	PROPN
ejpam-4383	265	4	i∈i	i∈i	ADJ
ejpam-4383	265	5	qi	qi	PROPN
ejpam-4383	265	6	is	be	AUX
ejpam-4383	265	7	a	a	DET
ejpam-4383	265	8	b	b	NOUN
ejpam-4383	265	9	-ideal	-ideal	NOUN
ejpam-4383	265	10	of	of	ADP
ejpam-4383	265	11	∏	∏	NUM
ejpam-4383	265	12	i∈i	i∈i	ADJ
ejpam-4383	265	13	pi	pi	NOUN
ejpam-4383	265	14	.	.	PUNCT
ejpam-4383	266	1	conversely	conversely	ADV
ejpam-4383	266	2	,	,	PUNCT
ejpam-4383	266	3	assume	assume	VERB
ejpam-4383	266	4	that	that	SCONJ
ejpam-4383	266	5	∏	∏	PROPN
ejpam-4383	266	6	i∈i	i∈i	NOUN
ejpam-4383	266	7	qi	qi	PROPN
ejpam-4383	266	8	is	be	AUX
ejpam-4383	266	9	a	a	DET
ejpam-4383	266	10	b	b	NOUN
ejpam-4383	266	11	-ideal	-ideal	NOUN
ejpam-4383	266	12	of	of	ADP
ejpam-4383	266	13	∏	∏	NUM
ejpam-4383	266	14	i∈i	i∈i	ADJ
ejpam-4383	266	15	pi	pi	NOUN
ejpam-4383	266	16	.	.	PUNCT
ejpam-4383	267	1	then	then	ADV
ejpam-4383	267	2	∏	∏	PROPN
ejpam-4383	267	3	i∈i	i∈i	ADJ
ejpam-4383	267	4	qi	qi	PROPN
ejpam-4383	267	5	̸=	̸=	PROPN
ejpam-4383	267	6	∅	∅	NOUN
ejpam-4383	267	7	,	,	PUNCT
ejpam-4383	267	8	so	so	ADV
ejpam-4383	267	9	qi	qi	PROPN
ejpam-4383	267	10	̸=	̸=	PROPN
ejpam-4383	267	11	∅	∅	NOUN
ejpam-4383	267	12	for	for	ADP
ejpam-4383	267	13	all	all	PRON
ejpam-4383	267	14	i	i	PRON
ejpam-4383	267	15	∈	∈	PROPN
ejpam-4383	267	16	i.	i.	NOUN
ejpam-4383	267	17	let	let	VERB
ejpam-4383	267	18	i	i	PRON
ejpam-4383	267	19	∈	∈	PROPN
ejpam-4383	267	20	i.	i.	NOUN
ejpam-4383	267	21	(	(	PUNCT
ejpam-4383	267	22	bi-1	bi-1	NOUN
ejpam-4383	267	23	)	)	PUNCT
ejpam-4383	267	24	by	by	ADP
ejpam-4383	267	25	(	(	PUNCT
ejpam-4383	267	26	bi-1	bi-1	NOUN
ejpam-4383	267	27	)	)	PUNCT
ejpam-4383	267	28	,	,	PUNCT
ejpam-4383	267	29	we	we	PRON
ejpam-4383	267	30	have	have	VERB
ejpam-4383	267	31	(	(	PUNCT
ejpam-4383	267	32	0i)i∈i	0i)i∈i	ADJ
ejpam-4383	267	33	∈	∈	PROPN
ejpam-4383	267	34	∏	∏	PROPN
ejpam-4383	267	35	i∈i	i∈i	PROPN
ejpam-4383	267	36	qi	qi	PROPN
ejpam-4383	267	37	.	.	PUNCT
ejpam-4383	268	1	then	then	ADV
ejpam-4383	268	2	0i	0i	PROPN
ejpam-4383	268	3	∈	∈	PROPN
ejpam-4383	268	4	qi	qi	PROPN
ejpam-4383	268	5	.	.	PUNCT
ejpam-4383	269	1	(	(	PUNCT
ejpam-4383	269	2	bi-2	bi-2	NUM
ejpam-4383	269	3	)	)	PUNCT
ejpam-4383	269	4	let	let	VERB
ejpam-4383	269	5	pi	pi	NOUN
ejpam-4383	269	6	,	,	PUNCT
ejpam-4383	269	7	qi	qi	PROPN
ejpam-4383	269	8	∈	∈	PROPN
ejpam-4383	269	9	pi	pi	NOUN
ejpam-4383	269	10	be	be	AUX
ejpam-4383	269	11	such	such	ADJ
ejpam-4383	269	12	that	that	DET
ejpam-4383	269	13	pi	pi	NOUN
ejpam-4383	269	14	∗i	∗i	PROPN
ejpam-4383	269	15	qi	qi	PROPN
ejpam-4383	269	16	∈	∈	PROPN
ejpam-4383	269	17	qi	qi	PROPN
ejpam-4383	269	18	and	and	CCONJ
ejpam-4383	269	19	qi	qi	PROPN
ejpam-4383	269	20	∈	∈	PROPN
ejpam-4383	269	21	qi	qi	PROPN
ejpam-4383	269	22	.	.	PUNCT
ejpam-4383	270	1	by	by	ADP
ejpam-4383	270	2	(	(	PUNCT
ejpam-4383	270	3	bi-1	bi-1	NOUN
ejpam-4383	270	4	)	)	PUNCT
ejpam-4383	270	5	,	,	PUNCT
ejpam-4383	270	6	we	we	PRON
ejpam-4383	270	7	have	have	VERB
ejpam-4383	270	8	0i	0i	NOUN
ejpam-4383	270	9	∈	∈	PROPN
ejpam-4383	270	10	qi	qi	PROPN
ejpam-4383	270	11	for	for	ADP
ejpam-4383	270	12	all	all	PRON
ejpam-4383	270	13	i	i	PRON
ejpam-4383	270	14	∈	∈	PROPN
ejpam-4383	270	15	i.	i.	NOUN
ejpam-4383	270	16	then	then	ADV
ejpam-4383	270	17	fpi	fpi	PROPN
ejpam-4383	270	18	∈	∈	PROPN
ejpam-4383	270	19	∏	∏	PROPN
ejpam-4383	270	20	i∈i	i∈i	ADJ
ejpam-4383	270	21	pi	pi	NOUN
ejpam-4383	270	22	and	and	CCONJ
ejpam-4383	270	23	fpi∗iqi	fpi∗iqi	NOUN
ejpam-4383	270	24	,	,	PUNCT
ejpam-4383	270	25	fqi	fqi	VERB
ejpam-4383	270	26	∈	∈	PROPN
ejpam-4383	270	27	∏	∏	PROPN
ejpam-4383	270	28	i∈i	i∈i	PROPN
ejpam-4383	270	29	qi	qi	PROPN
ejpam-4383	270	30	,	,	PUNCT
ejpam-4383	270	31	which	which	PRON
ejpam-4383	270	32	are	be	AUX
ejpam-4383	270	33	defined	define	VERB
ejpam-4383	270	34	by	by	ADP
ejpam-4383	270	35	(	(	PUNCT
ejpam-4383	270	36	2.2	2.2	NUM
ejpam-4383	270	37	)	)	PUNCT
ejpam-4383	270	38	.	.	PUNCT
ejpam-4383	271	1	by	by	ADP
ejpam-4383	271	2	lemma	lemma	PROPN
ejpam-4383	271	3	1	1	NUM
ejpam-4383	271	4	,	,	PUNCT
ejpam-4383	271	5	we	we	PRON
ejpam-4383	271	6	have	have	VERB
ejpam-4383	271	7	fpi	fpi	PROPN
ejpam-4383	271	8	⊗	⊗	PROPN
ejpam-4383	271	9	fqi	fqi	NOUN
ejpam-4383	272	1	=	=	PUNCT
ejpam-4383	272	2	fpi∗iqi	fpi∗iqi	PROPN
ejpam-4383	272	3	∈	∈	PROPN
ejpam-4383	272	4	∏	∏	PROPN
ejpam-4383	272	5	i∈i	i∈i	PROPN
ejpam-4383	272	6	qi	qi	PROPN
ejpam-4383	272	7	.	.	PUNCT
ejpam-4383	273	1	by	by	ADP
ejpam-4383	273	2	(	(	PUNCT
ejpam-4383	273	3	bi-2	bi-2	NUM
ejpam-4383	273	4	)	)	PUNCT
ejpam-4383	273	5	,	,	PUNCT
ejpam-4383	273	6	we	we	PRON
ejpam-4383	273	7	have	have	VERB
ejpam-4383	273	8	fpi	fpi	PROPN
ejpam-4383	273	9	∈	∈	PROPN
ejpam-4383	273	10	∏	∏	PROPN
ejpam-4383	273	11	i∈i	i∈i	PROPN
ejpam-4383	273	12	qi	qi	PROPN
ejpam-4383	273	13	.	.	PUNCT
ejpam-4383	274	1	by	by	ADP
ejpam-4383	274	2	(	(	PUNCT
ejpam-4383	274	3	2.2	2.2	NUM
ejpam-4383	274	4	)	)	PUNCT
ejpam-4383	274	5	,	,	PUNCT
ejpam-4383	274	6	we	we	PRON
ejpam-4383	274	7	have	have	VERB
ejpam-4383	274	8	pi	pi	PROPN
ejpam-4383	274	9	∈	∈	PROPN
ejpam-4383	274	10	qi	qi	PROPN
ejpam-4383	274	11	.	.	PUNCT
ejpam-4383	275	1	hence	hence	ADV
ejpam-4383	275	2	,	,	PUNCT
ejpam-4383	275	3	qi	qi	PROPN
ejpam-4383	275	4	is	be	AUX
ejpam-4383	275	5	a	a	DET
ejpam-4383	275	6	b	b	NOUN
ejpam-4383	275	7	-ideal	-ideal	NOUN
ejpam-4383	275	8	of	of	ADP
ejpam-4383	275	9	pi	pi	NOUN
ejpam-4383	275	10	for	for	ADP
ejpam-4383	275	11	all	all	DET
ejpam-4383	275	12	i	i	PRON
ejpam-4383	275	13	∈	∈	PROPN
ejpam-4383	275	14	i.	i.	NOUN
ejpam-4383	275	15	moreover	moreover	ADV
ejpam-4383	275	16	,	,	PUNCT
ejpam-4383	275	17	we	we	PRON
ejpam-4383	275	18	discuss	discuss	VERB
ejpam-4383	275	19	several	several	ADJ
ejpam-4383	275	20	homomorphism	homomorphism	NOUN
ejpam-4383	275	21	theorems	theorem	NOUN
ejpam-4383	275	22	in	in	ADP
ejpam-4383	275	23	view	view	NOUN
ejpam-4383	275	24	of	of	ADP
ejpam-4383	275	25	the	the	DET
ejpam-4383	275	26	external	external	ADJ
ejpam-4383	275	27	direct	direct	ADJ
ejpam-4383	275	28	product	product	NOUN
ejpam-4383	275	29	of	of	ADP
ejpam-4383	275	30	b	b	NOUN
ejpam-4383	275	31	-algebras	-algebras	PROPN
ejpam-4383	275	32	.	.	PUNCT
ejpam-4383	276	1	definition	definition	NOUN
ejpam-4383	276	2	10	10	NUM
ejpam-4383	276	3	.	.	PUNCT
ejpam-4383	277	1	let	let	AUX
ejpam-4383	277	2	pi	pi	NOUN
ejpam-4383	277	3	=	=	PUNCT
ejpam-4383	277	4	(	(	PUNCT
ejpam-4383	277	5	pi	pi	NOUN
ejpam-4383	277	6	;	;	PUNCT
ejpam-4383	277	7	∗i	∗i	PROPN
ejpam-4383	277	8	)	)	PUNCT
ejpam-4383	277	9	and	and	CCONJ
ejpam-4383	277	10	qi	qi	PROPN
ejpam-4383	277	11	=	=	SYM
ejpam-4383	277	12	(	(	PUNCT
ejpam-4383	277	13	qi	qi	NOUN
ejpam-4383	277	14	;	;	PUNCT
ejpam-4383	277	15	◦	◦	NOUN
ejpam-4383	277	16	i	i	NOUN
ejpam-4383	277	17	)	)	PUNCT
ejpam-4383	277	18	be	be	AUX
ejpam-4383	277	19	algebras	algebra	NOUN
ejpam-4383	277	20	and	and	CCONJ
ejpam-4383	277	21	ψi	ψi	ADP
ejpam-4383	277	22	:	:	PUNCT
ejpam-4383	277	23	pi	pi	NOUN
ejpam-4383	277	24	→	→	SYM
ejpam-4383	277	25	qi	qi	PROPN
ejpam-4383	277	26	be	be	AUX
ejpam-4383	277	27	a	a	DET
ejpam-4383	277	28	function	function	NOUN
ejpam-4383	277	29	for	for	SCONJ
ejpam-4383	277	30	all	all	PRON
ejpam-4383	277	31	i	i	PRON
ejpam-4383	277	32	∈	∈	PROPN
ejpam-4383	277	33	i.	i.	NOUN
ejpam-4383	277	34	define	define	VERB
ejpam-4383	277	35	the	the	DET
ejpam-4383	277	36	function	function	NOUN
ejpam-4383	277	37	ψ	ψ	NOUN
ejpam-4383	277	38	:	:	PUNCT
ejpam-4383	277	39	∏	∏	NUM
ejpam-4383	277	40	i∈i	i∈i	ADJ
ejpam-4383	277	41	pi	pi	NOUN
ejpam-4383	277	42	→	→	SYM
ejpam-4383	277	43	∏	∏	X
ejpam-4383	277	44	i∈i	i∈i	ADJ
ejpam-4383	277	45	qi	qi	NOUN
ejpam-4383	277	46	given	give	VERB
ejpam-4383	277	47	by	by	ADP
ejpam-4383	277	48	(	(	PUNCT
ejpam-4383	277	49	∀(pi)i∈i	∀(pi)i∈i	SYM
ejpam-4383	277	50	∈	∈	PROPN
ejpam-4383	277	51	∏	∏	PROPN
ejpam-4383	277	52	i∈i	i∈i	NOUN
ejpam-4383	277	53	pi)(ψ(pi)i∈i	pi)(ψ(pi)i∈i	NOUN
ejpam-4383	277	54	=	=	SYM
ejpam-4383	277	55	(	(	PUNCT
ejpam-4383	277	56	ψi(pi))i∈i	ψi(pi))i∈i	PROPN
ejpam-4383	277	57	)	)	PUNCT
ejpam-4383	277	58	.	.	PUNCT
ejpam-4383	278	1	(	(	PUNCT
ejpam-4383	278	2	2.3	2.3	NUM
ejpam-4383	278	3	)	)	PUNCT
ejpam-4383	278	4	we	we	PRON
ejpam-4383	278	5	shall	shall	AUX
ejpam-4383	278	6	show	show	VERB
ejpam-4383	278	7	that	that	SCONJ
ejpam-4383	278	8	ψ	ψ	X
ejpam-4383	278	9	:	:	PUNCT
ejpam-4383	278	10	∏	∏	NUM
ejpam-4383	278	11	i∈i	i∈i	ADJ
ejpam-4383	278	12	pi	pi	NOUN
ejpam-4383	278	13	→	→	SYM
ejpam-4383	278	14	∏	∏	X
ejpam-4383	278	15	i∈i	i∈i	NOUN
ejpam-4383	278	16	qi	qi	PROPN
ejpam-4383	278	17	is	be	AUX
ejpam-4383	278	18	a	a	DET
ejpam-4383	278	19	function	function	NOUN
ejpam-4383	278	20	.	.	PUNCT
ejpam-4383	279	1	let	let	VERB
ejpam-4383	279	2	(	(	PUNCT
ejpam-4383	279	3	pi)i∈i	pi)i∈i	NUM
ejpam-4383	279	4	∈	∈	PROPN
ejpam-4383	279	5	∏	∏	PROPN
ejpam-4383	279	6	i∈i	i∈i	ADJ
ejpam-4383	279	7	pi	pi	NOUN
ejpam-4383	279	8	.	.	PUNCT
ejpam-4383	280	1	since	since	SCONJ
ejpam-4383	280	2	ψi	ψi	NOUN
ejpam-4383	280	3	:	:	PUNCT
ejpam-4383	280	4	pi	pi	NOUN
ejpam-4383	280	5	→	→	SYM
ejpam-4383	280	6	qi	qi	PROPN
ejpam-4383	280	7	is	be	AUX
ejpam-4383	280	8	a	a	DET
ejpam-4383	280	9	function	function	NOUN
ejpam-4383	280	10	and	and	CCONJ
ejpam-4383	280	11	pi	pi	NOUN
ejpam-4383	280	12	∈	∈	PROPN
ejpam-4383	280	13	pi	pi	NOUN
ejpam-4383	280	14	for	for	ADP
ejpam-4383	280	15	all	all	PRON
ejpam-4383	280	16	i	i	PRON
ejpam-4383	280	17	∈	∈	PROPN
ejpam-4383	281	1	i	i	PRON
ejpam-4383	281	2	,	,	PUNCT
ejpam-4383	281	3	we	we	PRON
ejpam-4383	281	4	have	have	VERB
ejpam-4383	281	5	ψi(pi	ψi(pi	NOUN
ejpam-4383	281	6	)	)	PUNCT
ejpam-4383	281	7	∈	∈	PROPN
ejpam-4383	281	8	qi	qi	PROPN
ejpam-4383	281	9	for	for	ADP
ejpam-4383	281	10	all	all	DET
ejpam-4383	281	11	i	i	PRON
ejpam-4383	281	12	∈	∈	PROPN
ejpam-4383	281	13	i.	i.	NOUN
ejpam-4383	281	14	thus	thus	ADV
ejpam-4383	281	15	(	(	PUNCT
ejpam-4383	281	16	ψi(pi))i∈i	ψi(pi))i∈i	ADP
ejpam-4383	281	17	∈	∈	PROPN
ejpam-4383	281	18	∏	∏	PROPN
ejpam-4383	281	19	i∈i	i∈i	PROPN
ejpam-4383	281	20	qi	qi	PROPN
ejpam-4383	281	21	.	.	PUNCT
ejpam-4383	282	1	such	such	ADJ
ejpam-4383	282	2	that	that	DET
ejpam-4383	282	3	ψ(pi)i∈i	ψ(pi)i∈i	X
ejpam-4383	282	4	=	=	SYM
ejpam-4383	282	5	(	(	PUNCT
ejpam-4383	282	6	ψi(pi))i∈i	ψi(pi))i∈i	PROPN
ejpam-4383	282	7	.	.	PUNCT
ejpam-4383	283	1	let	let	VERB
ejpam-4383	283	2	(	(	PUNCT
ejpam-4383	283	3	pi)i∈i	pi)i∈i	NUM
ejpam-4383	283	4	,	,	PUNCT
ejpam-4383	283	5	(	(	PUNCT
ejpam-4383	283	6	p	p	NOUN
ejpam-4383	283	7	′	′	NUM
ejpam-4383	283	8	i)i∈i	i)i∈i	PROPN
ejpam-4383	283	9	∈	∈	PROPN
ejpam-4383	283	10	∏	∏	PROPN
ejpam-4383	283	11	i∈i	i∈i	ADJ
ejpam-4383	283	12	pi	pi	NOUN
ejpam-4383	283	13	be	be	AUX
ejpam-4383	283	14	such	such	ADJ
ejpam-4383	283	15	that	that	SCONJ
ejpam-4383	283	16	(	(	PUNCT
ejpam-4383	283	17	pi)i∈i	pi)i∈i	NUM
ejpam-4383	283	18	=	=	SYM
ejpam-4383	283	19	(	(	PUNCT
ejpam-4383	283	20	p′i)i∈i	p′i)i∈i	NOUN
ejpam-4383	283	21	.	.	PUNCT
ejpam-4383	284	1	then	then	ADV
ejpam-4383	284	2	pi	pi	NOUN
ejpam-4383	285	1	=	=	PROPN
ejpam-4383	285	2	p′i	p′i	NOUN
ejpam-4383	285	3	for	for	ADP
ejpam-4383	285	4	all	all	PRON
ejpam-4383	285	5	i	i	PRON
ejpam-4383	285	6	∈	∈	PROPN
ejpam-4383	286	1	i	i	PRON
ejpam-4383	286	2	,	,	PUNCT
ejpam-4383	286	3	so	so	ADV
ejpam-4383	286	4	ψi(pi	ψi(pi	NOUN
ejpam-4383	286	5	)	)	PUNCT
ejpam-4383	286	6	=	=	NOUN
ejpam-4383	286	7	ψi(p	ψi(p	X
ejpam-4383	286	8	′	′	NUM
ejpam-4383	286	9	i	i	NOUN
ejpam-4383	286	10	)	)	PUNCT
ejpam-4383	286	11	.	.	PUNCT
ejpam-4383	287	1	thus	thus	ADV
ejpam-4383	287	2	ψ(pi)i∈i	ψ(pi)i∈i	X
ejpam-4383	287	3	=	=	SYM
ejpam-4383	287	4	(	(	PUNCT
ejpam-4383	287	5	ψi(pi))i∈i	ψi(pi))i∈i	PROPN
ejpam-4383	287	6	=	=	SYM
ejpam-4383	287	7	(	(	PUNCT
ejpam-4383	287	8	ψi(p	ψi(p	NOUN
ejpam-4383	287	9	′	′	NUM
ejpam-4383	287	10	i))i∈i	i))i∈i	NOUN
ejpam-4383	287	11	=	=	NOUN
ejpam-4383	287	12	ψ(p′i)i∈i	ψ(p′i)i∈i	NUM
ejpam-4383	287	13	.	.	PUNCT
ejpam-4383	288	1	therefore	therefore	ADV
ejpam-4383	288	2	,	,	PUNCT
ejpam-4383	288	3	ψ	ψ	X
ejpam-4383	288	4	:	:	PUNCT
ejpam-4383	288	5	∏	∏	NUM
ejpam-4383	288	6	i∈i	i∈i	ADJ
ejpam-4383	288	7	pi	pi	NOUN
ejpam-4383	288	8	→	→	SYM
ejpam-4383	288	9	∏	∏	X
ejpam-4383	288	10	i∈i	i∈i	NOUN
ejpam-4383	288	11	qi	qi	PROPN
ejpam-4383	288	12	is	be	AUX
ejpam-4383	288	13	a	a	DET
ejpam-4383	288	14	function	function	NOUN
ejpam-4383	288	15	.	.	PUNCT
ejpam-4383	289	1	theorem	theorem	ADJ
ejpam-4383	289	2	10	10	NUM
ejpam-4383	289	3	.	.	PUNCT
ejpam-4383	290	1	let	let	AUX
ejpam-4383	290	2	pi	pi	NOUN
ejpam-4383	290	3	=	=	PUNCT
ejpam-4383	290	4	(	(	PUNCT
ejpam-4383	290	5	pi	pi	NOUN
ejpam-4383	290	6	;	;	PUNCT
ejpam-4383	290	7	∗i	∗i	PROPN
ejpam-4383	290	8	)	)	PUNCT
ejpam-4383	290	9	and	and	CCONJ
ejpam-4383	290	10	qi	qi	PROPN
ejpam-4383	290	11	=	=	SYM
ejpam-4383	290	12	(	(	PUNCT
ejpam-4383	290	13	qi	qi	NOUN
ejpam-4383	290	14	;	;	PUNCT
ejpam-4383	290	15	◦	◦	NOUN
ejpam-4383	290	16	i	i	NOUN
ejpam-4383	290	17	)	)	PUNCT
ejpam-4383	290	18	be	be	AUX
ejpam-4383	290	19	algebras	algebra	NOUN
ejpam-4383	290	20	and	and	CCONJ
ejpam-4383	290	21	ψi	ψi	ADP
ejpam-4383	290	22	:	:	PUNCT
ejpam-4383	290	23	pi	pi	NOUN
ejpam-4383	290	24	→	→	SYM
ejpam-4383	290	25	qi	qi	PROPN
ejpam-4383	290	26	be	be	AUX
ejpam-4383	290	27	a	a	DET
ejpam-4383	290	28	function	function	NOUN
ejpam-4383	290	29	for	for	ADP
ejpam-4383	290	30	all	all	PRON
ejpam-4383	290	31	i	i	PRON
ejpam-4383	290	32	∈	∈	PROPN
ejpam-4383	290	33	i.	i.	NOUN
ejpam-4383	290	34	(	(	PUNCT
ejpam-4383	290	35	i	i	NOUN
ejpam-4383	290	36	)	)	PUNCT
ejpam-4383	290	37	ψi	ψi	NOUN
ejpam-4383	290	38	is	be	AUX
ejpam-4383	290	39	injective	injective	ADJ
ejpam-4383	290	40	for	for	ADP
ejpam-4383	290	41	all	all	PRON
ejpam-4383	290	42	i	i	PRON
ejpam-4383	290	43	∈	∈	VERB
ejpam-4383	291	1	i	i	PRON
ejpam-4383	291	2	if	if	SCONJ
ejpam-4383	291	3	and	and	CCONJ
ejpam-4383	291	4	only	only	ADV
ejpam-4383	291	5	if	if	SCONJ
ejpam-4383	291	6	ψ	ψ	NOUN
ejpam-4383	291	7	is	be	AUX
ejpam-4383	291	8	injective	injective	ADJ
ejpam-4383	291	9	which	which	PRON
ejpam-4383	291	10	is	be	AUX
ejpam-4383	291	11	defined	define	VERB
ejpam-4383	291	12	in	in	ADP
ejpam-4383	291	13	definition	definition	NOUN
ejpam-4383	291	14	10	10	NUM
ejpam-4383	291	15	,	,	PUNCT
ejpam-4383	291	16	a.	a.	NOUN
ejpam-4383	291	17	iampan	iampan	NOUN
ejpam-4383	291	18	et	et	PROPN
ejpam-4383	291	19	al	al	PROPN
ejpam-4383	291	20	.	.	PUNCT
ejpam-4383	291	21	/	/	SYM
ejpam-4383	291	22	eur	eur	PROPN
ejpam-4383	291	23	.	.	PUNCT
ejpam-4383	292	1	j.	j.	PROPN
ejpam-4383	292	2	pure	pure	PROPN
ejpam-4383	292	3	appl	appl	PROPN
ejpam-4383	292	4	.	.	PROPN
ejpam-4383	292	5	math	math	PROPN
ejpam-4383	292	6	,	,	PUNCT
ejpam-4383	292	7	15	15	NUM
ejpam-4383	292	8	(	(	PUNCT
ejpam-4383	292	9	3	3	NUM
ejpam-4383	292	10	)	)	PUNCT
ejpam-4383	292	11	(	(	PUNCT
ejpam-4383	292	12	2022	2022	NUM
ejpam-4383	292	13	)	)	PUNCT
ejpam-4383	292	14	,	,	PUNCT
ejpam-4383	292	15	999	999	NUM
ejpam-4383	292	16	-	-	SYM
ejpam-4383	292	17	1014	1014	NUM
ejpam-4383	292	18	1009	1009	NUM
ejpam-4383	292	19	(	(	PUNCT
ejpam-4383	292	20	ii	ii	NOUN
ejpam-4383	292	21	)	)	PUNCT
ejpam-4383	292	22	ψi	ψi	NOUN
ejpam-4383	292	23	is	be	AUX
ejpam-4383	292	24	surjective	surjective	ADJ
ejpam-4383	292	25	for	for	ADP
ejpam-4383	292	26	all	all	PRON
ejpam-4383	292	27	i	i	PRON
ejpam-4383	292	28	∈	∈	VERB
ejpam-4383	293	1	i	i	PRON
ejpam-4383	293	2	if	if	SCONJ
ejpam-4383	293	3	and	and	CCONJ
ejpam-4383	293	4	only	only	ADV
ejpam-4383	293	5	if	if	SCONJ
ejpam-4383	293	6	ψ	ψ	NOUN
ejpam-4383	293	7	is	be	AUX
ejpam-4383	293	8	surjective	surjective	ADJ
ejpam-4383	293	9	,	,	PUNCT
ejpam-4383	293	10	(	(	PUNCT
ejpam-4383	293	11	iii	iii	NOUN
ejpam-4383	293	12	)	)	PUNCT
ejpam-4383	293	13	ψi	ψi	NOUN
ejpam-4383	293	14	is	be	AUX
ejpam-4383	293	15	bijective	bijective	ADJ
ejpam-4383	293	16	for	for	SCONJ
ejpam-4383	293	17	all	all	DET
ejpam-4383	293	18	i	i	PRON
ejpam-4383	293	19	∈	∈	VERB
ejpam-4383	294	1	i	i	PRON
ejpam-4383	294	2	if	if	SCONJ
ejpam-4383	294	3	and	and	CCONJ
ejpam-4383	294	4	only	only	ADV
ejpam-4383	294	5	if	if	SCONJ
ejpam-4383	294	6	ψ	ψ	NOUN
ejpam-4383	294	7	is	be	AUX
ejpam-4383	294	8	bijective	bijective	ADJ
ejpam-4383	294	9	.	.	PUNCT
ejpam-4383	295	1	proof	proof	NOUN
ejpam-4383	295	2	.	.	PUNCT
ejpam-4383	296	1	(	(	PUNCT
ejpam-4383	296	2	i	i	NOUN
ejpam-4383	296	3	)	)	PUNCT
ejpam-4383	296	4	assume	assume	VERB
ejpam-4383	296	5	that	that	SCONJ
ejpam-4383	296	6	ψi	ψi	NOUN
ejpam-4383	296	7	is	be	AUX
ejpam-4383	296	8	injective	injective	ADJ
ejpam-4383	296	9	for	for	SCONJ
ejpam-4383	296	10	all	all	PRON
ejpam-4383	296	11	i	i	PRON
ejpam-4383	296	12	∈	∈	PROPN
ejpam-4383	296	13	i.	i.	NOUN
ejpam-4383	296	14	let	let	VERB
ejpam-4383	296	15	(	(	PUNCT
ejpam-4383	296	16	pi)i∈i	pi)i∈i	NUM
ejpam-4383	296	17	,	,	PUNCT
ejpam-4383	296	18	(	(	PUNCT
ejpam-4383	296	19	qi)i∈i	qi)i∈i	NUM
ejpam-4383	296	20	∈	∈	PROPN
ejpam-4383	296	21	∏	∏	PROPN
ejpam-4383	296	22	i∈i	i∈i	ADJ
ejpam-4383	296	23	pi	pi	NOUN
ejpam-4383	296	24	be	be	AUX
ejpam-4383	296	25	such	such	ADJ
ejpam-4383	296	26	that	that	PRON
ejpam-4383	296	27	ψ(pi)i∈i	ψ(pi)i∈i	NOUN
ejpam-4383	296	28	=	=	SYM
ejpam-4383	296	29	ψ(qi)i∈i	ψ(qi)i∈i	NOUN
ejpam-4383	296	30	.	.	PUNCT
ejpam-4383	297	1	then	then	ADV
ejpam-4383	297	2	(	(	PUNCT
ejpam-4383	297	3	ψi(pi))i∈i	ψi(pi))i∈i	PROPN
ejpam-4383	297	4	=	=	SYM
ejpam-4383	297	5	(	(	PUNCT
ejpam-4383	297	6	ψi(qi))i∈i	ψi(qi))i∈i	X
ejpam-4383	297	7	.	.	PUNCT
ejpam-4383	298	1	thus	thus	ADV
ejpam-4383	298	2	ψi(pi	ψi(pi	NUM
ejpam-4383	298	3	)	)	PUNCT
ejpam-4383	298	4	=	=	SYM
ejpam-4383	298	5	ψi(qi	ψi(qi	PROPN
ejpam-4383	298	6	)	)	PUNCT
ejpam-4383	298	7	for	for	ADP
ejpam-4383	298	8	all	all	PRON
ejpam-4383	298	9	i	i	PRON
ejpam-4383	298	10	∈	∈	PROPN
ejpam-4383	298	11	i.	i.	NOUN
ejpam-4383	298	12	since	since	SCONJ
ejpam-4383	298	13	ψi	ψi	ADV
ejpam-4383	298	14	is	be	AUX
ejpam-4383	298	15	injective	injective	ADJ
ejpam-4383	298	16	for	for	ADP
ejpam-4383	298	17	all	all	PRON
ejpam-4383	298	18	i	i	PRON
ejpam-4383	298	19	∈	∈	PROPN
ejpam-4383	299	1	i	i	PRON
ejpam-4383	299	2	,	,	PUNCT
ejpam-4383	299	3	we	we	PRON
ejpam-4383	299	4	have	have	VERB
ejpam-4383	299	5	pi	pi	NOUN
ejpam-4383	299	6	=	=	PUNCT
ejpam-4383	299	7	qi	qi	PROPN
ejpam-4383	299	8	for	for	ADP
ejpam-4383	299	9	all	all	PRON
ejpam-4383	299	10	i	i	PRON
ejpam-4383	299	11	∈	∈	PROPN
ejpam-4383	299	12	i.	i.	NOUN
ejpam-4383	299	13	thus	thus	ADV
ejpam-4383	299	14	(	(	PUNCT
ejpam-4383	299	15	pi)i∈i	pi)i∈i	NUM
ejpam-4383	299	16	=	=	SYM
ejpam-4383	299	17	(	(	PUNCT
ejpam-4383	299	18	qi)i∈i	qi)i∈i	NUM
ejpam-4383	299	19	.	.	PUNCT
ejpam-4383	300	1	hence	hence	ADV
ejpam-4383	300	2	,	,	PUNCT
ejpam-4383	300	3	ψ	ψ	NOUN
ejpam-4383	300	4	is	be	AUX
ejpam-4383	300	5	injective	injective	ADJ
ejpam-4383	300	6	.	.	PUNCT
ejpam-4383	301	1	conversely	conversely	ADV
ejpam-4383	301	2	,	,	PUNCT
ejpam-4383	301	3	assume	assume	VERB
ejpam-4383	301	4	that	that	SCONJ
ejpam-4383	301	5	ψ	ψ	NOUN
ejpam-4383	301	6	is	be	AUX
ejpam-4383	301	7	injective	injective	ADJ
ejpam-4383	301	8	.	.	PUNCT
ejpam-4383	302	1	let	let	VERB
ejpam-4383	302	2	i	i	PRON
ejpam-4383	302	3	∈	∈	PROPN
ejpam-4383	302	4	i.	i.	NOUN
ejpam-4383	302	5	let	let	VERB
ejpam-4383	302	6	pi	pi	NOUN
ejpam-4383	302	7	,	,	PUNCT
ejpam-4383	302	8	p	p	NOUN
ejpam-4383	303	1	′	′	NUM
ejpam-4383	304	1	i	i	PRON
ejpam-4383	304	2	∈	∈	PROPN
ejpam-4383	304	3	pi	pi	NOUN
ejpam-4383	304	4	be	be	AUX
ejpam-4383	304	5	such	such	ADJ
ejpam-4383	304	6	that	that	DET
ejpam-4383	304	7	ψi(pi	ψi(pi	NOUN
ejpam-4383	304	8	)	)	PUNCT
ejpam-4383	304	9	=	=	NOUN
ejpam-4383	304	10	ψi(p	ψi(p	X
ejpam-4383	304	11	′	′	NUM
ejpam-4383	305	1	i	i	NOUN
ejpam-4383	305	2	)	)	PUNCT
ejpam-4383	305	3	.	.	PUNCT
ejpam-4383	306	1	let	let	VERB
ejpam-4383	306	2	pj	pj	PROPN
ejpam-4383	306	3	=	=	PUNCT
ejpam-4383	306	4	p′j	p′j	PROPN
ejpam-4383	306	5	∈	∈	PROPN
ejpam-4383	306	6	pj	pj	PROPN
ejpam-4383	306	7	for	for	ADP
ejpam-4383	306	8	all	all	DET
ejpam-4383	306	9	j	j	PROPN
ejpam-4383	306	10	∈	∈	PROPN
ejpam-4383	307	1	i	i	PRON
ejpam-4383	307	2	and	and	CCONJ
ejpam-4383	307	3	j	j	PROPN
ejpam-4383	307	4	̸=	̸=	PROPN
ejpam-4383	307	5	i.	i.	NOUN
ejpam-4383	307	6	then	then	ADV
ejpam-4383	307	7	ψj(pj	ψj(pj	PROPN
ejpam-4383	307	8	)	)	PUNCT
ejpam-4383	307	9	=	=	SYM
ejpam-4383	307	10	ψj(p	ψj(p	NUM
ejpam-4383	308	1	′	′	NUM
ejpam-4383	308	2	j	j	X
ejpam-4383	308	3	)	)	PUNCT
ejpam-4383	308	4	∈	∈	PROPN
ejpam-4383	308	5	qj	qj	PROPN
ejpam-4383	308	6	.	.	PUNCT
ejpam-4383	309	1	let	let	VERB
ejpam-4383	309	2	hψi(pi	hψi(pi	NOUN
ejpam-4383	309	3	)	)	PUNCT
ejpam-4383	309	4	:	:	PUNCT
ejpam-4383	310	1	i	i	PRON
ejpam-4383	310	2	→⋃	→⋃	VERB
ejpam-4383	310	3	i∈i	i∈i	ADJ
ejpam-4383	310	4	qi	qi	PROPN
ejpam-4383	310	5	and	and	CCONJ
ejpam-4383	310	6	hψi(p′i	hψi(p′i	PROPN
ejpam-4383	310	7	)	)	PUNCT
ejpam-4383	310	8	:	:	PUNCT
ejpam-4383	311	1	i	i	PRON
ejpam-4383	311	2	→	→	SYM
ejpam-4383	311	3	⋃	⋃	ADP
ejpam-4383	311	4	i∈i	i∈i	ADJ
ejpam-4383	311	5	qi	qi	NOUN
ejpam-4383	311	6	are	be	AUX
ejpam-4383	311	7	functions	function	NOUN
ejpam-4383	311	8	defined	define	VERB
ejpam-4383	311	9	by	by	ADP
ejpam-4383	311	10	(	(	PUNCT
ejpam-4383	311	11	∀j	∀j	PROPN
ejpam-4383	311	12	∈	∈	PROPN
ejpam-4383	311	13	i	i	PROPN
ejpam-4383	311	14	)	)	PUNCT
ejpam-4383	311	15	(	(	PUNCT
ejpam-4383	311	16	hψi(pi)(j	hψi(pi)(j	PROPN
ejpam-4383	311	17	)	)	PUNCT
ejpam-4383	311	18	=	=	PRON
ejpam-4383	311	19	{	{	PUNCT
ejpam-4383	311	20	ψi(pi	ψi(pi	NOUN
ejpam-4383	311	21	)	)	PUNCT
ejpam-4383	311	22	if	if	SCONJ
ejpam-4383	311	23	j	j	PROPN
ejpam-4383	312	1	=	=	VERB
ejpam-4383	312	2	i	i	PRON
ejpam-4383	312	3	ψj(pj	ψj(pj	VERB
ejpam-4383	312	4	)	)	PUNCT
ejpam-4383	312	5	otherwise	otherwise	ADV
ejpam-4383	312	6	)	)	PUNCT
ejpam-4383	312	7	(	(	PUNCT
ejpam-4383	312	8	2.4	2.4	NUM
ejpam-4383	312	9	)	)	PUNCT
ejpam-4383	312	10	and	and	CCONJ
ejpam-4383	312	11	(	(	PUNCT
ejpam-4383	312	12	∀j	∀j	PROPN
ejpam-4383	312	13	∈	∈	PROPN
ejpam-4383	312	14	i	i	NOUN
ejpam-4383	312	15	)	)	PUNCT
ejpam-4383	312	16	(	(	PUNCT
ejpam-4383	312	17	hψi(p′i	hψi(p′i	NOUN
ejpam-4383	312	18	)	)	PUNCT
ejpam-4383	312	19	(	(	PUNCT
ejpam-4383	312	20	j	j	NOUN
ejpam-4383	312	21	)	)	PUNCT
ejpam-4383	312	22	=	=	PRON
ejpam-4383	312	23	{	{	PUNCT
ejpam-4383	312	24	ψi(p	ψi(p	NOUN
ejpam-4383	312	25	′	′	NUM
ejpam-4383	313	1	i	i	NOUN
ejpam-4383	313	2	)	)	PUNCT
ejpam-4383	313	3	if	if	SCONJ
ejpam-4383	313	4	j	j	PROPN
ejpam-4383	314	1	=	=	PUNCT
ejpam-4383	314	2	i	i	PRON
ejpam-4383	314	3	ψj(p	ψj(p	VERB
ejpam-4383	314	4	′	′	NUM
ejpam-4383	314	5	j	j	NOUN
ejpam-4383	314	6	)	)	PUNCT
ejpam-4383	314	7	otherwise	otherwise	ADV
ejpam-4383	314	8	)	)	PUNCT
ejpam-4383	314	9	.	.	PUNCT
ejpam-4383	315	1	(	(	PUNCT
ejpam-4383	315	2	2.5	2.5	NUM
ejpam-4383	315	3	)	)	PUNCT
ejpam-4383	315	4	then	then	ADV
ejpam-4383	315	5	hψi(pi	hψi(pi	NOUN
ejpam-4383	315	6	)	)	PUNCT
ejpam-4383	315	7	,	,	PUNCT
ejpam-4383	315	8	hψi(p′i	hψi(p′i	NOUN
ejpam-4383	315	9	)	)	PUNCT
ejpam-4383	315	10	∈	∈	PROPN
ejpam-4383	315	11	∏	∏	PROPN
ejpam-4383	315	12	i∈i	i∈i	ADJ
ejpam-4383	315	13	qi	qi	PROPN
ejpam-4383	315	14	such	such	ADJ
ejpam-4383	315	15	that	that	PRON
ejpam-4383	315	16	ψ(pi)i∈i	ψ(pi)i∈i	X
ejpam-4383	315	17	=	=	SYM
ejpam-4383	315	18	hψi(pi	hψi(pi	NOUN
ejpam-4383	315	19	)	)	PUNCT
ejpam-4383	315	20	=	=	SYM
ejpam-4383	315	21	hψi(p′i	hψi(p′i	NOUN
ejpam-4383	315	22	)	)	PUNCT
ejpam-4383	315	23	=	=	NOUN
ejpam-4383	315	24	ψ(p′i)i∈i	ψ(p′i)i∈i	NOUN
ejpam-4383	315	25	.	.	PUNCT
ejpam-4383	316	1	since	since	SCONJ
ejpam-4383	316	2	ψ	ψ	NOUN
ejpam-4383	316	3	is	be	AUX
ejpam-4383	316	4	injective	injective	ADJ
ejpam-4383	316	5	,	,	PUNCT
ejpam-4383	316	6	we	we	PRON
ejpam-4383	316	7	have	have	VERB
ejpam-4383	316	8	(	(	PUNCT
ejpam-4383	316	9	pi)i∈i	pi)i∈i	NUM
ejpam-4383	316	10	=	=	SYM
ejpam-4383	316	11	(	(	PUNCT
ejpam-4383	316	12	p′i)i∈i	p′i)i∈i	NOUN
ejpam-4383	316	13	.	.	PUNCT
ejpam-4383	317	1	thus	thus	ADV
ejpam-4383	317	2	pi	pi	NOUN
ejpam-4383	317	3	=	=	PUNCT
ejpam-4383	317	4	p′i	p′i	NOUN
ejpam-4383	317	5	.	.	PUNCT
ejpam-4383	318	1	hence	hence	ADV
ejpam-4383	318	2	,	,	PUNCT
ejpam-4383	318	3	ψi	ψi	ADV
ejpam-4383	318	4	is	be	AUX
ejpam-4383	318	5	injective	injective	ADJ
ejpam-4383	318	6	for	for	ADP
ejpam-4383	318	7	all	all	PRON
ejpam-4383	318	8	i	i	PRON
ejpam-4383	318	9	∈	∈	PROPN
ejpam-4383	318	10	i.	i.	NOUN
ejpam-4383	318	11	(	(	PUNCT
ejpam-4383	318	12	ii	ii	PROPN
ejpam-4383	318	13	)	)	PUNCT
ejpam-4383	318	14	assume	assume	VERB
ejpam-4383	318	15	that	that	SCONJ
ejpam-4383	318	16	ψi	ψi	NOUN
ejpam-4383	318	17	is	be	AUX
ejpam-4383	318	18	surjective	surjective	ADJ
ejpam-4383	318	19	for	for	SCONJ
ejpam-4383	318	20	all	all	PRON
ejpam-4383	318	21	i	i	PRON
ejpam-4383	318	22	∈	∈	PROPN
ejpam-4383	318	23	i.	i.	NOUN
ejpam-4383	318	24	let	let	VERB
ejpam-4383	318	25	(	(	PUNCT
ejpam-4383	318	26	qi)i∈i	qi)i∈i	NUM
ejpam-4383	318	27	∈	∈	PROPN
ejpam-4383	318	28	∏	∏	PROPN
ejpam-4383	318	29	i∈i	i∈i	PROPN
ejpam-4383	318	30	qi	qi	PROPN
ejpam-4383	318	31	.	.	PUNCT
ejpam-4383	319	1	then	then	ADV
ejpam-4383	319	2	qi	qi	PROPN
ejpam-4383	319	3	∈	∈	PROPN
ejpam-4383	319	4	qi	qi	PROPN
ejpam-4383	319	5	for	for	ADP
ejpam-4383	319	6	all	all	PRON
ejpam-4383	319	7	i	i	PRON
ejpam-4383	319	8	∈	∈	PROPN
ejpam-4383	319	9	i.	i.	NOUN
ejpam-4383	319	10	since	since	SCONJ
ejpam-4383	319	11	ψi	ψi	ADV
ejpam-4383	319	12	is	be	AUX
ejpam-4383	319	13	surjective	surjective	ADJ
ejpam-4383	319	14	,	,	PUNCT
ejpam-4383	319	15	there	there	PRON
ejpam-4383	319	16	exists	exist	VERB
ejpam-4383	319	17	pi	pi	PROPN
ejpam-4383	319	18	∈	∈	PROPN
ejpam-4383	319	19	pi	pi	NOUN
ejpam-4383	319	20	such	such	ADJ
ejpam-4383	319	21	that	that	DET
ejpam-4383	319	22	ψi(pi	ψi(pi	NOUN
ejpam-4383	319	23	)	)	PUNCT
ejpam-4383	319	24	=	=	SYM
ejpam-4383	320	1	qi	qi	PROPN
ejpam-4383	320	2	for	for	ADP
ejpam-4383	320	3	all	all	DET
ejpam-4383	320	4	i	i	PRON
ejpam-4383	320	5	∈	∈	PROPN
ejpam-4383	320	6	i.	i.	NOUN
ejpam-4383	320	7	thus	thus	ADV
ejpam-4383	320	8	(	(	PUNCT
ejpam-4383	320	9	pi)i∈i	pi)i∈i	NUM
ejpam-4383	320	10	∈	∈	PROPN
ejpam-4383	320	11	∏	∏	PROPN
ejpam-4383	320	12	i∈i	i∈i	ADJ
ejpam-4383	320	13	pi	pi	NOUN
ejpam-4383	320	14	such	such	ADJ
ejpam-4383	320	15	that	that	PRON
ejpam-4383	320	16	ψ(pi)i∈i	ψ(pi)i∈i	X
ejpam-4383	321	1	=	=	SYM
ejpam-4383	321	2	(	(	PUNCT
ejpam-4383	321	3	ψi(pi))i∈i	ψi(pi))i∈i	PROPN
ejpam-4383	321	4	=	=	SYM
ejpam-4383	321	5	(	(	PUNCT
ejpam-4383	321	6	qi)i∈i	qi)i∈i	NUM
ejpam-4383	321	7	.	.	PUNCT
ejpam-4383	322	1	hence	hence	ADV
ejpam-4383	322	2	,	,	PUNCT
ejpam-4383	322	3	ψ	ψ	X
ejpam-4383	322	4	is	be	AUX
ejpam-4383	322	5	surjective	surjective	ADJ
ejpam-4383	322	6	.	.	PUNCT
ejpam-4383	323	1	conversely	conversely	ADV
ejpam-4383	323	2	,	,	PUNCT
ejpam-4383	323	3	assume	assume	VERB
ejpam-4383	323	4	that	that	SCONJ
ejpam-4383	323	5	ψ	ψ	NOUN
ejpam-4383	323	6	is	be	AUX
ejpam-4383	323	7	surjective	surjective	ADJ
ejpam-4383	323	8	.	.	PUNCT
ejpam-4383	324	1	let	let	VERB
ejpam-4383	324	2	i	i	PRON
ejpam-4383	324	3	∈	∈	PROPN
ejpam-4383	324	4	i.	i.	NOUN
ejpam-4383	324	5	let	let	VERB
ejpam-4383	324	6	ki	ki	PROPN
ejpam-4383	324	7	∈	∈	PROPN
ejpam-4383	324	8	qi	qi	PROPN
ejpam-4383	324	9	.	.	PUNCT
ejpam-4383	325	1	let	let	VERB
ejpam-4383	325	2	kj	kj	PROPN
ejpam-4383	325	3	∈	∈	PROPN
ejpam-4383	325	4	qj	qj	PROPN
ejpam-4383	325	5	for	for	ADP
ejpam-4383	325	6	all	all	DET
ejpam-4383	325	7	j	j	PROPN
ejpam-4383	325	8	∈	∈	PROPN
ejpam-4383	326	1	i	i	PRON
ejpam-4383	326	2	and	and	CCONJ
ejpam-4383	326	3	j	j	PROPN
ejpam-4383	326	4	̸=	̸=	PROPN
ejpam-4383	326	5	i.	i.	NOUN
ejpam-4383	326	6	then	then	ADV
ejpam-4383	326	7	(	(	PUNCT
ejpam-4383	326	8	ki)i∈i	ki)i∈i	X
ejpam-4383	326	9	∈	∈	PROPN
ejpam-4383	326	10	∏	∏	PROPN
ejpam-4383	326	11	i∈i	i∈i	PROPN
ejpam-4383	326	12	qi	qi	PROPN
ejpam-4383	326	13	.	.	PUNCT
ejpam-4383	327	1	since	since	SCONJ
ejpam-4383	327	2	ψ	ψ	NOUN
ejpam-4383	327	3	is	be	AUX
ejpam-4383	327	4	surjective	surjective	ADJ
ejpam-4383	327	5	,	,	PUNCT
ejpam-4383	327	6	there	there	PRON
ejpam-4383	327	7	exists	exist	VERB
ejpam-4383	327	8	(	(	PUNCT
ejpam-4383	327	9	pi)i∈i	pi)i∈i	NUM
ejpam-4383	327	10	∈	∈	PROPN
ejpam-4383	327	11	∏	∏	PROPN
ejpam-4383	327	12	i∈i	i∈i	ADJ
ejpam-4383	327	13	pi	pi	NOUN
ejpam-4383	327	14	such	such	ADJ
ejpam-4383	327	15	that	that	SCONJ
ejpam-4383	327	16	(	(	PUNCT
ejpam-4383	327	17	ki)i∈i	ki)i∈i	NUM
ejpam-4383	327	18	=	=	SYM
ejpam-4383	327	19	ψ(pi)i∈i	ψ(pi)i∈i	X
ejpam-4383	327	20	=	=	SYM
ejpam-4383	327	21	(	(	PUNCT
ejpam-4383	327	22	ψi(pi))i∈i	ψi(pi))i∈i	PROPN
ejpam-4383	327	23	.	.	PUNCT
ejpam-4383	328	1	thus	thus	ADV
ejpam-4383	328	2	ki	ki	PROPN
ejpam-4383	328	3	=	=	PUNCT
ejpam-4383	328	4	ψi(pi	ψi(pi	PROPN
ejpam-4383	328	5	)	)	PUNCT
ejpam-4383	328	6	.	.	PUNCT
ejpam-4383	329	1	hence	hence	ADV
ejpam-4383	329	2	,	,	PUNCT
ejpam-4383	329	3	ψi	ψi	ADV
ejpam-4383	329	4	is	be	AUX
ejpam-4383	329	5	surjective	surjective	ADJ
ejpam-4383	329	6	for	for	ADP
ejpam-4383	329	7	all	all	PRON
ejpam-4383	329	8	i	i	PRON
ejpam-4383	329	9	∈	∈	PROPN
ejpam-4383	329	10	i.	i.	NOUN
ejpam-4383	329	11	(	(	PUNCT
ejpam-4383	329	12	iii	iii	X
ejpam-4383	329	13	)	)	PUNCT
ejpam-4383	329	14	it	it	PRON
ejpam-4383	329	15	is	be	AUX
ejpam-4383	329	16	straightforward	straightforward	ADJ
ejpam-4383	329	17	from	from	ADP
ejpam-4383	329	18	(	(	PUNCT
ejpam-4383	329	19	i	i	NOUN
ejpam-4383	329	20	)	)	PUNCT
ejpam-4383	329	21	and	and	CCONJ
ejpam-4383	329	22	(	(	PUNCT
ejpam-4383	329	23	ii	ii	NOUN
ejpam-4383	329	24	)	)	PUNCT
ejpam-4383	329	25	.	.	PUNCT
ejpam-4383	330	1	theorem	theorem	VERB
ejpam-4383	330	2	11	11	NUM
ejpam-4383	330	3	.	.	PUNCT
ejpam-4383	331	1	let	let	AUX
ejpam-4383	331	2	pi	pi	NOUN
ejpam-4383	331	3	=	=	PUNCT
ejpam-4383	331	4	(	(	PUNCT
ejpam-4383	331	5	pi	pi	NOUN
ejpam-4383	331	6	;	;	PUNCT
ejpam-4383	331	7	∗i	∗i	PROPN
ejpam-4383	331	8	,	,	PUNCT
ejpam-4383	331	9	0i	0i	NOUN
ejpam-4383	331	10	)	)	PUNCT
ejpam-4383	331	11	and	and	CCONJ
ejpam-4383	331	12	qi	qi	PROPN
ejpam-4383	331	13	=	=	SYM
ejpam-4383	331	14	(	(	PUNCT
ejpam-4383	331	15	qi	qi	NOUN
ejpam-4383	331	16	;	;	PUNCT
ejpam-4383	331	17	◦	◦	NOUN
ejpam-4383	331	18	i	i	PROPN
ejpam-4383	331	19	,	,	PUNCT
ejpam-4383	331	20	1i	1i	NUM
ejpam-4383	331	21	)	)	PUNCT
ejpam-4383	331	22	be	be	VERB
ejpam-4383	331	23	b	b	NOUN
ejpam-4383	331	24	-	-	PUNCT
ejpam-4383	331	25	algebras	algebra	NOUN
ejpam-4383	331	26	and	and	CCONJ
ejpam-4383	331	27	ψi	ψi	ADP
ejpam-4383	331	28	:	:	PUNCT
ejpam-4383	331	29	pi	pi	NOUN
ejpam-4383	331	30	→	→	SYM
ejpam-4383	331	31	qi	qi	PROPN
ejpam-4383	331	32	be	be	AUX
ejpam-4383	331	33	a	a	DET
ejpam-4383	331	34	function	function	NOUN
ejpam-4383	331	35	for	for	ADP
ejpam-4383	331	36	all	all	PRON
ejpam-4383	331	37	i	i	PRON
ejpam-4383	331	38	∈	∈	PROPN
ejpam-4383	331	39	i.	i.	NOUN
ejpam-4383	331	40	then	then	ADV
ejpam-4383	331	41	(	(	PUNCT
ejpam-4383	331	42	i	i	NOUN
ejpam-4383	331	43	)	)	PUNCT
ejpam-4383	331	44	ψi	ψi	NOUN
ejpam-4383	331	45	is	be	AUX
ejpam-4383	331	46	a	a	DET
ejpam-4383	331	47	b	b	NOUN
ejpam-4383	331	48	-	-	PUNCT
ejpam-4383	331	49	homomorphism	homomorphism	NOUN
ejpam-4383	331	50	for	for	ADP
ejpam-4383	331	51	all	all	PRON
ejpam-4383	331	52	i	i	PRON
ejpam-4383	331	53	∈	∈	VERB
ejpam-4383	332	1	i	i	PRON
ejpam-4383	332	2	if	if	SCONJ
ejpam-4383	332	3	and	and	CCONJ
ejpam-4383	332	4	only	only	ADV
ejpam-4383	332	5	if	if	SCONJ
ejpam-4383	332	6	ψ	ψ	NOUN
ejpam-4383	332	7	is	be	AUX
ejpam-4383	332	8	a	a	DET
ejpam-4383	332	9	b	b	NOUN
ejpam-4383	332	10	-	-	PUNCT
ejpam-4383	332	11	homomorphism	homomorphism	NOUN
ejpam-4383	332	12	which	which	PRON
ejpam-4383	332	13	is	be	AUX
ejpam-4383	332	14	defined	define	VERB
ejpam-4383	332	15	in	in	ADP
ejpam-4383	332	16	definition	definition	NOUN
ejpam-4383	332	17	10	10	NUM
ejpam-4383	332	18	,	,	PUNCT
ejpam-4383	332	19	(	(	PUNCT
ejpam-4383	332	20	ii	ii	NOUN
ejpam-4383	332	21	)	)	PUNCT
ejpam-4383	332	22	ψi	ψi	NOUN
ejpam-4383	332	23	is	be	AUX
ejpam-4383	332	24	a	a	DET
ejpam-4383	332	25	b	b	NOUN
ejpam-4383	332	26	-	-	PUNCT
ejpam-4383	332	27	monomorphism	monomorphism	NOUN
ejpam-4383	332	28	for	for	ADP
ejpam-4383	332	29	all	all	PRON
ejpam-4383	332	30	i	i	PRON
ejpam-4383	332	31	∈	∈	VERB
ejpam-4383	333	1	i	i	PRON
ejpam-4383	333	2	if	if	SCONJ
ejpam-4383	333	3	and	and	CCONJ
ejpam-4383	333	4	only	only	ADV
ejpam-4383	333	5	if	if	SCONJ
ejpam-4383	333	6	ψ	ψ	NOUN
ejpam-4383	333	7	is	be	AUX
ejpam-4383	333	8	a	a	DET
ejpam-4383	333	9	b	b	NOUN
ejpam-4383	333	10	-	-	PUNCT
ejpam-4383	333	11	monomorphism	monomorphism	NOUN
ejpam-4383	333	12	,	,	PUNCT
ejpam-4383	333	13	a.	a.	NOUN
ejpam-4383	333	14	iampan	iampan	NOUN
ejpam-4383	333	15	et	et	PROPN
ejpam-4383	333	16	al	al	PROPN
ejpam-4383	333	17	.	.	PUNCT
ejpam-4383	333	18	/	/	SYM
ejpam-4383	333	19	eur	eur	PROPN
ejpam-4383	333	20	.	.	PUNCT
ejpam-4383	334	1	j.	j.	PROPN
ejpam-4383	334	2	pure	pure	PROPN
ejpam-4383	334	3	appl	appl	PROPN
ejpam-4383	334	4	.	.	PROPN
ejpam-4383	334	5	math	math	PROPN
ejpam-4383	334	6	,	,	PUNCT
ejpam-4383	334	7	15	15	NUM
ejpam-4383	334	8	(	(	PUNCT
ejpam-4383	334	9	3	3	NUM
ejpam-4383	334	10	)	)	PUNCT
ejpam-4383	334	11	(	(	PUNCT
ejpam-4383	334	12	2022	2022	NUM
ejpam-4383	334	13	)	)	PUNCT
ejpam-4383	334	14	,	,	PUNCT
ejpam-4383	334	15	999	999	NUM
ejpam-4383	334	16	-	-	SYM
ejpam-4383	334	17	1014	1014	NUM
ejpam-4383	334	18	1010	1010	NUM
ejpam-4383	334	19	(	(	PUNCT
ejpam-4383	334	20	iii	iii	NOUN
ejpam-4383	334	21	)	)	PUNCT
ejpam-4383	334	22	ψi	ψi	NOUN
ejpam-4383	334	23	is	be	AUX
ejpam-4383	334	24	a	a	DET
ejpam-4383	334	25	b	b	NOUN
ejpam-4383	334	26	-	-	PUNCT
ejpam-4383	334	27	epimorphism	epimorphism	NOUN
ejpam-4383	334	28	for	for	ADP
ejpam-4383	334	29	all	all	PRON
ejpam-4383	334	30	i	i	PRON
ejpam-4383	334	31	∈	∈	VERB
ejpam-4383	335	1	i	i	PRON
ejpam-4383	335	2	if	if	SCONJ
ejpam-4383	335	3	and	and	CCONJ
ejpam-4383	335	4	only	only	ADV
ejpam-4383	335	5	if	if	SCONJ
ejpam-4383	335	6	ψ	ψ	NOUN
ejpam-4383	335	7	is	be	AUX
ejpam-4383	335	8	a	a	DET
ejpam-4383	335	9	b	b	NOUN
ejpam-4383	335	10	-	-	PUNCT
ejpam-4383	335	11	epimorphism	epimorphism	NOUN
ejpam-4383	335	12	,	,	PUNCT
ejpam-4383	335	13	(	(	PUNCT
ejpam-4383	335	14	iv	iv	X
ejpam-4383	335	15	)	)	PUNCT
ejpam-4383	335	16	ψi	ψi	NOUN
ejpam-4383	335	17	is	be	AUX
ejpam-4383	335	18	a	a	DET
ejpam-4383	335	19	b	b	NOUN
ejpam-4383	335	20	-	-	PUNCT
ejpam-4383	335	21	isomorphism	isomorphism	NOUN
ejpam-4383	335	22	for	for	ADP
ejpam-4383	335	23	all	all	PRON
ejpam-4383	335	24	i	i	PRON
ejpam-4383	335	25	∈	∈	VERB
ejpam-4383	336	1	i	i	PRON
ejpam-4383	336	2	if	if	SCONJ
ejpam-4383	336	3	and	and	CCONJ
ejpam-4383	336	4	only	only	ADV
ejpam-4383	336	5	if	if	SCONJ
ejpam-4383	336	6	ψ	ψ	NOUN
ejpam-4383	336	7	is	be	AUX
ejpam-4383	336	8	a	a	DET
ejpam-4383	336	9	b	b	NOUN
ejpam-4383	336	10	-	-	PUNCT
ejpam-4383	336	11	isomorphism	isomorphism	NOUN
ejpam-4383	336	12	,	,	PUNCT
ejpam-4383	336	13	(	(	PUNCT
ejpam-4383	336	14	v	v	NOUN
ejpam-4383	336	15	)	)	PUNCT
ejpam-4383	336	16	kerψ	kerψ	NOUN
ejpam-4383	336	17	=	=	PUNCT
ejpam-4383	336	18	∏	∏	PROPN
ejpam-4383	336	19	i∈i	i∈i	ADJ
ejpam-4383	336	20	kerψi	kerψi	NOUN
ejpam-4383	336	21	and	and	CCONJ
ejpam-4383	336	22	ψ	ψ	PROPN
ejpam-4383	336	23	(	(	PUNCT
ejpam-4383	336	24	∏	∏	PROPN
ejpam-4383	336	25	i∈i	i∈i	ADJ
ejpam-4383	336	26	pi	pi	NOUN
ejpam-4383	336	27	)	)	PUNCT
ejpam-4383	336	28	=	=	SYM
ejpam-4383	336	29	∏	∏	PROPN
ejpam-4383	336	30	i∈i	i∈i	ADJ
ejpam-4383	336	31	ψi(pi	ψi(pi	NOUN
ejpam-4383	336	32	)	)	PUNCT
ejpam-4383	336	33	.	.	PUNCT
ejpam-4383	337	1	proof	proof	NOUN
ejpam-4383	337	2	.	.	PUNCT
ejpam-4383	338	1	(	(	PUNCT
ejpam-4383	338	2	i	i	NOUN
ejpam-4383	338	3	)	)	PUNCT
ejpam-4383	338	4	assume	assume	VERB
ejpam-4383	338	5	that	that	SCONJ
ejpam-4383	338	6	ψi	ψi	NOUN
ejpam-4383	338	7	is	be	AUX
ejpam-4383	338	8	a	a	DET
ejpam-4383	338	9	b	b	NOUN
ejpam-4383	338	10	-homomorphism	-homomorphism	NOUN
ejpam-4383	338	11	for	for	ADP
ejpam-4383	338	12	all	all	DET
ejpam-4383	338	13	i	i	PRON
ejpam-4383	338	14	∈	∈	PROPN
ejpam-4383	338	15	i.	i.	NOUN
ejpam-4383	338	16	let	let	VERB
ejpam-4383	338	17	(	(	PUNCT
ejpam-4383	338	18	pi)i∈i	pi)i∈i	NUM
ejpam-4383	338	19	,	,	PUNCT
ejpam-4383	338	20	(	(	PUNCT
ejpam-4383	338	21	p	p	NOUN
ejpam-4383	338	22	′	′	NUM
ejpam-4383	338	23	i)i∈i	i)i∈i	PROPN
ejpam-4383	338	24	∈∏	∈∏	PROPN
ejpam-4383	338	25	i∈i	i∈i	ADJ
ejpam-4383	338	26	pi	pi	PROPN
ejpam-4383	338	27	.	.	PUNCT
ejpam-4383	339	1	then	then	ADV
ejpam-4383	339	2	ψ((pi)i∈i	ψ((pi)i∈i	PROPN
ejpam-4383	339	3	⊗	⊗	PROPN
ejpam-4383	339	4	(	(	PUNCT
ejpam-4383	339	5	p′i)i∈i	p′i)i∈i	PROPN
ejpam-4383	339	6	)	)	PUNCT
ejpam-4383	339	7	=	=	NOUN
ejpam-4383	339	8	ψ(pi	ψ(pi	X
ejpam-4383	339	9	∗i	∗i	PROPN
ejpam-4383	339	10	p′i)i∈i	p′i)i∈i	NOUN
ejpam-4383	339	11	=	=	SYM
ejpam-4383	339	12	(	(	PUNCT
ejpam-4383	339	13	ψi(pi	ψi(pi	NOUN
ejpam-4383	339	14	∗i	∗i	ADJ
ejpam-4383	339	15	p′i))i∈i	p′i))i∈i	NOUN
ejpam-4383	339	16	=	=	SYM
ejpam-4383	339	17	(	(	PUNCT
ejpam-4383	339	18	ψi(pi	ψi(pi	NOUN
ejpam-4383	339	19	)	)	PUNCT
ejpam-4383	339	20	∗i	∗i	NOUN
ejpam-4383	339	21	ψi(p′i))i∈i	ψi(p′i))i∈i	VERB
ejpam-4383	339	22	=	=	PUNCT
ejpam-4383	339	23	(	(	PUNCT
ejpam-4383	339	24	ψi(pi))i∈i	ψi(pi))i∈i	ADP
ejpam-4383	339	25	⊗	⊗	PROPN
ejpam-4383	339	26	(	(	PUNCT
ejpam-4383	339	27	ψi(p	ψi(p	NOUN
ejpam-4383	339	28	′	′	NUM
ejpam-4383	339	29	i))i∈i	i))i∈i	NOUN
ejpam-4383	339	30	=	=	SYM
ejpam-4383	339	31	ψ(pi)i∈i	ψ(pi)i∈i	ADP
ejpam-4383	339	32	⊗	⊗	PROPN
ejpam-4383	339	33	ψ(p′i)i∈i	ψ(p′i)i∈i	NUM
ejpam-4383	339	34	.	.	PUNCT
ejpam-4383	340	1	hence	hence	ADV
ejpam-4383	340	2	,	,	PUNCT
ejpam-4383	340	3	ψ	ψ	X
ejpam-4383	340	4	is	be	AUX
ejpam-4383	340	5	a	a	DET
ejpam-4383	340	6	b	b	NOUN
ejpam-4383	340	7	-homomorphism	-homomorphism	NOUN
ejpam-4383	340	8	.	.	PUNCT
ejpam-4383	341	1	conversely	conversely	ADV
ejpam-4383	341	2	,	,	PUNCT
ejpam-4383	341	3	assume	assume	VERB
ejpam-4383	341	4	that	that	SCONJ
ejpam-4383	341	5	ψ	ψ	NOUN
ejpam-4383	341	6	is	be	AUX
ejpam-4383	341	7	a	a	DET
ejpam-4383	341	8	b	b	NOUN
ejpam-4383	341	9	-homomorphism	-homomorphism	NOUN
ejpam-4383	341	10	.	.	PUNCT
ejpam-4383	342	1	let	let	VERB
ejpam-4383	342	2	i	i	PRON
ejpam-4383	342	3	∈	∈	PROPN
ejpam-4383	342	4	i.	i.	NOUN
ejpam-4383	342	5	let	let	VERB
ejpam-4383	342	6	pi	pi	NOUN
ejpam-4383	342	7	,	,	PUNCT
ejpam-4383	342	8	qi	qi	PROPN
ejpam-4383	342	9	∈	∈	PROPN
ejpam-4383	342	10	pi	pi	NOUN
ejpam-4383	342	11	.	.	PUNCT
ejpam-4383	343	1	then	then	ADV
ejpam-4383	343	2	fpi	fpi	PROPN
ejpam-4383	343	3	,	,	PUNCT
ejpam-4383	343	4	fqi	fqi	VERB
ejpam-4383	343	5	∈	∈	PROPN
ejpam-4383	343	6	∏	∏	PROPN
ejpam-4383	343	7	i∈i	i∈i	ADJ
ejpam-4383	343	8	pi	pi	NOUN
ejpam-4383	343	9	,	,	PUNCT
ejpam-4383	343	10	which	which	PRON
ejpam-4383	343	11	is	be	AUX
ejpam-4383	343	12	defined	define	VERB
ejpam-4383	343	13	by	by	ADP
ejpam-4383	343	14	(	(	PUNCT
ejpam-4383	343	15	2.2	2.2	NUM
ejpam-4383	343	16	)	)	PUNCT
ejpam-4383	343	17	.	.	PUNCT
ejpam-4383	344	1	since	since	SCONJ
ejpam-4383	344	2	ψ	ψ	NOUN
ejpam-4383	344	3	is	be	AUX
ejpam-4383	344	4	a	a	DET
ejpam-4383	344	5	b	b	NOUN
ejpam-4383	344	6	-homomorphism	-homomorphism	NOUN
ejpam-4383	344	7	,	,	PUNCT
ejpam-4383	344	8	we	we	PRON
ejpam-4383	344	9	have	have	VERB
ejpam-4383	344	10	ψ(fpi	ψ(fpi	PROPN
ejpam-4383	344	11	⊗	⊗	PROPN
ejpam-4383	344	12	fqi	fqi	NOUN
ejpam-4383	344	13	)	)	PUNCT
ejpam-4383	345	1	=	=	PUNCT
ejpam-4383	345	2	ψ(fpi)⊗	ψ(fpi)⊗	SYM
ejpam-4383	345	3	ψ(fqi	ψ(fqi	NUM
ejpam-4383	345	4	)	)	PUNCT
ejpam-4383	345	5	.	.	PUNCT
ejpam-4383	346	1	since	since	SCONJ
ejpam-4383	346	2	(	(	PUNCT
ejpam-4383	346	3	∀j	∀j	PROPN
ejpam-4383	346	4	∈	∈	PROPN
ejpam-4383	346	5	i	i	NOUN
ejpam-4383	346	6	)	)	PUNCT
ejpam-4383	346	7	(	(	PUNCT
ejpam-4383	346	8	(	(	PUNCT
ejpam-4383	346	9	fpi	fpi	PROPN
ejpam-4383	346	10	⊗	⊗	PROPN
ejpam-4383	346	11	fqi)(j	fqi)(j	PROPN
ejpam-4383	346	12	)	)	PUNCT
ejpam-4383	346	13	=	=	PRON
ejpam-4383	346	14	{	{	PUNCT
ejpam-4383	346	15	pi	pi	NOUN
ejpam-4383	346	16	∗i	∗i	PROPN
ejpam-4383	346	17	qi	qi	PROPN
ejpam-4383	346	18	if	if	SCONJ
ejpam-4383	346	19	j	j	PROPN
ejpam-4383	346	20	=	=	PRON
ejpam-4383	346	21	i	i	PRON
ejpam-4383	346	22	0j	0j	VERB
ejpam-4383	346	23	∗j	∗j	PROPN
ejpam-4383	346	24	0j	0j	NOUN
ejpam-4383	346	25	otherwise	otherwise	ADV
ejpam-4383	346	26	)	)	PUNCT
ejpam-4383	346	27	,	,	PUNCT
ejpam-4383	346	28	we	we	PRON
ejpam-4383	346	29	have	have	VERB
ejpam-4383	346	30	(	(	PUNCT
ejpam-4383	346	31	∀j	∀j	PROPN
ejpam-4383	346	32	∈	∈	PROPN
ejpam-4383	346	33	i	i	PROPN
ejpam-4383	346	34	)	)	PUNCT
ejpam-4383	346	35	(	(	PUNCT
ejpam-4383	346	36	ψ(fpi	ψ(fpi	PROPN
ejpam-4383	346	37	⊗	⊗	PROPN
ejpam-4383	346	38	fqi)(j	fqi)(j	PROPN
ejpam-4383	346	39	)	)	PUNCT
ejpam-4383	346	40	=	=	PRON
ejpam-4383	346	41	{	{	PUNCT
ejpam-4383	346	42	ψi(pi	ψi(pi	NOUN
ejpam-4383	346	43	∗i	∗i	PROPN
ejpam-4383	346	44	qi	qi	PROPN
ejpam-4383	346	45	)	)	PUNCT
ejpam-4383	346	46	if	if	SCONJ
ejpam-4383	346	47	j	j	PROPN
ejpam-4383	347	1	=	=	PRON
ejpam-4383	347	2	i	i	PRON
ejpam-4383	347	3	ψj(0j	ψj(0j	VERB
ejpam-4383	347	4	∗j	∗j	PROPN
ejpam-4383	347	5	0j	0j	NOUN
ejpam-4383	347	6	)	)	PUNCT
ejpam-4383	347	7	otherwise	otherwise	ADV
ejpam-4383	347	8	)	)	PUNCT
ejpam-4383	347	9	.	.	PUNCT
ejpam-4383	348	1	(	(	PUNCT
ejpam-4383	348	2	2.6	2.6	NUM
ejpam-4383	348	3	)	)	PUNCT
ejpam-4383	348	4	since	since	SCONJ
ejpam-4383	348	5	(	(	PUNCT
ejpam-4383	348	6	∀j	∀j	PROPN
ejpam-4383	348	7	∈	∈	PROPN
ejpam-4383	348	8	i	i	NOUN
ejpam-4383	348	9	)	)	PUNCT
ejpam-4383	348	10	(	(	PUNCT
ejpam-4383	348	11	ψ(fpi)(j	ψ(fpi)(j	NOUN
ejpam-4383	348	12	)	)	PUNCT
ejpam-4383	348	13	=	=	SYM
ejpam-4383	348	14	{	{	PUNCT
ejpam-4383	348	15	ψi(pi	ψi(pi	NOUN
ejpam-4383	348	16	)	)	PUNCT
ejpam-4383	348	17	if	if	SCONJ
ejpam-4383	348	18	j	j	PROPN
ejpam-4383	349	1	=	=	VERB
ejpam-4383	349	2	i	i	PRON
ejpam-4383	349	3	ψj(0j	ψj(0j	VERB
ejpam-4383	349	4	)	)	PUNCT
ejpam-4383	349	5	otherwise	otherwise	ADV
ejpam-4383	349	6	)	)	PUNCT
ejpam-4383	349	7	and	and	CCONJ
ejpam-4383	349	8	(	(	PUNCT
ejpam-4383	349	9	∀j	∀j	PROPN
ejpam-4383	349	10	∈	∈	PROPN
ejpam-4383	349	11	i	i	PROPN
ejpam-4383	349	12	)	)	PUNCT
ejpam-4383	349	13	(	(	PUNCT
ejpam-4383	349	14	ψ(fqi)(j	ψ(fqi)(j	PROPN
ejpam-4383	349	15	)	)	PUNCT
ejpam-4383	349	16	=	=	PRON
ejpam-4383	349	17	{	{	PUNCT
ejpam-4383	349	18	ψi(qi	ψi(qi	PROPN
ejpam-4383	349	19	)	)	PUNCT
ejpam-4383	349	20	if	if	SCONJ
ejpam-4383	349	21	j	j	PROPN
ejpam-4383	350	1	=	=	VERB
ejpam-4383	350	2	i	i	PRON
ejpam-4383	350	3	ψj(0j	ψj(0j	VERB
ejpam-4383	350	4	)	)	PUNCT
ejpam-4383	350	5	otherwise	otherwise	ADV
ejpam-4383	350	6	)	)	PUNCT
ejpam-4383	350	7	,	,	PUNCT
ejpam-4383	350	8	we	we	PRON
ejpam-4383	350	9	have	have	VERB
ejpam-4383	350	10	(	(	PUNCT
ejpam-4383	350	11	∀j	∀j	PROPN
ejpam-4383	350	12	∈	∈	PROPN
ejpam-4383	350	13	i	i	NOUN
ejpam-4383	350	14	)	)	PUNCT
ejpam-4383	350	15	(	(	PUNCT
ejpam-4383	350	16	(	(	PUNCT
ejpam-4383	350	17	ψ(fpi)⊗	ψ(fpi)⊗	SYM
ejpam-4383	350	18	ψ(fqi))(j	ψ(fqi))(j	ADJ
ejpam-4383	350	19	)	)	PUNCT
ejpam-4383	351	1	=	=	PRON
ejpam-4383	351	2	{	{	PUNCT
ejpam-4383	351	3	ψi(pi	ψi(pi	NOUN
ejpam-4383	351	4	)	)	PUNCT
ejpam-4383	351	5	◦	◦	NOUN
ejpam-4383	351	6	i	i	NOUN
ejpam-4383	351	7	ψi(qi	ψi(qi	PROPN
ejpam-4383	351	8	)	)	PUNCT
ejpam-4383	352	1	if	if	SCONJ
ejpam-4383	352	2	j	j	PROPN
ejpam-4383	352	3	=	=	VERB
ejpam-4383	352	4	i	i	PRON
ejpam-4383	352	5	ψj(0j	ψj(0j	VERB
ejpam-4383	352	6	)	)	PUNCT
ejpam-4383	352	7	◦	◦	NOUN
ejpam-4383	352	8	j	j	NOUN
ejpam-4383	352	9	ψj(0j	ψj(0j	NOUN
ejpam-4383	352	10	)	)	PUNCT
ejpam-4383	352	11	otherwise	otherwise	ADV
ejpam-4383	352	12	)	)	PUNCT
ejpam-4383	353	1	.	.	PUNCT
ejpam-4383	354	1	(	(	PUNCT
ejpam-4383	354	2	2.7	2.7	NUM
ejpam-4383	354	3	)	)	PUNCT
ejpam-4383	354	4	by	by	ADP
ejpam-4383	354	5	(	(	PUNCT
ejpam-4383	354	6	2.6	2.6	NUM
ejpam-4383	354	7	)	)	PUNCT
ejpam-4383	354	8	and	and	CCONJ
ejpam-4383	354	9	(	(	PUNCT
ejpam-4383	354	10	2.7	2.7	NUM
ejpam-4383	354	11	)	)	PUNCT
ejpam-4383	354	12	,	,	PUNCT
ejpam-4383	354	13	we	we	PRON
ejpam-4383	354	14	have	have	VERB
ejpam-4383	354	15	ψi(pi	ψi(pi	NOUN
ejpam-4383	354	16	∗i	∗i	PROPN
ejpam-4383	354	17	qi	qi	PROPN
ejpam-4383	354	18	)	)	PUNCT
ejpam-4383	354	19	=	=	SYM
ejpam-4383	354	20	ψi(pi	ψi(pi	NOUN
ejpam-4383	354	21	)	)	PUNCT
ejpam-4383	354	22	◦	◦	NOUN
ejpam-4383	354	23	i	i	NOUN
ejpam-4383	354	24	ψi(qi	ψi(qi	PROPN
ejpam-4383	354	25	)	)	PUNCT
ejpam-4383	354	26	.	.	PUNCT
ejpam-4383	355	1	hence	hence	ADV
ejpam-4383	355	2	,	,	PUNCT
ejpam-4383	355	3	ψi	ψi	ADV
ejpam-4383	355	4	is	be	AUX
ejpam-4383	355	5	a	a	DET
ejpam-4383	355	6	b	b	NOUN
ejpam-4383	355	7	-homomorphism	-homomorphism	NOUN
ejpam-4383	355	8	for	for	ADP
ejpam-4383	355	9	all	all	PRON
ejpam-4383	355	10	i	i	PRON
ejpam-4383	355	11	∈	∈	PROPN
ejpam-4383	355	12	i.	i.	PROPN
ejpam-4383	355	13	a.	a.	PROPN
ejpam-4383	355	14	iampan	iampan	PROPN
ejpam-4383	355	15	et	et	PROPN
ejpam-4383	355	16	al	al	PROPN
ejpam-4383	355	17	.	.	PUNCT
ejpam-4383	355	18	/	/	SYM
ejpam-4383	355	19	eur	eur	PROPN
ejpam-4383	355	20	.	.	PUNCT
ejpam-4383	356	1	j.	j.	PROPN
ejpam-4383	356	2	pure	pure	PROPN
ejpam-4383	356	3	appl	appl	PROPN
ejpam-4383	356	4	.	.	PROPN
ejpam-4383	356	5	math	math	PROPN
ejpam-4383	356	6	,	,	PUNCT
ejpam-4383	356	7	15	15	NUM
ejpam-4383	356	8	(	(	PUNCT
ejpam-4383	356	9	3	3	NUM
ejpam-4383	356	10	)	)	PUNCT
ejpam-4383	356	11	(	(	PUNCT
ejpam-4383	356	12	2022	2022	NUM
ejpam-4383	356	13	)	)	PUNCT
ejpam-4383	356	14	,	,	PUNCT
ejpam-4383	356	15	999	999	NUM
ejpam-4383	356	16	-	-	SYM
ejpam-4383	356	17	1014	1014	NUM
ejpam-4383	356	18	1011	1011	NUM
ejpam-4383	356	19	(	(	PUNCT
ejpam-4383	356	20	ii	ii	NOUN
ejpam-4383	356	21	)	)	PUNCT
ejpam-4383	356	22	it	it	PRON
ejpam-4383	356	23	is	be	AUX
ejpam-4383	356	24	straightforward	straightforward	ADJ
ejpam-4383	356	25	from	from	ADP
ejpam-4383	356	26	(	(	PUNCT
ejpam-4383	356	27	i	i	NOUN
ejpam-4383	356	28	)	)	PUNCT
ejpam-4383	356	29	and	and	CCONJ
ejpam-4383	356	30	theorem	theorem	VERB
ejpam-4383	356	31	10	10	NUM
ejpam-4383	356	32	(	(	PUNCT
ejpam-4383	356	33	i	i	NOUN
ejpam-4383	356	34	)	)	PUNCT
ejpam-4383	356	35	.	.	PUNCT
ejpam-4383	357	1	(	(	PUNCT
ejpam-4383	357	2	iii	iii	X
ejpam-4383	357	3	)	)	PUNCT
ejpam-4383	357	4	it	it	PRON
ejpam-4383	357	5	is	be	AUX
ejpam-4383	357	6	straightforward	straightforward	ADJ
ejpam-4383	357	7	from	from	ADP
ejpam-4383	357	8	(	(	PUNCT
ejpam-4383	357	9	i	i	NOUN
ejpam-4383	357	10	)	)	PUNCT
ejpam-4383	357	11	and	and	CCONJ
ejpam-4383	357	12	theorem	theorem	VERB
ejpam-4383	357	13	10	10	NUM
ejpam-4383	357	14	(	(	PUNCT
ejpam-4383	357	15	ii	ii	NOUN
ejpam-4383	357	16	)	)	PUNCT
ejpam-4383	357	17	.	.	PUNCT
ejpam-4383	358	1	(	(	PUNCT
ejpam-4383	358	2	iv	iv	X
ejpam-4383	358	3	)	)	PUNCT
ejpam-4383	358	4	it	it	PRON
ejpam-4383	358	5	is	be	AUX
ejpam-4383	358	6	straightforward	straightforward	ADJ
ejpam-4383	358	7	from	from	ADP
ejpam-4383	358	8	(	(	PUNCT
ejpam-4383	358	9	i	i	NOUN
ejpam-4383	358	10	)	)	PUNCT
ejpam-4383	358	11	and	and	CCONJ
ejpam-4383	358	12	theorem	theorem	VERB
ejpam-4383	358	13	10	10	NUM
ejpam-4383	358	14	(	(	PUNCT
ejpam-4383	358	15	iii	iii	NOUN
ejpam-4383	358	16	)	)	PUNCT
ejpam-4383	358	17	.	.	PUNCT
ejpam-4383	359	1	(	(	PUNCT
ejpam-4383	359	2	v	v	X
ejpam-4383	359	3	)	)	PUNCT
ejpam-4383	359	4	let	let	VERB
ejpam-4383	359	5	(	(	PUNCT
ejpam-4383	359	6	pi)i∈i	pi)i∈i	NUM
ejpam-4383	359	7	∈	∈	PROPN
ejpam-4383	359	8	∏	∏	PROPN
ejpam-4383	359	9	i∈i	i∈i	ADJ
ejpam-4383	359	10	pi	pi	NOUN
ejpam-4383	359	11	.	.	PUNCT
ejpam-4383	360	1	then	then	ADV
ejpam-4383	360	2	(	(	PUNCT
ejpam-4383	360	3	pi)i∈i	pi)i∈i	NUM
ejpam-4383	360	4	∈	∈	PROPN
ejpam-4383	360	5	kerψ	kerψ	NOUN
ejpam-4383	360	6	⇔	⇔	PROPN
ejpam-4383	360	7	ψ(pi)i∈i	ψ(pi)i∈i	X
ejpam-4383	360	8	=	=	SYM
ejpam-4383	360	9	(	(	PUNCT
ejpam-4383	360	10	1i)i∈i	1i)i∈i	X
ejpam-4383	360	11	⇔	⇔	X
ejpam-4383	360	12	(	(	PUNCT
ejpam-4383	360	13	ψi(pi))i∈i	ψi(pi))i∈i	PROPN
ejpam-4383	360	14	=	=	SYM
ejpam-4383	360	15	(	(	PUNCT
ejpam-4383	360	16	1i)i∈i	1i)i∈i	NUM
ejpam-4383	360	17	⇔	⇔	NUM
ejpam-4383	360	18	ψi(pi	ψi(pi	NOUN
ejpam-4383	360	19	)	)	PUNCT
ejpam-4383	360	20	=	=	NOUN
ejpam-4383	360	21	1i	1i	NOUN
ejpam-4383	360	22	∀i	∀i	X
ejpam-4383	360	23	∈	∈	PROPN
ejpam-4383	360	24	i	i	PRON
ejpam-4383	360	25	⇔	⇔	PROPN
ejpam-4383	360	26	pi	pi	PROPN
ejpam-4383	360	27	∈	∈	PROPN
ejpam-4383	360	28	kerψi	kerψi	NOUN
ejpam-4383	360	29	∀i	∀i	NOUN
ejpam-4383	360	30	∈	∈	PROPN
ejpam-4383	360	31	i	i	PRON
ejpam-4383	360	32	⇔	⇔	X
ejpam-4383	360	33	(	(	PUNCT
ejpam-4383	360	34	pi)i∈i	pi)i∈i	NUM
ejpam-4383	360	35	∈	∈	PROPN
ejpam-4383	360	36	∏	∏	PROPN
ejpam-4383	360	37	i∈i	i∈i	ADJ
ejpam-4383	360	38	kerψi	kerψi	NOUN
ejpam-4383	360	39	.	.	PUNCT
ejpam-4383	361	1	hence	hence	ADV
ejpam-4383	361	2	,	,	PUNCT
ejpam-4383	361	3	kerψ	kerψ	PROPN
ejpam-4383	361	4	=	=	SYM
ejpam-4383	361	5	∏	∏	PROPN
ejpam-4383	361	6	i∈i	i∈i	ADJ
ejpam-4383	361	7	kerψi	kerψi	NOUN
ejpam-4383	361	8	.	.	PUNCT
ejpam-4383	362	1	now	now	ADV
ejpam-4383	362	2	,	,	PUNCT
ejpam-4383	362	3	(	(	PUNCT
ejpam-4383	362	4	qi)i∈i	qi)i∈i	NUM
ejpam-4383	362	5	∈	∈	PROPN
ejpam-4383	362	6	ψ	ψ	X
ejpam-4383	362	7	(	(	PUNCT
ejpam-4383	362	8	∏	∏	PROPN
ejpam-4383	362	9	i∈i	i∈i	ADJ
ejpam-4383	362	10	pi	pi	NOUN
ejpam-4383	362	11	)	)	PUNCT
ejpam-4383	362	12	⇔	⇔	X
ejpam-4383	362	13	∃(pi)i∈i	∃(pi)i∈i	X
ejpam-4383	362	14	∈	∈	PROPN
ejpam-4383	362	15	∏	∏	PROPN
ejpam-4383	362	16	i∈i	i∈i	ADJ
ejpam-4383	362	17	pi	pi	PROPN
ejpam-4383	362	18	s.t	s.t	PROPN
ejpam-4383	362	19	.	.	PROPN
ejpam-4383	362	20	(	(	PUNCT
ejpam-4383	362	21	qi)i∈i	qi)i∈i	NUM
ejpam-4383	362	22	=	=	SYM
ejpam-4383	362	23	ψ(pi)i∈i	ψ(pi)i∈i	X
ejpam-4383	362	24	⇔	⇔	PROPN
ejpam-4383	362	25	∃(pi)i∈i	∃(pi)i∈i	X
ejpam-4383	362	26	∈	∈	PROPN
ejpam-4383	362	27	∏	∏	PROPN
ejpam-4383	362	28	i∈i	i∈i	ADJ
ejpam-4383	362	29	pi	pi	PROPN
ejpam-4383	362	30	s.t	s.t	PROPN
ejpam-4383	362	31	.	.	PROPN
ejpam-4383	363	1	(	(	PUNCT
ejpam-4383	363	2	qi)i∈i	qi)i∈i	NUM
ejpam-4383	363	3	=	=	SYM
ejpam-4383	363	4	(	(	PUNCT
ejpam-4383	363	5	ψi(pi))i∈i	ψi(pi))i∈i	ADP
ejpam-4383	363	6	⇔	⇔	PROPN
ejpam-4383	363	7	∃pi	∃pi	NOUN
ejpam-4383	363	8	∈	∈	PROPN
ejpam-4383	363	9	pi	pi	PROPN
ejpam-4383	363	10	s.t	s.t	PROPN
ejpam-4383	363	11	.	.	PROPN
ejpam-4383	363	12	qi	qi	PROPN
ejpam-4383	363	13	=	=	PUNCT
ejpam-4383	363	14	ψi(pi	ψi(pi	PROPN
ejpam-4383	363	15	)	)	PUNCT
ejpam-4383	363	16	∈	∈	PROPN
ejpam-4383	363	17	ψ(pi	ψ(pi	NOUN
ejpam-4383	363	18	)	)	PUNCT
ejpam-4383	363	19	∀i	∀i	NOUN
ejpam-4383	363	20	∈	∈	PROPN
ejpam-4383	363	21	i	i	PRON
ejpam-4383	363	22	⇔	⇔	X
ejpam-4383	363	23	(	(	PUNCT
ejpam-4383	363	24	qi)i∈i	qi)i∈i	NUM
ejpam-4383	363	25	∈	∈	PROPN
ejpam-4383	363	26	∏	∏	PROPN
ejpam-4383	363	27	i∈i	i∈i	ADJ
ejpam-4383	363	28	ψi(pi	ψi(pi	NOUN
ejpam-4383	363	29	)	)	PUNCT
ejpam-4383	363	30	.	.	PUNCT
ejpam-4383	364	1	hence	hence	ADV
ejpam-4383	364	2	,	,	PUNCT
ejpam-4383	364	3	ψ	ψ	X
ejpam-4383	364	4	(	(	PUNCT
ejpam-4383	364	5	∏	∏	PROPN
ejpam-4383	364	6	i∈i	i∈i	ADJ
ejpam-4383	364	7	pi	pi	NOUN
ejpam-4383	364	8	)	)	PUNCT
ejpam-4383	364	9	=	=	SYM
ejpam-4383	364	10	∏	∏	PROPN
ejpam-4383	364	11	i∈i	i∈i	ADJ
ejpam-4383	364	12	ψi(pi	ψi(pi	NOUN
ejpam-4383	364	13	)	)	PUNCT
ejpam-4383	364	14	.	.	PUNCT
ejpam-4383	365	1	finally	finally	ADV
ejpam-4383	365	2	,	,	PUNCT
ejpam-4383	365	3	we	we	PRON
ejpam-4383	365	4	discuss	discuss	VERB
ejpam-4383	365	5	several	several	ADJ
ejpam-4383	365	6	anti	anti	ADJ
ejpam-4383	365	7	-	-	ADJ
ejpam-4383	365	8	b	b	ADJ
ejpam-4383	365	9	-homomorphism	-homomorphism	NOUN
ejpam-4383	365	10	theorems	theorem	NOUN
ejpam-4383	365	11	in	in	ADP
ejpam-4383	365	12	view	view	NOUN
ejpam-4383	365	13	of	of	ADP
ejpam-4383	365	14	the	the	DET
ejpam-4383	365	15	external	external	ADJ
ejpam-4383	365	16	direct	direct	ADJ
ejpam-4383	365	17	product	product	NOUN
ejpam-4383	365	18	of	of	ADP
ejpam-4383	365	19	b	b	PROPN
ejpam-4383	365	20	-algebras	-algebras	PROPN
ejpam-4383	365	21	.	.	PUNCT
ejpam-4383	365	22	theorem	theorem	NOUN
ejpam-4383	365	23	12	12	NUM
ejpam-4383	365	24	.	.	PUNCT
ejpam-4383	366	1	let	let	AUX
ejpam-4383	366	2	pi	pi	NOUN
ejpam-4383	366	3	=	=	PUNCT
ejpam-4383	366	4	(	(	PUNCT
ejpam-4383	366	5	pi	pi	NOUN
ejpam-4383	366	6	;	;	PUNCT
ejpam-4383	366	7	∗i	∗i	PROPN
ejpam-4383	366	8	,	,	PUNCT
ejpam-4383	366	9	0i	0i	NOUN
ejpam-4383	366	10	)	)	PUNCT
ejpam-4383	366	11	and	and	CCONJ
ejpam-4383	366	12	qi	qi	PROPN
ejpam-4383	366	13	=	=	SYM
ejpam-4383	366	14	(	(	PUNCT
ejpam-4383	366	15	qi	qi	NOUN
ejpam-4383	366	16	;	;	PUNCT
ejpam-4383	366	17	◦	◦	NOUN
ejpam-4383	366	18	i	i	PROPN
ejpam-4383	366	19	,	,	PUNCT
ejpam-4383	366	20	1i	1i	NUM
ejpam-4383	366	21	)	)	PUNCT
ejpam-4383	366	22	be	be	VERB
ejpam-4383	366	23	b	b	NOUN
ejpam-4383	366	24	-	-	PUNCT
ejpam-4383	366	25	algebras	algebra	NOUN
ejpam-4383	366	26	and	and	CCONJ
ejpam-4383	366	27	ψi	ψi	ADP
ejpam-4383	366	28	:	:	PUNCT
ejpam-4383	366	29	pi	pi	NOUN
ejpam-4383	366	30	→	→	SYM
ejpam-4383	366	31	qi	qi	PROPN
ejpam-4383	366	32	be	be	AUX
ejpam-4383	366	33	a	a	DET
ejpam-4383	366	34	function	function	NOUN
ejpam-4383	366	35	for	for	ADP
ejpam-4383	366	36	all	all	PRON
ejpam-4383	366	37	i	i	PRON
ejpam-4383	366	38	∈	∈	PROPN
ejpam-4383	366	39	i.	i.	NOUN
ejpam-4383	366	40	then	then	ADV
ejpam-4383	366	41	(	(	PUNCT
ejpam-4383	366	42	i	i	NOUN
ejpam-4383	366	43	)	)	PUNCT
ejpam-4383	366	44	ψi	ψi	NOUN
ejpam-4383	366	45	is	be	AUX
ejpam-4383	366	46	an	an	DET
ejpam-4383	366	47	anti	anti	ADJ
ejpam-4383	366	48	-	-	ADJ
ejpam-4383	366	49	b	b	NOUN
ejpam-4383	366	50	-	-	PUNCT
ejpam-4383	366	51	homomorphism	homomorphism	NOUN
ejpam-4383	366	52	for	for	ADP
ejpam-4383	366	53	all	all	PRON
ejpam-4383	366	54	i	i	PRON
ejpam-4383	366	55	∈	∈	VERB
ejpam-4383	367	1	i	i	PRON
ejpam-4383	367	2	if	if	SCONJ
ejpam-4383	367	3	and	and	CCONJ
ejpam-4383	367	4	only	only	ADV
ejpam-4383	367	5	if	if	SCONJ
ejpam-4383	367	6	ψ	ψ	NOUN
ejpam-4383	367	7	is	be	AUX
ejpam-4383	367	8	an	an	DET
ejpam-4383	367	9	anti	anti	ADJ
ejpam-4383	367	10	-	-	ADJ
ejpam-4383	367	11	b	b	NOUN
ejpam-4383	367	12	-	-	PUNCT
ejpam-4383	367	13	homomorphism	homomorphism	NOUN
ejpam-4383	367	14	which	which	PRON
ejpam-4383	367	15	is	be	AUX
ejpam-4383	367	16	defined	define	VERB
ejpam-4383	367	17	in	in	ADP
ejpam-4383	367	18	definition	definition	NOUN
ejpam-4383	367	19	10	10	NUM
ejpam-4383	367	20	,	,	PUNCT
ejpam-4383	367	21	(	(	PUNCT
ejpam-4383	367	22	ii	ii	NOUN
ejpam-4383	367	23	)	)	PUNCT
ejpam-4383	367	24	ψi	ψi	NOUN
ejpam-4383	367	25	is	be	AUX
ejpam-4383	367	26	an	an	DET
ejpam-4383	367	27	anti	anti	ADJ
ejpam-4383	367	28	-	-	ADJ
ejpam-4383	367	29	b	b	NOUN
ejpam-4383	367	30	-	-	PUNCT
ejpam-4383	367	31	monomorphism	monomorphism	NOUN
ejpam-4383	367	32	for	for	ADP
ejpam-4383	367	33	all	all	PRON
ejpam-4383	367	34	i	i	PRON
ejpam-4383	367	35	∈	∈	VERB
ejpam-4383	368	1	i	i	PRON
ejpam-4383	368	2	if	if	SCONJ
ejpam-4383	368	3	and	and	CCONJ
ejpam-4383	368	4	only	only	ADV
ejpam-4383	368	5	if	if	SCONJ
ejpam-4383	368	6	ψ	ψ	NOUN
ejpam-4383	368	7	is	be	AUX
ejpam-4383	368	8	an	an	DET
ejpam-4383	368	9	anti	anti	ADJ
ejpam-4383	368	10	-	-	ADJ
ejpam-4383	368	11	b	b	NOUN
ejpam-4383	368	12	-	-	PUNCT
ejpam-4383	368	13	monomorphism	monomorphism	NOUN
ejpam-4383	368	14	,	,	PUNCT
ejpam-4383	368	15	(	(	PUNCT
ejpam-4383	368	16	iii	iii	NOUN
ejpam-4383	368	17	)	)	PUNCT
ejpam-4383	368	18	ψi	ψi	NOUN
ejpam-4383	368	19	is	be	AUX
ejpam-4383	368	20	an	an	DET
ejpam-4383	368	21	anti	anti	ADJ
ejpam-4383	368	22	-	-	ADJ
ejpam-4383	368	23	b	b	NOUN
ejpam-4383	368	24	-	-	PUNCT
ejpam-4383	368	25	epimorphism	epimorphism	NOUN
ejpam-4383	368	26	for	for	ADP
ejpam-4383	368	27	all	all	PRON
ejpam-4383	368	28	i	i	PRON
ejpam-4383	368	29	∈	∈	VERB
ejpam-4383	369	1	i	i	PRON
ejpam-4383	369	2	if	if	SCONJ
ejpam-4383	369	3	and	and	CCONJ
ejpam-4383	369	4	only	only	ADV
ejpam-4383	369	5	if	if	SCONJ
ejpam-4383	369	6	ψ	ψ	NOUN
ejpam-4383	369	7	is	be	AUX
ejpam-4383	369	8	an	an	DET
ejpam-4383	369	9	anti	anti	ADJ
ejpam-4383	369	10	-	-	ADJ
ejpam-4383	369	11	b	b	NOUN
ejpam-4383	369	12	-	-	PUNCT
ejpam-4383	369	13	epimorphism	epimorphism	NOUN
ejpam-4383	369	14	,	,	PUNCT
ejpam-4383	369	15	(	(	PUNCT
ejpam-4383	369	16	iv	iv	X
ejpam-4383	369	17	)	)	PUNCT
ejpam-4383	369	18	ψi	ψi	NOUN
ejpam-4383	369	19	is	be	AUX
ejpam-4383	369	20	an	an	DET
ejpam-4383	369	21	anti	anti	ADJ
ejpam-4383	369	22	-	-	ADJ
ejpam-4383	369	23	b	b	NOUN
ejpam-4383	369	24	-	-	PUNCT
ejpam-4383	369	25	isomorphism	isomorphism	NOUN
ejpam-4383	369	26	for	for	ADP
ejpam-4383	369	27	all	all	PRON
ejpam-4383	369	28	i	i	PRON
ejpam-4383	369	29	∈	∈	VERB
ejpam-4383	370	1	i	i	PRON
ejpam-4383	370	2	if	if	SCONJ
ejpam-4383	370	3	and	and	CCONJ
ejpam-4383	370	4	only	only	ADV
ejpam-4383	370	5	if	if	SCONJ
ejpam-4383	370	6	ψ	ψ	NOUN
ejpam-4383	370	7	is	be	AUX
ejpam-4383	370	8	an	an	DET
ejpam-4383	370	9	anti	anti	ADJ
ejpam-4383	370	10	-	-	ADJ
ejpam-4383	370	11	b	b	NOUN
ejpam-4383	370	12	-	-	PUNCT
ejpam-4383	370	13	isomorphism	isomorphism	NOUN
ejpam-4383	370	14	.	.	PUNCT
ejpam-4383	371	1	proof	proof	NOUN
ejpam-4383	371	2	.	.	PUNCT
ejpam-4383	372	1	(	(	PUNCT
ejpam-4383	372	2	i	i	NOUN
ejpam-4383	372	3	)	)	PUNCT
ejpam-4383	372	4	assume	assume	VERB
ejpam-4383	372	5	that	that	SCONJ
ejpam-4383	372	6	ψi	ψi	NOUN
ejpam-4383	372	7	is	be	AUX
ejpam-4383	372	8	an	an	DET
ejpam-4383	372	9	anti	anti	ADJ
ejpam-4383	372	10	-	-	ADJ
ejpam-4383	372	11	b	b	ADJ
ejpam-4383	372	12	-homomorphism	-homomorphism	NOUN
ejpam-4383	372	13	for	for	ADP
ejpam-4383	372	14	all	all	DET
ejpam-4383	372	15	i	i	PRON
ejpam-4383	372	16	∈	∈	PROPN
ejpam-4383	372	17	i.	i.	NOUN
ejpam-4383	372	18	let	let	VERB
ejpam-4383	372	19	(	(	PUNCT
ejpam-4383	372	20	pi)i∈i	pi)i∈i	NUM
ejpam-4383	372	21	,	,	PUNCT
ejpam-4383	372	22	(	(	PUNCT
ejpam-4383	372	23	p	p	NOUN
ejpam-4383	372	24	′	′	NUM
ejpam-4383	372	25	i)i∈i	i)i∈i	PROPN
ejpam-4383	372	26	∈∏	∈∏	PROPN
ejpam-4383	372	27	i∈i	i∈i	ADJ
ejpam-4383	372	28	pi	pi	PROPN
ejpam-4383	372	29	.	.	PUNCT
ejpam-4383	373	1	then	then	ADV
ejpam-4383	373	2	ψ((pi)i∈i	ψ((pi)i∈i	PROPN
ejpam-4383	373	3	⊗	⊗	PROPN
ejpam-4383	373	4	(	(	PUNCT
ejpam-4383	373	5	p′i)i∈i	p′i)i∈i	PROPN
ejpam-4383	373	6	)	)	PUNCT
ejpam-4383	373	7	=	=	NOUN
ejpam-4383	373	8	ψ(pi	ψ(pi	X
ejpam-4383	373	9	∗i	∗i	PROPN
ejpam-4383	373	10	p′i)i∈i	p′i)i∈i	NOUN
ejpam-4383	373	11	=	=	SYM
ejpam-4383	373	12	(	(	PUNCT
ejpam-4383	373	13	ψi(pi	ψi(pi	NOUN
ejpam-4383	373	14	∗i	∗i	ADJ
ejpam-4383	373	15	p′i))i∈i	p′i))i∈i	NOUN
ejpam-4383	373	16	=	=	SYM
ejpam-4383	373	17	(	(	PUNCT
ejpam-4383	373	18	ψi(p	ψi(p	NOUN
ejpam-4383	373	19	′	′	NUM
ejpam-4383	374	1	i	i	NOUN
ejpam-4383	374	2	)	)	PUNCT
ejpam-4383	375	1	∗i	∗i	PROPN
ejpam-4383	375	2	ψi(pi))i∈i	ψi(pi))i∈i	ADP
ejpam-4383	375	3	a.	a.	NOUN
ejpam-4383	375	4	iampan	iampan	PROPN
ejpam-4383	375	5	et	et	PROPN
ejpam-4383	375	6	al	al	PROPN
ejpam-4383	375	7	.	.	PUNCT
ejpam-4383	375	8	/	/	SYM
ejpam-4383	375	9	eur	eur	PROPN
ejpam-4383	375	10	.	.	PUNCT
ejpam-4383	376	1	j.	j.	PROPN
ejpam-4383	376	2	pure	pure	PROPN
ejpam-4383	376	3	appl	appl	PROPN
ejpam-4383	376	4	.	.	PROPN
ejpam-4383	376	5	math	math	PROPN
ejpam-4383	376	6	,	,	PUNCT
ejpam-4383	376	7	15	15	NUM
ejpam-4383	376	8	(	(	PUNCT
ejpam-4383	376	9	3	3	NUM
ejpam-4383	376	10	)	)	PUNCT
ejpam-4383	376	11	(	(	PUNCT
ejpam-4383	376	12	2022	2022	NUM
ejpam-4383	376	13	)	)	PUNCT
ejpam-4383	376	14	,	,	PUNCT
ejpam-4383	376	15	999	999	NUM
ejpam-4383	376	16	-	-	SYM
ejpam-4383	376	17	1014	1014	NUM
ejpam-4383	376	18	1012	1012	NUM
ejpam-4383	376	19	=	=	SYM
ejpam-4383	376	20	(	(	PUNCT
ejpam-4383	376	21	ψi(p	ψi(p	NOUN
ejpam-4383	376	22	′	′	NUM
ejpam-4383	376	23	i))i∈i	i))i∈i	PROPN
ejpam-4383	376	24	⊗	⊗	PROPN
ejpam-4383	376	25	(	(	PUNCT
ejpam-4383	376	26	ψi(pi))i∈i	ψi(pi))i∈i	PROPN
ejpam-4383	376	27	=	=	PROPN
ejpam-4383	376	28	ψ(p′i)i∈i	ψ(p′i)i∈i	PRON
ejpam-4383	376	29	⊗	⊗	PROPN
ejpam-4383	376	30	ψ(pi)i∈i	ψ(pi)i∈i	NOUN
ejpam-4383	376	31	.	.	PUNCT
ejpam-4383	377	1	hence	hence	ADV
ejpam-4383	377	2	,	,	PUNCT
ejpam-4383	377	3	ψ	ψ	X
ejpam-4383	377	4	is	be	AUX
ejpam-4383	377	5	an	an	DET
ejpam-4383	377	6	anti	anti	ADJ
ejpam-4383	377	7	-	-	ADJ
ejpam-4383	377	8	b	b	ADJ
ejpam-4383	377	9	-homomorphism	-homomorphism	NOUN
ejpam-4383	377	10	.	.	PUNCT
ejpam-4383	378	1	conversely	conversely	ADV
ejpam-4383	378	2	,	,	PUNCT
ejpam-4383	378	3	assume	assume	VERB
ejpam-4383	378	4	that	that	SCONJ
ejpam-4383	378	5	ψ	ψ	NOUN
ejpam-4383	378	6	is	be	AUX
ejpam-4383	378	7	an	an	DET
ejpam-4383	378	8	anti	anti	ADJ
ejpam-4383	378	9	-	-	ADJ
ejpam-4383	378	10	b	b	ADJ
ejpam-4383	378	11	-homomorphism	-homomorphism	NOUN
ejpam-4383	378	12	.	.	PUNCT
ejpam-4383	379	1	let	let	VERB
ejpam-4383	379	2	i	i	PRON
ejpam-4383	379	3	∈	∈	PROPN
ejpam-4383	379	4	i.	i.	NOUN
ejpam-4383	379	5	let	let	VERB
ejpam-4383	379	6	pi	pi	NOUN
ejpam-4383	379	7	,	,	PUNCT
ejpam-4383	379	8	qi	qi	PROPN
ejpam-4383	379	9	∈	∈	PROPN
ejpam-4383	379	10	pi	pi	NOUN
ejpam-4383	379	11	.	.	PUNCT
ejpam-4383	380	1	then	then	ADV
ejpam-4383	380	2	fpi	fpi	PROPN
ejpam-4383	380	3	,	,	PUNCT
ejpam-4383	380	4	fqi	fqi	VERB
ejpam-4383	380	5	∈	∈	PROPN
ejpam-4383	380	6	∏	∏	PROPN
ejpam-4383	380	7	i∈i	i∈i	ADJ
ejpam-4383	380	8	pi	pi	NOUN
ejpam-4383	380	9	,	,	PUNCT
ejpam-4383	380	10	which	which	PRON
ejpam-4383	380	11	is	be	AUX
ejpam-4383	380	12	defined	define	VERB
ejpam-4383	380	13	by	by	ADP
ejpam-4383	380	14	(	(	PUNCT
ejpam-4383	380	15	2.2	2.2	NUM
ejpam-4383	380	16	)	)	PUNCT
ejpam-4383	380	17	.	.	PUNCT
ejpam-4383	381	1	since	since	SCONJ
ejpam-4383	381	2	ψ	ψ	NOUN
ejpam-4383	381	3	is	be	AUX
ejpam-4383	381	4	an	an	DET
ejpam-4383	381	5	anti	anti	ADJ
ejpam-4383	381	6	-	-	ADJ
ejpam-4383	381	7	b	b	ADJ
ejpam-4383	381	8	-homomorphism	-homomorphism	NOUN
ejpam-4383	381	9	,	,	PUNCT
ejpam-4383	381	10	we	we	PRON
ejpam-4383	381	11	have	have	VERB
ejpam-4383	381	12	ψ(fpi	ψ(fpi	PROPN
ejpam-4383	381	13	⊗	⊗	PROPN
ejpam-4383	381	14	fqi	fqi	NOUN
ejpam-4383	381	15	)	)	PUNCT
ejpam-4383	381	16	=	=	PUNCT
ejpam-4383	382	1	ψ(fqi)⊗	ψ(fqi)⊗	ADP
ejpam-4383	382	2	ψ(fpi	ψ(fpi	PROPN
ejpam-4383	382	3	)	)	PUNCT
ejpam-4383	382	4	.	.	PUNCT
ejpam-4383	383	1	since	since	SCONJ
ejpam-4383	383	2	(	(	PUNCT
ejpam-4383	383	3	∀j	∀j	PROPN
ejpam-4383	383	4	∈	∈	PROPN
ejpam-4383	383	5	i	i	NOUN
ejpam-4383	383	6	)	)	PUNCT
ejpam-4383	383	7	(	(	PUNCT
ejpam-4383	383	8	(	(	PUNCT
ejpam-4383	383	9	fpi	fpi	PROPN
ejpam-4383	383	10	⊗	⊗	PROPN
ejpam-4383	383	11	fqi)(j	fqi)(j	PROPN
ejpam-4383	383	12	)	)	PUNCT
ejpam-4383	383	13	=	=	PRON
ejpam-4383	383	14	{	{	PUNCT
ejpam-4383	383	15	pi	pi	NOUN
ejpam-4383	383	16	∗i	∗i	PROPN
ejpam-4383	383	17	qi	qi	PROPN
ejpam-4383	383	18	if	if	SCONJ
ejpam-4383	383	19	j	j	PROPN
ejpam-4383	383	20	=	=	PRON
ejpam-4383	383	21	i	i	PRON
ejpam-4383	383	22	0j	0j	VERB
ejpam-4383	383	23	∗j	∗j	PROPN
ejpam-4383	383	24	0j	0j	NOUN
ejpam-4383	383	25	otherwise	otherwise	ADV
ejpam-4383	383	26	)	)	PUNCT
ejpam-4383	383	27	,	,	PUNCT
ejpam-4383	383	28	we	we	PRON
ejpam-4383	383	29	have	have	VERB
ejpam-4383	383	30	(	(	PUNCT
ejpam-4383	383	31	∀j	∀j	PROPN
ejpam-4383	383	32	∈	∈	PROPN
ejpam-4383	383	33	i	i	PROPN
ejpam-4383	383	34	)	)	PUNCT
ejpam-4383	383	35	(	(	PUNCT
ejpam-4383	383	36	ψ(fpi	ψ(fpi	PROPN
ejpam-4383	383	37	⊗	⊗	PROPN
ejpam-4383	383	38	fqi)(j	fqi)(j	PROPN
ejpam-4383	383	39	)	)	PUNCT
ejpam-4383	383	40	=	=	PRON
ejpam-4383	383	41	{	{	PUNCT
ejpam-4383	383	42	ψi(pi	ψi(pi	NOUN
ejpam-4383	383	43	∗i	∗i	PROPN
ejpam-4383	383	44	qi	qi	PROPN
ejpam-4383	383	45	)	)	PUNCT
ejpam-4383	383	46	if	if	SCONJ
ejpam-4383	383	47	j	j	PROPN
ejpam-4383	384	1	=	=	PRON
ejpam-4383	384	2	i	i	PRON
ejpam-4383	384	3	ψj(0j	ψj(0j	VERB
ejpam-4383	384	4	∗j	∗j	PROPN
ejpam-4383	384	5	0j	0j	NOUN
ejpam-4383	384	6	)	)	PUNCT
ejpam-4383	384	7	otherwise	otherwise	ADV
ejpam-4383	384	8	)	)	PUNCT
ejpam-4383	384	9	.	.	PUNCT
ejpam-4383	385	1	(	(	PUNCT
ejpam-4383	385	2	2.8	2.8	NUM
ejpam-4383	385	3	)	)	PUNCT
ejpam-4383	385	4	since	since	SCONJ
ejpam-4383	385	5	(	(	PUNCT
ejpam-4383	385	6	∀j	∀j	PROPN
ejpam-4383	385	7	∈	∈	PROPN
ejpam-4383	385	8	i	i	NOUN
ejpam-4383	385	9	)	)	PUNCT
ejpam-4383	385	10	(	(	PUNCT
ejpam-4383	385	11	ψ(fqi)(j	ψ(fqi)(j	PROPN
ejpam-4383	385	12	)	)	PUNCT
ejpam-4383	385	13	=	=	PRON
ejpam-4383	385	14	{	{	PUNCT
ejpam-4383	385	15	ψi(qi	ψi(qi	PROPN
ejpam-4383	385	16	)	)	PUNCT
ejpam-4383	385	17	if	if	SCONJ
ejpam-4383	385	18	j	j	PROPN
ejpam-4383	386	1	=	=	VERB
ejpam-4383	386	2	i	i	PRON
ejpam-4383	386	3	ψj(0j	ψj(0j	VERB
ejpam-4383	386	4	)	)	PUNCT
ejpam-4383	386	5	otherwise	otherwise	ADV
ejpam-4383	386	6	)	)	PUNCT
ejpam-4383	386	7	and	and	CCONJ
ejpam-4383	386	8	(	(	PUNCT
ejpam-4383	386	9	∀j	∀j	PROPN
ejpam-4383	386	10	∈	∈	PROPN
ejpam-4383	386	11	i	i	NOUN
ejpam-4383	386	12	)	)	PUNCT
ejpam-4383	386	13	(	(	PUNCT
ejpam-4383	386	14	ψ(fpi)(j	ψ(fpi)(j	NOUN
ejpam-4383	386	15	)	)	PUNCT
ejpam-4383	386	16	=	=	SYM
ejpam-4383	386	17	{	{	PUNCT
ejpam-4383	386	18	ψi(pi	ψi(pi	NOUN
ejpam-4383	386	19	)	)	PUNCT
ejpam-4383	386	20	if	if	SCONJ
ejpam-4383	386	21	j	j	PROPN
ejpam-4383	387	1	=	=	VERB
ejpam-4383	387	2	i	i	PRON
ejpam-4383	387	3	ψj(0j	ψj(0j	VERB
ejpam-4383	387	4	)	)	PUNCT
ejpam-4383	387	5	otherwise	otherwise	ADV
ejpam-4383	387	6	)	)	PUNCT
ejpam-4383	387	7	,	,	PUNCT
ejpam-4383	387	8	we	we	PRON
ejpam-4383	387	9	have	have	VERB
ejpam-4383	387	10	(	(	PUNCT
ejpam-4383	387	11	∀j	∀j	PROPN
ejpam-4383	387	12	∈	∈	PROPN
ejpam-4383	387	13	i	i	NOUN
ejpam-4383	387	14	)	)	PUNCT
ejpam-4383	387	15	(	(	PUNCT
ejpam-4383	387	16	(	(	PUNCT
ejpam-4383	387	17	ψ(fqi)⊗	ψ(fqi)⊗	ADP
ejpam-4383	387	18	ψ(fpi))(j	ψ(fpi))(j	NOUN
ejpam-4383	387	19	)	)	PUNCT
ejpam-4383	388	1	=	=	PRON
ejpam-4383	388	2	{	{	PUNCT
ejpam-4383	388	3	ψi(qi	ψi(qi	PROPN
ejpam-4383	388	4	)	)	PUNCT
ejpam-4383	388	5	◦	◦	NOUN
ejpam-4383	388	6	i	i	PRON
ejpam-4383	388	7	ψi(pi	ψi(pi	NOUN
ejpam-4383	388	8	)	)	PUNCT
ejpam-4383	388	9	if	if	SCONJ
ejpam-4383	388	10	j	j	PROPN
ejpam-4383	389	1	=	=	VERB
ejpam-4383	389	2	i	i	PRON
ejpam-4383	389	3	ψj(0j	ψj(0j	VERB
ejpam-4383	389	4	)	)	PUNCT
ejpam-4383	389	5	◦	◦	NOUN
ejpam-4383	389	6	j	j	NOUN
ejpam-4383	389	7	ψj(0j	ψj(0j	NOUN
ejpam-4383	389	8	)	)	PUNCT
ejpam-4383	389	9	otherwise	otherwise	ADV
ejpam-4383	389	10	)	)	PUNCT
ejpam-4383	389	11	.	.	PUNCT
ejpam-4383	390	1	(	(	PUNCT
ejpam-4383	390	2	2.9	2.9	NUM
ejpam-4383	390	3	)	)	PUNCT
ejpam-4383	390	4	by	by	ADP
ejpam-4383	390	5	(	(	PUNCT
ejpam-4383	390	6	2.8	2.8	NUM
ejpam-4383	390	7	)	)	PUNCT
ejpam-4383	390	8	and	and	CCONJ
ejpam-4383	390	9	(	(	PUNCT
ejpam-4383	390	10	2.9	2.9	NUM
ejpam-4383	390	11	)	)	PUNCT
ejpam-4383	390	12	,	,	PUNCT
ejpam-4383	390	13	we	we	PRON
ejpam-4383	390	14	have	have	VERB
ejpam-4383	390	15	ψi(pi	ψi(pi	NOUN
ejpam-4383	390	16	∗i	∗i	PROPN
ejpam-4383	390	17	qi	qi	PROPN
ejpam-4383	390	18	)	)	PUNCT
ejpam-4383	390	19	=	=	SYM
ejpam-4383	390	20	ψi(qi	ψi(qi	PROPN
ejpam-4383	390	21	)	)	PUNCT
ejpam-4383	390	22	◦	◦	NOUN
ejpam-4383	390	23	i	i	PRON
ejpam-4383	390	24	ψi(pi	ψi(pi	NOUN
ejpam-4383	390	25	)	)	PUNCT
ejpam-4383	390	26	.	.	PUNCT
ejpam-4383	391	1	hence	hence	ADV
ejpam-4383	391	2	,	,	PUNCT
ejpam-4383	391	3	ψi	ψi	ADV
ejpam-4383	391	4	is	be	AUX
ejpam-4383	391	5	an	an	DET
ejpam-4383	391	6	anti	anti	ADJ
ejpam-4383	391	7	-	-	ADJ
ejpam-4383	391	8	b	b	ADJ
ejpam-4383	391	9	homomorphism	homomorphism	NOUN
ejpam-4383	391	10	for	for	ADP
ejpam-4383	391	11	all	all	PRON
ejpam-4383	391	12	i	i	PRON
ejpam-4383	391	13	∈	∈	PROPN
ejpam-4383	391	14	i.	i.	NOUN
ejpam-4383	391	15	(	(	PUNCT
ejpam-4383	391	16	ii	ii	PROPN
ejpam-4383	391	17	)	)	PUNCT
ejpam-4383	391	18	it	it	PRON
ejpam-4383	391	19	is	be	AUX
ejpam-4383	391	20	straightforward	straightforward	ADJ
ejpam-4383	391	21	from	from	ADP
ejpam-4383	391	22	(	(	PUNCT
ejpam-4383	391	23	i	i	NOUN
ejpam-4383	391	24	)	)	PUNCT
ejpam-4383	391	25	and	and	CCONJ
ejpam-4383	391	26	theorem	theorem	VERB
ejpam-4383	391	27	10	10	NUM
ejpam-4383	391	28	(	(	PUNCT
ejpam-4383	391	29	i	i	NOUN
ejpam-4383	391	30	)	)	PUNCT
ejpam-4383	391	31	.	.	PUNCT
ejpam-4383	392	1	(	(	PUNCT
ejpam-4383	392	2	iii	iii	X
ejpam-4383	392	3	)	)	PUNCT
ejpam-4383	392	4	it	it	PRON
ejpam-4383	392	5	is	be	AUX
ejpam-4383	392	6	straightforward	straightforward	ADJ
ejpam-4383	392	7	from	from	ADP
ejpam-4383	392	8	(	(	PUNCT
ejpam-4383	392	9	i	i	NOUN
ejpam-4383	392	10	)	)	PUNCT
ejpam-4383	392	11	and	and	CCONJ
ejpam-4383	392	12	theorem	theorem	VERB
ejpam-4383	392	13	10	10	NUM
ejpam-4383	392	14	(	(	PUNCT
ejpam-4383	392	15	ii	ii	NOUN
ejpam-4383	392	16	)	)	PUNCT
ejpam-4383	392	17	.	.	PUNCT
ejpam-4383	393	1	(	(	PUNCT
ejpam-4383	393	2	iv	iv	X
ejpam-4383	393	3	)	)	PUNCT
ejpam-4383	393	4	it	it	PRON
ejpam-4383	393	5	is	be	AUX
ejpam-4383	393	6	straightforward	straightforward	ADJ
ejpam-4383	393	7	from	from	ADP
ejpam-4383	393	8	(	(	PUNCT
ejpam-4383	393	9	i	i	NOUN
ejpam-4383	393	10	)	)	PUNCT
ejpam-4383	393	11	and	and	CCONJ
ejpam-4383	393	12	theorem	theorem	VERB
ejpam-4383	393	13	10	10	NUM
ejpam-4383	393	14	(	(	PUNCT
ejpam-4383	393	15	iii	iii	NOUN
ejpam-4383	393	16	)	)	PUNCT
ejpam-4383	393	17	.	.	PUNCT
ejpam-4383	394	1	3	3	X
ejpam-4383	394	2	.	.	X
ejpam-4383	394	3	conclusions	conclusion	NOUN
ejpam-4383	394	4	and	and	CCONJ
ejpam-4383	394	5	future	future	ADJ
ejpam-4383	394	6	work	work	NOUN
ejpam-4383	394	7	in	in	ADP
ejpam-4383	394	8	this	this	DET
ejpam-4383	394	9	paper	paper	NOUN
ejpam-4383	394	10	,	,	PUNCT
ejpam-4383	394	11	we	we	PRON
ejpam-4383	394	12	have	have	AUX
ejpam-4383	394	13	introduced	introduce	VERB
ejpam-4383	394	14	the	the	DET
ejpam-4383	394	15	concept	concept	NOUN
ejpam-4383	394	16	of	of	ADP
ejpam-4383	394	17	the	the	DET
ejpam-4383	394	18	direct	direct	ADJ
ejpam-4383	394	19	product	product	NOUN
ejpam-4383	394	20	of	of	ADP
ejpam-4383	394	21	infinite	infinite	ADJ
ejpam-4383	394	22	family	family	NOUN
ejpam-4383	394	23	of	of	ADP
ejpam-4383	394	24	b	b	PROPN
ejpam-4383	394	25	-algebras	-algebras	PROPN
ejpam-4383	394	26	,	,	PUNCT
ejpam-4383	394	27	we	we	PRON
ejpam-4383	394	28	call	call	VERB
ejpam-4383	394	29	the	the	DET
ejpam-4383	394	30	external	external	ADJ
ejpam-4383	394	31	direct	direct	ADJ
ejpam-4383	394	32	product	product	NOUN
ejpam-4383	394	33	,	,	PUNCT
ejpam-4383	394	34	which	which	PRON
ejpam-4383	394	35	is	be	AUX
ejpam-4383	394	36	a	a	DET
ejpam-4383	394	37	generalization	generalization	NOUN
ejpam-4383	394	38	of	of	ADP
ejpam-4383	394	39	the	the	DET
ejpam-4383	394	40	direct	direct	ADJ
ejpam-4383	394	41	product	product	NOUN
ejpam-4383	394	42	in	in	ADP
ejpam-4383	394	43	the	the	DET
ejpam-4383	394	44	sense	sense	NOUN
ejpam-4383	394	45	of	of	ADP
ejpam-4383	394	46	lingcong	lingcong	NOUN
ejpam-4383	394	47	and	and	CCONJ
ejpam-4383	394	48	endam	endam	NOUN
ejpam-4383	395	1	[	[	X
ejpam-4383	395	2	12	12	NUM
ejpam-4383	395	3	]	]	PUNCT
ejpam-4383	395	4	.	.	PUNCT
ejpam-4383	396	1	we	we	PRON
ejpam-4383	396	2	proved	prove	VERB
ejpam-4383	396	3	that	that	SCONJ
ejpam-4383	396	4	the	the	DET
ejpam-4383	396	5	external	external	ADJ
ejpam-4383	396	6	direct	direct	ADJ
ejpam-4383	396	7	product	product	NOUN
ejpam-4383	396	8	of	of	ADP
ejpam-4383	396	9	b	b	PROPN
ejpam-4383	396	10	-algebras	-algebras	PROPN
ejpam-4383	396	11	is	be	AUX
ejpam-4383	396	12	also	also	ADV
ejpam-4383	396	13	a	a	DET
ejpam-4383	396	14	b	b	NOUN
ejpam-4383	396	15	-algebra	-algebra	NOUN
ejpam-4383	396	16	.	.	PUNCT
ejpam-4383	397	1	also	also	ADV
ejpam-4383	397	2	,	,	PUNCT
ejpam-4383	397	3	we	we	PRON
ejpam-4383	397	4	have	have	AUX
ejpam-4383	397	5	introduced	introduce	VERB
ejpam-4383	397	6	the	the	DET
ejpam-4383	397	7	concept	concept	NOUN
ejpam-4383	397	8	of	of	ADP
ejpam-4383	397	9	the	the	DET
ejpam-4383	397	10	weak	weak	ADJ
ejpam-4383	397	11	direct	direct	ADJ
ejpam-4383	397	12	product	product	NOUN
ejpam-4383	397	13	of	of	ADP
ejpam-4383	397	14	b	b	NOUN
ejpam-4383	397	15	-algebras	-algebras	PROPN
ejpam-4383	397	16	.	.	PUNCT
ejpam-4383	398	1	we	we	PRON
ejpam-4383	398	2	proved	prove	VERB
ejpam-4383	398	3	that	that	SCONJ
ejpam-4383	398	4	the	the	DET
ejpam-4383	398	5	weak	weak	ADJ
ejpam-4383	398	6	direct	direct	ADJ
ejpam-4383	398	7	product	product	NOUN
ejpam-4383	398	8	of	of	ADP
ejpam-4383	398	9	b	b	NOUN
ejpam-4383	398	10	-algebras	-algebras	PROPN
ejpam-4383	398	11	references	reference	NOUN
ejpam-4383	398	12	1013	1013	NUM
ejpam-4383	398	13	is	be	AUX
ejpam-4383	398	14	a	a	DET
ejpam-4383	398	15	b	b	NOUN
ejpam-4383	398	16	-subalgebra	-subalgebra	NOUN
ejpam-4383	398	17	of	of	ADP
ejpam-4383	398	18	the	the	DET
ejpam-4383	398	19	external	external	ADJ
ejpam-4383	398	20	direct	direct	ADJ
ejpam-4383	398	21	product	product	NOUN
ejpam-4383	398	22	b	b	NOUN
ejpam-4383	398	23	-algebras	-algebra	NOUN
ejpam-4383	398	24	.	.	PUNCT
ejpam-4383	399	1	finally	finally	ADV
ejpam-4383	399	2	,	,	PUNCT
ejpam-4383	399	3	we	we	PRON
ejpam-4383	399	4	have	have	AUX
ejpam-4383	399	5	provided	provide	VERB
ejpam-4383	399	6	several	several	ADJ
ejpam-4383	399	7	fundamental	fundamental	ADJ
ejpam-4383	399	8	theorems	theorem	NOUN
ejpam-4383	399	9	of	of	ADP
ejpam-4383	399	10	(	(	PUNCT
ejpam-4383	399	11	anti-)b	anti-)b	INTJ
ejpam-4383	399	12	-homomorphisms	-homomorphism	NOUN
ejpam-4383	399	13	in	in	ADP
ejpam-4383	399	14	view	view	NOUN
ejpam-4383	399	15	of	of	ADP
ejpam-4383	399	16	the	the	DET
ejpam-4383	399	17	external	external	ADJ
ejpam-4383	399	18	direct	direct	ADJ
ejpam-4383	399	19	product	product	NOUN
ejpam-4383	399	20	b	b	PROPN
ejpam-4383	399	21	-algebras	-algebras	PROPN
ejpam-4383	399	22	.	.	PUNCT
ejpam-4383	400	1	based	base	VERB
ejpam-4383	400	2	on	on	ADP
ejpam-4383	400	3	the	the	DET
ejpam-4383	400	4	concept	concept	NOUN
ejpam-4383	400	5	of	of	ADP
ejpam-4383	400	6	the	the	DET
ejpam-4383	400	7	external	external	ADJ
ejpam-4383	400	8	direct	direct	ADJ
ejpam-4383	400	9	product	product	NOUN
ejpam-4383	400	10	of	of	ADP
ejpam-4383	400	11	b	b	NOUN
ejpam-4383	400	12	-algebras	-algebra	NOUN
ejpam-4383	400	13	in	in	ADP
ejpam-4383	400	14	this	this	DET
ejpam-4383	400	15	article	article	NOUN
ejpam-4383	400	16	,	,	PUNCT
ejpam-4383	400	17	we	we	PRON
ejpam-4383	400	18	can	can	AUX
ejpam-4383	400	19	apply	apply	VERB
ejpam-4383	400	20	it	it	PRON
ejpam-4383	400	21	to	to	ADP
ejpam-4383	400	22	the	the	DET
ejpam-4383	400	23	study	study	NOUN
ejpam-4383	400	24	of	of	ADP
ejpam-4383	400	25	the	the	DET
ejpam-4383	400	26	external	external	ADJ
ejpam-4383	400	27	direct	direct	ADJ
ejpam-4383	400	28	product	product	NOUN
ejpam-4383	400	29	in	in	ADP
ejpam-4383	400	30	other	other	ADJ
ejpam-4383	400	31	algebraic	algebraic	ADJ
ejpam-4383	400	32	systems	system	NOUN
ejpam-4383	400	33	.	.	PUNCT
ejpam-4383	401	1	researching	research	VERB
ejpam-4383	401	2	the	the	DET
ejpam-4383	401	3	external	external	ADJ
ejpam-4383	401	4	and	and	CCONJ
ejpam-4383	401	5	weak	weak	ADJ
ejpam-4383	401	6	direct	direct	ADJ
ejpam-4383	401	7	products	product	NOUN
ejpam-4383	401	8	that	that	PRON
ejpam-4383	401	9	we	we	PRON
ejpam-4383	401	10	will	will	AUX
ejpam-4383	401	11	study	study	VERB
ejpam-4383	401	12	in	in	ADP
ejpam-4383	401	13	the	the	DET
ejpam-4383	401	14	future	future	NOUN
ejpam-4383	401	15	will	will	AUX
ejpam-4383	401	16	be	be	AUX
ejpam-4383	401	17	up	up	ADV
ejpam-4383	401	18	-	-	PUNCT
ejpam-4383	401	19	algebras	algebras	X
ejpam-4383	401	20	.	.	PUNCT
ejpam-4383	402	1	the	the	DET
ejpam-4383	402	2	research	research	NOUN
ejpam-4383	402	3	topics	topic	NOUN
ejpam-4383	402	4	of	of	ADP
ejpam-4383	402	5	interest	interest	NOUN
ejpam-4383	402	6	by	by	ADP
ejpam-4383	402	7	our	our	PRON
ejpam-4383	402	8	research	research	NOUN
ejpam-4383	402	9	team	team	NOUN
ejpam-4383	402	10	being	be	AUX
ejpam-4383	402	11	studied	study	VERB
ejpam-4383	402	12	in	in	ADP
ejpam-4383	402	13	the	the	DET
ejpam-4383	402	14	external	external	ADJ
ejpam-4383	402	15	direct	direct	ADJ
ejpam-4383	402	16	product	product	NOUN
ejpam-4383	402	17	of	of	ADP
ejpam-4383	402	18	b	b	NOUN
ejpam-4383	402	19	-algebras	-algebra	NOUN
ejpam-4383	402	20	are	be	AUX
ejpam-4383	402	21	as	as	SCONJ
ejpam-4383	402	22	follows	follow	VERB
ejpam-4383	402	23	:	:	PUNCT
ejpam-4383	402	24	(	(	PUNCT
ejpam-4383	402	25	1	1	X
ejpam-4383	402	26	)	)	PUNCT
ejpam-4383	402	27	to	to	PART
ejpam-4383	402	28	study	study	VERB
ejpam-4383	402	29	fuzzy	fuzzy	ADJ
ejpam-4383	402	30	set	set	NOUN
ejpam-4383	402	31	theory	theory	NOUN
ejpam-4383	402	32	(	(	PUNCT
ejpam-4383	402	33	with	with	ADP
ejpam-4383	402	34	respect	respect	NOUN
ejpam-4383	402	35	to	to	ADP
ejpam-4383	402	36	a	a	DET
ejpam-4383	402	37	triangular	triangular	NOUN
ejpam-4383	402	38	norm	norm	NOUN
ejpam-4383	402	39	)	)	PUNCT
ejpam-4383	402	40	based	base	VERB
ejpam-4383	402	41	on	on	ADP
ejpam-4383	402	42	the	the	DET
ejpam-4383	402	43	concept	concept	NOUN
ejpam-4383	402	44	of	of	ADP
ejpam-4383	402	45	somjanta	somjanta	NOUN
ejpam-4383	402	46	et	et	PROPN
ejpam-4383	402	47	al	al	PROPN
ejpam-4383	402	48	.	.	PUNCT
ejpam-4383	403	1	[	[	X
ejpam-4383	403	2	20	20	NUM
ejpam-4383	403	3	]	]	PUNCT
ejpam-4383	403	4	and	and	CCONJ
ejpam-4383	403	5	thongarsa	thongarsa	PROPN
ejpam-4383	403	6	et	et	PROPN
ejpam-4383	403	7	al	al	PROPN
ejpam-4383	403	8	.	.	PUNCT
ejpam-4383	404	1	[	[	X
ejpam-4383	404	2	4	4	NUM
ejpam-4383	404	3	,	,	PUNCT
ejpam-4383	404	4	21	21	NUM
ejpam-4383	404	5	]	]	PUNCT
ejpam-4383	404	6	,	,	PUNCT
ejpam-4383	404	7	(	(	PUNCT
ejpam-4383	404	8	2	2	X
ejpam-4383	404	9	)	)	PUNCT
ejpam-4383	404	10	to	to	PART
ejpam-4383	404	11	study	study	VERB
ejpam-4383	404	12	bipolar	bipolar	ADJ
ejpam-4383	404	13	fuzzy	fuzzy	ADJ
ejpam-4383	404	14	set	set	NOUN
ejpam-4383	404	15	theory	theory	NOUN
ejpam-4383	404	16	based	base	VERB
ejpam-4383	404	17	on	on	ADP
ejpam-4383	404	18	the	the	DET
ejpam-4383	404	19	concept	concept	NOUN
ejpam-4383	404	20	of	of	ADP
ejpam-4383	404	21	muhiuddin	muhiuddin	VERB
ejpam-4383	404	22	[	[	X
ejpam-4383	404	23	14	14	NUM
ejpam-4383	404	24	]	]	PUNCT
ejpam-4383	404	25	,	,	PUNCT
ejpam-4383	404	26	(	(	PUNCT
ejpam-4383	404	27	3	3	X
ejpam-4383	404	28	)	)	PUNCT
ejpam-4383	404	29	to	to	PART
ejpam-4383	404	30	study	study	VERB
ejpam-4383	404	31	interval	interval	NOUN
ejpam-4383	404	32	-	-	PUNCT
ejpam-4383	404	33	valued	value	VERB
ejpam-4383	404	34	fuzzy	fuzzy	ADJ
ejpam-4383	404	35	set	set	NOUN
ejpam-4383	404	36	theory	theory	NOUN
ejpam-4383	404	37	based	base	VERB
ejpam-4383	404	38	on	on	ADP
ejpam-4383	404	39	the	the	DET
ejpam-4383	404	40	concept	concept	NOUN
ejpam-4383	404	41	of	of	ADP
ejpam-4383	404	42	muhiuddin	muhiuddin	PROPN
ejpam-4383	404	43	et	et	PROPN
ejpam-4383	404	44	al	al	PROPN
ejpam-4383	404	45	.	.	PUNCT
ejpam-4383	405	1	[	[	X
ejpam-4383	405	2	15	15	NUM
ejpam-4383	405	3	]	]	PUNCT
ejpam-4383	405	4	,	,	PUNCT
ejpam-4383	405	5	(	(	PUNCT
ejpam-4383	405	6	4	4	NUM
ejpam-4383	405	7	)	)	PUNCT
ejpam-4383	405	8	to	to	PART
ejpam-4383	405	9	study	study	VERB
ejpam-4383	405	10	interval	interval	NOUN
ejpam-4383	405	11	-	-	PUNCT
ejpam-4383	405	12	valued	value	VERB
ejpam-4383	405	13	intuitionistic	intuitionistic	ADJ
ejpam-4383	405	14	fuzzy	fuzzy	ADJ
ejpam-4383	405	15	set	set	NOUN
ejpam-4383	405	16	theory	theory	NOUN
ejpam-4383	405	17	based	base	VERB
ejpam-4383	405	18	on	on	ADP
ejpam-4383	405	19	the	the	DET
ejpam-4383	405	20	concept	concept	NOUN
ejpam-4383	405	21	of	of	ADP
ejpam-4383	405	22	senapati	senapati	PROPN
ejpam-4383	405	23	et	et	PROPN
ejpam-4383	405	24	al	al	PROPN
ejpam-4383	405	25	.	.	PUNCT
ejpam-4383	406	1	[	[	X
ejpam-4383	406	2	18	18	NUM
ejpam-4383	406	3	]	]	PUNCT
ejpam-4383	406	4	.	.	PUNCT
ejpam-4383	407	1	acknowledgements	acknowledgement	NOUN
ejpam-4383	407	2	this	this	DET
ejpam-4383	407	3	work	work	NOUN
ejpam-4383	407	4	was	be	AUX
ejpam-4383	407	5	supported	support	VERB
ejpam-4383	407	6	by	by	ADP
ejpam-4383	407	7	the	the	DET
ejpam-4383	407	8	revenue	revenue	NOUN
ejpam-4383	407	9	budget	budget	NOUN
ejpam-4383	407	10	in	in	ADP
ejpam-4383	407	11	2022	2022	NUM
ejpam-4383	407	12	,	,	PUNCT
ejpam-4383	407	13	school	school	NOUN
ejpam-4383	407	14	of	of	ADP
ejpam-4383	407	15	science	science	NOUN
ejpam-4383	407	16	,	,	PUNCT
ejpam-4383	407	17	university	university	NOUN
ejpam-4383	407	18	of	of	ADP
ejpam-4383	407	19	phayao	phayao	NOUN
ejpam-4383	407	20	,	,	PUNCT
ejpam-4383	407	21	thailand	thailand	PROPN
ejpam-4383	407	22	.	.	PUNCT
ejpam-4383	408	1	references	reference	NOUN
ejpam-4383	408	2	[	[	X
ejpam-4383	408	3	1	1	NUM
ejpam-4383	408	4	]	]	PUNCT
ejpam-4383	408	5	g.	g.	PROPN
ejpam-4383	408	6	a.	a.	NOUN
ejpam-4383	408	7	abebe	abebe	PROPN
ejpam-4383	408	8	.	.	PUNCT
ejpam-4383	409	1	on	on	ADP
ejpam-4383	409	2	the	the	DET
ejpam-4383	409	3	theory	theory	NOUN
ejpam-4383	409	4	of	of	ADP
ejpam-4383	409	5	brk	brk	PROPN
ejpam-4383	409	6	-algebras	-algebras	PROPN
ejpam-4383	409	7	.	.	PUNCT
ejpam-4383	410	1	[	[	X
ejpam-4383	410	2	doctoral	doctoral	ADJ
ejpam-4383	410	3	dissertation	dissertation	NOUN
ejpam-4383	410	4	]	]	PUNCT
ejpam-4383	410	5	.	.	PUNCT
ejpam-4383	411	1	department	department	NOUN
ejpam-4383	411	2	of	of	ADP
ejpam-4383	411	3	mathematics	mathematics	PROPN
ejpam-4383	411	4	,	,	PUNCT
ejpam-4383	411	5	addis	addis	PROPN
ejpam-4383	411	6	ababa	ababa	PROPN
ejpam-4383	411	7	university	university	PROPN
ejpam-4383	411	8	,	,	PUNCT
ejpam-4383	411	9	2018	2018	NUM
ejpam-4383	411	10	.	.	PUNCT
ejpam-4383	412	1	[	[	X
ejpam-4383	412	2	2	2	X
ejpam-4383	412	3	]	]	PUNCT
ejpam-4383	412	4	d.	d.	PROPN
ejpam-4383	412	5	al	al	PROPN
ejpam-4383	412	6	-	-	PUNCT
ejpam-4383	412	7	kadi	kadi	PROPN
ejpam-4383	412	8	.	.	PUNCT
ejpam-4383	413	1	anti	anti	ADJ
ejpam-4383	413	2	fuzzy	fuzzy	ADJ
ejpam-4383	413	3	ideals	ideal	NOUN
ejpam-4383	413	4	of	of	ADP
ejpam-4383	413	5	b	b	NOUN
ejpam-4383	413	6	-algebra	-algebra	NOUN
ejpam-4383	413	7	.	.	PUNCT
ejpam-4383	414	1	int	int	NOUN
ejpam-4383	414	2	.	.	PUNCT
ejpam-4383	415	1	j.	j.	PROPN
ejpam-4383	415	2	pure	pure	PROPN
ejpam-4383	415	3	appl	appl	PROPN
ejpam-4383	415	4	.	.	PUNCT
ejpam-4383	415	5	math	math	PROPN
ejpam-4383	415	6	.	.	PUNCT
ejpam-4383	415	7	,	,	PUNCT
ejpam-4383	415	8	117(3):437–445	117(3):437–445	PROPN
ejpam-4383	415	9	,	,	PUNCT
ejpam-4383	415	10	2017	2017	NUM
ejpam-4383	415	11	.	.	PUNCT
ejpam-4383	416	1	[	[	X
ejpam-4383	416	2	3	3	X
ejpam-4383	416	3	]	]	PUNCT
ejpam-4383	416	4	k.	k.	PROPN
ejpam-4383	416	5	e.	e.	PROPN
ejpam-4383	416	6	belleza	belleza	PROPN
ejpam-4383	416	7	and	and	CCONJ
ejpam-4383	416	8	j.	j.	PROPN
ejpam-4383	416	9	p.	p.	PROPN
ejpam-4383	416	10	vilela	vilela	PROPN
ejpam-4383	416	11	.	.	PUNCT
ejpam-4383	417	1	on	on	ADP
ejpam-4383	417	2	b	b	NOUN
ejpam-4383	417	3	-ideals	-ideal	NOUN
ejpam-4383	417	4	in	in	ADP
ejpam-4383	417	5	a	a	DET
ejpam-4383	417	6	topological	topological	ADJ
ejpam-4383	417	7	b	b	PROPN
ejpam-4383	417	8	-algebra	-algebra	PROPN
ejpam-4383	417	9	and	and	CCONJ
ejpam-4383	417	10	the	the	DET
ejpam-4383	417	11	uniform	uniform	ADJ
ejpam-4383	417	12	b	b	PROPN
ejpam-4383	417	13	-topological	-topological	ADJ
ejpam-4383	417	14	space	space	NOUN
ejpam-4383	417	15	.	.	PUNCT
ejpam-4383	418	1	eur	eur	PROPN
ejpam-4383	418	2	.	.	PUNCT
ejpam-4383	419	1	j.	j.	PROPN
ejpam-4383	419	2	pure	pure	PROPN
ejpam-4383	419	3	appl	appl	PROPN
ejpam-4383	419	4	.	.	PUNCT
ejpam-4383	419	5	math	math	PROPN
ejpam-4383	419	6	.	.	PUNCT
ejpam-4383	419	7	,	,	PUNCT
ejpam-4383	419	8	13(4):830–839	13(4):830–839	NUM
ejpam-4383	419	9	,	,	PUNCT
ejpam-4383	419	10	2020	2020	NUM
ejpam-4383	419	11	.	.	PUNCT
ejpam-4383	420	1	[	[	X
ejpam-4383	420	2	4	4	NUM
ejpam-4383	420	3	]	]	PUNCT
ejpam-4383	420	4	p.	p.	NOUN
ejpam-4383	420	5	burandate	burandate	NOUN
ejpam-4383	420	6	,	,	PUNCT
ejpam-4383	420	7	s.	s.	PROPN
ejpam-4383	420	8	thongarsa	thongarsa	PROPN
ejpam-4383	420	9	,	,	PUNCT
ejpam-4383	420	10	and	and	CCONJ
ejpam-4383	420	11	a.	a.	NOUN
ejpam-4383	420	12	iampan	iampan	PROPN
ejpam-4383	420	13	.	.	PUNCT
ejpam-4383	421	1	fuzzy	fuzzy	ADJ
ejpam-4383	421	2	sets	set	NOUN
ejpam-4383	421	3	in	in	ADP
ejpam-4383	421	4	up	up	ADV
ejpam-4383	421	5	-	-	PUNCT
ejpam-4383	421	6	algebras	algebra	VERB
ejpam-4383	421	7	with	with	ADP
ejpam-4383	421	8	respect	respect	NOUN
ejpam-4383	421	9	to	to	ADP
ejpam-4383	421	10	a	a	DET
ejpam-4383	421	11	triangular	triangular	NOUN
ejpam-4383	421	12	norm	norm	NOUN
ejpam-4383	421	13	.	.	PUNCT
ejpam-4383	422	1	konuralp	konuralp	PROPN
ejpam-4383	422	2	j.	j.	PROPN
ejpam-4383	422	3	math	math	PROPN
ejpam-4383	422	4	.	.	PUNCT
ejpam-4383	422	5	,	,	PUNCT
ejpam-4383	422	6	7(2):410–432	7(2):410–432	NOUN
ejpam-4383	422	7	,	,	PUNCT
ejpam-4383	422	8	2019	2019	NUM
ejpam-4383	422	9	.	.	PUNCT
ejpam-4383	423	1	[	[	X
ejpam-4383	423	2	5	5	X
ejpam-4383	423	3	]	]	PUNCT
ejpam-4383	423	4	j.	j.	PROPN
ejpam-4383	423	5	c.	c.	PROPN
ejpam-4383	423	6	endam	endam	PROPN
ejpam-4383	423	7	and	and	CCONJ
ejpam-4383	423	8	r.	r.	PROPN
ejpam-4383	423	9	c.	c.	PROPN
ejpam-4383	423	10	teves	teves	PROPN
ejpam-4383	423	11	.	.	PUNCT
ejpam-4383	424	1	direct	direct	ADJ
ejpam-4383	424	2	product	product	NOUN
ejpam-4383	424	3	of	of	ADP
ejpam-4383	424	4	bf	bf	NOUN
ejpam-4383	424	5	-algebras	-algebras	PROPN
ejpam-4383	424	6	.	.	PUNCT
ejpam-4383	425	1	int	int	NOUN
ejpam-4383	425	2	.	.	PUNCT
ejpam-4383	426	1	j.	j.	PROPN
ejpam-4383	426	2	algebra	algebra	PROPN
ejpam-4383	426	3	,	,	PUNCT
ejpam-4383	426	4	10(3):125–132	10(3):125–132	PROPN
ejpam-4383	426	5	,	,	PUNCT
ejpam-4383	426	6	2016	2016	NUM
ejpam-4383	426	7	.	.	PUNCT
ejpam-4383	427	1	[	[	X
ejpam-4383	427	2	6	6	NUM
ejpam-4383	427	3	]	]	PUNCT
ejpam-4383	427	4	a.	a.	NOUN
ejpam-4383	427	5	gan	gan	PROPN
ejpam-4383	427	6	,	,	PUNCT
ejpam-4383	427	7	a.	a.	NOUN
ejpam-4383	427	8	muzammal	muzammal	PROPN
ejpam-4383	427	9	,	,	PUNCT
ejpam-4383	427	10	and	and	CCONJ
ejpam-4383	427	11	y.	y.	PROPN
ejpam-4383	427	12	yang	yang	PROPN
ejpam-4383	427	13	.	.	PUNCT
ejpam-4383	428	1	limits	limit	NOUN
ejpam-4383	428	2	of	of	ADP
ejpam-4383	428	3	quantum	quantum	NOUN
ejpam-4383	428	4	b	b	PROPN
ejpam-4383	428	5	-algebras	-algebras	PROPN
ejpam-4383	428	6	.	.	PUNCT
ejpam-4383	429	1	mathematics	mathematic	NOUN
ejpam-4383	429	2	,	,	PUNCT
ejpam-4383	429	3	9:3184	9:3184	NUM
ejpam-4383	429	4	,	,	PUNCT
ejpam-4383	429	5	2021	2021	NUM
ejpam-4383	429	6	.	.	PUNCT
ejpam-4383	430	1	[	[	X
ejpam-4383	430	2	7	7	NUM
ejpam-4383	430	3	]	]	X
ejpam-4383	430	4	n.	n.	PROPN
ejpam-4383	430	5	c.	c.	PROPN
ejpam-4383	430	6	gonzaga	gonzaga	PROPN
ejpam-4383	430	7	,	,	PUNCT
ejpam-4383	430	8	jr	jr	PROPN
ejpam-4383	430	9	.	.	PROPN
ejpam-4383	430	10	and	and	CCONJ
ejpam-4383	430	11	j.	j.	PROPN
ejpam-4383	430	12	p.	p.	PROPN
ejpam-4383	430	13	vilela	vilela	PROPN
ejpam-4383	430	14	.	.	PUNCT
ejpam-4383	431	1	fuzzy	fuzzy	ADJ
ejpam-4383	431	2	order	order	NOUN
ejpam-4383	431	3	relative	relative	ADJ
ejpam-4383	431	4	to	to	ADP
ejpam-4383	431	5	fuzzy	fuzzy	ADJ
ejpam-4383	431	6	b	b	NOUN
ejpam-4383	431	7	-algebras	-algebras	PROPN
ejpam-4383	431	8	.	.	PUNCT
ejpam-4383	432	1	ital	ital	PROPN
ejpam-4383	432	2	.	.	PUNCT
ejpam-4383	433	1	j.	j.	PROPN
ejpam-4383	433	2	pure	pure	PROPN
ejpam-4383	433	3	appl	appl	PROPN
ejpam-4383	433	4	.	.	PUNCT
ejpam-4383	433	5	math	math	PROPN
ejpam-4383	433	6	.	.	PUNCT
ejpam-4383	433	7	,	,	PUNCT
ejpam-4383	433	8	42:485—-493	42:485—-493	NUM
ejpam-4383	433	9	,	,	PUNCT
ejpam-4383	433	10	2019	2019	NUM
ejpam-4383	433	11	.	.	PUNCT
ejpam-4383	434	1	references	reference	NOUN
ejpam-4383	434	2	1014	1014	NUM
ejpam-4383	435	1	[	[	X
ejpam-4383	435	2	8	8	NUM
ejpam-4383	435	3	]	]	X
ejpam-4383	435	4	y.	y.	PROPN
ejpam-4383	435	5	imai	imai	PROPN
ejpam-4383	435	6	and	and	CCONJ
ejpam-4383	435	7	k.	k.	PROPN
ejpam-4383	435	8	iséki	iséki	PROPN
ejpam-4383	435	9	.	.	PROPN
ejpam-4383	436	1	on	on	ADP
ejpam-4383	436	2	axiom	axiom	NOUN
ejpam-4383	436	3	systems	system	NOUN
ejpam-4383	436	4	of	of	ADP
ejpam-4383	436	5	propositional	propositional	ADJ
ejpam-4383	436	6	calculi	calculi	PROPN
ejpam-4383	436	7	,	,	PUNCT
ejpam-4383	436	8	xiv	xiv	PROPN
ejpam-4383	436	9	.	.	PUNCT
ejpam-4383	437	1	proc	proc	PROPN
ejpam-4383	437	2	.	.	PUNCT
ejpam-4383	438	1	japan	japan	PROPN
ejpam-4383	438	2	acad	acad	PROPN
ejpam-4383	438	3	.	.	PROPN
ejpam-4383	438	4	,	,	PUNCT
ejpam-4383	438	5	42(1):19–22	42(1):19–22	NUM
ejpam-4383	438	6	,	,	PUNCT
ejpam-4383	438	7	1966	1966	NUM
ejpam-4383	438	8	.	.	PUNCT
ejpam-4383	439	1	[	[	X
ejpam-4383	439	2	9	9	NUM
ejpam-4383	439	3	]	]	PUNCT
ejpam-4383	439	4	k.	k.	PROPN
ejpam-4383	439	5	iséki	iséki	PROPN
ejpam-4383	439	6	.	.	PUNCT
ejpam-4383	440	1	an	an	DET
ejpam-4383	440	2	algebra	algebra	NOUN
ejpam-4383	440	3	related	relate	VERB
ejpam-4383	440	4	with	with	ADP
ejpam-4383	440	5	a	a	DET
ejpam-4383	440	6	propositional	propositional	ADJ
ejpam-4383	440	7	calculus	calculus	NOUN
ejpam-4383	440	8	.	.	PUNCT
ejpam-4383	441	1	proc	proc	PROPN
ejpam-4383	441	2	.	.	PUNCT
ejpam-4383	442	1	japan	japan	PROPN
ejpam-4383	442	2	acad	acad	PROPN
ejpam-4383	442	3	.	.	PROPN
ejpam-4383	442	4	,	,	PUNCT
ejpam-4383	442	5	42(1):26–29	42(1):26–29	NUM
ejpam-4383	442	6	,	,	PUNCT
ejpam-4383	442	7	1966	1966	NUM
ejpam-4383	442	8	.	.	PUNCT
ejpam-4383	443	1	[	[	X
ejpam-4383	443	2	10	10	NUM
ejpam-4383	443	3	]	]	PUNCT
ejpam-4383	443	4	j.	j.	PROPN
ejpam-4383	443	5	kavitha	kavitha	PROPN
ejpam-4383	443	6	and	and	CCONJ
ejpam-4383	443	7	r.	r.	PROPN
ejpam-4383	443	8	gowri	gowri	PROPN
ejpam-4383	443	9	.	.	PUNCT
ejpam-4383	444	1	direct	direct	ADJ
ejpam-4383	444	2	product	product	NOUN
ejpam-4383	444	3	of	of	ADP
ejpam-4383	444	4	gk	gk	PROPN
ejpam-4383	444	5	algebra	algebra	PROPN
ejpam-4383	444	6	.	.	PUNCT
ejpam-4383	445	1	indian	indian	PROPN
ejpam-4383	445	2	j.	j.	PROPN
ejpam-4383	445	3	technol	technol	PROPN
ejpam-4383	445	4	.	.	PROPN
ejpam-4383	445	5	,	,	PUNCT
ejpam-4383	445	6	14(35):2802–2805	14(35):2802–2805	NUM
ejpam-4383	445	7	,	,	PUNCT
ejpam-4383	445	8	2018	2018	NUM
ejpam-4383	445	9	.	.	PUNCT
ejpam-4383	446	1	[	[	X
ejpam-4383	446	2	11	11	NUM
ejpam-4383	446	3	]	]	X
ejpam-4383	446	4	y.	y.	PROPN
ejpam-4383	446	5	h.	h.	PROPN
ejpam-4383	446	6	kim	kim	PROPN
ejpam-4383	446	7	.	.	PUNCT
ejpam-4383	447	1	on	on	ADP
ejpam-4383	447	2	medial	medial	ADJ
ejpam-4383	447	3	b	b	PROPN
ejpam-4383	447	4	-algebras	-algebras	PROPN
ejpam-4383	447	5	.	.	PUNCT
ejpam-4383	447	6	j.	j.	PROPN
ejpam-4383	447	7	appl	appl	PROPN
ejpam-4383	447	8	.	.	PROPN
ejpam-4383	447	9	math	math	PROPN
ejpam-4383	447	10	.	.	PUNCT
ejpam-4383	448	1	inform	inform	NOUN
ejpam-4383	448	2	.	.	PUNCT
ejpam-4383	448	3	,	,	PUNCT
ejpam-4383	448	4	32(5	32(5	PROPN
ejpam-4383	448	5	-	-	SYM
ejpam-4383	448	6	6):849–856	6):849–856	NUM
ejpam-4383	448	7	,	,	PUNCT
ejpam-4383	448	8	2014	2014	NUM
ejpam-4383	448	9	.	.	PUNCT
ejpam-4383	449	1	[	[	X
ejpam-4383	449	2	12	12	NUM
ejpam-4383	449	3	]	]	PUNCT
ejpam-4383	449	4	j.	j.	PROPN
ejpam-4383	449	5	a.	a.	PROPN
ejpam-4383	449	6	v.	v.	PROPN
ejpam-4383	449	7	lingcong	lingcong	PROPN
ejpam-4383	449	8	and	and	CCONJ
ejpam-4383	449	9	j.	j.	PROPN
ejpam-4383	449	10	c.	c.	PROPN
ejpam-4383	449	11	endam	endam	PROPN
ejpam-4383	449	12	.	.	PUNCT
ejpam-4383	450	1	direct	direct	ADJ
ejpam-4383	450	2	product	product	NOUN
ejpam-4383	450	3	of	of	ADP
ejpam-4383	450	4	b	b	NOUN
ejpam-4383	450	5	-algebras	-algebras	PROPN
ejpam-4383	450	6	.	.	PUNCT
ejpam-4383	451	1	int	int	NOUN
ejpam-4383	451	2	.	.	PUNCT
ejpam-4383	452	1	j.	j.	PROPN
ejpam-4383	452	2	algebra	algebra	PROPN
ejpam-4383	452	3	,	,	PUNCT
ejpam-4383	452	4	10(1):33–40	10(1):33–40	NUM
ejpam-4383	452	5	,	,	PUNCT
ejpam-4383	452	6	2016	2016	NUM
ejpam-4383	452	7	.	.	PUNCT
ejpam-4383	453	1	[	[	X
ejpam-4383	453	2	13	13	NUM
ejpam-4383	453	3	]	]	PUNCT
ejpam-4383	453	4	j.	j.	PROPN
ejpam-4383	453	5	a.	a.	PROPN
ejpam-4383	453	6	v.	v.	PROPN
ejpam-4383	453	7	lingcong	lingcong	PROPN
ejpam-4383	453	8	and	and	CCONJ
ejpam-4383	453	9	j.	j.	PROPN
ejpam-4383	453	10	c.	c.	PROPN
ejpam-4383	453	11	endam	endam	PROPN
ejpam-4383	453	12	.	.	PUNCT
ejpam-4383	454	1	mappings	mapping	NOUN
ejpam-4383	454	2	of	of	ADP
ejpam-4383	454	3	the	the	DET
ejpam-4383	454	4	direct	direct	ADJ
ejpam-4383	454	5	product	product	NOUN
ejpam-4383	454	6	of	of	ADP
ejpam-4383	454	7	b	b	NOUN
ejpam-4383	454	8	-algebras	-algebras	PROPN
ejpam-4383	454	9	.	.	PUNCT
ejpam-4383	455	1	int	int	NOUN
ejpam-4383	455	2	.	.	PUNCT
ejpam-4383	456	1	j.	j.	PROPN
ejpam-4383	456	2	algebra	algebra	PROPN
ejpam-4383	456	3	,	,	PUNCT
ejpam-4383	456	4	10(3):133–140	10(3):133–140	NUM
ejpam-4383	456	5	,	,	PUNCT
ejpam-4383	456	6	2016	2016	NUM
ejpam-4383	456	7	.	.	PUNCT
ejpam-4383	457	1	[	[	X
ejpam-4383	457	2	14	14	NUM
ejpam-4383	457	3	]	]	X
ejpam-4383	457	4	g.	g.	PROPN
ejpam-4383	457	5	muhiuddin	muhiuddin	PROPN
ejpam-4383	457	6	.	.	PUNCT
ejpam-4383	458	1	bipolar	bipolar	ADJ
ejpam-4383	458	2	fuzzy	fuzzy	ADJ
ejpam-4383	458	3	ku	ku	PROPN
ejpam-4383	458	4	-	-	PUNCT
ejpam-4383	458	5	subalgebras	subalgebras	PROPN
ejpam-4383	458	6	/	/	SYM
ejpam-4383	458	7	ideals	ideal	NOUN
ejpam-4383	458	8	of	of	ADP
ejpam-4383	458	9	ku	ku	PROPN
ejpam-4383	458	10	-	-	PUNCT
ejpam-4383	458	11	algebras	algebras	PROPN
ejpam-4383	458	12	.	.	PUNCT
ejpam-4383	459	1	ann	ann	PROPN
ejpam-4383	459	2	.	.	PUNCT
ejpam-4383	459	3	fuzzy	fuzzy	ADJ
ejpam-4383	459	4	math	math	NOUN
ejpam-4383	459	5	.	.	PUNCT
ejpam-4383	460	1	inform	inform	NOUN
ejpam-4383	460	2	.	.	PUNCT
ejpam-4383	460	3	,	,	PUNCT
ejpam-4383	460	4	8(3):409–418	8(3):409–418	NUM
ejpam-4383	460	5	,	,	PUNCT
ejpam-4383	460	6	2014	2014	NUM
ejpam-4383	460	7	.	.	PUNCT
ejpam-4383	461	1	[	[	X
ejpam-4383	461	2	15	15	NUM
ejpam-4383	461	3	]	]	X
ejpam-4383	461	4	g.	g.	PROPN
ejpam-4383	461	5	muhiuddin	muhiuddin	PROPN
ejpam-4383	461	6	,	,	PUNCT
ejpam-4383	461	7	d.	d.	PROPN
ejpam-4383	461	8	al	al	PROPN
ejpam-4383	461	9	-	-	PUNCT
ejpam-4383	461	10	kadi	kadi	PROPN
ejpam-4383	461	11	,	,	PUNCT
ejpam-4383	461	12	and	and	CCONJ
ejpam-4383	461	13	a.	a.	NOUN
ejpam-4383	461	14	mahboob	mahboob	PROPN
ejpam-4383	461	15	.	.	PUNCT
ejpam-4383	462	1	more	more	ADV
ejpam-4383	462	2	general	general	ADJ
ejpam-4383	462	3	form	form	NOUN
ejpam-4383	462	4	of	of	ADP
ejpam-4383	462	5	interval	interval	NOUN
ejpam-4383	462	6	-	-	PUNCT
ejpam-4383	462	7	valued	value	VERB
ejpam-4383	462	8	fuzzy	fuzzy	ADJ
ejpam-4383	462	9	ideals	ideal	NOUN
ejpam-4383	462	10	of	of	ADP
ejpam-4383	462	11	bck	bck	PROPN
ejpam-4383	462	12	/	/	SYM
ejpam-4383	462	13	bci	bci	PROPN
ejpam-4383	462	14	-algebras	-algebras	PROPN
ejpam-4383	462	15	.	.	PUNCT
ejpam-4383	462	16	secur	secur	PROPN
ejpam-4383	462	17	.	.	PUNCT
ejpam-4383	463	1	commun	commun	PROPN
ejpam-4383	463	2	.	.	PUNCT
ejpam-4383	464	1	netw	netw	PROPN
ejpam-4383	464	2	.	.	PUNCT
ejpam-4383	464	3	,	,	PUNCT
ejpam-4383	464	4	2021	2021	NUM
ejpam-4383	464	5	:	:	PUNCT
ejpam-4383	464	6	article	article	NOUN
ejpam-4383	464	7	i	i	PROPN
ejpam-4383	464	8	d	d	PROPN
ejpam-4383	464	9	9930467	9930467	NUM
ejpam-4383	464	10	,	,	PUNCT
ejpam-4383	464	11	10	10	NUM
ejpam-4383	464	12	pages	page	NOUN
ejpam-4383	464	13	,	,	PUNCT
ejpam-4383	464	14	2021	2021	NUM
ejpam-4383	464	15	.	.	PUNCT
ejpam-4383	465	1	[	[	X
ejpam-4383	465	2	16	16	NUM
ejpam-4383	465	3	]	]	X
ejpam-4383	465	4	j.	j.	PROPN
ejpam-4383	465	5	neggers	neggers	PROPN
ejpam-4383	465	6	and	and	CCONJ
ejpam-4383	465	7	h.	h.	PROPN
ejpam-4383	465	8	s.	s.	PROPN
ejpam-4383	465	9	kim	kim	PROPN
ejpam-4383	465	10	.	.	PUNCT
ejpam-4383	466	1	a	a	DET
ejpam-4383	466	2	fundamental	fundamental	ADJ
ejpam-4383	466	3	theorem	theorem	NOUN
ejpam-4383	466	4	of	of	ADP
ejpam-4383	466	5	b	b	PROPN
ejpam-4383	466	6	-homomorphism	-homomorphism	PROPN
ejpam-4383	466	7	for	for	ADP
ejpam-4383	466	8	b	b	PROPN
ejpam-4383	466	9	algebras	algebras	X
ejpam-4383	466	10	.	.	PUNCT
ejpam-4383	467	1	int	int	NOUN
ejpam-4383	467	2	.	.	PUNCT
ejpam-4383	468	1	math	math	NOUN
ejpam-4383	468	2	.	.	PUNCT
ejpam-4383	469	1	j.	j.	PROPN
ejpam-4383	469	2	,	,	PUNCT
ejpam-4383	469	3	3:207–214	3:207–214	NUM
ejpam-4383	469	4	,	,	PUNCT
ejpam-4383	469	5	2002	2002	NUM
ejpam-4383	469	6	.	.	PUNCT
ejpam-4383	470	1	[	[	X
ejpam-4383	470	2	17	17	NUM
ejpam-4383	470	3	]	]	X
ejpam-4383	470	4	j.	j.	PROPN
ejpam-4383	470	5	neggers	neggers	PROPN
ejpam-4383	470	6	and	and	CCONJ
ejpam-4383	470	7	h.	h.	PROPN
ejpam-4383	470	8	s.	s.	PROPN
ejpam-4383	470	9	kim	kim	PROPN
ejpam-4383	470	10	.	.	PUNCT
ejpam-4383	471	1	on	on	ADP
ejpam-4383	471	2	b	b	PROPN
ejpam-4383	471	3	-algebras	-algebras	PROPN
ejpam-4383	471	4	.	.	PUNCT
ejpam-4383	471	5	mat	mat	PROPN
ejpam-4383	471	6	.	.	PUNCT
ejpam-4383	471	7	vesnik	vesnik	PROPN
ejpam-4383	471	8	,	,	PUNCT
ejpam-4383	471	9	54:21–29	54:21–29	NUM
ejpam-4383	471	10	,	,	PUNCT
ejpam-4383	471	11	2002	2002	NUM
ejpam-4383	471	12	.	.	PUNCT
ejpam-4383	472	1	[	[	X
ejpam-4383	472	2	18	18	NUM
ejpam-4383	472	3	]	]	PUNCT
ejpam-4383	472	4	t.	t.	NOUN
ejpam-4383	472	5	senapati	senapati	PROPN
ejpam-4383	472	6	,	,	PUNCT
ejpam-4383	472	7	g.	g.	PROPN
ejpam-4383	472	8	muhiuddin	muhiuddin	PROPN
ejpam-4383	472	9	,	,	PUNCT
ejpam-4383	472	10	and	and	CCONJ
ejpam-4383	472	11	k.	k.	PROPN
ejpam-4383	472	12	p.	p.	PROPN
ejpam-4383	472	13	shum	shum	PROPN
ejpam-4383	472	14	.	.	PUNCT
ejpam-4383	473	1	representation	representation	NOUN
ejpam-4383	473	2	of	of	ADP
ejpam-4383	473	3	up	up	ADV
ejpam-4383	473	4	-	-	PUNCT
ejpam-4383	473	5	algebras	algebras	NOUN
ejpam-4383	473	6	in	in	ADP
ejpam-4383	473	7	interval	interval	NOUN
ejpam-4383	473	8	-	-	PUNCT
ejpam-4383	473	9	valued	value	VERB
ejpam-4383	473	10	intuitionistic	intuitionistic	ADJ
ejpam-4383	473	11	fuzzy	fuzzy	ADJ
ejpam-4383	473	12	environment	environment	NOUN
ejpam-4383	473	13	.	.	PUNCT
ejpam-4383	474	1	ital	ital	PROPN
ejpam-4383	474	2	.	.	PUNCT
ejpam-4383	475	1	j.	j.	PROPN
ejpam-4383	475	2	pure	pure	PROPN
ejpam-4383	475	3	appl	appl	PROPN
ejpam-4383	475	4	.	.	PUNCT
ejpam-4383	475	5	math	math	PROPN
ejpam-4383	475	6	.	.	PUNCT
ejpam-4383	475	7	,	,	PUNCT
ejpam-4383	475	8	38:497	38:497	NUM
ejpam-4383	475	9	–	–	PUNCT
ejpam-4383	475	10	517	517	NUM
ejpam-4383	475	11	,	,	PUNCT
ejpam-4383	475	12	2017	2017	NUM
ejpam-4383	475	13	.	.	PUNCT
ejpam-4383	476	1	[	[	X
ejpam-4383	476	2	19	19	NUM
ejpam-4383	476	3	]	]	PUNCT
ejpam-4383	476	4	a.	a.	NOUN
ejpam-4383	476	5	setiani	setiani	PROPN
ejpam-4383	476	6	,	,	PUNCT
ejpam-4383	476	7	s.	s.	PROPN
ejpam-4383	476	8	gemawati	gemawati	PROPN
ejpam-4383	476	9	,	,	PUNCT
ejpam-4383	476	10	and	and	CCONJ
ejpam-4383	476	11	l.	l.	PROPN
ejpam-4383	476	12	deswita	deswita	PROPN
ejpam-4383	476	13	.	.	PUNCT
ejpam-4383	477	1	direct	direct	ADJ
ejpam-4383	477	2	product	product	NOUN
ejpam-4383	477	3	of	of	ADP
ejpam-4383	477	4	bp	bp	PROPN
ejpam-4383	477	5	-algebra	-algebra	PROPN
ejpam-4383	477	6	.	.	PUNCT
ejpam-4383	478	1	int	int	NOUN
ejpam-4383	478	2	.	.	PUNCT
ejpam-4383	479	1	j.	j.	PROPN
ejpam-4383	479	2	math	math	PROPN
ejpam-4383	479	3	.	.	PUNCT
ejpam-4383	480	1	trends	trend	NOUN
ejpam-4383	480	2	technol	technol	ADJ
ejpam-4383	480	3	.	.	PUNCT
ejpam-4383	480	4	,	,	PUNCT
ejpam-4383	480	5	66(2):63–66	66(2):63–66	NUM
ejpam-4383	480	6	,	,	PUNCT
ejpam-4383	480	7	2020	2020	NUM
ejpam-4383	480	8	.	.	PUNCT
ejpam-4383	481	1	[	[	X
ejpam-4383	481	2	20	20	NUM
ejpam-4383	481	3	]	]	PUNCT
ejpam-4383	481	4	j.	j.	PROPN
ejpam-4383	481	5	somjanta	somjanta	PROPN
ejpam-4383	481	6	,	,	PUNCT
ejpam-4383	481	7	n.	n.	PROPN
ejpam-4383	481	8	thuekaew	thuekaew	PROPN
ejpam-4383	481	9	,	,	PUNCT
ejpam-4383	481	10	p.	p.	NOUN
ejpam-4383	481	11	kumpeangkeaw	kumpeangkeaw	PROPN
ejpam-4383	481	12	,	,	PUNCT
ejpam-4383	481	13	and	and	CCONJ
ejpam-4383	481	14	a.	a.	NOUN
ejpam-4383	481	15	iampan	iampan	PROPN
ejpam-4383	481	16	.	.	PUNCT
ejpam-4383	482	1	fuzzy	fuzzy	ADJ
ejpam-4383	482	2	sets	set	NOUN
ejpam-4383	482	3	in	in	ADP
ejpam-4383	482	4	upalgebras	upalgebra	NOUN
ejpam-4383	482	5	.	.	PUNCT
ejpam-4383	483	1	ann	ann	PROPN
ejpam-4383	483	2	.	.	PUNCT
ejpam-4383	483	3	fuzzy	fuzzy	ADJ
ejpam-4383	483	4	math	math	NOUN
ejpam-4383	483	5	.	.	PUNCT
ejpam-4383	484	1	inform	inform	NOUN
ejpam-4383	484	2	.	.	PUNCT
ejpam-4383	484	3	,	,	PUNCT
ejpam-4383	484	4	12(6):739–756	12(6):739–756	PROPN
ejpam-4383	484	5	,	,	PUNCT
ejpam-4383	484	6	2016	2016	NUM
ejpam-4383	484	7	.	.	PUNCT
ejpam-4383	485	1	[	[	X
ejpam-4383	485	2	21	21	NUM
ejpam-4383	485	3	]	]	X
ejpam-4383	485	4	s.	s.	PROPN
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ejpam-4383	485	6	,	,	PUNCT
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ejpam-4383	485	8	burandate	burandate	NOUN
ejpam-4383	485	9	,	,	PUNCT
ejpam-4383	485	10	and	and	CCONJ
ejpam-4383	485	11	a.	a.	NOUN
ejpam-4383	485	12	iampan	iampan	PROPN
ejpam-4383	485	13	.	.	PUNCT
ejpam-4383	486	1	some	some	DET
ejpam-4383	486	2	operations	operation	NOUN
ejpam-4383	486	3	of	of	ADP
ejpam-4383	486	4	fuzzy	fuzzy	ADJ
ejpam-4383	486	5	sets	set	NOUN
ejpam-4383	486	6	in	in	ADP
ejpam-4383	486	7	upalgebras	upalgebra	NOUN
ejpam-4383	486	8	with	with	ADP
ejpam-4383	486	9	respect	respect	NOUN
ejpam-4383	486	10	to	to	ADP
ejpam-4383	486	11	a	a	DET
ejpam-4383	486	12	triangular	triangular	NOUN
ejpam-4383	486	13	norm	norm	NOUN
ejpam-4383	486	14	.	.	PUNCT
ejpam-4383	487	1	ann	ann	AUX
ejpam-4383	487	2	.	.	PUNCT
ejpam-4383	487	3	commun	commun	PROPN
ejpam-4383	487	4	.	.	PUNCT
ejpam-4383	488	1	math	math	PROPN
ejpam-4383	488	2	.	.	PUNCT
ejpam-4383	488	3	,	,	PUNCT
ejpam-4383	488	4	2(1):1–10	2(1):1–10	NUM
ejpam-4383	488	5	,	,	PUNCT
ejpam-4383	488	6	2019	2019	NUM
ejpam-4383	488	7	.	.	PUNCT
ejpam-4383	489	1	[	[	X
ejpam-4383	489	2	22	22	NUM
ejpam-4383	489	3	]	]	PUNCT
ejpam-4383	489	4	a.	a.	NOUN
ejpam-4383	489	5	walendziak	walendziak	PROPN
ejpam-4383	489	6	.	.	PUNCT
ejpam-4383	490	1	some	some	DET
ejpam-4383	490	2	axiomatizations	axiomatization	NOUN
ejpam-4383	490	3	of	of	ADP
ejpam-4383	490	4	b	b	NOUN
ejpam-4383	490	5	-algebras	-algebras	PROPN
ejpam-4383	490	6	.	.	PUNCT
ejpam-4383	490	7	math	math	NOUN
ejpam-4383	490	8	.	.	PUNCT
ejpam-4383	491	1	slovaca	slovaca	PROPN
ejpam-4383	491	2	,	,	PUNCT
ejpam-4383	491	3	56(3):301–306	56(3):301–306	PROPN
ejpam-4383	491	4	,	,	PUNCT
ejpam-4383	491	5	2006	2006	NUM
ejpam-4383	491	6	.	.	PUNCT
ejpam-4383	492	1	[	[	X
ejpam-4383	492	2	23	23	NUM
ejpam-4383	492	3	]	]	X
ejpam-4383	492	4	s.	s.	PROPN
ejpam-4383	492	5	widianto	widianto	PROPN
ejpam-4383	492	6	,	,	PUNCT
ejpam-4383	492	7	sri	sri	PROPN
ejpam-4383	492	8	gemawati	gemawati	PROPN
ejpam-4383	492	9	,	,	PUNCT
ejpam-4383	492	10	and	and	CCONJ
ejpam-4383	492	11	kartini	kartini	NOUN
ejpam-4383	492	12	.	.	PUNCT
ejpam-4383	493	1	direct	direct	ADJ
ejpam-4383	493	2	product	product	NOUN
ejpam-4383	493	3	in	in	ADP
ejpam-4383	493	4	bg	bg	PROPN
ejpam-4383	493	5	-	-	PUNCT
ejpam-4383	493	6	algebras	algebras	PROPN
ejpam-4383	493	7	.	.	PUNCT
ejpam-4383	494	1	int	int	NOUN
ejpam-4383	494	2	.	.	PUNCT
ejpam-4383	495	1	j.	j.	PROPN
ejpam-4383	495	2	algebra	algebra	PROPN
ejpam-4383	495	3	,	,	PUNCT
ejpam-4383	495	4	13(5):239–247	13(5):239–247	PROPN
ejpam-4383	495	5	,	,	PUNCT
ejpam-4383	495	6	2019	2019	NUM
ejpam-4383	495	7	.	.	PUNCT