id	sid	tid	token	lemma	pos
ejpam-4918	1	1	european	european	PROPN
ejpam-4918	1	2	journal	journal	PROPN
ejpam-4918	1	3	of	of	ADP
ejpam-4918	1	4	pure	pure	ADJ
ejpam-4918	1	5	and	and	CCONJ
ejpam-4918	1	6	applied	apply	VERB
ejpam-4918	1	7	mathematics	mathematic	NOUN
ejpam-4918	1	8	vol	vol	NOUN
ejpam-4918	1	9	.	.	PUNCT
ejpam-4918	2	1	16	16	NUM
ejpam-4918	2	2	,	,	PUNCT
ejpam-4918	2	3	no	no	INTJ
ejpam-4918	2	4	.	.	NOUN
ejpam-4918	2	5	4	4	NUM
ejpam-4918	2	6	,	,	PUNCT
ejpam-4918	2	7	2023	2023	NUM
ejpam-4918	2	8	,	,	PUNCT
ejpam-4918	2	9	2156	2156	NUM
ejpam-4918	2	10	-	-	SYM
ejpam-4918	2	11	2168	2168	NUM
ejpam-4918	2	12	issn	issn	VERB
ejpam-4918	2	13	1307	1307	NUM
ejpam-4918	2	14	-	-	SYM
ejpam-4918	2	15	5543	5543	NUM
ejpam-4918	2	16	–	–	PUNCT
ejpam-4918	2	17	ejpam.com	ejpam.com	X
ejpam-4918	2	18	published	publish	VERB
ejpam-4918	2	19	by	by	ADP
ejpam-4918	2	20	new	new	PROPN
ejpam-4918	2	21	york	york	PROPN
ejpam-4918	2	22	business	business	PROPN
ejpam-4918	2	23	global	global	ADJ
ejpam-4918	2	24	generalized	generalize	VERB
ejpam-4918	2	25	reflexive	reflexive	ADJ
ejpam-4918	2	26	structures	structure	NOUN
ejpam-4918	2	27	properties	property	NOUN
ejpam-4918	2	28	of	of	ADP
ejpam-4918	2	29	crossed	cross	VERB
ejpam-4918	2	30	products	product	NOUN
ejpam-4918	2	31	type	type	NOUN
ejpam-4918	2	32	eltiyeb	eltiyeb	PROPN
ejpam-4918	2	33	ali1,2	ali1,2	PROPN
ejpam-4918	2	34	1	1	NUM
ejpam-4918	2	35	department	department	NOUN
ejpam-4918	2	36	of	of	ADP
ejpam-4918	2	37	mathematics	mathematic	NOUN
ejpam-4918	2	38	,	,	PUNCT
ejpam-4918	2	39	college	college	NOUN
ejpam-4918	2	40	of	of	ADP
ejpam-4918	2	41	science	science	NOUN
ejpam-4918	2	42	and	and	CCONJ
ejpam-4918	2	43	arts	art	NOUN
ejpam-4918	2	44	,	,	PUNCT
ejpam-4918	2	45	najran	najran	ADJ
ejpam-4918	2	46	university	university	PROPN
ejpam-4918	2	47	,	,	PUNCT
ejpam-4918	2	48	ksa	ksa	PROPN
ejpam-4918	2	49	2	2	NUM
ejpam-4918	2	50	department	department	NOUN
ejpam-4918	2	51	of	of	ADP
ejpam-4918	2	52	mathematics	mathematic	NOUN
ejpam-4918	2	53	,	,	PUNCT
ejpam-4918	2	54	faculty	faculty	NOUN
ejpam-4918	2	55	of	of	ADP
ejpam-4918	2	56	education	education	NOUN
ejpam-4918	2	57	,	,	PUNCT
ejpam-4918	2	58	university	university	NOUN
ejpam-4918	2	59	of	of	ADP
ejpam-4918	2	60	khartoum	khartoum	PROPN
ejpam-4918	2	61	,	,	PUNCT
ejpam-4918	2	62	sudan	sudan	PROPN
ejpam-4918	2	63	abstract	abstract	NOUN
ejpam-4918	2	64	.	.	PUNCT
ejpam-4918	3	1	let	let	VERB
ejpam-4918	3	2	r	r	PRON
ejpam-4918	3	3	be	be	AUX
ejpam-4918	3	4	a	a	DET
ejpam-4918	3	5	ring	ring	NOUN
ejpam-4918	3	6	and	and	CCONJ
ejpam-4918	3	7	m	m	AUX
ejpam-4918	3	8	be	be	AUX
ejpam-4918	3	9	a	a	DET
ejpam-4918	3	10	monoid	monoid	NOUN
ejpam-4918	3	11	with	with	ADP
ejpam-4918	3	12	a	a	DET
ejpam-4918	3	13	twisting	twisting	NOUN
ejpam-4918	3	14	map	map	NOUN
ejpam-4918	3	15	f	f	X
ejpam-4918	3	16	:	:	PUNCT
ejpam-4918	3	17	m	m	VERB
ejpam-4918	3	18	×m	×m	NOUN
ejpam-4918	3	19	→	→	SYM
ejpam-4918	3	20	u(r	u(r	NOUN
ejpam-4918	3	21	)	)	PUNCT
ejpam-4918	3	22	and	and	CCONJ
ejpam-4918	3	23	an	an	DET
ejpam-4918	3	24	action	action	NOUN
ejpam-4918	3	25	map	map	NOUN
ejpam-4918	3	26	ω	ω	NOUN
ejpam-4918	3	27	:	:	PUNCT
ejpam-4918	3	28	m	m	PROPN
ejpam-4918	3	29	→	→	SYM
ejpam-4918	3	30	aut(r	aut(r	PROPN
ejpam-4918	3	31	)	)	PUNCT
ejpam-4918	3	32	.	.	PUNCT
ejpam-4918	4	1	the	the	DET
ejpam-4918	4	2	objective	objective	NOUN
ejpam-4918	4	3	of	of	ADP
ejpam-4918	4	4	our	our	PRON
ejpam-4918	4	5	work	work	NOUN
ejpam-4918	4	6	is	be	AUX
ejpam-4918	4	7	to	to	PART
ejpam-4918	4	8	extend	extend	VERB
ejpam-4918	4	9	the	the	DET
ejpam-4918	4	10	reflexive	reflexive	ADJ
ejpam-4918	4	11	properties	property	NOUN
ejpam-4918	4	12	of	of	ADP
ejpam-4918	4	13	rings	ring	NOUN
ejpam-4918	4	14	by	by	ADP
ejpam-4918	4	15	focusing	focus	VERB
ejpam-4918	4	16	on	on	ADP
ejpam-4918	4	17	the	the	DET
ejpam-4918	4	18	crossed	cross	VERB
ejpam-4918	4	19	product	product	NOUN
ejpam-4918	4	20	r	r	NOUN
ejpam-4918	4	21	∗m	∗m	NOUN
ejpam-4918	4	22	over	over	ADP
ejpam-4918	4	23	r.	r.	PROPN
ejpam-4918	4	24	in	in	ADP
ejpam-4918	4	25	order	order	NOUN
ejpam-4918	4	26	to	to	PART
ejpam-4918	4	27	achieve	achieve	VERB
ejpam-4918	4	28	this	this	PRON
ejpam-4918	4	29	,	,	PUNCT
ejpam-4918	4	30	we	we	PRON
ejpam-4918	4	31	introduce	introduce	VERB
ejpam-4918	4	32	and	and	CCONJ
ejpam-4918	4	33	examine	examine	VERB
ejpam-4918	4	34	the	the	DET
ejpam-4918	4	35	concept	concept	NOUN
ejpam-4918	4	36	of	of	ADP
ejpam-4918	4	37	strongly	strongly	ADV
ejpam-4918	4	38	cm	cm	NOUN
ejpam-4918	4	39	-reflexive	-reflexive	NOUN
ejpam-4918	4	40	.	.	PUNCT
ejpam-4918	5	1	although	although	SCONJ
ejpam-4918	5	2	a	a	DET
ejpam-4918	5	3	monoid	monoid	NOUN
ejpam-4918	5	4	m	m	ADJ
ejpam-4918	5	5	and	and	CCONJ
ejpam-4918	5	6	any	any	DET
ejpam-4918	5	7	ring	ring	NOUN
ejpam-4918	5	8	r	r	NOUN
ejpam-4918	5	9	with	with	ADP
ejpam-4918	5	10	an	an	DET
ejpam-4918	5	11	idempotent	idempotent	NOUN
ejpam-4918	5	12	are	be	AUX
ejpam-4918	5	13	not	not	PART
ejpam-4918	5	14	strongly	strongly	ADV
ejpam-4918	5	15	cm	cm	NOUN
ejpam-4918	5	16	-reflexive	-reflexive	NOUN
ejpam-4918	5	17	in	in	ADP
ejpam-4918	5	18	general	general	ADJ
ejpam-4918	5	19	,	,	PUNCT
ejpam-4918	5	20	we	we	PRON
ejpam-4918	5	21	prove	prove	VERB
ejpam-4918	5	22	that	that	SCONJ
ejpam-4918	5	23	r	r	NOUN
ejpam-4918	5	24	is	be	AUX
ejpam-4918	5	25	strongly	strongly	ADV
ejpam-4918	5	26	cm	cm	NOUN
ejpam-4918	5	27	-reflexive	-reflexive	NOUN
ejpam-4918	5	28	under	under	ADP
ejpam-4918	5	29	some	some	DET
ejpam-4918	5	30	additional	additional	ADJ
ejpam-4918	5	31	conditions	condition	NOUN
ejpam-4918	5	32	.	.	PUNCT
ejpam-4918	6	1	moreover	moreover	ADV
ejpam-4918	6	2	,	,	PUNCT
ejpam-4918	6	3	we	we	PRON
ejpam-4918	6	4	prove	prove	VERB
ejpam-4918	6	5	that	that	SCONJ
ejpam-4918	6	6	if	if	SCONJ
ejpam-4918	6	7	r	r	NOUN
ejpam-4918	6	8	is	be	AUX
ejpam-4918	6	9	a	a	DET
ejpam-4918	6	10	left	left	ADJ
ejpam-4918	6	11	p.q.-baer	p.q.-baer	NOUN
ejpam-4918	6	12	(	(	PUNCT
ejpam-4918	6	13	semiprime	semiprime	NOUN
ejpam-4918	6	14	,	,	PUNCT
ejpam-4918	6	15	left	leave	VERB
ejpam-4918	6	16	app	app	NOUN
ejpam-4918	6	17	-ring	-ring	PROPN
ejpam-4918	6	18	,	,	PUNCT
ejpam-4918	6	19	respectively	respectively	ADV
ejpam-4918	6	20	)	)	PUNCT
ejpam-4918	6	21	,	,	PUNCT
ejpam-4918	6	22	then	then	ADV
ejpam-4918	6	23	r	r	NOUN
ejpam-4918	6	24	is	be	AUX
ejpam-4918	6	25	strongly	strongly	ADV
ejpam-4918	6	26	cm	cm	NOUN
ejpam-4918	6	27	-reflexive	-reflexive	NOUN
ejpam-4918	6	28	.	.	PUNCT
ejpam-4918	7	1	additionally	additionally	ADV
ejpam-4918	7	2	,	,	PUNCT
ejpam-4918	7	3	for	for	ADP
ejpam-4918	7	4	a	a	DET
ejpam-4918	7	5	right	right	ADJ
ejpam-4918	7	6	ore	ore	NOUN
ejpam-4918	7	7	ring	ring	NOUN
ejpam-4918	7	8	r	r	NOUN
ejpam-4918	7	9	with	with	ADP
ejpam-4918	7	10	a	a	DET
ejpam-4918	7	11	classical	classical	ADJ
ejpam-4918	7	12	right	right	ADJ
ejpam-4918	7	13	quotient	quotient	NOUN
ejpam-4918	7	14	ring	ring	NOUN
ejpam-4918	7	15	q	q	PROPN
ejpam-4918	7	16	,	,	PUNCT
ejpam-4918	7	17	we	we	PRON
ejpam-4918	7	18	prove	prove	VERB
ejpam-4918	7	19	r	r	NOUN
ejpam-4918	7	20	is	be	AUX
ejpam-4918	7	21	strongly	strongly	ADV
ejpam-4918	7	22	cm	cm	NOUN
ejpam-4918	7	23	-reflexive	-reflexive	NOUN
ejpam-4918	7	24	if	if	SCONJ
ejpam-4918	8	1	and	and	CCONJ
ejpam-4918	8	2	only	only	ADV
ejpam-4918	8	3	if	if	SCONJ
ejpam-4918	8	4	q	q	NOUN
ejpam-4918	8	5	is	be	AUX
ejpam-4918	8	6	strongly	strongly	ADV
ejpam-4918	8	7	cm	cm	NOUN
ejpam-4918	8	8	-reflexive	-reflexive	NOUN
ejpam-4918	8	9	.	.	PUNCT
ejpam-4918	9	1	finally	finally	ADV
ejpam-4918	9	2	,	,	PUNCT
ejpam-4918	9	3	we	we	PRON
ejpam-4918	9	4	discuss	discuss	VERB
ejpam-4918	9	5	some	some	DET
ejpam-4918	9	6	relevant	relevant	ADJ
ejpam-4918	9	7	results	result	NOUN
ejpam-4918	9	8	on	on	ADP
ejpam-4918	9	9	crossed	cross	VERB
ejpam-4918	9	10	products	product	NOUN
ejpam-4918	9	11	.	.	PUNCT
ejpam-4918	10	1	2020	2020	NUM
ejpam-4918	10	2	mathematics	mathematic	NOUN
ejpam-4918	10	3	subject	subject	NOUN
ejpam-4918	10	4	classifications	classification	NOUN
ejpam-4918	10	5	:	:	PUNCT
ejpam-4918	10	6	16s36	16s36	NUM
ejpam-4918	10	7	,	,	PUNCT
ejpam-4918	10	8	16n60	16n60	NUM
ejpam-4918	10	9	,	,	PUNCT
ejpam-4918	10	10	16u99	16u99	NUM
ejpam-4918	10	11	.	.	PUNCT
ejpam-4918	11	1	key	key	ADJ
ejpam-4918	11	2	words	word	NOUN
ejpam-4918	11	3	and	and	CCONJ
ejpam-4918	11	4	phrases	phrase	NOUN
ejpam-4918	11	5	:	:	PUNCT
ejpam-4918	11	6	left	leave	VERB
ejpam-4918	11	7	p.q.-baer	p.q.-baer	NOUN
ejpam-4918	11	8	-	-	PUNCT
ejpam-4918	11	9	ring	ring	NOUN
ejpam-4918	11	10	,	,	PUNCT
ejpam-4918	11	11	crossed	cross	VERB
ejpam-4918	11	12	product	product	NOUN
ejpam-4918	11	13	monoid	monoid	NOUN
ejpam-4918	11	14	r	r	NOUN
ejpam-4918	11	15	∗m	∗m	NOUN
ejpam-4918	11	16	,	,	PUNCT
ejpam-4918	11	17	cm	cm	NOUN
ejpam-4918	11	18	-quasi	-quasi	NOUN
ejpam-4918	11	19	armendariz	armendariz	NOUN
ejpam-4918	11	20	ring	ring	NOUN
ejpam-4918	11	21	,	,	PUNCT
ejpam-4918	11	22	strongly	strongly	ADV
ejpam-4918	11	23	cm	cm	NUM
ejpam-4918	11	24	-reflexive	-reflexive	ADJ
ejpam-4918	11	25	ring	ring	NOUN
ejpam-4918	11	26	.	.	PUNCT
ejpam-4918	12	1	1	1	X
ejpam-4918	12	2	.	.	X
ejpam-4918	12	3	introduction	introduction	NOUN
ejpam-4918	12	4	unless	unless	SCONJ
ejpam-4918	12	5	otherwise	otherwise	ADV
ejpam-4918	12	6	stated	state	VERB
ejpam-4918	12	7	,	,	PUNCT
ejpam-4918	12	8	we	we	PRON
ejpam-4918	12	9	assume	assume	VERB
ejpam-4918	12	10	that	that	SCONJ
ejpam-4918	12	11	r	r	NOUN
ejpam-4918	12	12	is	be	AUX
ejpam-4918	12	13	an	an	DET
ejpam-4918	12	14	associative	associative	ADJ
ejpam-4918	12	15	ring	ring	NOUN
ejpam-4918	12	16	with	with	ADP
ejpam-4918	12	17	identity	identity	NOUN
ejpam-4918	12	18	and	and	CCONJ
ejpam-4918	12	19	m	m	NOUN
ejpam-4918	12	20	is	be	AUX
ejpam-4918	12	21	a	a	DET
ejpam-4918	12	22	monoid	monoid	NOUN
ejpam-4918	12	23	.	.	PUNCT
ejpam-4918	13	1	the	the	DET
ejpam-4918	13	2	concept	concept	NOUN
ejpam-4918	13	3	of	of	ADP
ejpam-4918	13	4	reflexive	reflexive	ADJ
ejpam-4918	13	5	properties	property	NOUN
ejpam-4918	13	6	of	of	ADP
ejpam-4918	13	7	rings	ring	NOUN
ejpam-4918	13	8	was	be	AUX
ejpam-4918	13	9	first	first	ADV
ejpam-4918	13	10	studied	study	VERB
ejpam-4918	13	11	by	by	ADP
ejpam-4918	13	12	mason	mason	PROPN
ejpam-4918	14	1	[	[	X
ejpam-4918	14	2	1	1	NUM
ejpam-4918	14	3	]	]	PUNCT
ejpam-4918	14	4	.	.	PUNCT
ejpam-4918	15	1	in	in	ADP
ejpam-4918	15	2	particular	particular	ADJ
ejpam-4918	15	3	,	,	PUNCT
ejpam-4918	15	4	a	a	DET
ejpam-4918	15	5	right	right	ADJ
ejpam-4918	15	6	ideal	ideal	NOUN
ejpam-4918	15	7	i	i	PRON
ejpam-4918	15	8	of	of	ADP
ejpam-4918	15	9	r	r	NOUN
ejpam-4918	15	10	is	be	AUX
ejpam-4918	15	11	said	say	VERB
ejpam-4918	15	12	to	to	PART
ejpam-4918	15	13	be	be	AUX
ejpam-4918	15	14	reflexive	reflexive	ADJ
ejpam-4918	15	15	if	if	SCONJ
ejpam-4918	15	16	xry	xry	PROPN
ejpam-4918	15	17	⊆	⊆	NUM
ejpam-4918	15	18	i	i	PRON
ejpam-4918	15	19	implies	imply	VERB
ejpam-4918	15	20	yrx	yrx	PROPN
ejpam-4918	15	21	⊆	⊆	NUM
ejpam-4918	15	22	i	i	PRON
ejpam-4918	15	23	for	for	ADP
ejpam-4918	15	24	any	any	DET
ejpam-4918	15	25	x	x	NOUN
ejpam-4918	15	26	,	,	PUNCT
ejpam-4918	15	27	y	y	PROPN
ejpam-4918	15	28	∈	∈	PROPN
ejpam-4918	15	29	r.	r.	PROPN
ejpam-4918	15	30	this	this	DET
ejpam-4918	15	31	concept	concept	NOUN
ejpam-4918	15	32	is	be	AUX
ejpam-4918	15	33	also	also	ADV
ejpam-4918	15	34	specialized	specialize	VERB
ejpam-4918	15	35	to	to	ADP
ejpam-4918	15	36	the	the	DET
ejpam-4918	15	37	zero	zero	NUM
ejpam-4918	15	38	ideal	ideal	NOUN
ejpam-4918	15	39	of	of	ADP
ejpam-4918	15	40	a	a	DET
ejpam-4918	15	41	ring	ring	NOUN
ejpam-4918	15	42	,	,	PUNCT
ejpam-4918	15	43	where	where	SCONJ
ejpam-4918	15	44	a	a	DET
ejpam-4918	15	45	ring	ring	NOUN
ejpam-4918	15	46	r	r	NOUN
ejpam-4918	15	47	is	be	AUX
ejpam-4918	15	48	said	say	VERB
ejpam-4918	15	49	to	to	PART
ejpam-4918	15	50	be	be	AUX
ejpam-4918	15	51	reflexive	reflexive	ADJ
ejpam-4918	15	52	if	if	SCONJ
ejpam-4918	15	53	its	its	PRON
ejpam-4918	15	54	zero	zero	NUM
ejpam-4918	15	55	ideal	ideal	NOUN
ejpam-4918	15	56	is	be	AUX
ejpam-4918	15	57	reflexive	reflexive	ADJ
ejpam-4918	15	58	.	.	PUNCT
ejpam-4918	16	1	moreover	moreover	ADV
ejpam-4918	16	2	,	,	PUNCT
ejpam-4918	16	3	a	a	DET
ejpam-4918	16	4	ring	ring	NOUN
ejpam-4918	16	5	r	r	NOUN
ejpam-4918	16	6	is	be	AUX
ejpam-4918	16	7	called	call	VERB
ejpam-4918	16	8	completely	completely	ADV
ejpam-4918	16	9	reflexive	reflexive	ADJ
ejpam-4918	16	10	if	if	SCONJ
ejpam-4918	16	11	xy	xy	PROPN
ejpam-4918	16	12	=	=	SYM
ejpam-4918	16	13	0	0	NUM
ejpam-4918	16	14	implies	imply	VERB
ejpam-4918	16	15	yx	yx	NOUN
ejpam-4918	16	16	=	=	SYM
ejpam-4918	16	17	0	0	NUM
ejpam-4918	16	18	for	for	ADP
ejpam-4918	16	19	any	any	DET
ejpam-4918	16	20	x	x	NOUN
ejpam-4918	16	21	,	,	PUNCT
ejpam-4918	16	22	y	y	PROPN
ejpam-4918	16	23	∈	∈	PROPN
ejpam-4918	16	24	r.	r.	PROPN
ejpam-4918	16	25	it	it	PRON
ejpam-4918	16	26	is	be	AUX
ejpam-4918	16	27	worth	worth	ADJ
ejpam-4918	16	28	noting	note	VERB
ejpam-4918	16	29	that	that	SCONJ
ejpam-4918	16	30	reduced	reduce	VERB
ejpam-4918	16	31	rings	ring	NOUN
ejpam-4918	16	32	are	be	AUX
ejpam-4918	16	33	completely	completely	ADV
ejpam-4918	16	34	reflexive	reflexive	ADJ
ejpam-4918	16	35	,	,	PUNCT
ejpam-4918	16	36	and	and	CCONJ
ejpam-4918	16	37	every	every	DET
ejpam-4918	16	38	completely	completely	ADV
ejpam-4918	16	39	reflexive	reflexive	ADJ
ejpam-4918	16	40	ring	ring	NOUN
ejpam-4918	16	41	is	be	AUX
ejpam-4918	16	42	semicommutative	semicommutative	ADJ
ejpam-4918	16	43	,	,	PUNCT
ejpam-4918	16	44	as	as	SCONJ
ejpam-4918	16	45	shown	show	VERB
ejpam-4918	16	46	in	in	ADP
ejpam-4918	16	47	the	the	DET
ejpam-4918	16	48	literature	literature	NOUN
ejpam-4918	17	1	[	[	X
ejpam-4918	17	2	1	1	NUM
ejpam-4918	17	3	]	]	PUNCT
ejpam-4918	17	4	.	.	PUNCT
ejpam-4918	18	1	several	several	ADJ
ejpam-4918	18	2	authors	author	NOUN
ejpam-4918	18	3	have	have	AUX
ejpam-4918	18	4	discussed	discuss	VERB
ejpam-4918	18	5	extensions	extension	NOUN
ejpam-4918	18	6	of	of	ADP
ejpam-4918	18	7	reflexive	reflexive	ADJ
ejpam-4918	18	8	rings	ring	NOUN
ejpam-4918	18	9	,	,	PUNCT
ejpam-4918	18	10	including	include	VERB
ejpam-4918	18	11	strongly	strongly	ADV
ejpam-4918	18	12	reflexive	reflexive	ADJ
ejpam-4918	18	13	rings	ring	NOUN
ejpam-4918	18	14	,	,	PUNCT
ejpam-4918	18	15	strongly	strongly	ADV
ejpam-4918	18	16	m	m	VERB
ejpam-4918	18	17	-reflexive	-reflexive	ADJ
ejpam-4918	18	18	rings	ring	NOUN
ejpam-4918	18	19	,	,	PUNCT
ejpam-4918	18	20	armendariz	armendariz	ADJ
ejpam-4918	18	21	rings	ring	NOUN
ejpam-4918	18	22	,	,	PUNCT
ejpam-4918	18	23	reversible	reversible	ADJ
ejpam-4918	18	24	rings	ring	NOUN
ejpam-4918	18	25	,	,	PUNCT
ejpam-4918	18	26	and	and	CCONJ
ejpam-4918	18	27	reflexive	reflexive	VERB
ejpam-4918	18	28	on	on	ADP
ejpam-4918	18	29	skew	skew	ADJ
ejpam-4918	18	30	monoid	monoid	NOUN
ejpam-4918	18	31	rings	ring	NOUN
ejpam-4918	18	32	,	,	PUNCT
ejpam-4918	18	33	in	in	ADP
ejpam-4918	18	34	numerous	numerous	ADJ
ejpam-4918	18	35	publications	publication	NOUN
ejpam-4918	18	36	(	(	PUNCT
ejpam-4918	18	37	see	see	VERB
ejpam-4918	18	38	,	,	PUNCT
ejpam-4918	18	39	for	for	ADP
ejpam-4918	18	40	example	example	NOUN
ejpam-4918	18	41	,	,	PUNCT
ejpam-4918	18	42	[	[	X
ejpam-4918	18	43	2	2	NUM
ejpam-4918	18	44	]	]	PUNCT
ejpam-4918	18	45	,	,	PUNCT
ejpam-4918	18	46	[	[	X
ejpam-4918	18	47	3	3	NUM
ejpam-4918	18	48	]	]	PUNCT
ejpam-4918	18	49	,	,	PUNCT
ejpam-4918	18	50	[	[	X
ejpam-4918	18	51	4	4	X
ejpam-4918	18	52	]	]	PUNCT
ejpam-4918	18	53	and	and	CCONJ
ejpam-4918	18	54	[	[	X
ejpam-4918	18	55	5	5	NUM
ejpam-4918	18	56	]	]	PUNCT
ejpam-4918	18	57	)	)	PUNCT
ejpam-4918	18	58	.	.	PUNCT
ejpam-4918	19	1	according	accord	VERB
ejpam-4918	19	2	to	to	ADP
ejpam-4918	19	3	[	[	X
ejpam-4918	19	4	6	6	NUM
ejpam-4918	19	5	]	]	PUNCT
ejpam-4918	19	6	,	,	PUNCT
ejpam-4918	19	7	a	a	DET
ejpam-4918	19	8	ring	ring	NOUN
ejpam-4918	19	9	r	r	NOUN
ejpam-4918	19	10	is	be	AUX
ejpam-4918	19	11	said	say	VERB
ejpam-4918	19	12	to	to	PART
ejpam-4918	19	13	be	be	AUX
ejpam-4918	19	14	anm	anm	PROPN
ejpam-4918	19	15	-armendariz	-armendariz	PROPN
ejpam-4918	19	16	ring	ring	NOUN
ejpam-4918	19	17	of	of	ADP
ejpam-4918	19	18	crossed	cross	VERB
ejpam-4918	19	19	product	product	NOUN
ejpam-4918	19	20	type	type	NOUN
ejpam-4918	19	21	relative	relative	ADJ
ejpam-4918	19	22	to	to	ADP
ejpam-4918	19	23	the	the	DET
ejpam-4918	19	24	given	give	VERB
ejpam-4918	19	25	doi	doi	NOUN
ejpam-4918	19	26	:	:	PUNCT
ejpam-4918	19	27	https://doi.org/10.29020/nybg.ejpam.v16i4.4918	https://doi.org/10.29020/nybg.ejpam.v16i4.4918	PROPN
ejpam-4918	19	28	email	email	NOUN
ejpam-4918	19	29	addresses	address	NOUN
ejpam-4918	19	30	:	:	PUNCT
ejpam-4918	19	31	eltiyeb76@gmail.com	eltiyeb76@gmail.com	PROPN
ejpam-4918	19	32	,	,	PUNCT
ejpam-4918	19	33	emali@nu.edu.sa	emali@nu.edu.sa	PROPN
ejpam-4918	19	34	(	(	PUNCT
ejpam-4918	19	35	e.	e.	PROPN
ejpam-4918	19	36	ali	ali	PROPN
ejpam-4918	19	37	)	)	PUNCT
ejpam-4918	19	38	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4918	19	39	2156	2156	NUM
ejpam-4918	20	1	©	©	ADP
ejpam-4918	20	2	2023	2023	NUM
ejpam-4918	20	3	ejpam	ejpam	NOUN
ejpam-4918	20	4	all	all	DET
ejpam-4918	20	5	rights	right	NOUN
ejpam-4918	20	6	reserved	reserve	VERB
ejpam-4918	20	7	.	.	PUNCT
ejpam-4918	21	1	e.	e.	PROPN
ejpam-4918	21	2	ali	ali	PROPN
ejpam-4918	21	3	/	/	SYM
ejpam-4918	21	4	eur	eur	PROPN
ejpam-4918	21	5	.	.	PUNCT
ejpam-4918	22	1	j.	j.	PROPN
ejpam-4918	22	2	pure	pure	PROPN
ejpam-4918	22	3	appl	appl	PROPN
ejpam-4918	22	4	.	.	PROPN
ejpam-4918	22	5	math	math	PROPN
ejpam-4918	22	6	,	,	PUNCT
ejpam-4918	22	7	16	16	NUM
ejpam-4918	22	8	(	(	PUNCT
ejpam-4918	22	9	4	4	NUM
ejpam-4918	22	10	)	)	PUNCT
ejpam-4918	22	11	(	(	PUNCT
ejpam-4918	22	12	2023	2023	NUM
ejpam-4918	22	13	)	)	PUNCT
ejpam-4918	22	14	,	,	PUNCT
ejpam-4918	22	15	2156	2156	NUM
ejpam-4918	22	16	-	-	SYM
ejpam-4918	22	17	2168	2168	NUM
ejpam-4918	22	18	2157	2157	NUM
ejpam-4918	22	19	twisting	twisting	NOUN
ejpam-4918	22	20	f	f	NOUN
ejpam-4918	22	21	and	and	CCONJ
ejpam-4918	22	22	action	action	NOUN
ejpam-4918	22	23	ω	ω	PROPN
ejpam-4918	22	24	,	,	PUNCT
ejpam-4918	22	25	or	or	CCONJ
ejpam-4918	22	26	an	an	DET
ejpam-4918	22	27	m	m	NOUN
ejpam-4918	22	28	-quasi	-quasi	NOUN
ejpam-4918	22	29	armendariz	armendariz	NOUN
ejpam-4918	22	30	ring	ring	NOUN
ejpam-4918	22	31	(	(	PUNCT
ejpam-4918	22	32	or	or	CCONJ
ejpam-4918	22	33	simply	simply	ADV
ejpam-4918	22	34	a	a	DET
ejpam-4918	22	35	cm	cm	NOUN
ejpam-4918	22	36	-armendariz	-armendariz	NOUN
ejpam-4918	22	37	ring	ring	NOUN
ejpam-4918	22	38	or	or	CCONJ
ejpam-4918	22	39	cm	cm	NOUN
ejpam-4918	22	40	-quasi	-quasi	NOUN
ejpam-4918	22	41	armendariz	armendariz	NOUN
ejpam-4918	22	42	ring	ring	NOUN
ejpam-4918	22	43	,	,	PUNCT
ejpam-4918	22	44	respectively	respectively	ADV
ejpam-4918	22	45	)	)	PUNCT
ejpam-4918	22	46	,	,	PUNCT
ejpam-4918	22	47	if	if	SCONJ
ejpam-4918	22	48	for	for	ADP
ejpam-4918	22	49	any	any	DET
ejpam-4918	22	50	ϕ	ϕ	NOUN
ejpam-4918	22	51	=	=	PUNCT
ejpam-4918	22	52	∑n	∑n	PROPN
ejpam-4918	22	53	i=1	i=1	PROPN
ejpam-4918	22	54	aigi	aigi	NOUN
ejpam-4918	22	55	,	,	PUNCT
ejpam-4918	22	56	ψ	ψ	X
ejpam-4918	22	57	=	=	SYM
ejpam-4918	22	58	∑m	∑m	PROPN
ejpam-4918	22	59	j=1	j=1	PROPN
ejpam-4918	22	60	bjhj	bjhj	NOUN
ejpam-4918	22	61	∈	∈	PROPN
ejpam-4918	22	62	r	r	NOUN
ejpam-4918	22	63	∗m	∗m	NOUN
ejpam-4918	22	64	such	such	ADJ
ejpam-4918	22	65	that	that	SCONJ
ejpam-4918	22	66	ϕψ	ϕψ	ADP
ejpam-4918	22	67	=	=	SYM
ejpam-4918	22	68	0	0	PROPN
ejpam-4918	22	69	(	(	PUNCT
ejpam-4918	22	70	resp	resp	NOUN
ejpam-4918	22	71	.	.	PUNCT
ejpam-4918	22	72	,	,	PUNCT
ejpam-4918	23	1	ϕ(r	ϕ(r	PROPN
ejpam-4918	23	2	∗m)ψ	∗m)ψ	ADJ
ejpam-4918	23	3	=	=	PUNCT
ejpam-4918	24	1	0	0	NUM
ejpam-4918	24	2	)	)	PUNCT
ejpam-4918	24	3	,	,	PUNCT
ejpam-4918	24	4	it	it	PRON
ejpam-4918	24	5	follows	follow	VERB
ejpam-4918	24	6	that	that	SCONJ
ejpam-4918	24	7	aiωgi(bj	aiωgi(bj	PROPN
ejpam-4918	24	8	)	)	PUNCT
ejpam-4918	24	9	=	=	SYM
ejpam-4918	24	10	0	0	NUM
ejpam-4918	24	11	(	(	PUNCT
ejpam-4918	24	12	resp	resp	NOUN
ejpam-4918	24	13	.	.	PUNCT
ejpam-4918	24	14	,	,	PUNCT
ejpam-4918	24	15	airωgil(bj	airωgil(bj	PROPN
ejpam-4918	24	16	)	)	PUNCT
ejpam-4918	24	17	=	=	SYM
ejpam-4918	24	18	0	0	NUM
ejpam-4918	24	19	)	)	PUNCT
ejpam-4918	24	20	for	for	ADP
ejpam-4918	24	21	all	all	DET
ejpam-4918	24	22	i	i	PROPN
ejpam-4918	24	23	,	,	PUNCT
ejpam-4918	24	24	j	j	PROPN
ejpam-4918	24	25	and	and	CCONJ
ejpam-4918	24	26	all	all	DET
ejpam-4918	24	27	gi	gi	NOUN
ejpam-4918	24	28	,	,	PUNCT
ejpam-4918	24	29	hj	hj	PROPN
ejpam-4918	24	30	,	,	PUNCT
ejpam-4918	24	31	l	l	PROPN
ejpam-4918	24	32	∈m	∈m	NOUN
ejpam-4918	24	33	.	.	PUNCT
ejpam-4918	25	1	the	the	DET
ejpam-4918	25	2	focus	focus	NOUN
ejpam-4918	25	3	of	of	ADP
ejpam-4918	25	4	this	this	DET
ejpam-4918	25	5	paper	paper	NOUN
ejpam-4918	25	6	is	be	AUX
ejpam-4918	25	7	on	on	ADP
ejpam-4918	25	8	investigating	investigate	VERB
ejpam-4918	25	9	strongly	strongly	ADV
ejpam-4918	25	10	cm	cm	NUM
ejpam-4918	25	11	-reflexive	-reflexive	NOUN
ejpam-4918	25	12	rings	ring	NOUN
ejpam-4918	25	13	,	,	PUNCT
ejpam-4918	25	14	which	which	PRON
ejpam-4918	25	15	are	be	AUX
ejpam-4918	25	16	a	a	DET
ejpam-4918	25	17	reflexive	reflexive	ADJ
ejpam-4918	25	18	-	-	PUNCT
ejpam-4918	25	19	like	like	ADJ
ejpam-4918	25	20	property	property	NOUN
ejpam-4918	25	21	defined	define	VERB
ejpam-4918	25	22	for	for	ADP
ejpam-4918	25	23	the	the	DET
ejpam-4918	25	24	monoid	monoid	NOUN
ejpam-4918	25	25	crossed	cross	VERB
ejpam-4918	25	26	product	product	NOUN
ejpam-4918	25	27	r	r	NOUN
ejpam-4918	25	28	∗m	∗m	NOUN
ejpam-4918	25	29	with	with	ADP
ejpam-4918	25	30	respect	respect	NOUN
ejpam-4918	25	31	to	to	ADP
ejpam-4918	25	32	the	the	DET
ejpam-4918	25	33	given	give	VERB
ejpam-4918	25	34	twisting	twisting	NOUN
ejpam-4918	25	35	map	map	NOUN
ejpam-4918	25	36	f	f	NOUN
ejpam-4918	25	37	and	and	CCONJ
ejpam-4918	25	38	action	action	NOUN
ejpam-4918	25	39	map	map	NOUN
ejpam-4918	25	40	ω	ω	X
ejpam-4918	25	41	.	.	PUNCT
ejpam-4918	26	1	this	this	DET
ejpam-4918	26	2	concept	concept	NOUN
ejpam-4918	26	3	is	be	AUX
ejpam-4918	26	4	a	a	DET
ejpam-4918	26	5	generalization	generalization	NOUN
ejpam-4918	26	6	of	of	ADP
ejpam-4918	26	7	several	several	ADJ
ejpam-4918	26	8	other	other	ADJ
ejpam-4918	26	9	reflexive	reflexive	ADJ
ejpam-4918	26	10	properties	property	NOUN
ejpam-4918	26	11	,	,	PUNCT
ejpam-4918	26	12	including	include	VERB
ejpam-4918	26	13	reflexive	reflexive	ADJ
ejpam-4918	26	14	rings	ring	NOUN
ejpam-4918	26	15	,	,	PUNCT
ejpam-4918	26	16	strongly	strongly	ADV
ejpam-4918	26	17	reflexive	reflexive	ADJ
ejpam-4918	26	18	rings	ring	NOUN
ejpam-4918	26	19	,	,	PUNCT
ejpam-4918	26	20	strongly	strongly	ADV
ejpam-4918	26	21	m	m	VERB
ejpam-4918	26	22	-reflexive	-reflexive	ADJ
ejpam-4918	26	23	rings	ring	NOUN
ejpam-4918	26	24	,	,	PUNCT
ejpam-4918	26	25	and	and	CCONJ
ejpam-4918	26	26	skew	skew	VERB
ejpam-4918	26	27	monoid	monoid	NOUN
ejpam-4918	26	28	rings	ring	NOUN
ejpam-4918	26	29	.	.	PUNCT
ejpam-4918	27	1	the	the	DET
ejpam-4918	27	2	paper	paper	NOUN
ejpam-4918	27	3	is	be	AUX
ejpam-4918	27	4	devoted	devote	VERB
ejpam-4918	27	5	to	to	AUX
ejpam-4918	27	6	presents	present	VERB
ejpam-4918	27	7	several	several	ADJ
ejpam-4918	27	8	results	result	NOUN
ejpam-4918	27	9	,	,	PUNCT
ejpam-4918	27	10	including	include	VERB
ejpam-4918	27	11	,	,	PUNCT
ejpam-4918	27	12	if	if	SCONJ
ejpam-4918	27	13	r	r	NOUN
ejpam-4918	27	14	is	be	AUX
ejpam-4918	27	15	a	a	DET
ejpam-4918	27	16	semiprime	semiprime	NOUN
ejpam-4918	27	17	,	,	PUNCT
ejpam-4918	27	18	then	then	ADV
ejpam-4918	27	19	r	r	NOUN
ejpam-4918	27	20	is	be	AUX
ejpam-4918	27	21	strongly	strongly	ADV
ejpam-4918	27	22	cm	cm	NOUN
ejpam-4918	27	23	-reflexive	-reflexive	NOUN
ejpam-4918	27	24	for	for	ADP
ejpam-4918	27	25	a	a	DET
ejpam-4918	27	26	u.p.-monoid	u.p.-monoid	ADJ
ejpam-4918	27	27	m	m	NOUN
ejpam-4918	27	28	.	.	PUNCT
ejpam-4918	28	1	also	also	ADV
ejpam-4918	28	2	,	,	PUNCT
ejpam-4918	28	3	if	if	SCONJ
ejpam-4918	28	4	r	r	NOUN
ejpam-4918	28	5	is	be	AUX
ejpam-4918	28	6	a	a	DET
ejpam-4918	28	7	left	left	ADJ
ejpam-4918	28	8	p.q.-baer	p.q.-baer	NOUN
ejpam-4918	28	9	(	(	PUNCT
ejpam-4918	28	10	semiprime	semiprime	NOUN
ejpam-4918	28	11	,	,	PUNCT
ejpam-4918	28	12	left	leave	VERB
ejpam-4918	28	13	app	app	NOUN
ejpam-4918	28	14	-ring	-ring	PROPN
ejpam-4918	28	15	,	,	PUNCT
ejpam-4918	28	16	respectively	respectively	ADV
ejpam-4918	28	17	)	)	PUNCT
ejpam-4918	28	18	,	,	PUNCT
ejpam-4918	28	19	then	then	ADV
ejpam-4918	28	20	r	r	NOUN
ejpam-4918	28	21	is	be	AUX
ejpam-4918	28	22	strongly	strongly	ADV
ejpam-4918	28	23	cm	cm	NOUN
ejpam-4918	28	24	-reflexive	-reflexive	NOUN
ejpam-4918	28	25	for	for	ADP
ejpam-4918	28	26	a	a	DET
ejpam-4918	28	27	strictly	strictly	ADV
ejpam-4918	28	28	totally	totally	ADV
ejpam-4918	28	29	ordered	order	VERB
ejpam-4918	28	30	monoid	monoid	NOUN
ejpam-4918	28	31	.	.	PUNCT
ejpam-4918	29	1	additionally	additionally	ADV
ejpam-4918	29	2	,	,	PUNCT
ejpam-4918	29	3	if	if	SCONJ
ejpam-4918	29	4	r	r	NOUN
ejpam-4918	29	5	is	be	AUX
ejpam-4918	29	6	an	an	DET
ejpam-4918	29	7	m	m	NOUN
ejpam-4918	29	8	-compatible	-compatible	ADJ
ejpam-4918	29	9	ring	ring	NOUN
ejpam-4918	29	10	and	and	CCONJ
ejpam-4918	29	11	m	m	NOUN
ejpam-4918	29	12	is	be	AUX
ejpam-4918	29	13	a	a	DET
ejpam-4918	29	14	monoid	monoid	NOUN
ejpam-4918	29	15	with	with	ADP
ejpam-4918	29	16	twisting	twist	VERB
ejpam-4918	29	17	f	f	NOUN
ejpam-4918	29	18	and	and	CCONJ
ejpam-4918	29	19	action	action	NOUN
ejpam-4918	29	20	ω	ω	PROPN
ejpam-4918	29	21	as	as	ADP
ejpam-4918	29	22	above	above	ADV
ejpam-4918	29	23	,	,	PUNCT
ejpam-4918	29	24	then	then	ADV
ejpam-4918	29	25	for	for	SCONJ
ejpam-4918	29	26	any	any	DET
ejpam-4918	29	27	reduced	reduce	VERB
ejpam-4918	29	28	ideal	ideal	NOUN
ejpam-4918	29	29	i	i	PRON
ejpam-4918	29	30	of	of	ADP
ejpam-4918	29	31	r	r	NOUN
ejpam-4918	29	32	such	such	ADJ
ejpam-4918	29	33	that	that	PRON
ejpam-4918	29	34	r	r	NOUN
ejpam-4918	29	35	/	/	SYM
ejpam-4918	29	36	i	i	PRON
ejpam-4918	29	37	is	be	AUX
ejpam-4918	29	38	strongly	strongly	ADV
ejpam-4918	29	39	cm	cm	NUM
ejpam-4918	29	40	reflexive	reflexive	ADJ
ejpam-4918	29	41	,	,	PUNCT
ejpam-4918	29	42	then	then	ADV
ejpam-4918	29	43	r	r	NOUN
ejpam-4918	29	44	is	be	AUX
ejpam-4918	29	45	strongly	strongly	ADV
ejpam-4918	29	46	cm	cm	NOUN
ejpam-4918	29	47	-reflexive	-reflexive	NOUN
ejpam-4918	29	48	.	.	PUNCT
ejpam-4918	30	1	moreover	moreover	ADV
ejpam-4918	30	2	,	,	PUNCT
ejpam-4918	30	3	for	for	ADP
ejpam-4918	30	4	a	a	DET
ejpam-4918	30	5	right	right	ADJ
ejpam-4918	30	6	ore	ore	NOUN
ejpam-4918	30	7	ring	ring	NOUN
ejpam-4918	30	8	r	r	NOUN
ejpam-4918	30	9	with	with	ADP
ejpam-4918	30	10	classical	classical	ADJ
ejpam-4918	30	11	right	right	ADJ
ejpam-4918	30	12	quotient	quotient	NOUN
ejpam-4918	30	13	ring	ring	NOUN
ejpam-4918	30	14	q	q	PROPN
ejpam-4918	30	15	,	,	PUNCT
ejpam-4918	30	16	we	we	PRON
ejpam-4918	30	17	show	show	VERB
ejpam-4918	30	18	that	that	SCONJ
ejpam-4918	30	19	r	r	NOUN
ejpam-4918	30	20	is	be	AUX
ejpam-4918	30	21	strongly	strongly	ADV
ejpam-4918	30	22	cm	cm	NOUN
ejpam-4918	30	23	-reflexive	-reflexive	NOUN
ejpam-4918	30	24	if	if	SCONJ
ejpam-4918	31	1	and	and	CCONJ
ejpam-4918	31	2	only	only	ADV
ejpam-4918	31	3	if	if	SCONJ
ejpam-4918	31	4	q	q	NOUN
ejpam-4918	31	5	is	be	AUX
ejpam-4918	31	6	strongly	strongly	ADV
ejpam-4918	31	7	cm	cm	NOUN
ejpam-4918	31	8	-reflexive	-reflexive	NOUN
ejpam-4918	31	9	.	.	PUNCT
ejpam-4918	32	1	finally	finally	ADV
ejpam-4918	32	2	,	,	PUNCT
ejpam-4918	32	3	we	we	PRON
ejpam-4918	32	4	discuss	discuss	VERB
ejpam-4918	32	5	example	example	NOUN
ejpam-4918	32	6	and	and	CCONJ
ejpam-4918	32	7	some	some	DET
ejpam-4918	32	8	results	result	NOUN
ejpam-4918	32	9	in	in	ADP
ejpam-4918	32	10	the	the	DET
ejpam-4918	32	11	subject	subject	NOUN
ejpam-4918	32	12	.	.	PUNCT
ejpam-4918	33	1	to	to	PART
ejpam-4918	33	2	begin	begin	VERB
ejpam-4918	33	3	with	with	ADP
ejpam-4918	33	4	,	,	PUNCT
ejpam-4918	33	5	we	we	PRON
ejpam-4918	33	6	introduce	introduce	VERB
ejpam-4918	33	7	some	some	DET
ejpam-4918	33	8	notions	notion	NOUN
ejpam-4918	33	9	and	and	CCONJ
ejpam-4918	33	10	notations	notation	NOUN
ejpam-4918	33	11	relevant	relevant	ADJ
ejpam-4918	33	12	to	to	ADP
ejpam-4918	33	13	this	this	DET
ejpam-4918	33	14	paper	paper	NOUN
ejpam-4918	33	15	.	.	PUNCT
ejpam-4918	34	1	let	let	VERB
ejpam-4918	34	2	ω	ω	NOUN
ejpam-4918	34	3	:	:	PUNCT
ejpam-4918	34	4	m	m	PROPN
ejpam-4918	34	5	→	→	SYM
ejpam-4918	34	6	aut(r	aut(r	PROPN
ejpam-4918	34	7	)	)	PUNCT
ejpam-4918	34	8	be	be	AUX
ejpam-4918	34	9	a	a	DET
ejpam-4918	34	10	monoid	monoid	NOUN
ejpam-4918	34	11	homomorphism	homomorphism	NOUN
ejpam-4918	34	12	.	.	PUNCT
ejpam-4918	35	1	for	for	ADP
ejpam-4918	35	2	h	h	NOUN
ejpam-4918	35	3	∈m	∈m	NOUN
ejpam-4918	35	4	,	,	PUNCT
ejpam-4918	35	5	we	we	PRON
ejpam-4918	35	6	denote	denote	VERB
ejpam-4918	35	7	by	by	ADP
ejpam-4918	35	8	ωh	ωh	ADP
ejpam-4918	35	9	the	the	DET
ejpam-4918	35	10	automorphism	automorphism	NOUN
ejpam-4918	35	11	ω(h	ω(h	NUM
ejpam-4918	35	12	)	)	PUNCT
ejpam-4918	35	13	.	.	PUNCT
ejpam-4918	36	1	the	the	DET
ejpam-4918	36	2	crossed	cross	VERB
ejpam-4918	36	3	product	product	NOUN
ejpam-4918	36	4	r	r	NOUN
ejpam-4918	36	5	∗m	∗m	NOUN
ejpam-4918	36	6	over	over	ADP
ejpam-4918	36	7	r	r	NOUN
ejpam-4918	36	8	is	be	AUX
ejpam-4918	36	9	defined	define	VERB
ejpam-4918	36	10	as	as	ADP
ejpam-4918	36	11	the	the	DET
ejpam-4918	36	12	set	set	NOUN
ejpam-4918	36	13	of	of	ADP
ejpam-4918	36	14	all	all	DET
ejpam-4918	36	15	finite	finite	NOUN
ejpam-4918	36	16	sums	sum	NOUN
ejpam-4918	36	17	r	r	NOUN
ejpam-4918	36	18	∗m	∗m	NOUN
ejpam-4918	36	19	=	=	SYM
ejpam-4918	36	20	{	{	PUNCT
ejpam-4918	37	1	xhh|xh	xhh|xh	PROPN
ejpam-4918	37	2	∈	∈	PROPN
ejpam-4918	37	3	r	r	NOUN
ejpam-4918	37	4	,	,	PUNCT
ejpam-4918	37	5	h	h	NOUN
ejpam-4918	37	6	∈	∈	PROPN
ejpam-4918	37	7	m	m	PRON
ejpam-4918	37	8	}	}	PUNCT
ejpam-4918	37	9	,	,	PUNCT
ejpam-4918	37	10	where	where	SCONJ
ejpam-4918	37	11	addition	addition	NOUN
ejpam-4918	37	12	is	be	AUX
ejpam-4918	37	13	defined	define	VERB
ejpam-4918	37	14	component	component	NOUN
ejpam-4918	37	15	-	-	PUNCT
ejpam-4918	37	16	wise	wise	ADJ
ejpam-4918	37	17	and	and	CCONJ
ejpam-4918	37	18	multiplication	multiplication	NOUN
ejpam-4918	37	19	is	be	AUX
ejpam-4918	37	20	defined	define	VERB
ejpam-4918	37	21	using	use	VERB
ejpam-4918	37	22	the	the	DET
ejpam-4918	37	23	distributive	distributive	ADJ
ejpam-4918	37	24	law	law	NOUN
ejpam-4918	37	25	and	and	CCONJ
ejpam-4918	37	26	two	two	NUM
ejpam-4918	37	27	rules	rule	NOUN
ejpam-4918	37	28	known	know	VERB
ejpam-4918	37	29	as	as	ADP
ejpam-4918	37	30	action	action	NOUN
ejpam-4918	37	31	and	and	CCONJ
ejpam-4918	37	32	twisting	twisting	NOUN
ejpam-4918	37	33	.	.	PUNCT
ejpam-4918	38	1	specifically	specifically	ADV
ejpam-4918	38	2	,	,	PUNCT
ejpam-4918	38	3	for	for	ADP
ejpam-4918	38	4	l	l	NOUN
ejpam-4918	38	5	,	,	PUNCT
ejpam-4918	38	6	h	h	NOUN
ejpam-4918	38	7	∈	∈	PROPN
ejpam-4918	38	8	m	m	VERB
ejpam-4918	38	9	and	and	CCONJ
ejpam-4918	38	10	x	x	PUNCT
ejpam-4918	38	11	∈	∈	NOUN
ejpam-4918	38	12	r	r	NOUN
ejpam-4918	38	13	,	,	PUNCT
ejpam-4918	38	14	we	we	PRON
ejpam-4918	38	15	have	have	AUX
ejpam-4918	38	16	hx	hx	NOUN
ejpam-4918	38	17	=	=	SYM
ejpam-4918	38	18	ωh(x)h	ωh(x)h	PROPN
ejpam-4918	38	19	and	and	CCONJ
ejpam-4918	38	20	l	l	NOUN
ejpam-4918	38	21	h	h	NOUN
ejpam-4918	39	1	=	=	SYM
ejpam-4918	39	2	f(l	f(l	PROPN
ejpam-4918	39	3	,	,	PUNCT
ejpam-4918	39	4	h)l	h)l	ADJ
ejpam-4918	39	5	h	h	NOUN
ejpam-4918	39	6	,	,	PUNCT
ejpam-4918	39	7	where	where	SCONJ
ejpam-4918	39	8	f	f	X
ejpam-4918	39	9	:	:	PUNCT
ejpam-4918	39	10	m	m	VERB
ejpam-4918	39	11	×m	×m	NOUN
ejpam-4918	39	12	→	→	SYM
ejpam-4918	39	13	u(r	u(r	NOUN
ejpam-4918	39	14	)	)	PUNCT
ejpam-4918	39	15	is	be	AUX
ejpam-4918	39	16	a	a	DET
ejpam-4918	39	17	twisted	twisted	ADJ
ejpam-4918	39	18	function	function	NOUN
ejpam-4918	39	19	and	and	CCONJ
ejpam-4918	39	20	u(r	u(r	NOUN
ejpam-4918	39	21	)	)	PUNCT
ejpam-4918	39	22	denotes	denote	VERB
ejpam-4918	39	23	the	the	DET
ejpam-4918	39	24	set	set	NOUN
ejpam-4918	39	25	of	of	ADP
ejpam-4918	39	26	units	unit	NOUN
ejpam-4918	39	27	of	of	ADP
ejpam-4918	39	28	r.	r.	PROPN
ejpam-4918	39	29	here	here	ADV
ejpam-4918	39	30	,	,	PUNCT
ejpam-4918	39	31	the	the	DET
ejpam-4918	39	32	twisted	twisted	ADJ
ejpam-4918	39	33	function	function	NOUN
ejpam-4918	39	34	f	f	PROPN
ejpam-4918	39	35	and	and	CCONJ
ejpam-4918	39	36	the	the	DET
ejpam-4918	39	37	action	action	NOUN
ejpam-4918	39	38	ω	ω	NUM
ejpam-4918	39	39	of	of	ADP
ejpam-4918	39	40	m	m	PRON
ejpam-4918	39	41	on	on	ADP
ejpam-4918	39	42	r	r	NOUN
ejpam-4918	39	43	satisfy	satisfy	VERB
ejpam-4918	39	44	the	the	DET
ejpam-4918	39	45	following	follow	VERB
ejpam-4918	39	46	conditions	condition	NOUN
ejpam-4918	39	47	:	:	PUNCT
ejpam-4918	39	48	ωl(ωh(x	ωl(ωh(x	NUM
ejpam-4918	39	49	)	)	PUNCT
ejpam-4918	39	50	)	)	PUNCT
ejpam-4918	40	1	=	=	SYM
ejpam-4918	40	2	f(l	f(l	PROPN
ejpam-4918	40	3	,	,	PUNCT
ejpam-4918	40	4	h)ωl(ωh(x)f(l	h)ωl(ωh(x)f(l	VERB
ejpam-4918	40	5	,	,	PUNCT
ejpam-4918	40	6	h	h	NOUN
ejpam-4918	40	7	)	)	PUNCT
ejpam-4918	40	8	−1	−1	NOUN
ejpam-4918	40	9	)	)	PUNCT
ejpam-4918	40	10	,	,	PUNCT
ejpam-4918	40	11	ωl(f(h	ωl(f(h	NUM
ejpam-4918	40	12	,	,	PUNCT
ejpam-4918	40	13	k))f(l	k))f(l	PROPN
ejpam-4918	40	14	,	,	PUNCT
ejpam-4918	40	15	hk	hk	PROPN
ejpam-4918	40	16	)	)	PUNCT
ejpam-4918	40	17	=	=	SYM
ejpam-4918	41	1	f(l	f(l	PROPN
ejpam-4918	41	2	,	,	PUNCT
ejpam-4918	41	3	h)f(l	h)f(l	VERB
ejpam-4918	41	4	h	h	NOUN
ejpam-4918	41	5	,	,	PUNCT
ejpam-4918	41	6	k	k	NOUN
ejpam-4918	41	7	)	)	PUNCT
ejpam-4918	41	8	,	,	PUNCT
ejpam-4918	41	9	f(1	f(1	PROPN
ejpam-4918	41	10	,	,	PUNCT
ejpam-4918	41	11	l	l	NOUN
ejpam-4918	41	12	)	)	PUNCT
ejpam-4918	41	13	=	=	SYM
ejpam-4918	41	14	f(l	f(l	NOUN
ejpam-4918	41	15	,	,	PUNCT
ejpam-4918	41	16	1	1	NUM
ejpam-4918	41	17	)	)	PUNCT
ejpam-4918	41	18	=	=	SYM
ejpam-4918	41	19	1	1	NUM
ejpam-4918	41	20	for	for	ADP
ejpam-4918	41	21	all	all	DET
ejpam-4918	41	22	l	l	NOUN
ejpam-4918	41	23	,	,	PUNCT
ejpam-4918	41	24	h	h	NOUN
ejpam-4918	41	25	,	,	PUNCT
ejpam-4918	41	26	k	k	PROPN
ejpam-4918	41	27	∈	∈	PROPN
ejpam-4918	41	28	m	m	VERB
ejpam-4918	41	29	.	.	PUNCT
ejpam-4918	42	1	it	it	PRON
ejpam-4918	42	2	is	be	AUX
ejpam-4918	42	3	worth	worth	ADJ
ejpam-4918	42	4	noting	note	VERB
ejpam-4918	42	5	that	that	SCONJ
ejpam-4918	42	6	the	the	DET
ejpam-4918	42	7	monoid	monoid	NOUN
ejpam-4918	42	8	crossed	cross	VERB
ejpam-4918	42	9	product	product	NOUN
ejpam-4918	42	10	is	be	AUX
ejpam-4918	42	11	a	a	DET
ejpam-4918	42	12	general	general	ADJ
ejpam-4918	42	13	ring	ring	NOUN
ejpam-4918	42	14	construction	construction	NOUN
ejpam-4918	42	15	.	.	PUNCT
ejpam-4918	43	1	given	give	VERB
ejpam-4918	43	2	a	a	DET
ejpam-4918	43	3	monoid	monoid	NOUN
ejpam-4918	43	4	crossed	cross	VERB
ejpam-4918	43	5	product	product	NOUN
ejpam-4918	43	6	r	r	NOUN
ejpam-4918	43	7	∗m	∗m	NOUN
ejpam-4918	43	8	with	with	ADP
ejpam-4918	43	9	twisting	twist	VERB
ejpam-4918	43	10	f	f	NOUN
ejpam-4918	43	11	and	and	CCONJ
ejpam-4918	43	12	action	action	NOUN
ejpam-4918	43	13	ω	ω	NOUN
ejpam-4918	43	14	,	,	PUNCT
ejpam-4918	43	15	if	if	SCONJ
ejpam-4918	43	16	the	the	DET
ejpam-4918	43	17	twisting	twisting	NOUN
ejpam-4918	43	18	f	f	NOUN
ejpam-4918	43	19	is	be	AUX
ejpam-4918	43	20	trivial	trivial	ADJ
ejpam-4918	43	21	,	,	PUNCT
ejpam-4918	43	22	(	(	PUNCT
ejpam-4918	43	23	i.e.	i.e.	X
ejpam-4918	43	24	,	,	PUNCT
ejpam-4918	43	25	f(a	f(a	NOUN
ejpam-4918	43	26	,	,	PUNCT
ejpam-4918	43	27	b	b	NOUN
ejpam-4918	43	28	)	)	PUNCT
ejpam-4918	43	29	=	=	SYM
ejpam-4918	43	30	1	1	X
ejpam-4918	43	31	)	)	PUNCT
ejpam-4918	43	32	for	for	ADP
ejpam-4918	43	33	all	all	DET
ejpam-4918	43	34	a	a	DET
ejpam-4918	43	35	,	,	PUNCT
ejpam-4918	43	36	b	b	NOUN
ejpam-4918	43	37	∈m	∈m	NOUN
ejpam-4918	43	38	,	,	PUNCT
ejpam-4918	43	39	then	then	ADV
ejpam-4918	43	40	r	r	PROPN
ejpam-4918	43	41	∗m	∗m	PROPN
ejpam-4918	43	42	is	be	AUX
ejpam-4918	43	43	the	the	DET
ejpam-4918	43	44	skew	skew	ADJ
ejpam-4918	43	45	monoid	monoid	NOUN
ejpam-4918	43	46	ring	ring	NOUN
ejpam-4918	43	47	r	r	NOUN
ejpam-4918	43	48	∗m	∗m	NOUN
ejpam-4918	43	49	.	.	PUNCT
ejpam-4918	44	1	if	if	SCONJ
ejpam-4918	44	2	both	both	PRON
ejpam-4918	44	3	the	the	DET
ejpam-4918	44	4	twisting	twisting	NOUN
ejpam-4918	44	5	f	f	NOUN
ejpam-4918	44	6	and	and	CCONJ
ejpam-4918	44	7	the	the	DET
ejpam-4918	44	8	action	action	NOUN
ejpam-4918	44	9	ω	ω	NOUN
ejpam-4918	44	10	are	be	AUX
ejpam-4918	44	11	trivial	trivial	ADJ
ejpam-4918	44	12	,	,	PUNCT
ejpam-4918	44	13	then	then	ADV
ejpam-4918	44	14	r	r	NOUN
ejpam-4918	44	15	∗m	∗m	NOUN
ejpam-4918	44	16	is	be	AUX
ejpam-4918	44	17	a	a	DET
ejpam-4918	44	18	monoid	monoid	NOUN
ejpam-4918	44	19	ring	ring	NOUN
ejpam-4918	44	20	denoted	denote	VERB
ejpam-4918	44	21	by	by	ADP
ejpam-4918	44	22	r[m	r[m	NOUN
ejpam-4918	44	23	]	]	PUNCT
ejpam-4918	44	24	(	(	PUNCT
ejpam-4918	44	25	see	see	VERB
ejpam-4918	44	26	[	[	X
ejpam-4918	44	27	7	7	X
ejpam-4918	44	28	]	]	PUNCT
ejpam-4918	44	29	and	and	CCONJ
ejpam-4918	44	30	[	[	X
ejpam-4918	44	31	8	8	NUM
ejpam-4918	44	32	]	]	NUM
ejpam-4918	44	33	)	)	PUNCT
ejpam-4918	44	34	.	.	PUNCT
ejpam-4918	45	1	a	a	DET
ejpam-4918	45	2	monoid	monoid	NOUN
ejpam-4918	45	3	m	m	NOUN
ejpam-4918	45	4	is	be	AUX
ejpam-4918	45	5	said	say	VERB
ejpam-4918	45	6	to	to	PART
ejpam-4918	45	7	be	be	AUX
ejpam-4918	45	8	a	a	DET
ejpam-4918	45	9	u.p.-monoid	u.p.-monoid	ADJ
ejpam-4918	45	10	(	(	PUNCT
ejpam-4918	45	11	unique	unique	ADJ
ejpam-4918	45	12	product	product	NOUN
ejpam-4918	45	13	monoid	monoid	NOUN
ejpam-4918	45	14	)	)	PUNCT
ejpam-4918	45	15	if	if	SCONJ
ejpam-4918	45	16	,	,	PUNCT
ejpam-4918	45	17	for	for	ADP
ejpam-4918	45	18	any	any	DET
ejpam-4918	45	19	two	two	NUM
ejpam-4918	45	20	nonempty	nonempty	ADJ
ejpam-4918	45	21	finite	finite	ADJ
ejpam-4918	45	22	subsets	subset	NOUN
ejpam-4918	45	23	x	x	PUNCT
ejpam-4918	45	24	and	and	CCONJ
ejpam-4918	45	25	y	y	PROPN
ejpam-4918	45	26	of	of	ADP
ejpam-4918	45	27	m	m	PROPN
ejpam-4918	45	28	,	,	PUNCT
ejpam-4918	45	29	there	there	PRON
ejpam-4918	45	30	exists	exist	VERB
ejpam-4918	45	31	a	a	DET
ejpam-4918	45	32	unique	unique	ADJ
ejpam-4918	45	33	element	element	NOUN
ejpam-4918	45	34	h	h	NOUN
ejpam-4918	45	35	∈m	∈m	NOUN
ejpam-4918	45	36	that	that	PRON
ejpam-4918	45	37	can	can	AUX
ejpam-4918	45	38	be	be	AUX
ejpam-4918	45	39	written	write	VERB
ejpam-4918	45	40	in	in	ADP
ejpam-4918	45	41	the	the	DET
ejpam-4918	45	42	form	form	NOUN
ejpam-4918	45	43	h	h	NOUN
ejpam-4918	46	1	=	=	NOUN
ejpam-4918	46	2	uv	uv	NOUN
ejpam-4918	46	3	with	with	ADP
ejpam-4918	46	4	u	u	PROPN
ejpam-4918	46	5	∈	∈	PROPN
ejpam-4918	46	6	x	x	X
ejpam-4918	46	7	and	and	CCONJ
ejpam-4918	46	8	v	v	ADP
ejpam-4918	46	9	∈	∈	PROPN
ejpam-4918	46	10	y	y	PROPN
ejpam-4918	46	11	.	.	PUNCT
ejpam-4918	47	1	an	an	DET
ejpam-4918	47	2	ordered	order	VERB
ejpam-4918	47	3	monoid	monoid	NOUN
ejpam-4918	47	4	(	(	PUNCT
ejpam-4918	47	5	m,⪯	m,⪯	NOUN
ejpam-4918	47	6	)	)	PUNCT
ejpam-4918	47	7	is	be	AUX
ejpam-4918	47	8	said	say	VERB
ejpam-4918	47	9	to	to	PART
ejpam-4918	47	10	be	be	AUX
ejpam-4918	47	11	strictly	strictly	ADV
ejpam-4918	47	12	ordered	order	VERB
ejpam-4918	47	13	if	if	SCONJ
ejpam-4918	47	14	the	the	DET
ejpam-4918	47	15	following	follow	VERB
ejpam-4918	47	16	condition	condition	NOUN
ejpam-4918	47	17	holds	hold	VERB
ejpam-4918	47	18	:	:	PUNCT
ejpam-4918	47	19	whenever	whenever	SCONJ
ejpam-4918	47	20	g	g	NOUN
ejpam-4918	47	21	,	,	PUNCT
ejpam-4918	47	22	k	k	PROPN
ejpam-4918	47	23	,	,	PUNCT
ejpam-4918	47	24	h	h	NOUN
ejpam-4918	47	25	∈	∈	PROPN
ejpam-4918	47	26	m	m	VERB
ejpam-4918	47	27	with	with	ADP
ejpam-4918	47	28	g	g	PROPN
ejpam-4918	47	29	≺	≺	NOUN
ejpam-4918	47	30	k	k	NOUN
ejpam-4918	47	31	,	,	PUNCT
ejpam-4918	47	32	it	it	PRON
ejpam-4918	47	33	follows	follow	VERB
ejpam-4918	47	34	that	that	SCONJ
ejpam-4918	47	35	gh	gh	PROPN
ejpam-4918	47	36	≺	≺	NOUN
ejpam-4918	47	37	kh	kh	PROPN
ejpam-4918	47	38	and	and	CCONJ
ejpam-4918	47	39	hg	hg	PROPN
ejpam-4918	47	40	≺	≺	PROPN
ejpam-4918	47	41	hk	hk	PROPN
ejpam-4918	47	42	.	.	PROPN
ejpam-4918	47	43	2	2	NUM
ejpam-4918	47	44	.	.	NUM
ejpam-4918	47	45	generalized	generalize	VERB
ejpam-4918	47	46	reflexive	reflexive	ADJ
ejpam-4918	47	47	rings	ring	NOUN
ejpam-4918	47	48	of	of	ADP
ejpam-4918	47	49	crossed	cross	VERB
ejpam-4918	47	50	product	product	NOUN
ejpam-4918	47	51	type	type	NOUN
ejpam-4918	47	52	in	in	ADP
ejpam-4918	47	53	this	this	DET
ejpam-4918	47	54	section	section	NOUN
ejpam-4918	47	55	,	,	PUNCT
ejpam-4918	47	56	we	we	PRON
ejpam-4918	47	57	will	will	AUX
ejpam-4918	47	58	discuss	discuss	VERB
ejpam-4918	47	59	the	the	DET
ejpam-4918	47	60	concept	concept	NOUN
ejpam-4918	47	61	of	of	ADP
ejpam-4918	47	62	strongly	strongly	ADV
ejpam-4918	47	63	reflexive	reflexive	ADJ
ejpam-4918	47	64	properties	property	NOUN
ejpam-4918	47	65	in	in	ADP
ejpam-4918	47	66	the	the	DET
ejpam-4918	47	67	context	context	NOUN
ejpam-4918	47	68	of	of	ADP
ejpam-4918	47	69	a	a	DET
ejpam-4918	47	70	monoid	monoid	NOUN
ejpam-4918	47	71	of	of	ADP
ejpam-4918	47	72	crossed	cross	VERB
ejpam-4918	47	73	product	product	NOUN
ejpam-4918	47	74	r∗m	r∗m	NOUN
ejpam-4918	47	75	,	,	PUNCT
ejpam-4918	47	76	where	where	SCONJ
ejpam-4918	47	77	r	r	NOUN
ejpam-4918	47	78	is	be	AUX
ejpam-4918	47	79	a	a	DET
ejpam-4918	47	80	ring	ring	NOUN
ejpam-4918	47	81	and	and	CCONJ
ejpam-4918	47	82	m	m	NOUN
ejpam-4918	47	83	is	be	AUX
ejpam-4918	47	84	a	a	DET
ejpam-4918	47	85	monoid	monoid	NOUN
ejpam-4918	47	86	with	with	ADP
ejpam-4918	47	87	a	a	DET
ejpam-4918	47	88	twisting	twisting	NOUN
ejpam-4918	47	89	map	map	NOUN
ejpam-4918	47	90	f	f	X
ejpam-4918	47	91	:	:	PUNCT
ejpam-4918	47	92	m	m	VERB
ejpam-4918	47	93	×m	×m	NOUN
ejpam-4918	47	94	→	→	SYM
ejpam-4918	47	95	u(r	u(r	NOUN
ejpam-4918	47	96	)	)	PUNCT
ejpam-4918	47	97	and	and	CCONJ
ejpam-4918	47	98	an	an	DET
ejpam-4918	47	99	action	action	NOUN
ejpam-4918	47	100	map	map	NOUN
ejpam-4918	47	101	ω	ω	NOUN
ejpam-4918	47	102	:	:	PUNCT
ejpam-4918	47	103	m	m	PROPN
ejpam-4918	47	104	→	→	SYM
ejpam-4918	47	105	aut(r	aut(r	PROPN
ejpam-4918	47	106	)	)	PUNCT
ejpam-4918	47	107	.	.	PUNCT
ejpam-4918	48	1	e.	e.	PROPN
ejpam-4918	48	2	ali	ali	PROPN
ejpam-4918	48	3	/	/	SYM
ejpam-4918	48	4	eur	eur	PROPN
ejpam-4918	48	5	.	.	PUNCT
ejpam-4918	49	1	j.	j.	PROPN
ejpam-4918	49	2	pure	pure	PROPN
ejpam-4918	49	3	appl	appl	PROPN
ejpam-4918	49	4	.	.	PROPN
ejpam-4918	49	5	math	math	PROPN
ejpam-4918	49	6	,	,	PUNCT
ejpam-4918	49	7	16	16	NUM
ejpam-4918	49	8	(	(	PUNCT
ejpam-4918	49	9	4	4	NUM
ejpam-4918	49	10	)	)	PUNCT
ejpam-4918	49	11	(	(	PUNCT
ejpam-4918	49	12	2023	2023	NUM
ejpam-4918	49	13	)	)	PUNCT
ejpam-4918	49	14	,	,	PUNCT
ejpam-4918	49	15	2156	2156	NUM
ejpam-4918	49	16	-	-	SYM
ejpam-4918	49	17	2168	2168	NUM
ejpam-4918	49	18	2158	2158	NUM
ejpam-4918	49	19	definition	definition	NOUN
ejpam-4918	49	20	1	1	NUM
ejpam-4918	49	21	.	.	PUNCT
ejpam-4918	50	1	a	a	DET
ejpam-4918	50	2	ring	ring	NOUN
ejpam-4918	50	3	r	r	NOUN
ejpam-4918	50	4	is	be	AUX
ejpam-4918	50	5	said	say	VERB
ejpam-4918	50	6	to	to	PART
ejpam-4918	50	7	be	be	AUX
ejpam-4918	50	8	strongly	strongly	ADV
ejpam-4918	50	9	m	m	PRON
ejpam-4918	50	10	-reflexive	-reflexive	NOUN
ejpam-4918	50	11	of	of	ADP
ejpam-4918	50	12	crossed	cross	VERB
ejpam-4918	50	13	product	product	NOUN
ejpam-4918	50	14	type	type	NOUN
ejpam-4918	50	15	with	with	ADP
ejpam-4918	50	16	respect	respect	NOUN
ejpam-4918	50	17	to	to	ADP
ejpam-4918	50	18	the	the	DET
ejpam-4918	50	19	given	give	VERB
ejpam-4918	50	20	twisting	twisting	NOUN
ejpam-4918	50	21	map	map	NOUN
ejpam-4918	50	22	f	f	NOUN
ejpam-4918	50	23	and	and	CCONJ
ejpam-4918	50	24	action	action	NOUN
ejpam-4918	50	25	map	map	NOUN
ejpam-4918	50	26	ω	ω	X
ejpam-4918	50	27	(	(	PUNCT
ejpam-4918	50	28	or	or	CCONJ
ejpam-4918	50	29	simply	simply	ADV
ejpam-4918	50	30	,	,	PUNCT
ejpam-4918	50	31	strongly	strongly	ADV
ejpam-4918	50	32	cm	cm	NOUN
ejpam-4918	50	33	-reflexive	-reflexive	NOUN
ejpam-4918	50	34	)	)	PUNCT
ejpam-4918	50	35	if	if	SCONJ
ejpam-4918	50	36	for	for	ADP
ejpam-4918	50	37	any	any	DET
ejpam-4918	50	38	ϕ	ϕ	NOUN
ejpam-4918	50	39	=	=	SYM
ejpam-4918	50	40	c1l1	c1l1	X
ejpam-4918	50	41	+	+	CCONJ
ejpam-4918	50	42	c2l2	c2l2	ADJ
ejpam-4918	50	43	+	+	X
ejpam-4918	50	44	·	·	PUNCT
ejpam-4918	50	45	·	·	PUNCT
ejpam-4918	50	46	·	·	PUNCT
ejpam-4918	51	1	+	+	NUM
ejpam-4918	51	2	cnln	cnln	NOUN
ejpam-4918	51	3	and	and	CCONJ
ejpam-4918	51	4	ψ	ψ	NOUN
ejpam-4918	51	5	=	=	PUNCT
ejpam-4918	51	6	a1h1+a2h2	a1h1+a2h2	PROPN
ejpam-4918	51	7	+	+	X
ejpam-4918	51	8	·	·	PUNCT
ejpam-4918	51	9	·	·	PUNCT
ejpam-4918	51	10	·	·	PUNCT
ejpam-4918	52	1	+	+	NUM
ejpam-4918	52	2	amhm	amhm	NOUN
ejpam-4918	52	3	∈	∈	PROPN
ejpam-4918	52	4	r	r	NOUN
ejpam-4918	52	5	∗m	∗m	NOUN
ejpam-4918	52	6	satisfying	satisfy	VERB
ejpam-4918	52	7	that	that	SCONJ
ejpam-4918	52	8	ϕ(r	ϕ(r	PROPN
ejpam-4918	52	9	∗m)ψ	∗m)ψ	VERB
ejpam-4918	52	10	=	=	PUNCT
ejpam-4918	52	11	0	0	NUM
ejpam-4918	52	12	implies	imply	VERB
ejpam-4918	52	13	that	that	SCONJ
ejpam-4918	52	14	ciωli(ωg(raj	ciωli(ωg(raj	VERB
ejpam-4918	52	15	)	)	PUNCT
ejpam-4918	52	16	)	)	PUNCT
ejpam-4918	53	1	=	=	SYM
ejpam-4918	53	2	0	0	NUM
ejpam-4918	53	3	,	,	PUNCT
ejpam-4918	53	4	then	then	ADV
ejpam-4918	53	5	ψ(r	ψ(r	PROPN
ejpam-4918	53	6	∗m)ϕ	∗m)ϕ	NOUN
ejpam-4918	53	7	=	=	SYM
ejpam-4918	53	8	0	0	NUM
ejpam-4918	53	9	for	for	ADP
ejpam-4918	53	10	each	each	DET
ejpam-4918	53	11	i	i	PROPN
ejpam-4918	53	12	,	,	PUNCT
ejpam-4918	53	13	j	j	PROPN
ejpam-4918	53	14	and	and	CCONJ
ejpam-4918	53	15	for	for	ADP
ejpam-4918	53	16	all	all	DET
ejpam-4918	53	17	g	g	PROPN
ejpam-4918	53	18	,	,	PUNCT
ejpam-4918	53	19	li	li	PROPN
ejpam-4918	53	20	,	,	PUNCT
ejpam-4918	53	21	hj	hj	PROPN
ejpam-4918	53	22	∈m	∈m	NOUN
ejpam-4918	53	23	.	.	PUNCT
ejpam-4918	53	24	remark	remark	PROPN
ejpam-4918	53	25	1	1	NUM
ejpam-4918	53	26	.	.	PUNCT
ejpam-4918	54	1	(	(	PUNCT
ejpam-4918	54	2	1	1	X
ejpam-4918	54	3	)	)	PUNCT
ejpam-4918	54	4	if	if	SCONJ
ejpam-4918	54	5	a	a	DET
ejpam-4918	54	6	ring	ring	NOUN
ejpam-4918	54	7	r	r	NOUN
ejpam-4918	54	8	is	be	AUX
ejpam-4918	54	9	strongly	strongly	ADV
ejpam-4918	54	10	cm	cm	NOUN
ejpam-4918	54	11	-reflexive	-reflexive	NOUN
ejpam-4918	54	12	with	with	ADP
ejpam-4918	54	13	a	a	DET
ejpam-4918	54	14	trivial	trivial	ADJ
ejpam-4918	54	15	twisting	twisting	NOUN
ejpam-4918	54	16	map	map	NOUN
ejpam-4918	54	17	f	f	NOUN
ejpam-4918	54	18	,	,	PUNCT
ejpam-4918	54	19	then	then	ADV
ejpam-4918	54	20	we	we	PRON
ejpam-4918	54	21	refer	refer	VERB
ejpam-4918	54	22	to	to	ADP
ejpam-4918	54	23	the	the	DET
ejpam-4918	54	24	monoid	monoid	NOUN
ejpam-4918	54	25	m	m	NOUN
ejpam-4918	54	26	as	as	ADP
ejpam-4918	54	27	a	a	DET
ejpam-4918	54	28	skew	skew	NOUN
ejpam-4918	54	29	strongly	strongly	ADV
ejpam-4918	54	30	m	m	VERB
ejpam-4918	54	31	-reflexive	-reflexive	ADJ
ejpam-4918	54	32	ring	ring	NOUN
ejpam-4918	54	33	.	.	PUNCT
ejpam-4918	55	1	if	if	SCONJ
ejpam-4918	55	2	r	r	NOUN
ejpam-4918	55	3	is	be	AUX
ejpam-4918	55	4	strongly	strongly	ADV
ejpam-4918	55	5	cm	cm	NOUN
ejpam-4918	55	6	reflexive	reflexive	ADJ
ejpam-4918	55	7	with	with	ADP
ejpam-4918	55	8	a	a	DET
ejpam-4918	55	9	trivial	trivial	ADJ
ejpam-4918	55	10	action	action	NOUN
ejpam-4918	55	11	map	map	NOUN
ejpam-4918	55	12	ω	ω	NOUN
ejpam-4918	55	13	,	,	PUNCT
ejpam-4918	55	14	then	then	ADV
ejpam-4918	55	15	we	we	PRON
ejpam-4918	55	16	call	call	VERB
ejpam-4918	55	17	r	r	NOUN
ejpam-4918	55	18	a	a	DET
ejpam-4918	55	19	strongly	strongly	ADV
ejpam-4918	55	20	tm	tm	ADJ
ejpam-4918	55	21	-reflexive	-reflexive	ADJ
ejpam-4918	55	22	(	(	PUNCT
ejpam-4918	55	23	i.e.	i.e.	X
ejpam-4918	55	24	,	,	PUNCT
ejpam-4918	55	25	twisted	twist	VERB
ejpam-4918	55	26	strongly	strongly	ADV
ejpam-4918	55	27	m	m	VERB
ejpam-4918	55	28	-reflexive	-reflexive	ADJ
ejpam-4918	55	29	)	)	PUNCT
ejpam-4918	55	30	ring	ring	NOUN
ejpam-4918	55	31	.	.	PUNCT
ejpam-4918	56	1	note	note	VERB
ejpam-4918	56	2	that	that	SCONJ
ejpam-4918	56	3	when	when	SCONJ
ejpam-4918	56	4	both	both	DET
ejpam-4918	56	5	f	f	PROPN
ejpam-4918	56	6	and	and	CCONJ
ejpam-4918	56	7	ω	ω	PROPN
ejpam-4918	56	8	are	be	AUX
ejpam-4918	56	9	trivial	trivial	ADJ
ejpam-4918	56	10	,	,	PUNCT
ejpam-4918	56	11	r	r	NOUN
ejpam-4918	56	12	is	be	AUX
ejpam-4918	56	13	simply	simply	ADV
ejpam-4918	56	14	strongly	strongly	ADV
ejpam-4918	56	15	m	m	NOUN
ejpam-4918	56	16	-reflexive	-reflexive	ADJ
ejpam-4918	56	17	.	.	PUNCT
ejpam-4918	57	1	in	in	ADP
ejpam-4918	57	2	particular	particular	ADJ
ejpam-4918	57	3	,	,	PUNCT
ejpam-4918	57	4	if	if	SCONJ
ejpam-4918	57	5	m	m	VERB
ejpam-4918	57	6	=	=	SYM
ejpam-4918	57	7	(	(	PUNCT
ejpam-4918	57	8	n	n	CCONJ
ejpam-4918	57	9	∪	∪	X
ejpam-4918	57	10	{	{	PUNCT
ejpam-4918	57	11	0},+	0},+	NUM
ejpam-4918	57	12	)	)	PUNCT
ejpam-4918	57	13	and	and	CCONJ
ejpam-4918	57	14	both	both	DET
ejpam-4918	57	15	f	f	PROPN
ejpam-4918	57	16	and	and	CCONJ
ejpam-4918	57	17	ω	ω	PROPN
ejpam-4918	57	18	are	be	AUX
ejpam-4918	57	19	trivial	trivial	ADJ
ejpam-4918	57	20	,	,	PUNCT
ejpam-4918	57	21	then	then	ADV
ejpam-4918	57	22	r	r	NOUN
ejpam-4918	57	23	is	be	AUX
ejpam-4918	57	24	strongly	strongly	ADV
ejpam-4918	57	25	cm	cm	NOUN
ejpam-4918	57	26	-reflexive	-reflexive	NOUN
ejpam-4918	57	27	if	if	SCONJ
ejpam-4918	58	1	and	and	CCONJ
ejpam-4918	58	2	only	only	ADV
ejpam-4918	58	3	if	if	SCONJ
ejpam-4918	58	4	r	r	NOUN
ejpam-4918	58	5	is	be	AUX
ejpam-4918	58	6	strongly	strongly	ADV
ejpam-4918	58	7	reflexive	reflexive	ADJ
ejpam-4918	58	8	.	.	PUNCT
ejpam-4918	59	1	(	(	PUNCT
ejpam-4918	59	2	2	2	X
ejpam-4918	59	3	)	)	PUNCT
ejpam-4918	59	4	if	if	SCONJ
ejpam-4918	59	5	r	r	NOUN
ejpam-4918	59	6	is	be	AUX
ejpam-4918	59	7	a	a	DET
ejpam-4918	59	8	strongly	strongly	ADV
ejpam-4918	59	9	cm	cm	NUM
ejpam-4918	59	10	-reflexive	-reflexive	ADJ
ejpam-4918	59	11	ring	ring	NOUN
ejpam-4918	59	12	with	with	ADP
ejpam-4918	59	13	a	a	DET
ejpam-4918	59	14	trivial	trivial	ADJ
ejpam-4918	59	15	twisting	twisting	NOUN
ejpam-4918	59	16	map	map	NOUN
ejpam-4918	59	17	f	f	NOUN
ejpam-4918	59	18	,	,	PUNCT
ejpam-4918	59	19	then	then	ADV
ejpam-4918	59	20	any	any	DET
ejpam-4918	59	21	m	m	NOUN
ejpam-4918	59	22	invariant	invariant	ADJ
ejpam-4918	59	23	subring	subre	VERB
ejpam-4918	59	24	s	s	X
ejpam-4918	59	25	(	(	PUNCT
ejpam-4918	59	26	i.e.	i.e.	X
ejpam-4918	59	27	,	,	PUNCT
ejpam-4918	59	28	ωg(s	ωg(s	NUM
ejpam-4918	59	29	)	)	PUNCT
ejpam-4918	59	30	⊆	⊆	NUM
ejpam-4918	59	31	s	s	NOUN
ejpam-4918	59	32	for	for	ADP
ejpam-4918	59	33	all	all	DET
ejpam-4918	59	34	g	g	NOUN
ejpam-4918	59	35	∈m	∈m	NOUN
ejpam-4918	59	36	)	)	PUNCT
ejpam-4918	59	37	of	of	ADP
ejpam-4918	59	38	r	r	NOUN
ejpam-4918	59	39	is	be	AUX
ejpam-4918	59	40	strongly	strongly	ADV
ejpam-4918	59	41	cm	cm	NOUN
ejpam-4918	59	42	-reflexive	-reflexive	NOUN
ejpam-4918	59	43	.	.	PUNCT
ejpam-4918	60	1	an	an	DET
ejpam-4918	60	2	ideal	ideal	ADJ
ejpam-4918	60	3	i	i	PRON
ejpam-4918	60	4	of	of	ADP
ejpam-4918	60	5	a	a	DET
ejpam-4918	60	6	ring	ring	NOUN
ejpam-4918	60	7	r	r	NOUN
ejpam-4918	60	8	is	be	AUX
ejpam-4918	60	9	considered	consider	VERB
ejpam-4918	60	10	to	to	PART
ejpam-4918	60	11	be	be	AUX
ejpam-4918	60	12	right	right	ADJ
ejpam-4918	60	13	s	s	NOUN
ejpam-4918	60	14	-	-	NOUN
ejpam-4918	60	15	unital	unital	ADJ
ejpam-4918	60	16	if	if	SCONJ
ejpam-4918	60	17	there	there	PRON
ejpam-4918	60	18	exists	exist	VERB
ejpam-4918	60	19	an	an	DET
ejpam-4918	60	20	element	element	NOUN
ejpam-4918	60	21	e	e	NOUN
ejpam-4918	60	22	∈	∈	PROPN
ejpam-4918	60	23	i	i	PRON
ejpam-4918	60	24	for	for	ADP
ejpam-4918	60	25	every	every	DET
ejpam-4918	60	26	t	t	NOUN
ejpam-4918	60	27	∈	∈	PROPN
ejpam-4918	60	28	i	i	PRON
ejpam-4918	60	29	such	such	ADJ
ejpam-4918	60	30	that	that	SCONJ
ejpam-4918	60	31	te	te	PROPN
ejpam-4918	60	32	=	=	PUNCT
ejpam-4918	60	33	t.	t.	NOUN
ejpam-4918	60	34	a	a	DET
ejpam-4918	60	35	ring	ring	NOUN
ejpam-4918	60	36	is	be	AUX
ejpam-4918	60	37	referred	refer	VERB
ejpam-4918	60	38	to	to	ADP
ejpam-4918	60	39	as	as	ADP
ejpam-4918	60	40	a	a	DET
ejpam-4918	60	41	left	left	ADJ
ejpam-4918	60	42	app	app	NOUN
ejpam-4918	60	43	-ring	-ring	PROPN
ejpam-4918	60	44	if	if	SCONJ
ejpam-4918	60	45	the	the	DET
ejpam-4918	60	46	left	left	ADJ
ejpam-4918	60	47	annihilator	annihilator	NOUN
ejpam-4918	60	48	lr(rt	lr(rt	NOUN
ejpam-4918	60	49	)	)	PUNCT
ejpam-4918	60	50	is	be	AUX
ejpam-4918	60	51	right	right	ADJ
ejpam-4918	60	52	s	s	NOUN
ejpam-4918	60	53	-	-	NOUN
ejpam-4918	60	54	unital	unital	ADJ
ejpam-4918	60	55	as	as	ADP
ejpam-4918	60	56	an	an	DET
ejpam-4918	60	57	ideal	ideal	NOUN
ejpam-4918	60	58	of	of	ADP
ejpam-4918	60	59	r	r	NOUN
ejpam-4918	60	60	for	for	ADP
ejpam-4918	60	61	any	any	DET
ejpam-4918	60	62	element	element	NOUN
ejpam-4918	60	63	t	t	PROPN
ejpam-4918	60	64	∈	∈	PROPN
ejpam-4918	60	65	r.	r.	PROPN
ejpam-4918	60	66	in	in	ADP
ejpam-4918	60	67	their	their	PRON
ejpam-4918	60	68	work	work	NOUN
ejpam-4918	61	1	[	[	X
ejpam-4918	61	2	9	9	NUM
ejpam-4918	61	3	]	]	PUNCT
ejpam-4918	61	4	,	,	PUNCT
ejpam-4918	61	5	nasr	nasr	PROPN
ejpam-4918	61	6	-	-	PUNCT
ejpam-4918	61	7	isfahani	isfahani	PROPN
ejpam-4918	61	8	and	and	CCONJ
ejpam-4918	61	9	moussavi	moussavi	NOUN
ejpam-4918	61	10	introduced	introduce	VERB
ejpam-4918	61	11	a	a	DET
ejpam-4918	61	12	ring	ring	NOUN
ejpam-4918	61	13	r	r	NOUN
ejpam-4918	61	14	with	with	ADP
ejpam-4918	61	15	an	an	DET
ejpam-4918	61	16	endomorphism	endomorphism	PROPN
ejpam-4918	61	17	ω	ω	PROPN
ejpam-4918	61	18	and	and	CCONJ
ejpam-4918	61	19	defined	define	VERB
ejpam-4918	61	20	it	it	PRON
ejpam-4918	61	21	as	as	ADP
ejpam-4918	61	22	ω	ω	NOUN
ejpam-4918	61	23	-	-	ADJ
ejpam-4918	61	24	weakly	weakly	ADV
ejpam-4918	61	25	rigid	rigid	ADJ
ejpam-4918	61	26	if	if	SCONJ
ejpam-4918	61	27	the	the	DET
ejpam-4918	61	28	condition	condition	NOUN
ejpam-4918	61	29	crt	crt	NOUN
ejpam-4918	61	30	=	=	SYM
ejpam-4918	61	31	0	0	NUM
ejpam-4918	61	32	holds	hold	VERB
ejpam-4918	61	33	if	if	SCONJ
ejpam-4918	61	34	and	and	CCONJ
ejpam-4918	61	35	only	only	ADV
ejpam-4918	61	36	if	if	SCONJ
ejpam-4918	61	37	c	c	PROPN
ejpam-4918	61	38	ω(rt	ω(rt	NUM
ejpam-4918	61	39	)	)	PUNCT
ejpam-4918	61	40	=	=	SYM
ejpam-4918	61	41	0	0	NUM
ejpam-4918	61	42	for	for	ADP
ejpam-4918	61	43	any	any	DET
ejpam-4918	61	44	c	c	NOUN
ejpam-4918	61	45	,	,	PUNCT
ejpam-4918	61	46	t	t	PROPN
ejpam-4918	61	47	∈	∈	PROPN
ejpam-4918	61	48	r.	r.	PROPN
ejpam-4918	61	49	it	it	PRON
ejpam-4918	61	50	is	be	AUX
ejpam-4918	61	51	worth	worth	ADJ
ejpam-4918	61	52	noting	note	VERB
ejpam-4918	61	53	that	that	SCONJ
ejpam-4918	61	54	the	the	DET
ejpam-4918	61	55	category	category	NOUN
ejpam-4918	61	56	of	of	ADP
ejpam-4918	61	57	ω	ω	ADJ
ejpam-4918	61	58	-	-	ADJ
ejpam-4918	61	59	rigid	rigid	ADJ
ejpam-4918	61	60	rings	ring	NOUN
ejpam-4918	61	61	and	and	CCONJ
ejpam-4918	61	62	ω	ω	VERB
ejpam-4918	61	63	-	-	ADJ
ejpam-4918	61	64	compatible	compatible	ADJ
ejpam-4918	61	65	rings	ring	NOUN
ejpam-4918	61	66	is	be	AUX
ejpam-4918	61	67	a	a	DET
ejpam-4918	61	68	limited	limited	ADJ
ejpam-4918	61	69	one	one	NUM
ejpam-4918	61	70	,	,	PUNCT
ejpam-4918	61	71	and	and	CCONJ
ejpam-4918	61	72	it	it	PRON
ejpam-4918	61	73	is	be	AUX
ejpam-4918	61	74	evident	evident	ADJ
ejpam-4918	61	75	that	that	SCONJ
ejpam-4918	61	76	every	every	DET
ejpam-4918	61	77	ω	ω	ADJ
ejpam-4918	61	78	-	-	ADJ
ejpam-4918	61	79	compatible	compatible	ADJ
ejpam-4918	61	80	ring	ring	NOUN
ejpam-4918	61	81	falls	fall	VERB
ejpam-4918	61	82	under	under	ADP
ejpam-4918	61	83	the	the	DET
ejpam-4918	61	84	category	category	NOUN
ejpam-4918	61	85	of	of	ADP
ejpam-4918	61	86	ω	ω	ADJ
ejpam-4918	61	87	-	-	ADJ
ejpam-4918	61	88	weakly	weakly	ADJ
ejpam-4918	61	89	rigid	rigid	ADJ
ejpam-4918	61	90	rings	ring	NOUN
ejpam-4918	61	91	.	.	PUNCT
ejpam-4918	62	1	however	however	ADV
ejpam-4918	62	2	,	,	PUNCT
ejpam-4918	62	3	there	there	PRON
ejpam-4918	62	4	exist	exist	VERB
ejpam-4918	62	5	several	several	ADJ
ejpam-4918	62	6	classes	class	NOUN
ejpam-4918	62	7	of	of	ADP
ejpam-4918	62	8	ω	ω	ADJ
ejpam-4918	62	9	-	-	ADJ
ejpam-4918	62	10	weakly	weakly	ADJ
ejpam-4918	62	11	rigid	rigid	ADJ
ejpam-4918	62	12	rings	ring	NOUN
ejpam-4918	62	13	that	that	PRON
ejpam-4918	62	14	do	do	AUX
ejpam-4918	62	15	not	not	PART
ejpam-4918	62	16	belong	belong	VERB
ejpam-4918	62	17	to	to	ADP
ejpam-4918	62	18	the	the	DET
ejpam-4918	62	19	category	category	NOUN
ejpam-4918	62	20	of	of	ADP
ejpam-4918	62	21	ω	ω	VERB
ejpam-4918	62	22	-	-	PUNCT
ejpam-4918	62	23	compatible	compatible	ADJ
ejpam-4918	62	24	rings	ring	NOUN
ejpam-4918	62	25	.	.	PUNCT
ejpam-4918	63	1	by	by	ADP
ejpam-4918	63	2	[	[	X
ejpam-4918	63	3	10	10	NUM
ejpam-4918	63	4	]	]	PUNCT
ejpam-4918	63	5	,	,	PUNCT
ejpam-4918	63	6	r	r	NOUN
ejpam-4918	63	7	is	be	AUX
ejpam-4918	63	8	αrigid	αrigid	ADJ
ejpam-4918	63	9	if	if	SCONJ
ejpam-4918	63	10	and	and	CCONJ
ejpam-4918	63	11	only	only	ADV
ejpam-4918	63	12	if	if	SCONJ
ejpam-4918	63	13	r	r	NOUN
ejpam-4918	63	14	is	be	AUX
ejpam-4918	63	15	α	α	NOUN
ejpam-4918	63	16	-	-	ADJ
ejpam-4918	63	17	compatible	compatible	ADJ
ejpam-4918	63	18	and	and	CCONJ
ejpam-4918	63	19	reduced	reduce	VERB
ejpam-4918	63	20	.	.	PUNCT
ejpam-4918	64	1	according	accord	VERB
ejpam-4918	64	2	to	to	ADP
ejpam-4918	64	3	[	[	X
ejpam-4918	64	4	9	9	NUM
ejpam-4918	64	5	]	]	PUNCT
ejpam-4918	64	6	,	,	PUNCT
ejpam-4918	64	7	any	any	DET
ejpam-4918	64	8	prime	prime	ADJ
ejpam-4918	64	9	ring	ring	NOUN
ejpam-4918	64	10	that	that	PRON
ejpam-4918	64	11	has	have	VERB
ejpam-4918	64	12	an	an	DET
ejpam-4918	64	13	automorphism	automorphism	NOUN
ejpam-4918	64	14	ω	ω	NOUN
ejpam-4918	64	15	is	be	AUX
ejpam-4918	64	16	considered	consider	VERB
ejpam-4918	64	17	to	to	PART
ejpam-4918	64	18	be	be	AUX
ejpam-4918	64	19	ω	ω	VERB
ejpam-4918	64	20	-	-	ADJ
ejpam-4918	64	21	weakly	weakly	ADV
ejpam-4918	64	22	rigid	rigid	ADJ
ejpam-4918	64	23	.	.	PUNCT
ejpam-4918	65	1	if	if	SCONJ
ejpam-4918	65	2	a	a	DET
ejpam-4918	65	3	monoid	monoid	NOUN
ejpam-4918	65	4	homomorphism	homomorphism	PROPN
ejpam-4918	65	5	ω	ω	X
ejpam-4918	65	6	:	:	PUNCT
ejpam-4918	65	7	m	m	PROPN
ejpam-4918	65	8	→	→	SYM
ejpam-4918	65	9	aut(r	aut(r	PROPN
ejpam-4918	65	10	)	)	PUNCT
ejpam-4918	65	11	is	be	AUX
ejpam-4918	65	12	weakly	weakly	ADV
ejpam-4918	65	13	-	-	PUNCT
ejpam-4918	65	14	rigid	rigid	ADJ
ejpam-4918	65	15	(	(	PUNCT
ejpam-4918	65	16	compatible	compatible	ADJ
ejpam-4918	65	17	)	)	PUNCT
ejpam-4918	65	18	,	,	PUNCT
ejpam-4918	65	19	it	it	PRON
ejpam-4918	65	20	means	mean	VERB
ejpam-4918	65	21	that	that	SCONJ
ejpam-4918	65	22	the	the	DET
ejpam-4918	65	23	ring	ring	NOUN
ejpam-4918	65	24	r	r	NOUN
ejpam-4918	65	25	is	be	AUX
ejpam-4918	65	26	also	also	ADV
ejpam-4918	65	27	weakly	weakly	ADV
ejpam-4918	65	28	rigid	rigid	ADJ
ejpam-4918	65	29	(	(	PUNCT
ejpam-4918	65	30	compatible	compatible	ADJ
ejpam-4918	65	31	)	)	PUNCT
ejpam-4918	65	32	with	with	ADP
ejpam-4918	65	33	respect	respect	NOUN
ejpam-4918	65	34	to	to	ADP
ejpam-4918	65	35	each	each	DET
ejpam-4918	65	36	g	g	NOUN
ejpam-4918	65	37	∈m	∈m	NOUN
ejpam-4918	65	38	under	under	ADP
ejpam-4918	65	39	the	the	DET
ejpam-4918	65	40	automorphism	automorphism	NOUN
ejpam-4918	65	41	ωg	ωg	NOUN
ejpam-4918	65	42	.	.	PUNCT
ejpam-4918	65	43	lemma	lemma	PROPN
ejpam-4918	66	1	1	1	NUM
ejpam-4918	66	2	.	.	PUNCT
ejpam-4918	67	1	[	[	X
ejpam-4918	67	2	11	11	NUM
ejpam-4918	67	3	,	,	PUNCT
ejpam-4918	67	4	lemma	lemma	PROPN
ejpam-4918	67	5	1.1	1.1	NUM
ejpam-4918	67	6	]	]	PUNCT
ejpam-4918	67	7	.	.	PUNCT
ejpam-4918	68	1	if	if	SCONJ
ejpam-4918	68	2	m	m	NOUN
ejpam-4918	68	3	is	be	AUX
ejpam-4918	68	4	a	a	DET
ejpam-4918	68	5	u.p.-monoid	u.p.-monoid	ADJ
ejpam-4918	68	6	,	,	PUNCT
ejpam-4918	68	7	then	then	ADV
ejpam-4918	68	8	m	m	VERB
ejpam-4918	68	9	is	be	AUX
ejpam-4918	68	10	cancellative	cancellative	ADJ
ejpam-4918	68	11	(	(	PUNCT
ejpam-4918	68	12	i.e.	i.e.	X
ejpam-4918	68	13	,	,	PUNCT
ejpam-4918	68	14	for	for	ADP
ejpam-4918	68	15	ℓ	ℓ	PROPN
ejpam-4918	68	16	,	,	PUNCT
ejpam-4918	68	17	h	h	NOUN
ejpam-4918	68	18	,	,	PUNCT
ejpam-4918	68	19	λ	λ	NOUN
ejpam-4918	68	20	∈m	∈m	NOUN
ejpam-4918	68	21	,	,	PUNCT
ejpam-4918	68	22	if	if	SCONJ
ejpam-4918	68	23	ℓλ	ℓλ	NOUN
ejpam-4918	68	24	=	=	VERB
ejpam-4918	68	25	hλ	hλ	NOUN
ejpam-4918	68	26	or	or	CCONJ
ejpam-4918	68	27	λℓ	λℓ	ADP
ejpam-4918	68	28	=	=	SYM
ejpam-4918	68	29	λh	λh	PROPN
ejpam-4918	68	30	,	,	PUNCT
ejpam-4918	68	31	then	then	ADV
ejpam-4918	68	32	ℓ	ℓ	PROPN
ejpam-4918	68	33	=	=	SYM
ejpam-4918	68	34	h	h	PROPN
ejpam-4918	68	35	)	)	PUNCT
ejpam-4918	68	36	.	.	PUNCT
ejpam-4918	69	1	lemma	lemma	PROPN
ejpam-4918	69	2	2	2	X
ejpam-4918	69	3	.	.	PUNCT
ejpam-4918	69	4	suppose	suppose	VERB
ejpam-4918	69	5	r	r	NOUN
ejpam-4918	69	6	is	be	AUX
ejpam-4918	69	7	a	a	DET
ejpam-4918	69	8	ring	ring	NOUN
ejpam-4918	69	9	and	and	CCONJ
ejpam-4918	69	10	m	m	NOUN
ejpam-4918	69	11	is	be	AUX
ejpam-4918	69	12	a	a	DET
ejpam-4918	69	13	u.p.-monoid	u.p.-monoid	NOUN
ejpam-4918	69	14	with	with	ADP
ejpam-4918	69	15	a	a	DET
ejpam-4918	69	16	twisting	twisting	NOUN
ejpam-4918	69	17	map	map	NOUN
ejpam-4918	69	18	f	f	X
ejpam-4918	69	19	:	:	PUNCT
ejpam-4918	69	20	m×m	m×m	ADJ
ejpam-4918	69	21	→	→	SYM
ejpam-4918	69	22	u(r	u(r	NOUN
ejpam-4918	69	23	)	)	PUNCT
ejpam-4918	69	24	and	and	CCONJ
ejpam-4918	69	25	an	an	DET
ejpam-4918	69	26	action	action	NOUN
ejpam-4918	69	27	map	map	NOUN
ejpam-4918	69	28	ω	ω	NOUN
ejpam-4918	69	29	:	:	PUNCT
ejpam-4918	69	30	m	m	PROPN
ejpam-4918	69	31	→	→	SYM
ejpam-4918	69	32	aut(r	aut(r	PROPN
ejpam-4918	69	33	)	)	PUNCT
ejpam-4918	69	34	.	.	PUNCT
ejpam-4918	70	1	if	if	SCONJ
ejpam-4918	70	2	r	r	NOUN
ejpam-4918	70	3	is	be	AUX
ejpam-4918	70	4	an	an	DET
ejpam-4918	70	5	m	m	NOUN
ejpam-4918	70	6	-rigid	-rigid	PROPN
ejpam-4918	70	7	ring	ring	NOUN
ejpam-4918	70	8	,	,	PUNCT
ejpam-4918	70	9	then	then	ADV
ejpam-4918	70	10	the	the	DET
ejpam-4918	70	11	monoid	monoid	NOUN
ejpam-4918	70	12	ring	ring	NOUN
ejpam-4918	70	13	r	r	NOUN
ejpam-4918	70	14	∗m	∗m	NOUN
ejpam-4918	70	15	is	be	AUX
ejpam-4918	70	16	reduced	reduce	VERB
ejpam-4918	70	17	.	.	PUNCT
ejpam-4918	71	1	proof	proof	NOUN
ejpam-4918	71	2	.	.	PUNCT
ejpam-4918	72	1	assume	assume	VERB
ejpam-4918	72	2	that	that	SCONJ
ejpam-4918	72	3	ϕ	ϕ	NOUN
ejpam-4918	72	4	=	=	X
ejpam-4918	72	5	c1h1	c1h1	X
ejpam-4918	72	6	+	+	X
ejpam-4918	72	7	·	·	PUNCT
ejpam-4918	72	8	·	·	PUNCT
ejpam-4918	72	9	·	·	PUNCT
ejpam-4918	73	1	+	+	NUM
ejpam-4918	73	2	cnhn	cnhn	ADJ
ejpam-4918	73	3	∈	∈	PROPN
ejpam-4918	73	4	r	r	NOUN
ejpam-4918	73	5	∗m	∗m	NOUN
ejpam-4918	73	6	satisfies	satisfy	VERB
ejpam-4918	73	7	ϕ2	ϕ2	ADV
ejpam-4918	73	8	=	=	SYM
ejpam-4918	73	9	0	0	NUM
ejpam-4918	73	10	.	.	PUNCT
ejpam-4918	74	1	according	accord	VERB
ejpam-4918	74	2	to	to	ADP
ejpam-4918	74	3	proposition	proposition	NOUN
ejpam-4918	74	4	2.2	2.2	NUM
ejpam-4918	74	5	[	[	X
ejpam-4918	74	6	6	6	NUM
ejpam-4918	74	7	]	]	PUNCT
ejpam-4918	74	8	,	,	PUNCT
ejpam-4918	74	9	r	r	NOUN
ejpam-4918	74	10	is	be	AUX
ejpam-4918	74	11	cm	cm	NOUN
ejpam-4918	74	12	-armendariz	-armendariz	NOUN
ejpam-4918	74	13	,	,	PUNCT
ejpam-4918	74	14	this	this	PRON
ejpam-4918	74	15	implies	imply	VERB
ejpam-4918	74	16	ciωhi(bj)f(li	ciωhi(bj)f(li	PROPN
ejpam-4918	74	17	,	,	PUNCT
ejpam-4918	74	18	hj))(lihj	hj))(lihj	X
ejpam-4918	74	19	)	)	PUNCT
ejpam-4918	74	20	=	=	SYM
ejpam-4918	74	21	0	0	NUM
ejpam-4918	75	1	for	for	ADP
ejpam-4918	75	2	all	all	DET
ejpam-4918	75	3	i	i	PROPN
ejpam-4918	75	4	and	and	CCONJ
ejpam-4918	75	5	j	j	PROPN
ejpam-4918	75	6	,	,	PUNCT
ejpam-4918	75	7	by	by	ADP
ejpam-4918	75	8	lemma	lemma	PROPN
ejpam-4918	75	9	1	1	NUM
ejpam-4918	75	10	,	,	PUNCT
ejpam-4918	75	11	m	m	VERB
ejpam-4918	75	12	is	be	AUX
ejpam-4918	75	13	a	a	DET
ejpam-4918	75	14	cancellative	cancellative	ADJ
ejpam-4918	76	1	so	so	SCONJ
ejpam-4918	76	2	ciωhi(bj	ciωhi(bj	NOUN
ejpam-4918	76	3	)	)	PUNCT
ejpam-4918	76	4	=	=	SYM
ejpam-4918	76	5	0	0	NUM
ejpam-4918	76	6	.	.	PUNCT
ejpam-4918	77	1	as	as	SCONJ
ejpam-4918	77	2	r	r	NOUN
ejpam-4918	77	3	is	be	AUX
ejpam-4918	77	4	an	an	DET
ejpam-4918	77	5	m	m	NOUN
ejpam-4918	77	6	-rigid	-rigid	ADJ
ejpam-4918	77	7	,	,	PUNCT
ejpam-4918	77	8	then	then	ADV
ejpam-4918	77	9	r	r	NOUN
ejpam-4918	77	10	is	be	AUX
ejpam-4918	77	11	a	a	DET
ejpam-4918	77	12	reduced	reduce	VERB
ejpam-4918	77	13	,	,	PUNCT
ejpam-4918	77	14	we	we	PRON
ejpam-4918	77	15	can	can	AUX
ejpam-4918	77	16	conclude	conclude	VERB
ejpam-4918	77	17	that	that	DET
ejpam-4918	77	18	ci	ci	NOUN
ejpam-4918	77	19	=	=	NOUN
ejpam-4918	77	20	0	0	PROPN
ejpam-4918	77	21	for	for	ADP
ejpam-4918	77	22	all	all	DET
ejpam-4918	77	23	1	1	NUM
ejpam-4918	77	24	≤	≤	NUM
ejpam-4918	77	25	i	i	PRON
ejpam-4918	77	26	≤	≤	PROPN
ejpam-4918	77	27	n.	n.	NOUN
ejpam-4918	77	28	consequently	consequently	ADV
ejpam-4918	77	29	,	,	PUNCT
ejpam-4918	77	30	ϕ	ϕ	X
ejpam-4918	77	31	=	=	SYM
ejpam-4918	77	32	0	0	NUM
ejpam-4918	77	33	,	,	PUNCT
ejpam-4918	77	34	and	and	CCONJ
ejpam-4918	77	35	hence	hence	ADV
ejpam-4918	77	36	r	r	NOUN
ejpam-4918	77	37	∗m	∗m	NOUN
ejpam-4918	77	38	is	be	AUX
ejpam-4918	77	39	a	a	DET
ejpam-4918	77	40	reduced	reduce	VERB
ejpam-4918	77	41	.	.	PUNCT
ejpam-4918	78	1	theorem	theorem	NOUN
ejpam-4918	78	2	1	1	NUM
ejpam-4918	78	3	.	.	PUNCT
ejpam-4918	79	1	let	let	VERB
ejpam-4918	79	2	r	r	PRON
ejpam-4918	79	3	be	be	AUX
ejpam-4918	79	4	a	a	DET
ejpam-4918	79	5	semiprime	semiprime	NOUN
ejpam-4918	79	6	ring	ring	NOUN
ejpam-4918	79	7	and	and	CCONJ
ejpam-4918	79	8	m	m	AUX
ejpam-4918	79	9	be	be	AUX
ejpam-4918	79	10	a	a	DET
ejpam-4918	79	11	u.p.-monoid	u.p.-monoid	NOUN
ejpam-4918	79	12	with	with	ADP
ejpam-4918	79	13	a	a	DET
ejpam-4918	79	14	twisting	twisting	NOUN
ejpam-4918	79	15	map	map	NOUN
ejpam-4918	79	16	f	f	X
ejpam-4918	79	17	:	:	PUNCT
ejpam-4918	79	18	m	m	VERB
ejpam-4918	79	19	×m	×m	NOUN
ejpam-4918	79	20	→	→	SYM
ejpam-4918	79	21	u(r	u(r	NOUN
ejpam-4918	79	22	)	)	PUNCT
ejpam-4918	79	23	and	and	CCONJ
ejpam-4918	79	24	an	an	DET
ejpam-4918	79	25	action	action	NOUN
ejpam-4918	79	26	map	map	NOUN
ejpam-4918	79	27	ω	ω	NOUN
ejpam-4918	79	28	:	:	PUNCT
ejpam-4918	79	29	m	m	PROPN
ejpam-4918	79	30	→	→	SYM
ejpam-4918	79	31	aut(r	aut(r	PROPN
ejpam-4918	79	32	)	)	PUNCT
ejpam-4918	79	33	.	.	PUNCT
ejpam-4918	80	1	if	if	SCONJ
ejpam-4918	80	2	r	r	NOUN
ejpam-4918	80	3	is	be	AUX
ejpam-4918	80	4	an	an	DET
ejpam-4918	80	5	m	m	NUM
ejpam-4918	80	6	-compatible	-compatible	ADJ
ejpam-4918	80	7	ring	ring	NOUN
ejpam-4918	80	8	,	,	PUNCT
ejpam-4918	80	9	then	then	ADV
ejpam-4918	80	10	r	r	NOUN
ejpam-4918	80	11	is	be	AUX
ejpam-4918	80	12	strongly	strongly	ADV
ejpam-4918	80	13	cm	cm	NOUN
ejpam-4918	80	14	-reflexive	-reflexive	NOUN
ejpam-4918	80	15	.	.	PUNCT
ejpam-4918	81	1	e.	e.	PROPN
ejpam-4918	81	2	ali	ali	PROPN
ejpam-4918	81	3	/	/	SYM
ejpam-4918	81	4	eur	eur	PROPN
ejpam-4918	81	5	.	.	PUNCT
ejpam-4918	82	1	j.	j.	PROPN
ejpam-4918	82	2	pure	pure	PROPN
ejpam-4918	82	3	appl	appl	PROPN
ejpam-4918	82	4	.	.	PROPN
ejpam-4918	82	5	math	math	PROPN
ejpam-4918	82	6	,	,	PUNCT
ejpam-4918	82	7	16	16	NUM
ejpam-4918	82	8	(	(	PUNCT
ejpam-4918	82	9	4	4	NUM
ejpam-4918	82	10	)	)	PUNCT
ejpam-4918	82	11	(	(	PUNCT
ejpam-4918	82	12	2023	2023	NUM
ejpam-4918	82	13	)	)	PUNCT
ejpam-4918	82	14	,	,	PUNCT
ejpam-4918	82	15	2156	2156	NUM
ejpam-4918	82	16	-	-	SYM
ejpam-4918	82	17	2168	2168	NUM
ejpam-4918	82	18	2159	2159	NUM
ejpam-4918	82	19	proof	proof	NOUN
ejpam-4918	82	20	.	.	PUNCT
ejpam-4918	83	1	the	the	DET
ejpam-4918	83	2	evidence	evidence	NOUN
ejpam-4918	83	3	has	have	AUX
ejpam-4918	83	4	been	be	AUX
ejpam-4918	83	5	modified	modify	VERB
ejpam-4918	83	6	from	from	ADP
ejpam-4918	83	7	the	the	DET
ejpam-4918	83	8	theorem	theorem	ADJ
ejpam-4918	83	9	1.1	1.1	NUM
ejpam-4918	83	10	of	of	ADP
ejpam-4918	83	11	[	[	X
ejpam-4918	83	12	12	12	NUM
ejpam-4918	83	13	]	]	PUNCT
ejpam-4918	83	14	.	.	PUNCT
ejpam-4918	84	1	let	let	VERB
ejpam-4918	84	2	ϕ	ϕ	NOUN
ejpam-4918	84	3	=	=	PUNCT
ejpam-4918	84	4	c1l1	c1l1	X
ejpam-4918	85	1	+	+	CCONJ
ejpam-4918	85	2	c2l2	c2l2	X
ejpam-4918	85	3	+	+	NUM
ejpam-4918	85	4	·	·	PUNCT
ejpam-4918	85	5	·	·	PUNCT
ejpam-4918	85	6	·	·	PUNCT
ejpam-4918	85	7	+	+	NUM
ejpam-4918	85	8	cnln	cnln	NOUN
ejpam-4918	85	9	,	,	PUNCT
ejpam-4918	85	10	ψ	ψ	X
ejpam-4918	85	11	=	=	PUNCT
ejpam-4918	85	12	a1h1	a1h1	X
ejpam-4918	85	13	+	+	NOUN
ejpam-4918	85	14	a2h2	a2h2	X
ejpam-4918	85	15	+	+	X
ejpam-4918	85	16	·	·	PUNCT
ejpam-4918	85	17	·	·	PUNCT
ejpam-4918	85	18	·	·	PUNCT
ejpam-4918	85	19	+	+	NUM
ejpam-4918	85	20	amhm	amhm	NOUN
ejpam-4918	85	21	∈	∈	PROPN
ejpam-4918	85	22	r	r	NOUN
ejpam-4918	85	23	∗m	∗m	NOUN
ejpam-4918	85	24	satisfy	satisfy	VERB
ejpam-4918	85	25	ϕ(r	ϕ(r	PROPN
ejpam-4918	85	26	∗m)ψ	∗m)ψ	ADJ
ejpam-4918	85	27	=	=	PUNCT
ejpam-4918	86	1	0	0	X
ejpam-4918	86	2	.	.	PUNCT
ejpam-4918	87	1	then	then	ADV
ejpam-4918	87	2	for	for	ADP
ejpam-4918	87	3	any	any	DET
ejpam-4918	87	4	r	r	NOUN
ejpam-4918	87	5	∈	∈	NOUN
ejpam-4918	87	6	r	r	NOUN
ejpam-4918	87	7	and	and	CCONJ
ejpam-4918	87	8	g	g	NOUN
ejpam-4918	87	9	∈m	∈m	NOUN
ejpam-4918	87	10	,	,	PUNCT
ejpam-4918	87	11	we	we	PRON
ejpam-4918	87	12	have	have	AUX
ejpam-4918	87	13	(	(	PUNCT
ejpam-4918	87	14	c1l1	c1l1	X
ejpam-4918	87	15	+	+	X
ejpam-4918	87	16	c2l2	c2l2	X
ejpam-4918	87	17	+	+	NUM
ejpam-4918	87	18	·	·	PUNCT
ejpam-4918	87	19	·	·	PUNCT
ejpam-4918	87	20	·	·	PUNCT
ejpam-4918	87	21	+	+	NUM
ejpam-4918	87	22	cnln)gr(a1h1	cnln)gr(a1h1	NOUN
ejpam-4918	87	23	+	+	X
ejpam-4918	87	24	a2h2	a2h2	X
ejpam-4918	87	25	+	+	X
ejpam-4918	87	26	·	·	PUNCT
ejpam-4918	87	27	·	·	PUNCT
ejpam-4918	87	28	·	·	PUNCT
ejpam-4918	87	29	+	+	NUM
ejpam-4918	87	30	amhm	amhm	NOUN
ejpam-4918	87	31	)	)	PUNCT
ejpam-4918	87	32	=	=	SYM
ejpam-4918	87	33	0	0	X
ejpam-4918	87	34	.	.	PUNCT
ejpam-4918	88	1	(	(	PUNCT
ejpam-4918	88	2	2.1	2.1	NUM
ejpam-4918	88	3	)	)	PUNCT
ejpam-4918	88	4	we	we	PRON
ejpam-4918	88	5	will	will	AUX
ejpam-4918	88	6	employ	employ	VERB
ejpam-4918	88	7	mathematical	mathematical	ADJ
ejpam-4918	88	8	induction	induction	NOUN
ejpam-4918	88	9	on	on	ADP
ejpam-4918	88	10	n	n	PART
ejpam-4918	88	11	to	to	PART
ejpam-4918	88	12	demonstrate	demonstrate	VERB
ejpam-4918	88	13	that	that	SCONJ
ejpam-4918	88	14	cirωli(ωg(aj	cirωli(ωg(aj	ADJ
ejpam-4918	88	15	)	)	PUNCT
ejpam-4918	88	16	)	)	PUNCT
ejpam-4918	89	1	=	=	SYM
ejpam-4918	89	2	0	0	NUM
ejpam-4918	90	1	for	for	ADP
ejpam-4918	90	2	all	all	DET
ejpam-4918	90	3	1	1	NUM
ejpam-4918	90	4	≤	≤	NUM
ejpam-4918	90	5	i	i	PRON
ejpam-4918	90	6	≤	≤	PROPN
ejpam-4918	90	7	n	n	CCONJ
ejpam-4918	90	8	,	,	PUNCT
ejpam-4918	90	9	1	1	NUM
ejpam-4918	90	10	≤	≤	NUM
ejpam-4918	90	11	j	j	PROPN
ejpam-4918	90	12	≤	≤	NUM
ejpam-4918	90	13	m	m	PROPN
ejpam-4918	90	14	,	,	PUNCT
ejpam-4918	90	15	and	and	CCONJ
ejpam-4918	90	16	for	for	ADP
ejpam-4918	90	17	any	any	DET
ejpam-4918	90	18	g	g	NOUN
ejpam-4918	90	19	∈m	∈m	NOUN
ejpam-4918	90	20	.	.	PUNCT
ejpam-4918	91	1	this	this	PRON
ejpam-4918	91	2	can	can	AUX
ejpam-4918	91	3	be	be	AUX
ejpam-4918	91	4	achieved	achieve	VERB
ejpam-4918	91	5	by	by	ADP
ejpam-4918	91	6	utilizing	utilize	VERB
ejpam-4918	91	7	the	the	DET
ejpam-4918	91	8	fact	fact	NOUN
ejpam-4918	91	9	thatm	thatm	NOUN
ejpam-4918	91	10	is	be	AUX
ejpam-4918	91	11	a	a	DET
ejpam-4918	91	12	compatible	compatible	ADJ
ejpam-4918	91	13	monoid	monoid	NOUN
ejpam-4918	91	14	.	.	PUNCT
ejpam-4918	92	1	if	if	SCONJ
ejpam-4918	92	2	we	we	PRON
ejpam-4918	92	3	take	take	VERB
ejpam-4918	92	4	n	n	NOUN
ejpam-4918	92	5	=	=	SYM
ejpam-4918	92	6	1	1	NUM
ejpam-4918	92	7	,	,	PUNCT
ejpam-4918	92	8	then	then	ADV
ejpam-4918	92	9	we	we	PRON
ejpam-4918	92	10	have	have	VERB
ejpam-4918	92	11	(	(	PUNCT
ejpam-4918	92	12	c1l1)gr(a1h1+a2h2	c1l1)gr(a1h1+a2h2	X
ejpam-4918	92	13	+	+	PROPN
ejpam-4918	92	14	·	·	PUNCT
ejpam-4918	92	15	·	·	PUNCT
ejpam-4918	92	16	·	·	PUNCT
ejpam-4918	93	1	+	+	NUM
ejpam-4918	93	2	amhm	amhm	NOUN
ejpam-4918	93	3	)	)	PUNCT
ejpam-4918	93	4	=	=	SYM
ejpam-4918	94	1	0	0	X
ejpam-4918	94	2	.	.	PUNCT
ejpam-4918	95	1	therefore	therefore	ADV
ejpam-4918	95	2	,	,	PUNCT
ejpam-4918	95	3	for	for	ADP
ejpam-4918	95	4	each	each	DET
ejpam-4918	95	5	1	1	NUM
ejpam-4918	95	6	≤	≤	NUM
ejpam-4918	95	7	j	j	PROPN
ejpam-4918	95	8	≤	≤	PROPN
ejpam-4918	95	9	m	m	PROPN
ejpam-4918	95	10	,	,	PUNCT
ejpam-4918	95	11	we	we	PRON
ejpam-4918	95	12	have	have	VERB
ejpam-4918	95	13	c1rωl1(ωg(aj))f(li	c1rωl1(ωg(aj))f(li	NOUN
ejpam-4918	95	14	,	,	PUNCT
ejpam-4918	95	15	hj))(lihj	hj))(lihj	X
ejpam-4918	95	16	)	)	PUNCT
ejpam-4918	95	17	=	=	SYM
ejpam-4918	95	18	0	0	X
ejpam-4918	95	19	.	.	PUNCT
ejpam-4918	96	1	by	by	ADP
ejpam-4918	96	2	lemma	lemma	PROPN
ejpam-4918	96	3	1	1	NUM
ejpam-4918	96	4	,	,	PUNCT
ejpam-4918	96	5	m	m	VERB
ejpam-4918	96	6	is	be	AUX
ejpam-4918	96	7	a	a	DET
ejpam-4918	96	8	cancellative	cancellative	ADJ
ejpam-4918	96	9	,	,	PUNCT
ejpam-4918	96	10	this	this	PRON
ejpam-4918	96	11	means	mean	VERB
ejpam-4918	96	12	l1hi	l1hi	PUNCT
ejpam-4918	96	13	̸=	̸=	PROPN
ejpam-4918	96	14	l1hj	l1hj	PUNCT
ejpam-4918	96	15	for	for	ADP
ejpam-4918	96	16	any	any	DET
ejpam-4918	96	17	i	i	PROPN
ejpam-4918	96	18	and	and	CCONJ
ejpam-4918	96	19	j	j	PROPN
ejpam-4918	96	20	with	with	ADP
ejpam-4918	96	21	1	1	NUM
ejpam-4918	96	22	≤	≤	NOUN
ejpam-4918	96	23	i	i	PRON
ejpam-4918	96	24	̸=	̸=	PROPN
ejpam-4918	96	25	j	j	PROPN
ejpam-4918	96	26	≤	≤	PROPN
ejpam-4918	96	27	m.	m.	NOUN
ejpam-4918	96	28	thus	thus	ADV
ejpam-4918	96	29	,	,	PUNCT
ejpam-4918	96	30	c1rωl1(ωg(aj	c1rωl1(ωg(aj	PROPN
ejpam-4918	96	31	)	)	PUNCT
ejpam-4918	96	32	)	)	PUNCT
ejpam-4918	97	1	=	=	PUNCT
ejpam-4918	97	2	0	0	X
ejpam-4918	97	3	.	.	X
ejpam-4918	98	1	for	for	ADP
ejpam-4918	98	2	the	the	DET
ejpam-4918	98	3	case	case	NOUN
ejpam-4918	98	4	where	where	SCONJ
ejpam-4918	98	5	n	n	NUM
ejpam-4918	98	6	≥	≥	NOUN
ejpam-4918	98	7	2	2	NUM
ejpam-4918	98	8	,	,	PUNCT
ejpam-4918	98	9	we	we	PRON
ejpam-4918	98	10	can	can	AUX
ejpam-4918	98	11	use	use	VERB
ejpam-4918	98	12	the	the	DET
ejpam-4918	98	13	assumption	assumption	NOUN
ejpam-4918	98	14	that	that	SCONJ
ejpam-4918	98	15	m	m	NOUN
ejpam-4918	98	16	is	be	AUX
ejpam-4918	98	17	a	a	DET
ejpam-4918	98	18	uniquely	uniquely	ADV
ejpam-4918	98	19	presented	present	VERB
ejpam-4918	98	20	monoid	monoid	NOUN
ejpam-4918	98	21	to	to	PART
ejpam-4918	98	22	find	find	VERB
ejpam-4918	98	23	s	s	PRON
ejpam-4918	98	24	and	and	CCONJ
ejpam-4918	98	25	t	t	X
ejpam-4918	98	26	with	with	ADP
ejpam-4918	98	27	1	1	NUM
ejpam-4918	98	28	≤	≤	NUM
ejpam-4918	98	29	s	s	PART
ejpam-4918	98	30	≤	≤	NUM
ejpam-4918	98	31	n	n	PRON
ejpam-4918	98	32	and	and	CCONJ
ejpam-4918	98	33	1	1	NUM
ejpam-4918	98	34	≤	≤	NOUN
ejpam-4918	98	35	t	t	PROPN
ejpam-4918	98	36	≤	≤	NUM
ejpam-4918	98	37	m	m	VERB
ejpam-4918	98	38	such	such	ADJ
ejpam-4918	98	39	that	that	SCONJ
ejpam-4918	98	40	lsght	lsght	NOUN
ejpam-4918	98	41	is	be	AUX
ejpam-4918	98	42	uniquely	uniquely	ADV
ejpam-4918	98	43	represented	represent	VERB
ejpam-4918	98	44	by	by	ADP
ejpam-4918	98	45	considering	consider	VERB
ejpam-4918	98	46	two	two	NUM
ejpam-4918	98	47	subsets	subset	NOUN
ejpam-4918	98	48	k	k	X
ejpam-4918	99	1	=	=	PUNCT
ejpam-4918	99	2	{	{	PUNCT
ejpam-4918	99	3	l1	l1	PROPN
ejpam-4918	99	4	g	g	PROPN
ejpam-4918	99	5	,	,	PUNCT
ejpam-4918	99	6	l2	l2	VERB
ejpam-4918	99	7	g	g	NOUN
ejpam-4918	99	8	,	,	PUNCT
ejpam-4918	99	9	.	.	PUNCT
ejpam-4918	99	10	.	.	PUNCT
ejpam-4918	100	1	.	.	PUNCT
ejpam-4918	101	1	,	,	PUNCT
ejpam-4918	101	2	lng	lng	PROPN
ejpam-4918	101	3	}	}	PUNCT
ejpam-4918	101	4	and	and	CCONJ
ejpam-4918	101	5	h	h	NOUN
ejpam-4918	101	6	=	=	PRON
ejpam-4918	101	7	{	{	PUNCT
ejpam-4918	101	8	h1	h1	PROPN
ejpam-4918	101	9	,	,	PUNCT
ejpam-4918	101	10	h2	h2	PROPN
ejpam-4918	101	11	,	,	PUNCT
ejpam-4918	101	12	.	.	PUNCT
ejpam-4918	101	13	.	.	PUNCT
ejpam-4918	102	1	.	.	PUNCT
ejpam-4918	103	1	,	,	PUNCT
ejpam-4918	103	2	hm	hm	INTJ
ejpam-4918	103	3	}	}	PUNCT
ejpam-4918	103	4	of	of	ADP
ejpam-4918	103	5	the	the	DET
ejpam-4918	103	6	monoid	monoid	NOUN
ejpam-4918	103	7	m	m	PROPN
ejpam-4918	103	8	.	.	PUNCT
ejpam-4918	104	1	without	without	ADP
ejpam-4918	104	2	loss	loss	NOUN
ejpam-4918	104	3	of	of	ADP
ejpam-4918	104	4	generality	generality	NOUN
ejpam-4918	104	5	,	,	PUNCT
ejpam-4918	104	6	we	we	PRON
ejpam-4918	104	7	may	may	AUX
ejpam-4918	104	8	assume	assume	VERB
ejpam-4918	104	9	that	that	SCONJ
ejpam-4918	104	10	s	s	VERB
ejpam-4918	104	11	=	=	SYM
ejpam-4918	104	12	1	1	NUM
ejpam-4918	104	13	and	and	CCONJ
ejpam-4918	104	14	t	t	NOUN
ejpam-4918	105	1	=	=	SYM
ejpam-4918	105	2	1	1	X
ejpam-4918	105	3	.	.	PUNCT
ejpam-4918	106	1	from	from	ADP
ejpam-4918	106	2	eq	eq	ADP
ejpam-4918	106	3	.	.	PUNCT
ejpam-4918	107	1	(	(	PUNCT
ejpam-4918	107	2	2.1	2.1	NUM
ejpam-4918	107	3	)	)	PUNCT
ejpam-4918	107	4	,	,	PUNCT
ejpam-4918	107	5	we	we	PRON
ejpam-4918	107	6	can	can	AUX
ejpam-4918	107	7	deduce	deduce	VERB
ejpam-4918	107	8	that	that	PRON
ejpam-4918	107	9	c1ωl1(ωg(ra1))f(l1	c1ωl1(ωg(ra1))f(l1	NOUN
ejpam-4918	107	10	,	,	PUNCT
ejpam-4918	107	11	h1)(l1h1	h1)(l1h1	PROPN
ejpam-4918	107	12	)	)	PUNCT
ejpam-4918	107	13	=	=	SYM
ejpam-4918	108	1	0	0	NUM
ejpam-4918	108	2	,	,	PUNCT
ejpam-4918	108	3	which	which	PRON
ejpam-4918	108	4	implies	imply	VERB
ejpam-4918	108	5	that	that	PRON
ejpam-4918	108	6	c1rωl1(ωg(a1	c1rωl1(ωg(a1	NOUN
ejpam-4918	108	7	)	)	PUNCT
ejpam-4918	108	8	)	)	PUNCT
ejpam-4918	109	1	=	=	PUNCT
ejpam-4918	109	2	0	0	X
ejpam-4918	109	3	.	.	PUNCT
ejpam-4918	110	1	since	since	SCONJ
ejpam-4918	110	2	ωg	ωg	PRON
ejpam-4918	110	3	and	and	CCONJ
ejpam-4918	110	4	ωl1	ωl1	NOUN
ejpam-4918	110	5	are	be	AUX
ejpam-4918	110	6	automorphisms	automorphism	NOUN
ejpam-4918	110	7	of	of	ADP
ejpam-4918	110	8	r	r	NOUN
ejpam-4918	110	9	,	,	PUNCT
ejpam-4918	110	10	we	we	PRON
ejpam-4918	110	11	have	have	VERB
ejpam-4918	110	12	c1rωl1(ωg(a1	c1rωl1(ωg(a1	VERB
ejpam-4918	110	13	)	)	PUNCT
ejpam-4918	110	14	)	)	PUNCT
ejpam-4918	111	1	=	=	PUNCT
ejpam-4918	111	2	0	0	X
ejpam-4918	111	3	.	.	PUNCT
ejpam-4918	112	1	as	as	ADP
ejpam-4918	112	2	a	a	DET
ejpam-4918	112	3	result	result	NOUN
ejpam-4918	112	4	,	,	PUNCT
ejpam-4918	112	5	for	for	ADP
ejpam-4918	112	6	every	every	DET
ejpam-4918	112	7	z	z	NOUN
ejpam-4918	112	8	∈	∈	PROPN
ejpam-4918	112	9	r	r	NOUN
ejpam-4918	112	10	,	,	PUNCT
ejpam-4918	112	11	we	we	PRON
ejpam-4918	112	12	have	have	VERB
ejpam-4918	112	13	c1rωl1(ωg(a1za1))f(l1	c1rωl1(ωg(a1za1))f(l1	PROPN
ejpam-4918	112	14	,	,	PUNCT
ejpam-4918	112	15	h1	h1	PROPN
ejpam-4918	112	16	)	)	PUNCT
ejpam-4918	112	17	=	=	SYM
ejpam-4918	112	18	0	0	NUM
ejpam-4918	112	19	,	,	PUNCT
ejpam-4918	112	20	which	which	PRON
ejpam-4918	112	21	implies	imply	VERB
ejpam-4918	112	22	that	that	SCONJ
ejpam-4918	112	23	0	0	X
ejpam-4918	113	1	=	=	SYM
ejpam-4918	113	2	(	(	PUNCT
ejpam-4918	113	3	c1l1	c1l1	X
ejpam-4918	113	4	+	+	CCONJ
ejpam-4918	113	5	c2l2	c2l2	X
ejpam-4918	113	6	+	+	NUM
ejpam-4918	113	7	·	·	PUNCT
ejpam-4918	113	8	·	·	PUNCT
ejpam-4918	113	9	·	·	PUNCT
ejpam-4918	113	10	+	+	NUM
ejpam-4918	113	11	cnln)gra1z(gra1z(a1h1	cnln)gra1z(gra1z(a1h1	PROPN
ejpam-4918	113	12	+	+	CCONJ
ejpam-4918	113	13	a2h2	a2h2	X
ejpam-4918	113	14	+	+	X
ejpam-4918	113	15	·	·	PUNCT
ejpam-4918	113	16	·	·	PUNCT
ejpam-4918	113	17	·	·	PUNCT
ejpam-4918	113	18	+	+	NUM
ejpam-4918	113	19	amhm	amhm	NOUN
ejpam-4918	113	20	)	)	PUNCT
ejpam-4918	113	21	=	=	PUNCT
ejpam-4918	113	22	(	(	PUNCT
ejpam-4918	113	23	c2l2	c2l2	X
ejpam-4918	113	24	+	+	X
ejpam-4918	113	25	·	·	PUNCT
ejpam-4918	113	26	·	·	PUNCT
ejpam-4918	113	27	·	·	PUNCT
ejpam-4918	113	28	+	+	NUM
ejpam-4918	113	29	cnln)gr(a1za1h1	cnln)gr(a1za1h1	PROPN
ejpam-4918	113	30	+	+	CCONJ
ejpam-4918	113	31	a1za2h2	a1za2h2	PROPN
ejpam-4918	113	32	+	+	CCONJ
ejpam-4918	113	33	·	·	PUNCT
ejpam-4918	113	34	·	·	PUNCT
ejpam-4918	113	35	·	·	PUNCT
ejpam-4918	113	36	+	+	CCONJ
ejpam-4918	113	37	a1zamhm	a1zamhm	ADJ
ejpam-4918	113	38	)	)	PUNCT
ejpam-4918	113	39	.	.	PUNCT
ejpam-4918	114	1	by	by	ADP
ejpam-4918	114	2	applying	apply	VERB
ejpam-4918	114	3	the	the	DET
ejpam-4918	114	4	induction	induction	NOUN
ejpam-4918	114	5	hypothesis	hypothesis	NOUN
ejpam-4918	114	6	,	,	PUNCT
ejpam-4918	114	7	it	it	PRON
ejpam-4918	114	8	follows	follow	VERB
ejpam-4918	114	9	that	that	SCONJ
ejpam-4918	114	10	ciωli(ωg(ra1zaj	ciωli(ωg(ra1zaj	NOUN
ejpam-4918	114	11	)	)	PUNCT
ejpam-4918	114	12	)	)	PUNCT
ejpam-4918	115	1	=	=	SYM
ejpam-4918	115	2	0	0	NUM
ejpam-4918	116	1	for	for	ADP
ejpam-4918	116	2	all	all	DET
ejpam-4918	116	3	2	2	NUM
ejpam-4918	116	4	≤	≤	NUM
ejpam-4918	116	5	i	i	PRON
ejpam-4918	116	6	≤	≤	ADJ
ejpam-4918	116	7	n	n	CCONJ
ejpam-4918	116	8	and	and	CCONJ
ejpam-4918	116	9	1	1	NUM
ejpam-4918	116	10	≤	≤	NUM
ejpam-4918	116	11	j	j	PROPN
ejpam-4918	116	12	≤	≤	PROPN
ejpam-4918	116	13	m.	m.	NOUN
ejpam-4918	116	14	thus	thus	ADV
ejpam-4918	116	15	,	,	PUNCT
ejpam-4918	116	16	we	we	PRON
ejpam-4918	116	17	have	have	AUX
ejpam-4918	116	18	cirωli(ωg(a1))rωli(ωg(a1	cirωli(ωg(a1))rωli(ωg(a1	VERB
ejpam-4918	116	19	)	)	PUNCT
ejpam-4918	116	20	)	)	PUNCT
ejpam-4918	117	1	=	=	PUNCT
ejpam-4918	117	2	0	0	NUM
ejpam-4918	117	3	,	,	PUNCT
ejpam-4918	117	4	which	which	PRON
ejpam-4918	117	5	implies	imply	VERB
ejpam-4918	117	6	that	that	SCONJ
ejpam-4918	117	7	cirωli(ωg(a1	cirωli(ωg(a1	NOUN
ejpam-4918	117	8	)	)	PUNCT
ejpam-4918	117	9	)	)	PUNCT
ejpam-4918	118	1	=	=	SYM
ejpam-4918	118	2	0	0	NUM
ejpam-4918	119	1	for	for	ADP
ejpam-4918	119	2	all	all	DET
ejpam-4918	119	3	2	2	NUM
ejpam-4918	119	4	≤	≤	NUM
ejpam-4918	119	5	i	i	PRON
ejpam-4918	119	6	≤	≤	NOUN
ejpam-4918	119	7	n	n	CCONJ
ejpam-4918	119	8	,	,	PUNCT
ejpam-4918	119	9	as	as	SCONJ
ejpam-4918	119	10	r	r	NOUN
ejpam-4918	119	11	is	be	AUX
ejpam-4918	119	12	a	a	DET
ejpam-4918	119	13	semiprime	semiprime	NOUN
ejpam-4918	119	14	ring	ring	NOUN
ejpam-4918	119	15	.	.	PUNCT
ejpam-4918	120	1	therefore	therefore	ADV
ejpam-4918	120	2	,	,	PUNCT
ejpam-4918	120	3	we	we	PRON
ejpam-4918	120	4	have	have	VERB
ejpam-4918	120	5	cirωli(ωg(a1	cirωli(ωg(a1	NOUN
ejpam-4918	120	6	)	)	PUNCT
ejpam-4918	120	7	)	)	PUNCT
ejpam-4918	121	1	=	=	SYM
ejpam-4918	121	2	0	0	NUM
ejpam-4918	122	1	for	for	ADP
ejpam-4918	122	2	all	all	DET
ejpam-4918	122	3	1	1	NUM
ejpam-4918	122	4	≤	≤	NUM
ejpam-4918	122	5	i	i	PRON
ejpam-4918	122	6	≤	≤	ADJ
ejpam-4918	122	7	n.	n.	NOUN
ejpam-4918	122	8	as	as	ADP
ejpam-4918	122	9	a	a	DET
ejpam-4918	122	10	result	result	NOUN
ejpam-4918	122	11	,	,	PUNCT
ejpam-4918	122	12	the	the	DET
ejpam-4918	122	13	eq	eq	NOUN
ejpam-4918	122	14	.	.	PUNCT
ejpam-4918	122	15	(	(	PUNCT
ejpam-4918	122	16	2.1	2.1	NUM
ejpam-4918	122	17	)	)	PUNCT
ejpam-4918	122	18	becomes	become	VERB
ejpam-4918	122	19	(	(	PUNCT
ejpam-4918	122	20	c1l1	c1l1	X
ejpam-4918	122	21	+	+	CCONJ
ejpam-4918	122	22	c2l2	c2l2	X
ejpam-4918	122	23	+	+	NUM
ejpam-4918	122	24	·	·	PUNCT
ejpam-4918	122	25	·	·	PUNCT
ejpam-4918	122	26	·	·	PUNCT
ejpam-4918	123	1	+	+	CCONJ
ejpam-4918	123	2	cnln)gr(a2h2	cnln)gr(a2h2	X
ejpam-4918	123	3	+	+	X
ejpam-4918	123	4	·	·	PUNCT
ejpam-4918	123	5	·	·	PUNCT
ejpam-4918	123	6	·	·	PUNCT
ejpam-4918	124	1	+	+	CCONJ
ejpam-4918	124	2	amhm	amhm	NOUN
ejpam-4918	124	3	)	)	PUNCT
ejpam-4918	124	4	=	=	SYM
ejpam-4918	125	1	0	0	X
ejpam-4918	125	2	.	.	PUNCT
ejpam-4918	126	1	we	we	PRON
ejpam-4918	126	2	can	can	AUX
ejpam-4918	126	3	repeat	repeat	VERB
ejpam-4918	126	4	this	this	DET
ejpam-4918	126	5	process	process	NOUN
ejpam-4918	126	6	to	to	PART
ejpam-4918	126	7	show	show	VERB
ejpam-4918	126	8	that	that	SCONJ
ejpam-4918	126	9	ciωli(ωg(raj	ciωli(ωg(raj	NOUN
ejpam-4918	126	10	)	)	PUNCT
ejpam-4918	126	11	)	)	PUNCT
ejpam-4918	127	1	=	=	SYM
ejpam-4918	127	2	0	0	NUM
ejpam-4918	128	1	for	for	ADP
ejpam-4918	128	2	all	all	DET
ejpam-4918	128	3	g	g	PROPN
ejpam-4918	128	4	∈	∈	PROPN
ejpam-4918	128	5	m	m	VERB
ejpam-4918	128	6	and	and	CCONJ
ejpam-4918	128	7	all	all	PRON
ejpam-4918	128	8	i	i	PROPN
ejpam-4918	128	9	,	,	PUNCT
ejpam-4918	128	10	j.	j.	PROPN
ejpam-4918	128	11	this	this	PRON
ejpam-4918	128	12	shows	show	VERB
ejpam-4918	128	13	that	that	SCONJ
ejpam-4918	128	14	cirωli(ωg(aj	cirωli(ωg(aj	ADJ
ejpam-4918	128	15	)	)	PUNCT
ejpam-4918	128	16	)	)	PUNCT
ejpam-4918	129	1	=	=	PUNCT
ejpam-4918	129	2	0	0	X
ejpam-4918	129	3	.	.	PUNCT
ejpam-4918	130	1	consequently	consequently	ADV
ejpam-4918	130	2	,	,	PUNCT
ejpam-4918	130	3	we	we	PRON
ejpam-4918	130	4	can	can	AUX
ejpam-4918	130	5	see	see	VERB
ejpam-4918	130	6	that	that	DET
ejpam-4918	130	7	ajrωhj	ajrωhj	NOUN
ejpam-4918	130	8	(	(	PUNCT
ejpam-4918	130	9	ωg(ci	ωg(ci	ADJ
ejpam-4918	130	10	)	)	PUNCT
ejpam-4918	130	11	)	)	PUNCT
ejpam-4918	131	1	=	=	SYM
ejpam-4918	131	2	0	0	NUM
ejpam-4918	131	3	for	for	ADP
ejpam-4918	131	4	all	all	DET
ejpam-4918	131	5	g	g	NOUN
ejpam-4918	131	6	∈m	∈m	NOUN
ejpam-4918	131	7	,	,	PUNCT
ejpam-4918	131	8	1	1	NUM
ejpam-4918	131	9	≤	≤	NUM
ejpam-4918	131	10	j	j	PROPN
ejpam-4918	131	11	≤	≤	NUM
ejpam-4918	131	12	m	m	PROPN
ejpam-4918	131	13	,	,	PUNCT
ejpam-4918	131	14	and	and	CCONJ
ejpam-4918	131	15	1	1	NUM
ejpam-4918	131	16	≤	≤	NUM
ejpam-4918	131	17	i	i	PRON
ejpam-4918	131	18	≤	≤	PROPN
ejpam-4918	131	19	n.	n.	NOUN
ejpam-4918	131	20	therefore	therefore	ADV
ejpam-4918	131	21	,	,	PUNCT
ejpam-4918	131	22	r	r	NOUN
ejpam-4918	131	23	is	be	AUX
ejpam-4918	131	24	strongly	strongly	ADV
ejpam-4918	131	25	cm	cm	NOUN
ejpam-4918	131	26	-reflexive	-reflexive	NOUN
ejpam-4918	131	27	.	.	PUNCT
ejpam-4918	132	1	the	the	DET
ejpam-4918	132	2	following	follow	VERB
ejpam-4918	132	3	example	example	NOUN
ejpam-4918	132	4	demonstrates	demonstrate	VERB
ejpam-4918	132	5	the	the	DET
ejpam-4918	132	6	existence	existence	NOUN
ejpam-4918	132	7	of	of	ADP
ejpam-4918	132	8	a	a	DET
ejpam-4918	132	9	ring	ring	NOUN
ejpam-4918	132	10	r	r	NOUN
ejpam-4918	132	11	over	over	ADP
ejpam-4918	132	12	a	a	DET
ejpam-4918	132	13	field	field	NOUN
ejpam-4918	132	14	f	f	NOUN
ejpam-4918	133	1	that	that	PRON
ejpam-4918	133	2	is	be	AUX
ejpam-4918	133	3	not	not	PART
ejpam-4918	133	4	strongly	strongly	ADV
ejpam-4918	133	5	cm	cm	NOUN
ejpam-4918	133	6	-reflexive	-reflexive	NOUN
ejpam-4918	133	7	.	.	PUNCT
ejpam-4918	133	8	example	example	NOUN
ejpam-4918	134	1	1	1	NUM
ejpam-4918	134	2	.	.	PUNCT
ejpam-4918	135	1	let	let	VERB
ejpam-4918	135	2	m	m	PRON
ejpam-4918	135	3	be	be	AUX
ejpam-4918	135	4	a	a	DET
ejpam-4918	135	5	monoid	monoid	NOUN
ejpam-4918	135	6	with	with	ADP
ejpam-4918	135	7	at	at	ADV
ejpam-4918	135	8	least	least	ADV
ejpam-4918	135	9	two	two	NUM
ejpam-4918	135	10	elements	element	NOUN
ejpam-4918	135	11	,	,	PUNCT
ejpam-4918	135	12	and	and	CCONJ
ejpam-4918	135	13	let	let	VERB
ejpam-4918	135	14	s	s	PRON
ejpam-4918	135	15	=	=	SYM
ejpam-4918	135	16	m2(f	m2(f	PROPN
ejpam-4918	135	17	)	)	PUNCT
ejpam-4918	135	18	be	be	AUX
ejpam-4918	135	19	the	the	DET
ejpam-4918	135	20	matrix	matrix	NOUN
ejpam-4918	135	21	ring	ring	NOUN
ejpam-4918	135	22	over	over	ADP
ejpam-4918	135	23	a	a	DET
ejpam-4918	135	24	field	field	NOUN
ejpam-4918	135	25	f	f	NOUN
ejpam-4918	135	26	with	with	ADP
ejpam-4918	135	27	a	a	DET
ejpam-4918	135	28	twisting	twisting	NOUN
ejpam-4918	135	29	map	map	NOUN
ejpam-4918	136	1	f	f	X
ejpam-4918	136	2	:	:	PUNCT
ejpam-4918	136	3	m	m	PROPN
ejpam-4918	136	4	×m	×m	NOUN
ejpam-4918	136	5	→	→	SYM
ejpam-4918	136	6	u(r	u(r	NOUN
ejpam-4918	136	7	)	)	PUNCT
ejpam-4918	136	8	,	,	PUNCT
ejpam-4918	136	9	then	then	ADV
ejpam-4918	136	10	s	s	VERB
ejpam-4918	136	11	is	be	AUX
ejpam-4918	136	12	not	not	PART
ejpam-4918	136	13	strongly	strongly	ADV
ejpam-4918	136	14	cm	cm	NOUN
ejpam-4918	136	15	-reflexive	-reflexive	NOUN
ejpam-4918	136	16	.	.	PUNCT
ejpam-4918	137	1	solution	solution	NOUN
ejpam-4918	137	2	.	.	PUNCT
ejpam-4918	138	1	take	take	VERB
ejpam-4918	138	2	e	e	NOUN
ejpam-4918	138	3	̸=	̸=	PROPN
ejpam-4918	138	4	h	h	NOUN
ejpam-4918	138	5	∈m	∈m	NOUN
ejpam-4918	138	6	,	,	PUNCT
ejpam-4918	138	7	we	we	PRON
ejpam-4918	138	8	define	define	VERB
ejpam-4918	138	9	ω	ω	NOUN
ejpam-4918	138	10	:	:	PUNCT
ejpam-4918	138	11	m	m	PROPN
ejpam-4918	138	12	→	→	SYM
ejpam-4918	138	13	aut(s	aut(s	PROPN
ejpam-4918	138	14	)	)	PUNCT
ejpam-4918	138	15	by	by	ADP
ejpam-4918	138	16	ωh	ωh	INTJ
ejpam-4918	138	17	(	(	PUNCT
ejpam-4918	138	18	(	(	PUNCT
ejpam-4918	138	19	a	a	DET
ejpam-4918	138	20	d	d	NOUN
ejpam-4918	138	21	0	0	NUM
ejpam-4918	138	22	c	c	NOUN
ejpam-4918	138	23	)	)	PUNCT
ejpam-4918	138	24	)	)	PUNCT
ejpam-4918	139	1	=	=	PRON
ejpam-4918	139	2	(	(	PUNCT
ejpam-4918	139	3	a	a	DET
ejpam-4918	139	4	−d	−d	ADJ
ejpam-4918	139	5	0	0	PUNCT
ejpam-4918	139	6	c	c	NOUN
ejpam-4918	139	7	)	)	PUNCT
ejpam-4918	139	8	.	.	PUNCT
ejpam-4918	140	1	if	if	SCONJ
ejpam-4918	140	2	the	the	DET
ejpam-4918	140	3	twisting	twisting	NOUN
ejpam-4918	140	4	map	map	NOUN
ejpam-4918	140	5	f	f	PROPN
ejpam-4918	140	6	is	be	AUX
ejpam-4918	140	7	trivial	trivial	ADJ
ejpam-4918	140	8	(	(	PUNCT
ejpam-4918	140	9	i.e.	i.e.	X
ejpam-4918	140	10	,	,	PUNCT
ejpam-4918	140	11	f(x	f(x	PROPN
ejpam-4918	140	12	,	,	PUNCT
ejpam-4918	140	13	y	y	NOUN
ejpam-4918	140	14	)	)	PUNCT
ejpam-4918	140	15	=	=	SYM
ejpam-4918	140	16	1	1	NUM
ejpam-4918	140	17	for	for	ADP
ejpam-4918	140	18	all	all	DET
ejpam-4918	140	19	x	x	NOUN
ejpam-4918	140	20	,	,	PUNCT
ejpam-4918	140	21	y	y	PROPN
ejpam-4918	140	22	∈	∈	PROPN
ejpam-4918	140	23	m	m	PROPN
ejpam-4918	140	24	)	)	PUNCT
ejpam-4918	140	25	,	,	PUNCT
ejpam-4918	140	26	then	then	ADV
ejpam-4918	140	27	the	the	DET
ejpam-4918	140	28	ring	ring	NOUN
ejpam-4918	140	29	s	s	VERB
ejpam-4918	140	30	is	be	AUX
ejpam-4918	140	31	not	not	PART
ejpam-4918	140	32	strongly	strongly	ADV
ejpam-4918	140	33	cm	cm	NOUN
ejpam-4918	140	34	-reflexive	-reflexive	NOUN
ejpam-4918	140	35	.	.	PUNCT
ejpam-4918	141	1	to	to	PART
ejpam-4918	141	2	see	see	VERB
ejpam-4918	141	3	this	this	PRON
ejpam-4918	141	4	,	,	PUNCT
ejpam-4918	141	5	consider	consider	VERB
ejpam-4918	141	6	ϕ	ϕ	NOUN
ejpam-4918	141	7	=	=	X
ejpam-4918	141	8	e12e+e11h	e12e+e11h	ADV
ejpam-4918	141	9	and	and	CCONJ
ejpam-4918	141	10	ψ	ψ	X
ejpam-4918	141	11	=	=	PUNCT
ejpam-4918	141	12	(	(	PUNCT
ejpam-4918	141	13	e11+e12)h	e11+e12)h	NOUN
ejpam-4918	141	14	∈	∈	PROPN
ejpam-4918	141	15	s∗m	s∗m	PROPN
ejpam-4918	141	16	.	.	PUNCT
ejpam-4918	142	1	for	for	ADP
ejpam-4918	142	2	φ	φ	PROPN
ejpam-4918	142	3	=	=	SYM
ejpam-4918	142	4	(	(	PUNCT
ejpam-4918	142	5	e11	e11	X
ejpam-4918	142	6	+	+	CCONJ
ejpam-4918	142	7	e22)h	e22)h	VERB
ejpam-4918	142	8	∈	∈	PROPN
ejpam-4918	142	9	s	s	PART
ejpam-4918	142	10	∗m	∗m	NOUN
ejpam-4918	142	11	,	,	PUNCT
ejpam-4918	142	12	we	we	PRON
ejpam-4918	142	13	can	can	AUX
ejpam-4918	142	14	easily	easily	ADV
ejpam-4918	142	15	verify	verify	VERB
ejpam-4918	142	16	that	that	PRON
ejpam-4918	142	17	ϕφψ	ϕφψ	ADV
ejpam-4918	142	18	=	=	NOUN
ejpam-4918	142	19	0	0	X
ejpam-4918	142	20	.	.	PUNCT
ejpam-4918	143	1	however	however	ADV
ejpam-4918	143	2	,	,	PUNCT
ejpam-4918	143	3	we	we	PRON
ejpam-4918	143	4	have	have	AUX
ejpam-4918	143	5	ψφϕ	ψφϕ	VERB
ejpam-4918	143	6	̸=	̸=	PROPN
ejpam-4918	143	7	0	0	NUM
ejpam-4918	143	8	,	,	PUNCT
ejpam-4918	143	9	which	which	PRON
ejpam-4918	143	10	implies	imply	VERB
ejpam-4918	143	11	that	that	SCONJ
ejpam-4918	143	12	s	s	VERB
ejpam-4918	143	13	is	be	AUX
ejpam-4918	143	14	not	not	PART
ejpam-4918	143	15	strongly	strongly	ADV
ejpam-4918	143	16	cm	cm	NOUN
ejpam-4918	143	17	-reflexive	-reflexive	NOUN
ejpam-4918	143	18	.	.	PUNCT
ejpam-4918	144	1	e.	e.	PROPN
ejpam-4918	144	2	ali	ali	PROPN
ejpam-4918	144	3	/	/	SYM
ejpam-4918	144	4	eur	eur	PROPN
ejpam-4918	144	5	.	.	PUNCT
ejpam-4918	145	1	j.	j.	PROPN
ejpam-4918	145	2	pure	pure	PROPN
ejpam-4918	145	3	appl	appl	PROPN
ejpam-4918	145	4	.	.	PROPN
ejpam-4918	145	5	math	math	PROPN
ejpam-4918	145	6	,	,	PUNCT
ejpam-4918	145	7	16	16	NUM
ejpam-4918	145	8	(	(	PUNCT
ejpam-4918	145	9	4	4	NUM
ejpam-4918	145	10	)	)	PUNCT
ejpam-4918	145	11	(	(	PUNCT
ejpam-4918	145	12	2023	2023	NUM
ejpam-4918	145	13	)	)	PUNCT
ejpam-4918	145	14	,	,	PUNCT
ejpam-4918	145	15	2156	2156	NUM
ejpam-4918	145	16	-	-	SYM
ejpam-4918	145	17	2168	2168	NUM
ejpam-4918	145	18	2160	2160	NUM
ejpam-4918	145	19	a	a	DET
ejpam-4918	145	20	ring	ring	NOUN
ejpam-4918	145	21	r	r	NOUN
ejpam-4918	145	22	is	be	AUX
ejpam-4918	145	23	categorized	categorize	VERB
ejpam-4918	145	24	as	as	ADP
ejpam-4918	145	25	a	a	DET
ejpam-4918	145	26	right	right	NOUN
ejpam-4918	145	27	pp	pp	ADP
ejpam-4918	145	28	-ring	-ring	ADJ
ejpam-4918	145	29	or	or	CCONJ
ejpam-4918	145	30	left	leave	VERB
ejpam-4918	145	31	pp	pp	ADV
ejpam-4918	145	32	-ring	-ring	ADJ
ejpam-4918	145	33	if	if	SCONJ
ejpam-4918	145	34	the	the	DET
ejpam-4918	145	35	right	right	NOUN
ejpam-4918	145	36	or	or	CCONJ
ejpam-4918	145	37	left	leave	VERB
ejpam-4918	145	38	annihilator	annihilator	NOUN
ejpam-4918	145	39	of	of	ADP
ejpam-4918	145	40	an	an	DET
ejpam-4918	145	41	element	element	NOUN
ejpam-4918	145	42	in	in	ADP
ejpam-4918	145	43	r	r	NOUN
ejpam-4918	145	44	,	,	PUNCT
ejpam-4918	145	45	respectively	respectively	ADV
ejpam-4918	145	46	,	,	PUNCT
ejpam-4918	145	47	is	be	AUX
ejpam-4918	145	48	generated	generate	VERB
ejpam-4918	145	49	by	by	ADP
ejpam-4918	145	50	an	an	DET
ejpam-4918	145	51	idempotent	idempotent	NOUN
ejpam-4918	145	52	.	.	PUNCT
ejpam-4918	146	1	a	a	DET
ejpam-4918	146	2	(	(	PUNCT
ejpam-4918	146	3	quasi-	quasi-	X
ejpam-4918	146	4	)	)	PUNCT
ejpam-4918	146	5	baer	baer	PROPN
ejpam-4918	146	6	ring	ring	NOUN
ejpam-4918	146	7	is	be	AUX
ejpam-4918	146	8	one	one	NUM
ejpam-4918	146	9	where	where	SCONJ
ejpam-4918	146	10	the	the	DET
ejpam-4918	146	11	right	right	ADJ
ejpam-4918	146	12	annihilator	annihilator	NOUN
ejpam-4918	146	13	of	of	ADP
ejpam-4918	146	14	every	every	DET
ejpam-4918	146	15	nonempty	nonempty	NOUN
ejpam-4918	146	16	subset	subset	NOUN
ejpam-4918	146	17	or	or	CCONJ
ejpam-4918	146	18	every	every	DET
ejpam-4918	146	19	right	right	ADJ
ejpam-4918	146	20	ideal	ideal	NOUN
ejpam-4918	146	21	of	of	ADP
ejpam-4918	146	22	r	r	NOUN
ejpam-4918	146	23	is	be	AUX
ejpam-4918	146	24	generated	generate	VERB
ejpam-4918	146	25	by	by	ADP
ejpam-4918	146	26	an	an	DET
ejpam-4918	146	27	idempotent	idempotent	NOUN
ejpam-4918	146	28	.	.	PUNCT
ejpam-4918	147	1	principally	principally	ADV
ejpam-4918	147	2	quasi	quasi	ADJ
ejpam-4918	147	3	-	-	PROPN
ejpam-4918	147	4	baer	baer	PROPN
ejpam-4918	147	5	rings	ring	NOUN
ejpam-4918	147	6	,	,	PUNCT
ejpam-4918	147	7	introduced	introduce	VERB
ejpam-4918	147	8	by	by	ADP
ejpam-4918	147	9	birkenmeier	birkenmeier	PROPN
ejpam-4918	147	10	et	et	PROPN
ejpam-4918	147	11	al	al	PROPN
ejpam-4918	147	12	.	.	PUNCT
ejpam-4918	148	1	[	[	X
ejpam-4918	148	2	13	13	NUM
ejpam-4918	148	3	]	]	PUNCT
ejpam-4918	148	4	,	,	PUNCT
ejpam-4918	148	5	extend	extend	VERB
ejpam-4918	148	6	the	the	DET
ejpam-4918	148	7	concept	concept	NOUN
ejpam-4918	148	8	of	of	ADP
ejpam-4918	148	9	quasi	quasi	ADJ
ejpam-4918	148	10	-	-	PROPN
ejpam-4918	148	11	baer	baer	PROPN
ejpam-4918	148	12	rings	ring	NOUN
ejpam-4918	148	13	.	.	PUNCT
ejpam-4918	149	1	a	a	DET
ejpam-4918	149	2	ring	ring	NOUN
ejpam-4918	149	3	r	r	NOUN
ejpam-4918	149	4	is	be	AUX
ejpam-4918	149	5	referred	refer	VERB
ejpam-4918	149	6	to	to	ADP
ejpam-4918	149	7	as	as	SCONJ
ejpam-4918	149	8	left	leave	VERB
ejpam-4918	149	9	principally	principally	ADV
ejpam-4918	149	10	quasibaer	quasibaer	PROPN
ejpam-4918	149	11	or	or	CCONJ
ejpam-4918	149	12	simply	simply	ADV
ejpam-4918	149	13	left	leave	VERB
ejpam-4918	149	14	p.q.-baer	p.q.-baer	NOUN
ejpam-4918	149	15	if	if	SCONJ
ejpam-4918	149	16	the	the	DET
ejpam-4918	149	17	left	left	ADJ
ejpam-4918	149	18	annihilator	annihilator	NOUN
ejpam-4918	149	19	of	of	ADP
ejpam-4918	149	20	a	a	DET
ejpam-4918	149	21	principal	principal	NOUN
ejpam-4918	149	22	left	leave	VERB
ejpam-4918	149	23	ideal	ideal	NOUN
ejpam-4918	149	24	in	in	SCONJ
ejpam-4918	149	25	r	r	NOUN
ejpam-4918	149	26	is	be	AUX
ejpam-4918	149	27	generated	generate	VERB
ejpam-4918	149	28	by	by	ADP
ejpam-4918	149	29	an	an	DET
ejpam-4918	149	30	idempotent	idempotent	NOUN
ejpam-4918	149	31	.	.	PUNCT
ejpam-4918	150	1	it	it	PRON
ejpam-4918	150	2	is	be	AUX
ejpam-4918	150	3	important	important	ADJ
ejpam-4918	150	4	to	to	PART
ejpam-4918	150	5	note	note	VERB
ejpam-4918	150	6	that	that	SCONJ
ejpam-4918	150	7	biregular	biregular	ADJ
ejpam-4918	150	8	rings	ring	NOUN
ejpam-4918	150	9	and	and	CCONJ
ejpam-4918	150	10	quasi	quasi	PROPN
ejpam-4918	150	11	-	-	PROPN
ejpam-4918	150	12	baer	baer	PROPN
ejpam-4918	150	13	rings	ring	NOUN
ejpam-4918	150	14	are	be	AUX
ejpam-4918	150	15	examples	example	NOUN
ejpam-4918	150	16	of	of	ADP
ejpam-4918	150	17	left	left	ADJ
ejpam-4918	150	18	p.q.-baer	p.q.-baer	PROPN
ejpam-4918	150	19	rings	ring	NOUN
ejpam-4918	150	20	.	.	PUNCT
ejpam-4918	151	1	for	for	ADP
ejpam-4918	151	2	more	more	ADJ
ejpam-4918	151	3	information	information	NOUN
ejpam-4918	151	4	and	and	CCONJ
ejpam-4918	151	5	examples	example	NOUN
ejpam-4918	151	6	of	of	ADP
ejpam-4918	151	7	left	left	ADJ
ejpam-4918	151	8	p.q.-baer	p.q.-baer	NOUN
ejpam-4918	151	9	rings	ring	NOUN
ejpam-4918	151	10	,	,	PUNCT
ejpam-4918	151	11	see	see	VERB
ejpam-4918	151	12	birkenmeier	birkenmeier	PROPN
ejpam-4918	151	13	et	et	PROPN
ejpam-4918	151	14	al	al	PROPN
ejpam-4918	151	15	.	.	PUNCT
ejpam-4918	152	1	(	(	PUNCT
ejpam-4918	152	2	[	[	X
ejpam-4918	152	3	13	13	NUM
ejpam-4918	152	4	]	]	PUNCT
ejpam-4918	152	5	,	,	PUNCT
ejpam-4918	152	6	[	[	X
ejpam-4918	152	7	14	14	NUM
ejpam-4918	152	8	]	]	PUNCT
ejpam-4918	152	9	)	)	PUNCT
ejpam-4918	152	10	and	and	CCONJ
ejpam-4918	152	11	liu	liu	PROPN
ejpam-4918	153	1	[	[	X
ejpam-4918	153	2	15	15	NUM
ejpam-4918	153	3	]	]	PUNCT
ejpam-4918	153	4	.	.	PUNCT
ejpam-4918	154	1	since	since	SCONJ
ejpam-4918	154	2	right	right	ADV
ejpam-4918	154	3	pp	pp	ADP
ejpam-4918	154	4	-rings	-ring	NOUN
ejpam-4918	154	5	and	and	CCONJ
ejpam-4918	154	6	left	leave	VERB
ejpam-4918	154	7	p.q.-baer	p.q.-baer	NOUN
ejpam-4918	154	8	rings	ring	NOUN
ejpam-4918	154	9	both	both	PRON
ejpam-4918	154	10	fall	fall	VERB
ejpam-4918	154	11	under	under	ADP
ejpam-4918	154	12	the	the	DET
ejpam-4918	154	13	category	category	NOUN
ejpam-4918	154	14	of	of	ADP
ejpam-4918	154	15	left	leave	VERB
ejpam-4918	154	16	app	app	NOUN
ejpam-4918	155	1	[	[	X
ejpam-4918	155	2	16	16	NUM
ejpam-4918	155	3	]	]	PUNCT
ejpam-4918	155	4	,	,	PUNCT
ejpam-4918	155	5	the	the	DET
ejpam-4918	155	6	following	follow	VERB
ejpam-4918	155	7	results	result	NOUN
ejpam-4918	155	8	can	can	AUX
ejpam-4918	155	9	be	be	AUX
ejpam-4918	155	10	deduced	deduce	VERB
ejpam-4918	155	11	.	.	PUNCT
ejpam-4918	156	1	theorem	theorem	NOUN
ejpam-4918	156	2	2	2	NUM
ejpam-4918	156	3	.	.	PUNCT
ejpam-4918	156	4	suppose	suppose	VERB
ejpam-4918	156	5	r	r	NOUN
ejpam-4918	156	6	is	be	AUX
ejpam-4918	156	7	a	a	DET
ejpam-4918	156	8	reduced	reduced	ADJ
ejpam-4918	156	9	ring	ring	NOUN
ejpam-4918	156	10	,	,	PUNCT
ejpam-4918	156	11	m	m	VERB
ejpam-4918	156	12	is	be	AUX
ejpam-4918	156	13	a	a	DET
ejpam-4918	156	14	strictly	strictly	ADV
ejpam-4918	156	15	totally	totally	ADV
ejpam-4918	156	16	ordered	order	VERB
ejpam-4918	156	17	monoid	monoid	NOUN
ejpam-4918	156	18	with	with	ADP
ejpam-4918	156	19	a	a	DET
ejpam-4918	156	20	twisting	twisting	NOUN
ejpam-4918	156	21	map	map	NOUN
ejpam-4918	156	22	f	f	X
ejpam-4918	156	23	:	:	PUNCT
ejpam-4918	156	24	m	m	VERB
ejpam-4918	156	25	×m	×m	NOUN
ejpam-4918	156	26	→	→	SYM
ejpam-4918	156	27	u(r	u(r	NOUN
ejpam-4918	156	28	)	)	PUNCT
ejpam-4918	156	29	and	and	CCONJ
ejpam-4918	156	30	an	an	DET
ejpam-4918	156	31	action	action	NOUN
ejpam-4918	156	32	map	map	NOUN
ejpam-4918	156	33	ω	ω	NOUN
ejpam-4918	156	34	:	:	PUNCT
ejpam-4918	156	35	m	m	PROPN
ejpam-4918	156	36	→	→	SYM
ejpam-4918	156	37	aut(r	aut(r	PROPN
ejpam-4918	156	38	)	)	PUNCT
ejpam-4918	156	39	that	that	PRON
ejpam-4918	156	40	is	be	AUX
ejpam-4918	156	41	compatible	compatible	ADJ
ejpam-4918	156	42	with	with	ADP
ejpam-4918	156	43	the	the	DET
ejpam-4918	156	44	multiplication	multiplication	NOUN
ejpam-4918	156	45	in	in	ADP
ejpam-4918	156	46	m	m	PROPN
ejpam-4918	156	47	.	.	PUNCT
ejpam-4918	157	1	if	if	SCONJ
ejpam-4918	157	2	r	r	NOUN
ejpam-4918	157	3	is	be	AUX
ejpam-4918	157	4	a	a	DET
ejpam-4918	157	5	left	left	ADJ
ejpam-4918	157	6	p.q.-baer	p.q.-baer	NOUN
ejpam-4918	157	7	ring	ring	NOUN
ejpam-4918	157	8	,	,	PUNCT
ejpam-4918	157	9	then	then	ADV
ejpam-4918	157	10	r	r	NOUN
ejpam-4918	157	11	is	be	AUX
ejpam-4918	157	12	strongly	strongly	ADV
ejpam-4918	157	13	cm	cm	NOUN
ejpam-4918	157	14	-reflexive	-reflexive	NOUN
ejpam-4918	157	15	.	.	PUNCT
ejpam-4918	158	1	proof	proof	NOUN
ejpam-4918	158	2	.	.	PUNCT
ejpam-4918	159	1	the	the	DET
ejpam-4918	159	2	proof	proof	NOUN
ejpam-4918	159	3	is	be	AUX
ejpam-4918	159	4	a	a	DET
ejpam-4918	159	5	variant	variant	NOUN
ejpam-4918	159	6	of	of	ADP
ejpam-4918	159	7	the	the	DET
ejpam-4918	159	8	proof	proof	NOUN
ejpam-4918	159	9	given	give	VERB
ejpam-4918	159	10	in	in	ADP
ejpam-4918	159	11	proposition	proposition	NOUN
ejpam-4918	159	12	2.9	2.9	NUM
ejpam-4918	159	13	[	[	X
ejpam-4918	159	14	17	17	NUM
ejpam-4918	159	15	]	]	PUNCT
ejpam-4918	159	16	.	.	PUNCT
ejpam-4918	160	1	let	let	VERB
ejpam-4918	160	2	ϕ	ϕ	X
ejpam-4918	160	3	=	=	VERB
ejpam-4918	160	4	c1l1+c2l2	c1l1+c2l2	PROPN
ejpam-4918	160	5	+	+	PROPN
ejpam-4918	160	6	·	·	PUNCT
ejpam-4918	160	7	·	·	PUNCT
ejpam-4918	160	8	·	·	PUNCT
ejpam-4918	160	9	+	+	NOUN
ejpam-4918	160	10	cnln	cnln	NOUN
ejpam-4918	160	11	,	,	PUNCT
ejpam-4918	160	12	ψ	ψ	X
ejpam-4918	160	13	=	=	PUNCT
ejpam-4918	160	14	a1h1+a2h2	a1h1+a2h2	PROPN
ejpam-4918	160	15	+	+	X
ejpam-4918	160	16	·	·	PUNCT
ejpam-4918	160	17	·	·	PUNCT
ejpam-4918	160	18	·	·	PUNCT
ejpam-4918	161	1	+	+	NUM
ejpam-4918	161	2	amhm	amhm	NOUN
ejpam-4918	161	3	∈	∈	PROPN
ejpam-4918	161	4	r∗m	r∗m	PUNCT
ejpam-4918	161	5	satisfy	satisfy	VERB
ejpam-4918	161	6	ϕ(r∗m)ψ	ϕ(r∗m)ψ	NOUN
ejpam-4918	161	7	=	=	SYM
ejpam-4918	161	8	0	0	X
ejpam-4918	161	9	.	.	PUNCT
ejpam-4918	162	1	since	since	SCONJ
ejpam-4918	162	2	m	m	PROPN
ejpam-4918	162	3	is	be	AUX
ejpam-4918	162	4	a	a	DET
ejpam-4918	162	5	strictly	strictly	ADV
ejpam-4918	162	6	totally	totally	ADV
ejpam-4918	162	7	ordered	order	VERB
ejpam-4918	162	8	monoid	monoid	NOUN
ejpam-4918	162	9	,	,	PUNCT
ejpam-4918	162	10	we	we	PRON
ejpam-4918	162	11	can	can	AUX
ejpam-4918	162	12	assume	assume	VERB
ejpam-4918	162	13	that	that	SCONJ
ejpam-4918	162	14	li	li	PROPN
ejpam-4918	162	15	⪯	⪯	PROPN
ejpam-4918	162	16	lj	lj	PROPN
ejpam-4918	162	17	and	and	CCONJ
ejpam-4918	162	18	hi	hi	INTJ
ejpam-4918	162	19	⪯	⪯	NOUN
ejpam-4918	162	20	hj	hj	INTJ
ejpam-4918	162	21	whenever	whenever	SCONJ
ejpam-4918	162	22	i	i	PRON
ejpam-4918	162	23	<	<	X
ejpam-4918	162	24	j.	j.	PROPN
ejpam-4918	163	1	now	now	ADV
ejpam-4918	163	2	,	,	PUNCT
ejpam-4918	163	3	we	we	PRON
ejpam-4918	163	4	claim	claim	VERB
ejpam-4918	163	5	ciωli(ωg(raj	ciωli(ωg(raj	NOUN
ejpam-4918	163	6	)	)	PUNCT
ejpam-4918	163	7	)	)	PUNCT
ejpam-4918	164	1	=	=	SYM
ejpam-4918	164	2	0	0	NUM
ejpam-4918	164	3	for	for	ADP
ejpam-4918	164	4	all	all	DET
ejpam-4918	164	5	i	i	PROPN
ejpam-4918	164	6	,	,	PUNCT
ejpam-4918	164	7	j.	j.	PROPN
ejpam-4918	164	8	let	let	VERB
ejpam-4918	164	9	r	r	PRON
ejpam-4918	164	10	be	be	AUX
ejpam-4918	164	11	an	an	DET
ejpam-4918	164	12	element	element	NOUN
ejpam-4918	164	13	of	of	ADP
ejpam-4918	164	14	r.	r.	PROPN
ejpam-4918	164	15	then	then	ADV
ejpam-4918	164	16	,	,	PUNCT
ejpam-4918	164	17	we	we	PRON
ejpam-4918	164	18	have	have	VERB
ejpam-4918	164	19	ϕ(re)ψ	ϕ(re)ψ	ADV
ejpam-4918	164	20	=	=	NOUN
ejpam-4918	164	21	0	0	PUNCT
ejpam-4918	164	22	since	since	SCONJ
ejpam-4918	164	23	ϕ(r	ϕ(r	PROPN
ejpam-4918	164	24	∗m)ψ	∗m)ψ	VERB
ejpam-4918	164	25	=	=	PUNCT
ejpam-4918	164	26	0	0	X
ejpam-4918	164	27	.	.	PUNCT
ejpam-4918	165	1	thus	thus	ADV
ejpam-4918	165	2	,	,	PUNCT
ejpam-4918	165	3	we	we	PRON
ejpam-4918	165	4	have	have	VERB
ejpam-4918	165	5	0	0	NUM
ejpam-4918	166	1	=	=	NUM
ejpam-4918	166	2	ϕ(re)ψ	ϕ(re)ψ	PROPN
ejpam-4918	166	3	=	=	SYM
ejpam-4918	166	4	c1rf(l1	c1rf(l1	PROPN
ejpam-4918	166	5	,	,	PUNCT
ejpam-4918	166	6	e)a1f(l1	e)a1f(l1	PROPN
ejpam-4918	166	7	,	,	PUNCT
ejpam-4918	166	8	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	166	9	+	+	PRON
ejpam-4918	166	10	·	·	PUNCT
ejpam-4918	166	11	·	·	PUNCT
ejpam-4918	166	12	·	·	PUNCT
ejpam-4918	166	13	+	+	PUNCT
ejpam-4918	167	1	[	[	X
ejpam-4918	167	2	cnrf(ln	cnrf(ln	NOUN
ejpam-4918	167	3	,	,	PUNCT
ejpam-4918	167	4	e)am−2f(ln	e)am−2f(ln	NOUN
ejpam-4918	167	5	,	,	PUNCT
ejpam-4918	167	6	hm−2)lnhm−2	hm−2)lnhm−2	PROPN
ejpam-4918	167	7	+	+	NUM
ejpam-4918	167	8	cn−1rf(ln−1	cn−1rf(ln−1	PROPN
ejpam-4918	167	9	,	,	PUNCT
ejpam-4918	167	10	e)am−1f(ln−1	e)am−1f(ln−1	PROPN
ejpam-4918	167	11	,	,	PUNCT
ejpam-4918	167	12	hm−1)ln−1hm−1	hm−1)ln−1hm−1	X
ejpam-4918	167	13	+	+	CCONJ
ejpam-4918	167	14	cn−2rf(ln−2	cn−2rf(ln−2	PROPN
ejpam-4918	167	15	,	,	PUNCT
ejpam-4918	167	16	e)lmf(ln−2	e)lmf(ln−2	PROPN
ejpam-4918	167	17	,	,	PUNCT
ejpam-4918	167	18	hm)ln−2hm	hm)ln−2hm	NOUN
ejpam-4918	167	19	]	]	X
ejpam-4918	168	1	+	+	CCONJ
ejpam-4918	168	2	[	[	X
ejpam-4918	168	3	cnrf(ln	cnrf(ln	NOUN
ejpam-4918	168	4	,	,	PUNCT
ejpam-4918	168	5	e)am−1f(ln	e)am−1f(ln	PROPN
ejpam-4918	168	6	,	,	PUNCT
ejpam-4918	168	7	hm−1)lnhm−1	hm−1)lnhm−1	PROPN
ejpam-4918	168	8	+	+	CCONJ
ejpam-4918	168	9	an−1rf(ln−1	an−1rf(ln−1	PROPN
ejpam-4918	168	10	,	,	PUNCT
ejpam-4918	168	11	e)amf(ln−1	e)amf(ln−1	PROPN
ejpam-4918	168	12	,	,	PUNCT
ejpam-4918	168	13	hm)ln−1hm	hm)ln−1hm	NOUN
ejpam-4918	168	14	]	]	X
ejpam-4918	168	15	+	+	X
ejpam-4918	168	16	cnrf(ln	cnrf(ln	PROPN
ejpam-4918	168	17	,	,	PUNCT
ejpam-4918	168	18	e)amf(ln	e)amf(ln	PROPN
ejpam-4918	168	19	,	,	PUNCT
ejpam-4918	168	20	hm)lnhm	hm)lnhm	PROPN
ejpam-4918	168	21	.	.	PUNCT
ejpam-4918	169	1	(	(	PUNCT
ejpam-4918	169	2	2.2	2.2	NUM
ejpam-4918	169	3	)	)	PUNCT
ejpam-4918	169	4	it	it	PRON
ejpam-4918	169	5	follows	follow	VERB
ejpam-4918	169	6	that	that	SCONJ
ejpam-4918	169	7	cnrf(ln	cnrf(ln	PROPN
ejpam-4918	169	8	,	,	PUNCT
ejpam-4918	169	9	e)amf(ln	e)amf(ln	PROPN
ejpam-4918	169	10	,	,	PUNCT
ejpam-4918	169	11	hm	hm	INTJ
ejpam-4918	169	12	)	)	PUNCT
ejpam-4918	169	13	=	=	SYM
ejpam-4918	169	14	0	0	NUM
ejpam-4918	169	15	since	since	SCONJ
ejpam-4918	169	16	lnhm	lnhm	NOUN
ejpam-4918	169	17	is	be	AUX
ejpam-4918	169	18	of	of	ADP
ejpam-4918	169	19	highest	high	ADJ
ejpam-4918	169	20	order	order	NOUN
ejpam-4918	169	21	in	in	ADP
ejpam-4918	169	22	the	the	DET
ejpam-4918	169	23	lih	lih	INTJ
ejpam-4918	169	24	′	′	NUM
ejpam-4918	169	25	js	js	PROPN
ejpam-4918	169	26	.	.	PUNCT
ejpam-4918	170	1	hence	hence	ADV
ejpam-4918	170	2	cnrf(ln	cnrf(ln	PROPN
ejpam-4918	170	3	,	,	PUNCT
ejpam-4918	170	4	e)am	e)am	PROPN
ejpam-4918	170	5	=	=	NOUN
ejpam-4918	170	6	0	0	X
ejpam-4918	170	7	.	.	PUNCT
ejpam-4918	171	1	this	this	PRON
ejpam-4918	171	2	shows	show	VERB
ejpam-4918	171	3	that	that	SCONJ
ejpam-4918	171	4	cn	cn	PROPN
ejpam-4918	171	5	∈	∈	PROPN
ejpam-4918	171	6	ℓr(rf(ln	ℓr(rf(ln	PROPN
ejpam-4918	171	7	,	,	PUNCT
ejpam-4918	171	8	e)am	e)am	PROPN
ejpam-4918	171	9	)	)	PUNCT
ejpam-4918	171	10	=	=	SYM
ejpam-4918	171	11	ℓr(ram	ℓr(ram	PROPN
ejpam-4918	171	12	)	)	PUNCT
ejpam-4918	171	13	.	.	PUNCT
ejpam-4918	172	1	hence	hence	ADV
ejpam-4918	172	2	,	,	PUNCT
ejpam-4918	172	3	ℓr(ram	ℓr(ram	PROPN
ejpam-4918	172	4	)	)	PUNCT
ejpam-4918	172	5	=	=	PUNCT
ejpam-4918	173	1	rem	rem	VERB
ejpam-4918	173	2	for	for	ADP
ejpam-4918	173	3	some	some	DET
ejpam-4918	173	4	idempotent	idempotent	NOUN
ejpam-4918	173	5	em	em	PRON
ejpam-4918	173	6	by	by	ADP
ejpam-4918	173	7	hypothesis	hypothesis	NOUN
ejpam-4918	173	8	.	.	PUNCT
ejpam-4918	174	1	replacing	replace	VERB
ejpam-4918	174	2	r	r	NOUN
ejpam-4918	174	3	by	by	ADP
ejpam-4918	174	4	rem	rem	NOUN
ejpam-4918	174	5	in	in	ADP
ejpam-4918	174	6	eq	eq	ADP
ejpam-4918	174	7	.	.	PUNCT
ejpam-4918	175	1	(	(	PUNCT
ejpam-4918	175	2	2.2	2.2	NUM
ejpam-4918	175	3	)	)	PUNCT
ejpam-4918	175	4	we	we	PRON
ejpam-4918	175	5	obtain	obtain	VERB
ejpam-4918	175	6	0	0	NUM
ejpam-4918	176	1	=	=	SYM
ejpam-4918	177	1	c1remf(l1	c1remf(l1	PROPN
ejpam-4918	177	2	,	,	PUNCT
ejpam-4918	177	3	e)a1f(l1	e)a1f(l1	PROPN
ejpam-4918	177	4	,	,	PUNCT
ejpam-4918	177	5	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	177	6	+	+	PRON
ejpam-4918	177	7	·	·	PUNCT
ejpam-4918	177	8	·	·	PUNCT
ejpam-4918	177	9	·	·	PUNCT
ejpam-4918	177	10	+	+	PUNCT
ejpam-4918	178	1	[	[	X
ejpam-4918	178	2	cnremf(ln	cnremf(ln	NOUN
ejpam-4918	178	3	,	,	PUNCT
ejpam-4918	178	4	e)am−2f(ln	e)am−2f(ln	NOUN
ejpam-4918	178	5	,	,	PUNCT
ejpam-4918	178	6	hm−2)lnhm−2	hm−2)lnhm−2	PROPN
ejpam-4918	178	7	+	+	NOUN
ejpam-4918	178	8	cn−1remf(ln−1	cn−1remf(ln−1	ADJ
ejpam-4918	178	9	,	,	PUNCT
ejpam-4918	178	10	e)am−1f(ln−1	e)am−1f(ln−1	PROPN
ejpam-4918	178	11	,	,	PUNCT
ejpam-4918	178	12	hm−1)ln−1hm−1]+cnremf(ln	hm−1)ln−1hm−1]+cnremf(ln	PROPN
ejpam-4918	178	13	,	,	PUNCT
ejpam-4918	178	14	e)am−1f(ln	e)am−1f(ln	NOUN
ejpam-4918	178	15	,	,	PUNCT
ejpam-4918	178	16	hm−1)lnhm−1(2.3	hm−1)lnhm−1(2.3	NOUN
ejpam-4918	178	17	)	)	PUNCT
ejpam-4918	178	18	so	so	ADV
ejpam-4918	178	19	cnremf(ln	cnremf(ln	PROPN
ejpam-4918	178	20	,	,	PUNCT
ejpam-4918	178	21	e)am−1f(ln	e)am−1f(ln	PROPN
ejpam-4918	178	22	,	,	PUNCT
ejpam-4918	178	23	hm−1	hm−1	NOUN
ejpam-4918	178	24	)	)	PUNCT
ejpam-4918	178	25	=	=	PUNCT
ejpam-4918	178	26	0	0	NUM
ejpam-4918	178	27	,	,	PUNCT
ejpam-4918	178	28	because	because	SCONJ
ejpam-4918	178	29	lnhm−1	lnhm−1	PROPN
ejpam-4918	178	30	is	be	AUX
ejpam-4918	178	31	of	of	ADP
ejpam-4918	178	32	highest	high	ADJ
ejpam-4918	178	33	order	order	NOUN
ejpam-4918	178	34	in	in	ADP
ejpam-4918	178	35	{	{	PUNCT
ejpam-4918	178	36	lihj	lihj	PROPN
ejpam-4918	178	37	|1	|1	NUM
ejpam-4918	178	38	≤	≤	NUM
ejpam-4918	178	39	i	i	PRON
ejpam-4918	178	40	≤	≤	PROPN
ejpam-4918	178	41	n	n	CCONJ
ejpam-4918	178	42	,	,	PUNCT
ejpam-4918	178	43	1	1	NUM
ejpam-4918	178	44	≤	≤	NUM
ejpam-4918	178	45	j	j	PROPN
ejpam-4918	178	46	≤	≤	PROPN
ejpam-4918	178	47	m	m	VERB
ejpam-4918	178	48	}	}	PUNCT
ejpam-4918	178	49	{	{	PUNCT
ejpam-4918	178	50	ln−1hm	ln−1hm	ADV
ejpam-4918	178	51	,	,	PUNCT
ejpam-4918	178	52	lnhm	lnhm	NOUN
ejpam-4918	178	53	}	}	PUNCT
ejpam-4918	178	54	.	.	PUNCT
ejpam-4918	179	1	hence	hence	ADV
ejpam-4918	179	2	cnremf(ln	cnremf(ln	PROPN
ejpam-4918	179	3	,	,	PUNCT
ejpam-4918	179	4	e)am−1	e)am−1	NOUN
ejpam-4918	179	5	=	=	PUNCT
ejpam-4918	179	6	0	0	X
ejpam-4918	179	7	.	.	PUNCT
ejpam-4918	180	1	since	since	SCONJ
ejpam-4918	180	2	rem	rem	X
ejpam-4918	180	3	is	be	AUX
ejpam-4918	180	4	an	an	DET
ejpam-4918	180	5	ideal	ideal	NOUN
ejpam-4918	180	6	of	of	ADP
ejpam-4918	180	7	r	r	NOUN
ejpam-4918	180	8	and	and	CCONJ
ejpam-4918	180	9	em	em	PRON
ejpam-4918	180	10	∈	∈	PROPN
ejpam-4918	180	11	rem	rem	PROPN
ejpam-4918	180	12	,	,	PUNCT
ejpam-4918	180	13	we	we	PRON
ejpam-4918	180	14	have	have	VERB
ejpam-4918	180	15	emr	emr	PROPN
ejpam-4918	180	16	∈	∈	PROPN
ejpam-4918	180	17	rem	rem	X
ejpam-4918	180	18	and	and	CCONJ
ejpam-4918	180	19	thus	thus	ADV
ejpam-4918	180	20	emr	emr	PROPN
ejpam-4918	180	21	=	=	PUNCT
ejpam-4918	180	22	emrem	emrem	VERB
ejpam-4918	180	23	for	for	ADP
ejpam-4918	180	24	all	all	DET
ejpam-4918	180	25	r	r	PROPN
ejpam-4918	180	26	∈	∈	PROPN
ejpam-4918	180	27	r.	r.	NOUN
ejpam-4918	180	28	on	on	ADP
ejpam-4918	180	29	the	the	DET
ejpam-4918	180	30	other	other	ADJ
ejpam-4918	180	31	hand	hand	NOUN
ejpam-4918	180	32	,	,	PUNCT
ejpam-4918	180	33	we	we	PRON
ejpam-4918	180	34	also	also	ADV
ejpam-4918	180	35	have	have	VERB
ejpam-4918	180	36	cn	cn	PROPN
ejpam-4918	180	37	=	=	SYM
ejpam-4918	180	38	cnem	cnem	PROPN
ejpam-4918	180	39	since	since	SCONJ
ejpam-4918	180	40	cn	cn	PROPN
ejpam-4918	180	41	∈	∈	PROPN
ejpam-4918	180	42	ℓr(ram	ℓr(ram	PROPN
ejpam-4918	180	43	)	)	PUNCT
ejpam-4918	180	44	=	=	PUNCT
ejpam-4918	181	1	rem	rem	X
ejpam-4918	181	2	.	.	PROPN
ejpam-4918	181	3	hence	hence	ADV
ejpam-4918	181	4	cnrf(ln	cnrf(ln	PROPN
ejpam-4918	181	5	,	,	PUNCT
ejpam-4918	181	6	e)am−1	e)am−1	NOUN
ejpam-4918	181	7	=	=	SYM
ejpam-4918	181	8	cnemrf(ln	cnemrf(ln	PROPN
ejpam-4918	181	9	,	,	PUNCT
ejpam-4918	181	10	e)am−1	e)am−1	NOUN
ejpam-4918	181	11	=	=	SYM
ejpam-4918	181	12	cnemremf(ln	cnemremf(ln	PROPN
ejpam-4918	181	13	,	,	PUNCT
ejpam-4918	181	14	e)am−1	e)am−1	NOUN
ejpam-4918	181	15	=	=	SYM
ejpam-4918	181	16	cnremf(ln	cnremf(ln	NOUN
ejpam-4918	181	17	,	,	PUNCT
ejpam-4918	181	18	e)am−1	e)am−1	NOUN
ejpam-4918	181	19	=	=	PUNCT
ejpam-4918	181	20	0	0	X
ejpam-4918	181	21	.	.	PUNCT
ejpam-4918	182	1	this	this	PRON
ejpam-4918	182	2	implies	imply	VERB
ejpam-4918	182	3	that	that	SCONJ
ejpam-4918	182	4	cn	cn	PROPN
ejpam-4918	182	5	∈	∈	PROPN
ejpam-4918	182	6	ℓr(ram	ℓr(ram	PROPN
ejpam-4918	182	7	+	+	CCONJ
ejpam-4918	182	8	ram−1	ram−1	PROPN
ejpam-4918	182	9	)	)	PUNCT
ejpam-4918	182	10	,	,	PUNCT
ejpam-4918	182	11	and	and	CCONJ
ejpam-4918	182	12	hence	hence	ADV
ejpam-4918	182	13	ℓr(ram	ℓr(ram	PROPN
ejpam-4918	182	14	+	+	CCONJ
ejpam-4918	182	15	ram−1	ram−1	PROPN
ejpam-4918	182	16	)	)	PUNCT
ejpam-4918	182	17	=	=	VERB
ejpam-4918	183	1	rem−1	rem−1	PROPN
ejpam-4918	183	2	for	for	ADP
ejpam-4918	183	3	some	some	DET
ejpam-4918	183	4	idempotent	idempotent	NOUN
ejpam-4918	183	5	em−1	em−1	PROPN
ejpam-4918	183	6	∈	∈	PROPN
ejpam-4918	183	7	r	r	NOUN
ejpam-4918	183	8	since	since	SCONJ
ejpam-4918	183	9	r	r	NOUN
ejpam-4918	183	10	is	be	AUX
ejpam-4918	183	11	a	a	DET
ejpam-4918	183	12	left	left	ADJ
ejpam-4918	183	13	p.q.-baer	p.q.-baer	NOUN
ejpam-4918	183	14	ring	ring	NOUN
ejpam-4918	183	15	.	.	PUNCT
ejpam-4918	184	1	replacing	replace	VERB
ejpam-4918	184	2	r	r	NOUN
ejpam-4918	184	3	by	by	ADP
ejpam-4918	184	4	rem−1	rem−1	PROPN
ejpam-4918	184	5	in	in	ADP
ejpam-4918	184	6	equation	equation	NOUN
ejpam-4918	184	7	(	(	PUNCT
ejpam-4918	184	8	2.3	2.3	NUM
ejpam-4918	184	9	)	)	PUNCT
ejpam-4918	184	10	we	we	PRON
ejpam-4918	184	11	obtain	obtain	VERB
ejpam-4918	184	12	cnrem−1f(ln	cnrem−1f(ln	NOUN
ejpam-4918	184	13	,	,	PUNCT
ejpam-4918	184	14	e)am−2f(ln	e)am−2f(ln	NOUN
ejpam-4918	184	15	,	,	PUNCT
ejpam-4918	184	16	hm−2	hm−2	ADJ
ejpam-4918	184	17	)	)	PUNCT
ejpam-4918	184	18	=	=	SYM
ejpam-4918	184	19	0	0	NUM
ejpam-4918	184	20	in	in	ADP
ejpam-4918	184	21	the	the	DET
ejpam-4918	184	22	same	same	ADJ
ejpam-4918	184	23	way	way	NOUN
ejpam-4918	184	24	as	as	ADP
ejpam-4918	184	25	above	above	ADV
ejpam-4918	184	26	.	.	PUNCT
ejpam-4918	185	1	this	this	PRON
ejpam-4918	185	2	shows	show	VERB
ejpam-4918	185	3	that	that	SCONJ
ejpam-4918	185	4	cn	cn	PROPN
ejpam-4918	185	5	∈	∈	PROPN
ejpam-4918	185	6	ℓr(ram+ram−1	ℓr(ram+ram−1	PROPN
ejpam-4918	186	1	+	+	ADJ
ejpam-4918	187	1	ram−2	ram−2	NOUN
ejpam-4918	187	2	)	)	PUNCT
ejpam-4918	187	3	.	.	PUNCT
ejpam-4918	188	1	continuing	continue	VERB
ejpam-4918	188	2	this	this	DET
ejpam-4918	188	3	process	process	NOUN
ejpam-4918	188	4	we	we	PRON
ejpam-4918	188	5	obtain	obtain	VERB
ejpam-4918	188	6	cnrat	cnrat	NOUN
ejpam-4918	188	7	=	=	NOUN
ejpam-4918	188	8	0	0	NUM
ejpam-4918	188	9	for	for	ADP
ejpam-4918	188	10	all	all	DET
ejpam-4918	188	11	t	t	NOUN
ejpam-4918	188	12	=	=	SYM
ejpam-4918	188	13	1	1	NUM
ejpam-4918	188	14	,	,	PUNCT
ejpam-4918	188	15	2	2	NUM
ejpam-4918	188	16	,	,	PUNCT
ejpam-4918	188	17	.	.	PUNCT
ejpam-4918	188	18	.	.	PUNCT
ejpam-4918	189	1	.	.	PUNCT
ejpam-4918	190	1	,	,	PUNCT
ejpam-4918	190	2	m.	m.	NOUN
ejpam-4918	191	1	so	so	ADV
ejpam-4918	191	2	,	,	PUNCT
ejpam-4918	191	3	we	we	PRON
ejpam-4918	191	4	have	have	VERB
ejpam-4918	191	5	(	(	PUNCT
ejpam-4918	191	6	c1l1+c2l2	c1l1+c2l2	PROPN
ejpam-4918	191	7	+	+	NUM
ejpam-4918	191	8	·	·	PUNCT
ejpam-4918	191	9	·	·	PUNCT
ejpam-4918	191	10	·	·	PUNCT
ejpam-4918	192	1	+	+	ADJ
ejpam-4918	192	2	cn−1ln−1)(r∗m)(a1h1+a2h2	cn−1ln−1)(r∗m)(a1h1+a2h2	X
ejpam-4918	192	3	+	+	X
ejpam-4918	192	4	·	·	PUNCT
ejpam-4918	192	5	·	·	PUNCT
ejpam-4918	192	6	·	·	PUNCT
ejpam-4918	192	7	+	+	NUM
ejpam-4918	192	8	amhm	amhm	NOUN
ejpam-4918	192	9	)	)	PUNCT
ejpam-4918	192	10	=	=	SYM
ejpam-4918	193	1	0	0	X
ejpam-4918	193	2	.	.	PUNCT
ejpam-4918	193	3	using	use	VERB
ejpam-4918	193	4	induction	induction	NOUN
ejpam-4918	193	5	on	on	ADP
ejpam-4918	193	6	m+n	m+n	PROPN
ejpam-4918	193	7	,	,	PUNCT
ejpam-4918	193	8	we	we	PRON
ejpam-4918	193	9	obtain	obtain	VERB
ejpam-4918	193	10	ciωli(ωg(raj	ciωli(ωg(raj	NOUN
ejpam-4918	193	11	)	)	PUNCT
ejpam-4918	193	12	)	)	PUNCT
ejpam-4918	194	1	=	=	SYM
ejpam-4918	194	2	0	0	NUM
ejpam-4918	194	3	for	for	ADP
ejpam-4918	194	4	all	all	DET
ejpam-4918	194	5	i	i	PROPN
ejpam-4918	194	6	,	,	PUNCT
ejpam-4918	194	7	j.	j.	PROPN
ejpam-4918	195	1	so	so	ADV
ejpam-4918	195	2	it	it	PRON
ejpam-4918	195	3	is	be	AUX
ejpam-4918	195	4	easy	easy	ADJ
ejpam-4918	195	5	to	to	PART
ejpam-4918	195	6	see	see	VERB
ejpam-4918	195	7	that	that	DET
ejpam-4918	195	8	ajωhj	ajωhj	NOUN
ejpam-4918	195	9	(	(	PUNCT
ejpam-4918	195	10	ωg(rci	ωg(rci	NOUN
ejpam-4918	195	11	)	)	PUNCT
ejpam-4918	195	12	)	)	PUNCT
ejpam-4918	196	1	=	=	PUNCT
ejpam-4918	196	2	0	0	NUM
ejpam-4918	196	3	by	by	ADP
ejpam-4918	196	4	a	a	DET
ejpam-4918	196	5	reduced	reduce	VERB
ejpam-4918	196	6	ness	ness	NOUN
ejpam-4918	196	7	.	.	PUNCT
ejpam-4918	197	1	therefore	therefore	ADV
ejpam-4918	197	2	,	,	PUNCT
ejpam-4918	197	3	r	r	NOUN
ejpam-4918	197	4	is	be	AUX
ejpam-4918	197	5	strongly	strongly	ADV
ejpam-4918	197	6	cm	cm	NOUN
ejpam-4918	197	7	-reflexive	-reflexive	NOUN
ejpam-4918	197	8	.	.	PUNCT
ejpam-4918	198	1	if	if	SCONJ
ejpam-4918	198	2	n	n	PRON
ejpam-4918	198	3	is	be	AUX
ejpam-4918	198	4	an	an	DET
ejpam-4918	198	5	ideal	ideal	NOUN
ejpam-4918	198	6	of	of	ADP
ejpam-4918	198	7	the	the	DET
ejpam-4918	198	8	monoid	monoid	NOUN
ejpam-4918	198	9	m	m	VERB
ejpam-4918	198	10	with	with	ADP
ejpam-4918	198	11	twisting	twist	VERB
ejpam-4918	198	12	f	f	X
ejpam-4918	198	13	:	:	PUNCT
ejpam-4918	198	14	m	m	VERB
ejpam-4918	198	15	×	×	NOUN
ejpam-4918	198	16	m	m	INTJ
ejpam-4918	198	17	→	→	SYM
ejpam-4918	198	18	u(r	u(r	NOUN
ejpam-4918	198	19	)	)	PUNCT
ejpam-4918	198	20	and	and	CCONJ
ejpam-4918	198	21	action	action	NOUN
ejpam-4918	198	22	e.	e.	PROPN
ejpam-4918	198	23	ali	ali	PROPN
ejpam-4918	198	24	/	/	SYM
ejpam-4918	198	25	eur	eur	PROPN
ejpam-4918	198	26	.	.	PUNCT
ejpam-4918	199	1	j.	j.	PROPN
ejpam-4918	199	2	pure	pure	PROPN
ejpam-4918	199	3	appl	appl	PROPN
ejpam-4918	199	4	.	.	PROPN
ejpam-4918	199	5	math	math	PROPN
ejpam-4918	199	6	,	,	PUNCT
ejpam-4918	199	7	16	16	NUM
ejpam-4918	199	8	(	(	PUNCT
ejpam-4918	199	9	4	4	NUM
ejpam-4918	199	10	)	)	PUNCT
ejpam-4918	199	11	(	(	PUNCT
ejpam-4918	199	12	2023	2023	NUM
ejpam-4918	199	13	)	)	PUNCT
ejpam-4918	199	14	,	,	PUNCT
ejpam-4918	199	15	2156	2156	NUM
ejpam-4918	199	16	-	-	SYM
ejpam-4918	199	17	2168	2168	NUM
ejpam-4918	199	18	2161	2161	NUM
ejpam-4918	199	19	ω	ω	NOUN
ejpam-4918	199	20	:	:	PUNCT
ejpam-4918	199	21	m	m	PROPN
ejpam-4918	199	22	→	→	SYM
ejpam-4918	199	23	aut(r	aut(r	PROPN
ejpam-4918	199	24	)	)	PUNCT
ejpam-4918	199	25	,	,	PUNCT
ejpam-4918	199	26	then	then	ADV
ejpam-4918	199	27	the	the	DET
ejpam-4918	199	28	restrictions	restriction	NOUN
ejpam-4918	199	29	f	f	X
ejpam-4918	200	1	|n×n	|n×n	PROPN
ejpam-4918	200	2	:	:	PUNCT
ejpam-4918	200	3	n	n	PROPN
ejpam-4918	200	4	×n	×n	PROPN
ejpam-4918	200	5	→	→	SYM
ejpam-4918	200	6	u(r	u(r	NOUN
ejpam-4918	200	7	)	)	PUNCT
ejpam-4918	200	8	and	and	CCONJ
ejpam-4918	200	9	ω|n	ω|n	ADV
ejpam-4918	200	10	:	:	PUNCT
ejpam-4918	200	11	n	n	X
ejpam-4918	200	12	→	→	SYM
ejpam-4918	200	13	aut(r	aut(r	PROPN
ejpam-4918	200	14	)	)	PUNCT
ejpam-4918	200	15	are	be	AUX
ejpam-4918	200	16	induced	induce	VERB
ejpam-4918	200	17	twisting	twisting	NOUN
ejpam-4918	200	18	and	and	CCONJ
ejpam-4918	200	19	action	action	NOUN
ejpam-4918	200	20	.	.	PUNCT
ejpam-4918	201	1	proposition	proposition	NOUN
ejpam-4918	201	2	1	1	NUM
ejpam-4918	201	3	.	.	PUNCT
ejpam-4918	202	1	let	let	VERB
ejpam-4918	202	2	r	r	PRON
ejpam-4918	202	3	be	be	AUX
ejpam-4918	202	4	an	an	DET
ejpam-4918	202	5	m	m	NOUN
ejpam-4918	202	6	-compatible	-compatible	ADJ
ejpam-4918	202	7	ring	ring	NOUN
ejpam-4918	202	8	and	and	CCONJ
ejpam-4918	202	9	m	m	AUX
ejpam-4918	202	10	be	be	AUX
ejpam-4918	202	11	a	a	DET
ejpam-4918	202	12	commutative	commutative	ADJ
ejpam-4918	202	13	,	,	PUNCT
ejpam-4918	202	14	cancellative	cancellative	ADJ
ejpam-4918	202	15	monoid	monoid	NOUN
ejpam-4918	202	16	and	and	CCONJ
ejpam-4918	202	17	n	n	CCONJ
ejpam-4918	202	18	be	be	VERB
ejpam-4918	202	19	an	an	DET
ejpam-4918	202	20	ideal	ideal	NOUN
ejpam-4918	202	21	of	of	ADP
ejpam-4918	202	22	m	m	PROPN
ejpam-4918	202	23	with	with	ADP
ejpam-4918	202	24	a	a	DET
ejpam-4918	202	25	center	center	NOUN
ejpam-4918	202	26	element	element	NOUN
ejpam-4918	202	27	λ	λ	NOUN
ejpam-4918	202	28	.	.	PUNCT
ejpam-4918	203	1	if	if	SCONJ
ejpam-4918	203	2	r	r	NOUN
ejpam-4918	203	3	is	be	AUX
ejpam-4918	203	4	strongly	strongly	ADV
ejpam-4918	203	5	cn	cn	ADJ
ejpam-4918	203	6	-reflexive	-reflexive	NOUN
ejpam-4918	203	7	,	,	PUNCT
ejpam-4918	203	8	then	then	ADV
ejpam-4918	203	9	r	r	NOUN
ejpam-4918	203	10	is	be	AUX
ejpam-4918	203	11	strongly	strongly	ADV
ejpam-4918	203	12	cm	cm	NOUN
ejpam-4918	203	13	-reflexive	-reflexive	NOUN
ejpam-4918	203	14	.	.	PUNCT
ejpam-4918	204	1	proof	proof	NOUN
ejpam-4918	204	2	.	.	PUNCT
ejpam-4918	205	1	let	let	VERB
ejpam-4918	205	2	ϕ	ϕ	NOUN
ejpam-4918	205	3	=	=	PUNCT
ejpam-4918	205	4	∑n	∑n	PROPN
ejpam-4918	205	5	i=1	i=1	PROPN
ejpam-4918	205	6	cili	cili	NOUN
ejpam-4918	205	7	,	,	PUNCT
ejpam-4918	205	8	ψ	ψ	X
ejpam-4918	205	9	=	=	SYM
ejpam-4918	205	10	∑m	∑m	ADJ
ejpam-4918	205	11	j=1	j=1	ADJ
ejpam-4918	205	12	ajhj	ajhj	NOUN
ejpam-4918	205	13	∈	∈	PROPN
ejpam-4918	205	14	r	r	NOUN
ejpam-4918	205	15	∗m	∗m	NOUN
ejpam-4918	205	16	satisfying	satisfy	VERB
ejpam-4918	205	17	ϕφψ	ϕφψ	PROPN
ejpam-4918	205	18	=	=	NOUN
ejpam-4918	205	19	0	0	NUM
ejpam-4918	205	20	for	for	ADP
ejpam-4918	205	21	any	any	DET
ejpam-4918	205	22	φ	φ	PROPN
ejpam-4918	205	23	=	=	PROPN
ejpam-4918	205	24	∑v	∑v	PROPN
ejpam-4918	205	25	r=1	r=1	NOUN
ejpam-4918	205	26	ℓrgr	ℓrgr	NOUN
ejpam-4918	205	27	∈	∈	NOUN
ejpam-4918	205	28	r	r	NOUN
ejpam-4918	205	29	∗m	∗m	NOUN
ejpam-4918	205	30	.	.	PUNCT
ejpam-4918	206	1	since	since	SCONJ
ejpam-4918	206	2	λ	λ	PROPN
ejpam-4918	206	3	∈	∈	PROPN
ejpam-4918	206	4	n	n	PRON
ejpam-4918	206	5	is	be	AUX
ejpam-4918	206	6	a	a	DET
ejpam-4918	206	7	center	center	NOUN
ejpam-4918	206	8	element	element	NOUN
ejpam-4918	206	9	,	,	PUNCT
ejpam-4918	206	10	this	this	PRON
ejpam-4918	206	11	implies	imply	VERB
ejpam-4918	206	12	that	that	SCONJ
ejpam-4918	206	13	λl1	λl1	VERB
ejpam-4918	206	14	,	,	PUNCT
ejpam-4918	206	15	λl2	λl2	ADJ
ejpam-4918	206	16	,	,	PUNCT
ejpam-4918	206	17	.	.	PUNCT
ejpam-4918	206	18	.	.	PUNCT
ejpam-4918	206	19	.	.	PUNCT
ejpam-4918	207	1	,	,	PUNCT
ejpam-4918	207	2	λln	λln	PROPN
ejpam-4918	207	3	,	,	PUNCT
ejpam-4918	207	4	λg1λ	λg1λ	PROPN
ejpam-4918	207	5	,	,	PUNCT
ejpam-4918	207	6	λg2λ	λg2λ	PROPN
ejpam-4918	207	7	,	,	PUNCT
ejpam-4918	207	8	.	.	PUNCT
ejpam-4918	207	9	.	.	PUNCT
ejpam-4918	207	10	.	.	PUNCT
ejpam-4918	208	1	,	,	PUNCT
ejpam-4918	208	2	λgvλ	λgvλ	NOUN
ejpam-4918	208	3	,	,	PUNCT
ejpam-4918	208	4	h1λ	h1λ	PROPN
ejpam-4918	208	5	,	,	PUNCT
ejpam-4918	208	6	h2λ	h2λ	PROPN
ejpam-4918	208	7	,	,	PUNCT
ejpam-4918	208	8	.	.	PUNCT
ejpam-4918	208	9	.	.	PUNCT
ejpam-4918	208	10	.	.	PUNCT
ejpam-4918	209	1	,	,	PUNCT
ejpam-4918	209	2	hmλ	hmλ	PROPN
ejpam-4918	209	3	∈	∈	PROPN
ejpam-4918	209	4	n	n	CCONJ
ejpam-4918	209	5	,	,	PUNCT
ejpam-4918	209	6	such	such	ADJ
ejpam-4918	209	7	that	that	DET
ejpam-4918	209	8	λli	λli	PROPN
ejpam-4918	209	9	̸=	̸=	PROPN
ejpam-4918	209	10	λlj	λlj	NOUN
ejpam-4918	209	11	,	,	PUNCT
ejpam-4918	209	12	λgiλ	λgiλ	VERB
ejpam-4918	209	13	̸=	̸=	PROPN
ejpam-4918	209	14	λgjλ	λgjλ	NOUN
ejpam-4918	209	15	and	and	CCONJ
ejpam-4918	209	16	hiλ	hiλ	VERB
ejpam-4918	209	17	̸=	̸=	PROPN
ejpam-4918	209	18	hjλ	hjλ	NOUN
ejpam-4918	209	19	for	for	ADP
ejpam-4918	209	20	all	all	DET
ejpam-4918	209	21	i	i	PRON
ejpam-4918	209	22	̸=	̸=	PROPN
ejpam-4918	209	23	j.	j.	PROPN
ejpam-4918	209	24	then	then	ADV
ejpam-4918	209	25	,	,	PUNCT
ejpam-4918	209	26	we	we	PRON
ejpam-4918	209	27	have	have	VERB
ejpam-4918	209	28	ϕ1φ1ψ1	ϕ1φ1ψ1	ADJ
ejpam-4918	210	1	=	=	SYM
ejpam-4918	210	2	n∑	n∑	INTJ
ejpam-4918	210	3	i=1	i=1	PROPN
ejpam-4918	211	1	m∑	m∑	ADV
ejpam-4918	211	2	j=1	j=1	NOUN
ejpam-4918	211	3	v∑	v∑	NUM
ejpam-4918	211	4	r=1	r=1	ADJ
ejpam-4918	211	5	(	(	PUNCT
ejpam-4918	211	6	ciωli(ℓrωλ(aj)))f(liλ	ciωli(ℓrωλ(aj)))f(liλ	PROPN
ejpam-4918	211	7	,	,	PUNCT
ejpam-4918	211	8	hj)(λ	hj)(λ	NOUN
ejpam-4918	211	9	2ligrhjλ	2ligrhjλ	NUM
ejpam-4918	211	10	2	2	NUM
ejpam-4918	211	11	)	)	PUNCT
ejpam-4918	211	12	=	=	SYM
ejpam-4918	211	13	0	0	X
ejpam-4918	211	14	.	.	PUNCT
ejpam-4918	212	1	since	since	SCONJ
ejpam-4918	212	2	φ	φ	PROPN
ejpam-4918	212	3	,	,	PUNCT
ejpam-4918	212	4	ϕ	ϕ	PROPN
ejpam-4918	212	5	and	and	CCONJ
ejpam-4918	212	6	ψ	ψ	NOUN
ejpam-4918	212	7	are	be	AUX
ejpam-4918	212	8	nonzero	nonzero	NOUN
ejpam-4918	212	9	in	in	ADP
ejpam-4918	212	10	r∗m	r∗m	NOUN
ejpam-4918	212	11	,	,	PUNCT
ejpam-4918	212	12	so	so	ADV
ejpam-4918	212	13	ϕ1	ϕ1	NOUN
ejpam-4918	212	14	and	and	CCONJ
ejpam-4918	212	15	ψ1	ψ1	NOUN
ejpam-4918	212	16	are	be	AUX
ejpam-4918	212	17	nonzero	nonzero	NOUN
ejpam-4918	212	18	elements	element	NOUN
ejpam-4918	212	19	in	in	ADP
ejpam-4918	212	20	(	(	PUNCT
ejpam-4918	212	21	r∗m)[n	r∗m)[n	PROPN
ejpam-4918	212	22	]	]	PUNCT
ejpam-4918	212	23	.	.	PUNCT
ejpam-4918	213	1	moreover	moreover	ADV
ejpam-4918	213	2	,	,	PUNCT
ejpam-4918	213	3	from	from	ADP
ejpam-4918	213	4	ϕφψ	ϕφψ	NOUN
ejpam-4918	213	5	=	=	SYM
ejpam-4918	213	6	0	0	NUM
ejpam-4918	213	7	and	and	CCONJ
ejpam-4918	213	8	ω	ω	NUM
ejpam-4918	213	9	compatible	compatible	ADJ
ejpam-4918	213	10	automorphism	automorphism	NOUN
ejpam-4918	213	11	,	,	PUNCT
ejpam-4918	213	12	λ	λ	X
ejpam-4918	213	13	a	a	DET
ejpam-4918	213	14	center	center	NOUN
ejpam-4918	213	15	element	element	NOUN
ejpam-4918	213	16	of	of	ADP
ejpam-4918	213	17	n	n	DET
ejpam-4918	213	18	one	one	NOUN
ejpam-4918	213	19	can	can	AUX
ejpam-4918	213	20	easily	easily	ADV
ejpam-4918	213	21	obtain	obtain	VERB
ejpam-4918	213	22	that	that	PRON
ejpam-4918	213	23	ϕ1φ1ψ1	ϕ1φ1ψ1	ADJ
ejpam-4918	213	24	=	=	SYM
ejpam-4918	213	25	0	0	NUM
ejpam-4918	213	26	for	for	ADP
ejpam-4918	213	27	any	any	DET
ejpam-4918	213	28	φ1	φ1	PROPN
ejpam-4918	213	29	∈	∈	PROPN
ejpam-4918	213	30	(	(	PUNCT
ejpam-4918	213	31	r	r	NOUN
ejpam-4918	213	32	∗m)[n	∗m)[n	NOUN
ejpam-4918	213	33	]	]	X
ejpam-4918	213	34	.	.	PUNCT
ejpam-4918	214	1	since	since	SCONJ
ejpam-4918	214	2	r	r	NOUN
ejpam-4918	214	3	is	be	AUX
ejpam-4918	214	4	strongly	strongly	ADV
ejpam-4918	214	5	cn	cn	VERB
ejpam-4918	214	6	reflexive	reflexive	ADJ
ejpam-4918	214	7	.	.	PUNCT
ejpam-4918	215	1	then	then	ADV
ejpam-4918	215	2	,	,	PUNCT
ejpam-4918	215	3	ciωli(ωλ(r	ciωli(ωλ(r	NOUN
ejpam-4918	215	4	aj))f(li	aj))f(li	NOUN
ejpam-4918	215	5	,	,	PUNCT
ejpam-4918	215	6	hj)(lihj	hj)(lihj	X
ejpam-4918	215	7	)	)	PUNCT
ejpam-4918	215	8	=	=	SYM
ejpam-4918	216	1	0	0	X
ejpam-4918	216	2	.	.	PUNCT
ejpam-4918	217	1	so	so	ADV
ejpam-4918	217	2	ciωli(ωλ(raj	ciωli(ωλ(raj	PROPN
ejpam-4918	217	3	)	)	PUNCT
ejpam-4918	217	4	)	)	PUNCT
ejpam-4918	218	1	=	=	PUNCT
ejpam-4918	218	2	0	0	X
ejpam-4918	218	3	.	.	PUNCT
ejpam-4918	219	1	by	by	ADP
ejpam-4918	219	2	a	a	DET
ejpam-4918	219	3	compatible	compatible	ADJ
ejpam-4918	219	4	automorphism	automorphism	NOUN
ejpam-4918	219	5	,	,	PUNCT
ejpam-4918	219	6	we	we	PRON
ejpam-4918	219	7	have	have	VERB
ejpam-4918	219	8	ajωhj	ajωhj	NOUN
ejpam-4918	219	9	(	(	PUNCT
ejpam-4918	219	10	ωλ(rci	ωλ(rci	ADJ
ejpam-4918	219	11	)	)	PUNCT
ejpam-4918	219	12	)	)	PUNCT
ejpam-4918	220	1	=	=	PUNCT
ejpam-4918	220	2	0	0	X
ejpam-4918	220	3	.	.	PUNCT
ejpam-4918	221	1	therefore	therefore	ADV
ejpam-4918	221	2	,	,	PUNCT
ejpam-4918	221	3	r	r	NOUN
ejpam-4918	221	4	is	be	AUX
ejpam-4918	221	5	strongly	strongly	ADV
ejpam-4918	221	6	cm	cm	NOUN
ejpam-4918	221	7	-reflexive	-reflexive	NOUN
ejpam-4918	221	8	.	.	PUNCT
ejpam-4918	222	1	corollary	corollary	ADJ
ejpam-4918	222	2	1	1	NUM
ejpam-4918	222	3	.	.	PUNCT
ejpam-4918	223	1	[	[	X
ejpam-4918	223	2	4	4	NUM
ejpam-4918	223	3	,	,	PUNCT
ejpam-4918	223	4	proposition	proposition	NOUN
ejpam-4918	223	5	3.1	3.1	NUM
ejpam-4918	223	6	]	]	PUNCT
ejpam-4918	223	7	let	let	VERB
ejpam-4918	223	8	m	m	PRON
ejpam-4918	223	9	be	be	AUX
ejpam-4918	223	10	a	a	DET
ejpam-4918	223	11	cancellative	cancellative	ADJ
ejpam-4918	223	12	monoid	monoid	NOUN
ejpam-4918	223	13	and	and	CCONJ
ejpam-4918	223	14	n	n	DET
ejpam-4918	223	15	an	an	DET
ejpam-4918	223	16	ideal	ideal	NOUN
ejpam-4918	223	17	of	of	ADP
ejpam-4918	223	18	m.	m.	NOUN
ejpam-4918	223	19	if	if	SCONJ
ejpam-4918	223	20	r	r	NOUN
ejpam-4918	223	21	is	be	AUX
ejpam-4918	223	22	strongly	strongly	ADV
ejpam-4918	223	23	n	n	DET
ejpam-4918	223	24	-reflexive	-reflexive	NOUN
ejpam-4918	223	25	,	,	PUNCT
ejpam-4918	223	26	then	then	ADV
ejpam-4918	223	27	r	r	NOUN
ejpam-4918	223	28	is	be	AUX
ejpam-4918	223	29	strongly	strongly	ADV
ejpam-4918	223	30	m	m	PRON
ejpam-4918	223	31	-reflexive	-reflexive	ADJ
ejpam-4918	223	32	.	.	PUNCT
ejpam-4918	224	1	suppose	suppose	VERB
ejpam-4918	224	2	i	i	PRON
ejpam-4918	224	3	is	be	AUX
ejpam-4918	224	4	an	an	DET
ejpam-4918	224	5	ideal	ideal	NOUN
ejpam-4918	224	6	of	of	ADP
ejpam-4918	224	7	r	r	NOUN
ejpam-4918	224	8	and	and	CCONJ
ejpam-4918	224	9	ω	ω	NUM
ejpam-4918	224	10	:	:	PUNCT
ejpam-4918	224	11	m	m	PROPN
ejpam-4918	224	12	→	→	SYM
ejpam-4918	224	13	aut(r	aut(r	PROPN
ejpam-4918	224	14	)	)	PUNCT
ejpam-4918	224	15	is	be	AUX
ejpam-4918	224	16	a	a	DET
ejpam-4918	224	17	monoid	monoid	NOUN
ejpam-4918	224	18	homomorphism	homomorphism	NOUN
ejpam-4918	224	19	.	.	PUNCT
ejpam-4918	225	1	we	we	PRON
ejpam-4918	225	2	define	define	VERB
ejpam-4918	225	3	ω̄	ω̄	ADP
ejpam-4918	225	4	:	:	PUNCT
ejpam-4918	225	5	m	m	PROPN
ejpam-4918	225	6	→	→	SYM
ejpam-4918	225	7	aut(r	aut(r	PROPN
ejpam-4918	225	8	/	/	SYM
ejpam-4918	225	9	i	i	PROPN
ejpam-4918	225	10	)	)	PUNCT
ejpam-4918	225	11	as	as	ADP
ejpam-4918	225	12	ω̄g(d	ω̄g(d	PROPN
ejpam-4918	225	13	+	+	X
ejpam-4918	225	14	i	i	NOUN
ejpam-4918	225	15	)	)	PUNCT
ejpam-4918	225	16	=	=	SYM
ejpam-4918	225	17	ωg(d	ωg(d	X
ejpam-4918	225	18	)	)	PUNCT
ejpam-4918	226	1	+	+	CCONJ
ejpam-4918	226	2	i	i	PRON
ejpam-4918	226	3	,	,	PUNCT
ejpam-4918	226	4	where	where	SCONJ
ejpam-4918	226	5	d	d	X
ejpam-4918	226	6	∈	∈	PROPN
ejpam-4918	226	7	r	r	NOUN
ejpam-4918	226	8	and	and	CCONJ
ejpam-4918	226	9	g	g	PROPN
ejpam-4918	226	10	∈	∈	PROPN
ejpam-4918	226	11	m	m	VERB
ejpam-4918	226	12	.	.	PUNCT
ejpam-4918	227	1	it	it	PRON
ejpam-4918	227	2	can	can	AUX
ejpam-4918	227	3	be	be	AUX
ejpam-4918	227	4	shown	show	VERB
ejpam-4918	227	5	that	that	SCONJ
ejpam-4918	227	6	ω̄	ω̄	NOUN
ejpam-4918	227	7	is	be	AUX
ejpam-4918	227	8	a	a	DET
ejpam-4918	227	9	monoid	monoid	NOUN
ejpam-4918	227	10	homomorphism	homomorphism	NOUN
ejpam-4918	227	11	.	.	PUNCT
ejpam-4918	228	1	additionally	additionally	ADV
ejpam-4918	228	2	,	,	PUNCT
ejpam-4918	228	3	the	the	DET
ejpam-4918	228	4	twisting	twisting	NOUN
ejpam-4918	228	5	map	map	NOUN
ejpam-4918	228	6	f	f	X
ejpam-4918	228	7	:	:	PUNCT
ejpam-4918	228	8	m	m	VERB
ejpam-4918	228	9	×m	×m	NOUN
ejpam-4918	228	10	→	→	SYM
ejpam-4918	228	11	u(r	u(r	NOUN
ejpam-4918	228	12	)	)	PUNCT
ejpam-4918	228	13	induces	induce	VERB
ejpam-4918	228	14	a	a	DET
ejpam-4918	228	15	twisting	twisting	NOUN
ejpam-4918	228	16	map	map	NOUN
ejpam-4918	228	17	f̄	f̄	NOUN
ejpam-4918	228	18	:	:	PUNCT
ejpam-4918	228	19	m×m	m×m	ADJ
ejpam-4918	228	20	→	→	SYM
ejpam-4918	228	21	u(r	u(r	NOUN
ejpam-4918	228	22	/	/	SYM
ejpam-4918	228	23	i	i	NOUN
ejpam-4918	228	24	)	)	PUNCT
ejpam-4918	228	25	given	give	VERB
ejpam-4918	228	26	by	by	ADP
ejpam-4918	228	27	f̄(x	f̄(x	PROPN
ejpam-4918	228	28	,	,	PUNCT
ejpam-4918	228	29	y	y	NOUN
ejpam-4918	228	30	)	)	PUNCT
ejpam-4918	229	1	=	=	SYM
ejpam-4918	229	2	f(x	f(x	PROPN
ejpam-4918	229	3	,	,	PUNCT
ejpam-4918	229	4	y)+i	y)+i	PROPN
ejpam-4918	229	5	.	.	PUNCT
ejpam-4918	230	1	furthermore	furthermore	ADV
ejpam-4918	230	2	,	,	PUNCT
ejpam-4918	230	3	for	for	ADP
ejpam-4918	230	4	every	every	DET
ejpam-4918	230	5	ϕ	ϕ	NOUN
ejpam-4918	230	6	=	=	PUNCT
ejpam-4918	230	7	∑n	∑n	PROPN
ejpam-4918	230	8	i=1	i=1	PROPN
ejpam-4918	230	9	cili	cili	NOUN
ejpam-4918	230	10	∈	∈	PROPN
ejpam-4918	230	11	r	r	NOUN
ejpam-4918	230	12	∗m	∗m	NOUN
ejpam-4918	230	13	,	,	PUNCT
ejpam-4918	230	14	we	we	PRON
ejpam-4918	230	15	denote	denote	VERB
ejpam-4918	230	16	ϕ̄	ϕ̄	NOUN
ejpam-4918	231	1	=	=	PUNCT
ejpam-4918	231	2	∑n	∑n	PROPN
ejpam-4918	231	3	i=1	i=1	PROPN
ejpam-4918	231	4	c̄ili	c̄ili	PROPN
ejpam-4918	231	5	∈	∈	PROPN
ejpam-4918	231	6	(	(	PUNCT
ejpam-4918	231	7	r	r	NOUN
ejpam-4918	231	8	/	/	SYM
ejpam-4918	231	9	i	i	NOUN
ejpam-4918	231	10	)	)	PUNCT
ejpam-4918	231	11	∗m	∗m	PROPN
ejpam-4918	231	12	,	,	PUNCT
ejpam-4918	231	13	where	where	SCONJ
ejpam-4918	231	14	c̄i	c̄i	PROPN
ejpam-4918	231	15	=	=	SYM
ejpam-4918	231	16	ci	ci	PROPN
ejpam-4918	231	17	+	+	CCONJ
ejpam-4918	231	18	i	i	PRON
ejpam-4918	231	19	for	for	ADP
ejpam-4918	231	20	1	1	NUM
ejpam-4918	231	21	≤	≤	NUM
ejpam-4918	232	1	i	i	PRON
ejpam-4918	232	2	≤	≤	NUM
ejpam-4918	232	3	n.	n.	NOUN
ejpam-4918	232	4	it	it	PRON
ejpam-4918	232	5	can	can	AUX
ejpam-4918	232	6	be	be	AUX
ejpam-4918	232	7	easily	easily	ADV
ejpam-4918	232	8	verified	verify	VERB
ejpam-4918	232	9	that	that	SCONJ
ejpam-4918	232	10	the	the	DET
ejpam-4918	232	11	mapping	mapping	NOUN
ejpam-4918	232	12	θ	θ	NOUN
ejpam-4918	232	13	:	:	PUNCT
ejpam-4918	233	1	r×m	r×m	NOUN
ejpam-4918	233	2	→	→	SYM
ejpam-4918	233	3	(	(	PUNCT
ejpam-4918	233	4	r	r	X
ejpam-4918	233	5	/	/	SYM
ejpam-4918	233	6	i)×m	i)×m	ADJ
ejpam-4918	233	7	defined	define	VERB
ejpam-4918	233	8	as	as	ADP
ejpam-4918	233	9	θ(ϕ	θ(ϕ	PROPN
ejpam-4918	233	10	)	)	PUNCT
ejpam-4918	233	11	=	=	SYM
ejpam-4918	234	1	ϕ̄	ϕ̄	PROPN
ejpam-4918	234	2	is	be	AUX
ejpam-4918	234	3	a	a	DET
ejpam-4918	234	4	ring	ring	NOUN
ejpam-4918	234	5	homomorphism	homomorphism	NOUN
ejpam-4918	234	6	.	.	PUNCT
ejpam-4918	235	1	in	in	ADP
ejpam-4918	235	2	a	a	DET
ejpam-4918	235	3	proof	proof	NOUN
ejpam-4918	235	4	presented	present	VERB
ejpam-4918	235	5	[	[	X
ejpam-4918	235	6	4	4	NUM
ejpam-4918	235	7	]	]	PUNCT
ejpam-4918	235	8	,	,	PUNCT
ejpam-4918	235	9	it	it	PRON
ejpam-4918	235	10	was	be	AUX
ejpam-4918	235	11	shown	show	VERB
ejpam-4918	235	12	that	that	SCONJ
ejpam-4918	235	13	when	when	SCONJ
ejpam-4918	235	14	i	i	PRON
ejpam-4918	235	15	is	be	AUX
ejpam-4918	235	16	a	a	DET
ejpam-4918	235	17	reduced	reduce	VERB
ejpam-4918	235	18	ideal	ideal	NOUN
ejpam-4918	235	19	of	of	ADP
ejpam-4918	235	20	r	r	NOUN
ejpam-4918	235	21	and	and	CCONJ
ejpam-4918	235	22	r	r	NOUN
ejpam-4918	235	23	/	/	SYM
ejpam-4918	235	24	i	i	PRON
ejpam-4918	235	25	is	be	AUX
ejpam-4918	235	26	strongly	strongly	ADV
ejpam-4918	235	27	m	m	VERB
ejpam-4918	235	28	-reflexive	-reflexive	ADJ
ejpam-4918	235	29	,	,	PUNCT
ejpam-4918	235	30	then	then	ADV
ejpam-4918	235	31	r	r	NOUN
ejpam-4918	235	32	is	be	AUX
ejpam-4918	235	33	strongly	strongly	ADV
ejpam-4918	235	34	m	m	PRON
ejpam-4918	235	35	-reflexive	-reflexive	ADJ
ejpam-4918	235	36	.	.	PUNCT
ejpam-4918	236	1	similarly	similarly	ADV
ejpam-4918	236	2	,	,	PUNCT
ejpam-4918	236	3	we	we	PRON
ejpam-4918	236	4	can	can	AUX
ejpam-4918	236	5	establish	establish	VERB
ejpam-4918	236	6	the	the	DET
ejpam-4918	236	7	following	follow	VERB
ejpam-4918	236	8	result	result	NOUN
ejpam-4918	236	9	.	.	PUNCT
ejpam-4918	237	1	theorem	theorem	NOUN
ejpam-4918	237	2	3	3	X
ejpam-4918	237	3	.	.	PUNCT
ejpam-4918	238	1	let	let	VERB
ejpam-4918	238	2	m	m	PRON
ejpam-4918	238	3	be	be	AUX
ejpam-4918	238	4	a	a	DET
ejpam-4918	238	5	u.p.-monoid	u.p.-monoid	NOUN
ejpam-4918	239	1	and	and	CCONJ
ejpam-4918	239	2	i	i	PRON
ejpam-4918	239	3	an	an	DET
ejpam-4918	239	4	ideal	ideal	NOUN
ejpam-4918	239	5	of	of	ADP
ejpam-4918	239	6	r	r	NOUN
ejpam-4918	239	7	with	with	ADP
ejpam-4918	239	8	twisting	twist	VERB
ejpam-4918	239	9	f	f	X
ejpam-4918	239	10	:	:	PUNCT
ejpam-4918	239	11	m×m	m×m	ADJ
ejpam-4918	239	12	→	→	SYM
ejpam-4918	239	13	u(r	u(r	NOUN
ejpam-4918	239	14	)	)	PUNCT
ejpam-4918	239	15	and	and	CCONJ
ejpam-4918	239	16	action	action	NOUN
ejpam-4918	239	17	ω	ω	NOUN
ejpam-4918	239	18	:	:	PUNCT
ejpam-4918	240	1	m	m	PROPN
ejpam-4918	240	2	→	→	SYM
ejpam-4918	240	3	aut(r	aut(r	PROPN
ejpam-4918	240	4	)	)	PUNCT
ejpam-4918	240	5	.	.	PUNCT
ejpam-4918	241	1	if	if	SCONJ
ejpam-4918	241	2	i	i	PRON
ejpam-4918	241	3	is	be	AUX
ejpam-4918	241	4	a	a	DET
ejpam-4918	241	5	reduced	reduced	ADJ
ejpam-4918	241	6	and	and	CCONJ
ejpam-4918	241	7	r	r	X
ejpam-4918	241	8	/	/	SYM
ejpam-4918	241	9	i	i	PRON
ejpam-4918	241	10	is	be	AUX
ejpam-4918	241	11	strongly	strongly	ADV
ejpam-4918	241	12	cm	cm	NOUN
ejpam-4918	241	13	-reflexive	-reflexive	NOUN
ejpam-4918	241	14	,	,	PUNCT
ejpam-4918	241	15	then	then	ADV
ejpam-4918	241	16	r	r	NOUN
ejpam-4918	241	17	is	be	AUX
ejpam-4918	241	18	strongly	strongly	ADV
ejpam-4918	241	19	cm	cm	NOUN
ejpam-4918	241	20	-reflexive	-reflexive	NOUN
ejpam-4918	241	21	.	.	PUNCT
ejpam-4918	242	1	proof	proof	NOUN
ejpam-4918	242	2	.	.	PUNCT
ejpam-4918	243	1	let	let	VERB
ejpam-4918	243	2	ϕ	ϕ	NOUN
ejpam-4918	243	3	=	=	PROPN
ejpam-4918	243	4	σni=1cili	σni=1cili	PROPN
ejpam-4918	243	5	,	,	PUNCT
ejpam-4918	243	6	ψ	ψ	X
ejpam-4918	243	7	=	=	SYM
ejpam-4918	243	8	σmj=1ajhj	σmj=1ajhj	NOUN
ejpam-4918	243	9	∈	∈	NOUN
ejpam-4918	243	10	r	r	NOUN
ejpam-4918	243	11	∗m	∗m	NOUN
ejpam-4918	243	12	satisfying	satisfy	VERB
ejpam-4918	243	13	ϕ(r	ϕ(r	PROPN
ejpam-4918	243	14	∗m)ψ	∗m)ψ	ADJ
ejpam-4918	243	15	=	=	PUNCT
ejpam-4918	244	1	0	0	X
ejpam-4918	244	2	.	.	PUNCT
ejpam-4918	245	1	we	we	PRON
ejpam-4918	245	2	will	will	AUX
ejpam-4918	245	3	show	show	VERB
ejpam-4918	245	4	that	that	SCONJ
ejpam-4918	245	5	ciωli(ωg(raj	ciωli(ωg(raj	NOUN
ejpam-4918	245	6	)	)	PUNCT
ejpam-4918	245	7	)	)	PUNCT
ejpam-4918	246	1	=	=	SYM
ejpam-4918	246	2	0	0	NUM
ejpam-4918	246	3	for	for	ADP
ejpam-4918	246	4	any	any	DET
ejpam-4918	246	5	i	i	PROPN
ejpam-4918	246	6	and	and	CCONJ
ejpam-4918	246	7	j.	j.	PROPN
ejpam-4918	246	8	e.	e.	PROPN
ejpam-4918	246	9	ali	ali	PROPN
ejpam-4918	246	10	/	/	SYM
ejpam-4918	246	11	eur	eur	PROPN
ejpam-4918	246	12	.	.	PUNCT
ejpam-4918	247	1	j.	j.	PROPN
ejpam-4918	247	2	pure	pure	PROPN
ejpam-4918	247	3	appl	appl	PROPN
ejpam-4918	247	4	.	.	PROPN
ejpam-4918	247	5	math	math	PROPN
ejpam-4918	247	6	,	,	PUNCT
ejpam-4918	247	7	16	16	NUM
ejpam-4918	247	8	(	(	PUNCT
ejpam-4918	247	9	4	4	NUM
ejpam-4918	247	10	)	)	PUNCT
ejpam-4918	247	11	(	(	PUNCT
ejpam-4918	247	12	2023	2023	NUM
ejpam-4918	247	13	)	)	PUNCT
ejpam-4918	247	14	,	,	PUNCT
ejpam-4918	247	15	2156	2156	NUM
ejpam-4918	247	16	-	-	SYM
ejpam-4918	247	17	2168	2168	NUM
ejpam-4918	247	18	2162	2162	NUM
ejpam-4918	247	19	note	note	VERB
ejpam-4918	247	20	that	that	SCONJ
ejpam-4918	247	21	in	in	ADP
ejpam-4918	247	22	(	(	PUNCT
ejpam-4918	247	23	r	r	X
ejpam-4918	247	24	/	/	SYM
ejpam-4918	247	25	i	i	NOUN
ejpam-4918	247	26	)	)	PUNCT
ejpam-4918	247	27	∗m	∗m	NOUN
ejpam-4918	247	28	,	,	PUNCT
ejpam-4918	247	29	ϕ̄	ϕ̄	PROPN
ejpam-4918	247	30	=	=	SYM
ejpam-4918	247	31	σni=1c̄ili	σni=1c̄ili	PROPN
ejpam-4918	247	32	,	,	PUNCT
ejpam-4918	247	33	ψ̄	ψ̄	PUNCT
ejpam-4918	247	34	=	=	SYM
ejpam-4918	247	35	σmj=1ājhj	σmj=1ājhj	X
ejpam-4918	247	36	∈	∈	NOUN
ejpam-4918	247	37	(	(	PUNCT
ejpam-4918	247	38	r	r	NOUN
ejpam-4918	247	39	/	/	SYM
ejpam-4918	247	40	i	i	NOUN
ejpam-4918	247	41	)	)	PUNCT
ejpam-4918	247	42	∗m	∗m	NOUN
ejpam-4918	247	43	,	,	PUNCT
ejpam-4918	247	44	we	we	PRON
ejpam-4918	247	45	have	have	VERB
ejpam-4918	247	46	0̄	0̄	NUM
ejpam-4918	247	47	=	=	SYM
ejpam-4918	247	48	ϕ̄((r	ϕ̄((r	PROPN
ejpam-4918	247	49	/	/	SYM
ejpam-4918	247	50	i	i	NOUN
ejpam-4918	247	51	)	)	PUNCT
ejpam-4918	248	1	∗m)ψ̄	∗m)ψ̄	PROPN
ejpam-4918	248	2	=	=	SYM
ejpam-4918	248	3	(	(	PUNCT
ejpam-4918	248	4	c̄1l1	c̄1l1	X
ejpam-4918	248	5	+	+	SYM
ejpam-4918	248	6	c̄2l2	c̄2l2	PROPN
ejpam-4918	248	7	+	+	CCONJ
ejpam-4918	248	8	·	·	PUNCT
ejpam-4918	248	9	·	·	PUNCT
ejpam-4918	248	10	·	·	PUNCT
ejpam-4918	248	11	+	+	CCONJ
ejpam-4918	248	12	c̄nln)r̄gωli(ωg(ā1h1	c̄nln)r̄gωli(ωg(ā1h1	NOUN
ejpam-4918	249	1	+	+	CCONJ
ejpam-4918	249	2	ā2h2	ā2h2	PROPN
ejpam-4918	249	3	+	+	X
ejpam-4918	249	4	·	·	PUNCT
ejpam-4918	249	5	·	·	PUNCT
ejpam-4918	249	6	·	·	PUNCT
ejpam-4918	249	7	+	+	PUNCT
ejpam-4918	249	8	āmhm))f(li	āmhm))f(li	PROPN
ejpam-4918	249	9	,	,	PUNCT
ejpam-4918	249	10	hj)lihj	hj)lihj	PROPN
ejpam-4918	249	11	=	=	PRON
ejpam-4918	249	12	(	(	PUNCT
ejpam-4918	249	13	c1	c1	PROPN
ejpam-4918	249	14	+	+	CCONJ
ejpam-4918	249	15	i)r̄gω̄l1(ωg(a1	i)r̄gω̄l1(ωg(a1	PROPN
ejpam-4918	249	16	+	+	PROPN
ejpam-4918	249	17	i))f(l1	i))f(l1	PROPN
ejpam-4918	249	18	,	,	PUNCT
ejpam-4918	249	19	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	249	20	+	+	CCONJ
ejpam-4918	249	21	(	(	PUNCT
ejpam-4918	249	22	c2	c2	PROPN
ejpam-4918	249	23	+	+	CCONJ
ejpam-4918	249	24	i)r̄gω̄l2(ωg(a2	i)r̄gω̄l2(ωg(a2	PROPN
ejpam-4918	249	25	+	+	CCONJ
ejpam-4918	249	26	i))f(l2	i))f(l2	ADJ
ejpam-4918	249	27	,	,	PUNCT
ejpam-4918	249	28	h2)l2h2	h2)l2h2	PROPN
ejpam-4918	249	29	+	+	CCONJ
ejpam-4918	249	30	·	·	PUNCT
ejpam-4918	249	31	·	·	PUNCT
ejpam-4918	249	32	·	·	PUNCT
ejpam-4918	249	33	+	+	CCONJ
ejpam-4918	249	34	(	(	PUNCT
ejpam-4918	249	35	cn	cn	X
ejpam-4918	249	36	+	+	CCONJ
ejpam-4918	249	37	i)r̄gω̄ln(ωg(am	i)r̄gω̄ln(ωg(am	PROPN
ejpam-4918	249	38	+	+	CCONJ
ejpam-4918	249	39	i))f(ln	i))f(ln	NOUN
ejpam-4918	249	40	,	,	PUNCT
ejpam-4918	249	41	hm)lnhm	hm)lnhm	PROPN
ejpam-4918	249	42	.	.	PUNCT
ejpam-4918	250	1	thus	thus	ADV
ejpam-4918	250	2	we	we	PRON
ejpam-4918	250	3	have	have	AUX
ejpam-4918	250	4	ciωli(ωg(raj))f(li	ciωli(ωg(raj))f(li	NOUN
ejpam-4918	250	5	,	,	PUNCT
ejpam-4918	250	6	hj)(lihj	hj)(lihj	X
ejpam-4918	250	7	)	)	PUNCT
ejpam-4918	250	8	⊆	⊆	NUM
ejpam-4918	250	9	i	i	PRON
ejpam-4918	250	10	for	for	ADP
ejpam-4918	250	11	all	all	DET
ejpam-4918	250	12	i	i	PRON
ejpam-4918	250	13	and	and	CCONJ
ejpam-4918	250	14	j	j	PROPN
ejpam-4918	250	15	with	with	ADP
ejpam-4918	250	16	1	1	NUM
ejpam-4918	250	17	≤	≤	NUM
ejpam-4918	250	18	i	i	PRON
ejpam-4918	250	19	≤	≤	ADJ
ejpam-4918	250	20	n	n	CCONJ
ejpam-4918	250	21	and	and	CCONJ
ejpam-4918	250	22	1	1	NUM
ejpam-4918	250	23	≤	≤	NUM
ejpam-4918	250	24	j	j	PROPN
ejpam-4918	250	25	≤	≤	PROPN
ejpam-4918	250	26	m	m	VERB
ejpam-4918	250	27	since	since	SCONJ
ejpam-4918	250	28	r	r	NOUN
ejpam-4918	250	29	/	/	SYM
ejpam-4918	250	30	i	i	PRON
ejpam-4918	250	31	is	be	AUX
ejpam-4918	250	32	strongly	strongly	ADV
ejpam-4918	250	33	cm	cm	NOUN
ejpam-4918	250	34	-reflexive	-reflexive	NOUN
ejpam-4918	250	35	.	.	PUNCT
ejpam-4918	251	1	by	by	ADP
ejpam-4918	251	2	induction	induction	NOUN
ejpam-4918	251	3	on	on	ADP
ejpam-4918	251	4	both	both	CCONJ
ejpam-4918	251	5	n	n	PROPN
ejpam-4918	251	6	and	and	CCONJ
ejpam-4918	251	7	m	m	PROPN
ejpam-4918	251	8	,	,	PUNCT
ejpam-4918	251	9	considering	consider	VERB
ejpam-4918	251	10	every	every	DET
ejpam-4918	251	11	g	g	NOUN
ejpam-4918	251	12	in	in	ADP
ejpam-4918	251	13	m	m	PROPN
ejpam-4918	251	14	,	,	PUNCT
ejpam-4918	251	15	and	and	CCONJ
ejpam-4918	251	16	for	for	ADP
ejpam-4918	251	17	1	1	NUM
ejpam-4918	251	18	≤	≤	NUM
ejpam-4918	251	19	i	i	PRON
ejpam-4918	251	20	≤	≤	ADJ
ejpam-4918	251	21	n	n	CCONJ
ejpam-4918	251	22	and	and	CCONJ
ejpam-4918	251	23	1	1	NUM
ejpam-4918	251	24	≤	≤	NUM
ejpam-4918	251	25	j	j	PROPN
ejpam-4918	251	26	≤	≤	PROPN
ejpam-4918	251	27	m.	m.	NOUN
ejpam-4918	251	28	if	if	SCONJ
ejpam-4918	251	29	we	we	PRON
ejpam-4918	251	30	take	take	VERB
ejpam-4918	251	31	n	n	NOUN
ejpam-4918	251	32	=	=	SYM
ejpam-4918	251	33	1	1	X
ejpam-4918	251	34	.	.	PUNCT
ejpam-4918	252	1	then	then	ADV
ejpam-4918	252	2	(	(	PUNCT
ejpam-4918	252	3	c1l1)(r	c1l1)(r	NOUN
ejpam-4918	252	4	∗m)(a1h1	∗m)(a1h1	VERB
ejpam-4918	252	5	+	+	CCONJ
ejpam-4918	252	6	a2h2	a2h2	X
ejpam-4918	252	7	+	+	X
ejpam-4918	252	8	·	·	PUNCT
ejpam-4918	252	9	·	·	PUNCT
ejpam-4918	252	10	·	·	PUNCT
ejpam-4918	253	1	+	+	CCONJ
ejpam-4918	253	2	amhm	amhm	NOUN
ejpam-4918	253	3	)	)	PUNCT
ejpam-4918	253	4	=	=	SYM
ejpam-4918	254	1	0	0	X
ejpam-4918	254	2	.	.	PUNCT
ejpam-4918	255	1	thus	thus	ADV
ejpam-4918	255	2	,	,	PUNCT
ejpam-4918	255	3	(	(	PUNCT
ejpam-4918	255	4	c1l1)(rg)(a1h1	c1l1)(rg)(a1h1	X
ejpam-4918	255	5	)	)	PUNCT
ejpam-4918	255	6	+	+	CCONJ
ejpam-4918	255	7	(	(	PUNCT
ejpam-4918	255	8	c1l1)rg(a2h2	c1l1)rg(a2h2	NOUN
ejpam-4918	255	9	)	)	PUNCT
ejpam-4918	256	1	+	+	CCONJ
ejpam-4918	256	2	·	·	PUNCT
ejpam-4918	256	3	·	·	PUNCT
ejpam-4918	256	4	·	·	PUNCT
ejpam-4918	256	5	+	+	CCONJ
ejpam-4918	256	6	(	(	PUNCT
ejpam-4918	256	7	c1l1)rg(amhm	c1l1)rg(amhm	NOUN
ejpam-4918	256	8	)	)	PUNCT
ejpam-4918	256	9	=	=	SYM
ejpam-4918	256	10	c1ωl1(ωg(ra1))f(l1	c1ωl1(ωg(ra1))f(l1	NOUN
ejpam-4918	256	11	,	,	PUNCT
ejpam-4918	256	12	h1)(l1h1	h1)(l1h1	PROPN
ejpam-4918	256	13	)	)	PUNCT
ejpam-4918	256	14	+	+	CCONJ
ejpam-4918	256	15	c1ωl1(ωg(ra2))f(l1	c1ωl1(ωg(ra2))f(l1	PROPN
ejpam-4918	256	16	,	,	PUNCT
ejpam-4918	256	17	h2)(l1h2	h2)(l1h2	NOUN
ejpam-4918	256	18	)	)	PUNCT
ejpam-4918	256	19	+	+	NUM
ejpam-4918	256	20	·	·	PUNCT
ejpam-4918	256	21	·	·	PUNCT
ejpam-4918	256	22	·	·	PUNCT
ejpam-4918	256	23	+	+	CCONJ
ejpam-4918	256	24	c1ωl1(ωg(ram))f(l1	c1ωl1(ωg(ram))f(l1	PROPN
ejpam-4918	256	25	,	,	PUNCT
ejpam-4918	256	26	hm)(l1hm	hm)(l1hm	X
ejpam-4918	256	27	)	)	PUNCT
ejpam-4918	256	28	=	=	SYM
ejpam-4918	256	29	0	0	NUM
ejpam-4918	256	30	for	for	ADP
ejpam-4918	256	31	any	any	DET
ejpam-4918	256	32	r	r	NOUN
ejpam-4918	256	33	∈	∈	NOUN
ejpam-4918	256	34	r	r	NOUN
ejpam-4918	256	35	,	,	PUNCT
ejpam-4918	256	36	g	g	PROPN
ejpam-4918	256	37	∈	∈	PROPN
ejpam-4918	256	38	m.	m.	NOUN
ejpam-4918	256	39	by	by	ADP
ejpam-4918	256	40	lemma	lemma	PROPN
ejpam-4918	256	41	1	1	NUM
ejpam-4918	256	42	,	,	PUNCT
ejpam-4918	256	43	m	m	VERB
ejpam-4918	256	44	is	be	AUX
ejpam-4918	256	45	cancellative	cancellative	ADJ
ejpam-4918	256	46	we	we	PRON
ejpam-4918	256	47	have	have	VERB
ejpam-4918	256	48	l1hi	l1hi	VERB
ejpam-4918	257	1	̸=	̸=	PROPN
ejpam-4918	257	2	l1hj	l1hj	PUNCT
ejpam-4918	257	3	for	for	ADP
ejpam-4918	257	4	any	any	DET
ejpam-4918	257	5	i	i	PROPN
ejpam-4918	257	6	and	and	CCONJ
ejpam-4918	257	7	j	j	PROPN
ejpam-4918	257	8	with	with	ADP
ejpam-4918	257	9	1	1	NUM
ejpam-4918	257	10	≤	≤	NOUN
ejpam-4918	257	11	i	i	PRON
ejpam-4918	257	12	̸=	̸=	PROPN
ejpam-4918	257	13	j	j	PROPN
ejpam-4918	257	14	≤	≤	PROPN
ejpam-4918	257	15	m.	m.	NOUN
ejpam-4918	257	16	then	then	ADV
ejpam-4918	257	17	c1ωl1(ωg(raj))f(l1	c1ωl1(ωg(raj))f(l1	PROPN
ejpam-4918	257	18	,	,	PUNCT
ejpam-4918	257	19	hj)(l1hj	hj)(l1hj	PRON
ejpam-4918	257	20	)	)	PUNCT
ejpam-4918	257	21	=	=	SYM
ejpam-4918	257	22	0	0	NUM
ejpam-4918	257	23	,	,	PUNCT
ejpam-4918	257	24	j	j	PROPN
ejpam-4918	257	25	=	=	SYM
ejpam-4918	257	26	1	1	NUM
ejpam-4918	257	27	,	,	PUNCT
ejpam-4918	257	28	2	2	NUM
ejpam-4918	257	29	,	,	PUNCT
ejpam-4918	257	30	.	.	PUNCT
ejpam-4918	257	31	.	.	PUNCT
ejpam-4918	257	32	.	.	PUNCT
ejpam-4918	258	1	,	,	PUNCT
ejpam-4918	258	2	m.	m.	NOUN
ejpam-4918	258	3	thus	thus	ADV
ejpam-4918	258	4	,	,	PUNCT
ejpam-4918	258	5	c1ωl1(ωg(raj	c1ωl1(ωg(raj	ADJ
ejpam-4918	258	6	)	)	PUNCT
ejpam-4918	258	7	)	)	PUNCT
ejpam-4918	259	1	=	=	SYM
ejpam-4918	259	2	0	0	NUM
ejpam-4918	260	1	for	for	ADP
ejpam-4918	260	2	any	any	DET
ejpam-4918	260	3	j.	j.	NOUN
ejpam-4918	260	4	if	if	SCONJ
ejpam-4918	260	5	m	m	VERB
ejpam-4918	260	6	=	=	NOUN
ejpam-4918	260	7	1	1	NUM
ejpam-4918	260	8	,	,	PUNCT
ejpam-4918	260	9	then	then	ADV
ejpam-4918	260	10	proof	proof	NOUN
ejpam-4918	260	11	is	be	AUX
ejpam-4918	260	12	similar	similar	ADJ
ejpam-4918	260	13	.	.	PUNCT
ejpam-4918	261	1	now	now	ADV
ejpam-4918	261	2	suppose	suppose	VERB
ejpam-4918	261	3	that	that	SCONJ
ejpam-4918	261	4	n	n	PROPN
ejpam-4918	261	5	≥	≥	NUM
ejpam-4918	261	6	2	2	NUM
ejpam-4918	261	7	and	and	CCONJ
ejpam-4918	261	8	m	m	PROPN
ejpam-4918	261	9	≥	≥	NOUN
ejpam-4918	261	10	2	2	NUM
ejpam-4918	261	11	.	.	PUNCT
ejpam-4918	262	1	since	since	SCONJ
ejpam-4918	262	2	m	m	PROPN
ejpam-4918	262	3	is	be	AUX
ejpam-4918	262	4	a	a	DET
ejpam-4918	262	5	u.p.-monoid	u.p.-monoid	ADJ
ejpam-4918	262	6	,	,	PUNCT
ejpam-4918	262	7	there	there	PRON
ejpam-4918	262	8	exist	exist	VERB
ejpam-4918	262	9	i	i	PRON
ejpam-4918	262	10	,	,	PUNCT
ejpam-4918	262	11	j	j	PROPN
ejpam-4918	262	12	with	with	ADP
ejpam-4918	262	13	1	1	NUM
ejpam-4918	262	14	≤	≤	NUM
ejpam-4918	262	15	i	i	PRON
ejpam-4918	262	16	≤	≤	ADJ
ejpam-4918	262	17	n	n	CCONJ
ejpam-4918	262	18	and	and	CCONJ
ejpam-4918	262	19	1	1	NUM
ejpam-4918	262	20	≤	≤	NUM
ejpam-4918	262	21	j	j	PROPN
ejpam-4918	262	22	≤	≤	NUM
ejpam-4918	262	23	m	m	VERB
ejpam-4918	262	24	such	such	ADJ
ejpam-4918	262	25	that	that	DET
ejpam-4918	262	26	lighj	lighj	NOUN
ejpam-4918	262	27	is	be	AUX
ejpam-4918	262	28	uniquely	uniquely	ADV
ejpam-4918	262	29	presented	present	VERB
ejpam-4918	262	30	by	by	ADP
ejpam-4918	262	31	considering	consider	VERB
ejpam-4918	262	32	two	two	NUM
ejpam-4918	262	33	subsets	subset	NOUN
ejpam-4918	262	34	k	k	X
ejpam-4918	263	1	=	=	PUNCT
ejpam-4918	263	2	{	{	PUNCT
ejpam-4918	263	3	l1	l1	PROPN
ejpam-4918	263	4	g	g	PROPN
ejpam-4918	263	5	,	,	PUNCT
ejpam-4918	263	6	l2	l2	VERB
ejpam-4918	263	7	g	g	NOUN
ejpam-4918	263	8	,	,	PUNCT
ejpam-4918	263	9	.	.	PUNCT
ejpam-4918	263	10	.	.	PUNCT
ejpam-4918	264	1	.	.	PUNCT
ejpam-4918	265	1	,	,	PUNCT
ejpam-4918	265	2	lng	lng	PROPN
ejpam-4918	265	3	}	}	PUNCT
ejpam-4918	265	4	and	and	CCONJ
ejpam-4918	265	5	h	h	NOUN
ejpam-4918	265	6	=	=	PRON
ejpam-4918	265	7	{	{	PUNCT
ejpam-4918	265	8	h1	h1	PROPN
ejpam-4918	265	9	,	,	PUNCT
ejpam-4918	265	10	h2	h2	PROPN
ejpam-4918	265	11	,	,	PUNCT
ejpam-4918	265	12	.	.	PUNCT
ejpam-4918	265	13	.	.	PUNCT
ejpam-4918	266	1	.	.	PUNCT
ejpam-4918	267	1	,	,	PUNCT
ejpam-4918	267	2	hm	hm	INTJ
ejpam-4918	267	3	}	}	PUNCT
ejpam-4918	267	4	of	of	ADP
ejpam-4918	267	5	the	the	DET
ejpam-4918	267	6	monoid	monoid	NOUN
ejpam-4918	267	7	m	m	PROPN
ejpam-4918	267	8	.	.	PUNCT
ejpam-4918	268	1	without	without	ADP
ejpam-4918	268	2	loss	loss	NOUN
ejpam-4918	268	3	of	of	ADP
ejpam-4918	268	4	generality	generality	NOUN
ejpam-4918	268	5	,	,	PUNCT
ejpam-4918	268	6	we	we	PRON
ejpam-4918	268	7	may	may	AUX
ejpam-4918	268	8	assume	assume	VERB
ejpam-4918	268	9	that	that	SCONJ
ejpam-4918	269	1	i	i	PRON
ejpam-4918	269	2	=	=	NOUN
ejpam-4918	269	3	1	1	NUM
ejpam-4918	269	4	and	and	CCONJ
ejpam-4918	269	5	j	j	NOUN
ejpam-4918	269	6	=	=	NOUN
ejpam-4918	269	7	1	1	X
ejpam-4918	269	8	.	.	PUNCT
ejpam-4918	270	1	we	we	PRON
ejpam-4918	270	2	can	can	AUX
ejpam-4918	270	3	deduce	deduce	VERB
ejpam-4918	270	4	that	that	PRON
ejpam-4918	270	5	c1ωl1(ωg(ra1))f(l1	c1ωl1(ωg(ra1))f(l1	NOUN
ejpam-4918	270	6	g	g	NOUN
ejpam-4918	270	7	,	,	PUNCT
ejpam-4918	270	8	h1)l1(gh1	h1)l1(gh1	ADJ
ejpam-4918	270	9	)	)	PUNCT
ejpam-4918	270	10	=	=	SYM
ejpam-4918	270	11	0	0	NUM
ejpam-4918	270	12	,	,	PUNCT
ejpam-4918	270	13	which	which	PRON
ejpam-4918	270	14	implies	imply	VERB
ejpam-4918	270	15	that	that	DET
ejpam-4918	270	16	c1ωl1(ωg(ra1	c1ωl1(ωg(ra1	NOUN
ejpam-4918	270	17	)	)	PUNCT
ejpam-4918	270	18	)	)	PUNCT
ejpam-4918	271	1	=	=	PUNCT
ejpam-4918	271	2	0	0	X
ejpam-4918	271	3	.	.	PUNCT
ejpam-4918	272	1	since	since	SCONJ
ejpam-4918	272	2	ωg	ωg	PRON
ejpam-4918	272	3	and	and	CCONJ
ejpam-4918	272	4	ωl1	ωl1	NOUN
ejpam-4918	272	5	are	be	AUX
ejpam-4918	272	6	automorphisms	automorphism	NOUN
ejpam-4918	272	7	of	of	ADP
ejpam-4918	272	8	r	r	NOUN
ejpam-4918	272	9	,	,	PUNCT
ejpam-4918	272	10	we	we	PRON
ejpam-4918	272	11	have	have	VERB
ejpam-4918	272	12	c1ωl1(ra1	c1ωl1(ra1	NOUN
ejpam-4918	272	13	)	)	PUNCT
ejpam-4918	273	1	=	=	PUNCT
ejpam-4918	273	2	0	0	X
ejpam-4918	273	3	.	.	PUNCT
ejpam-4918	274	1	let	let	VERB
ejpam-4918	274	2	b	b	NOUN
ejpam-4918	274	3	=	=	SYM
ejpam-4918	274	4	ckraq	ckraq	PROPN
ejpam-4918	274	5	,	,	PUNCT
ejpam-4918	274	6	where	where	SCONJ
ejpam-4918	274	7	r	r	NOUN
ejpam-4918	274	8	∈	∈	PROPN
ejpam-4918	274	9	r	r	NOUN
ejpam-4918	274	10	,	,	PUNCT
ejpam-4918	274	11	1	1	NUM
ejpam-4918	274	12	≤	≤	NUM
ejpam-4918	274	13	k	k	NOUN
ejpam-4918	274	14	≤	≤	PROPN
ejpam-4918	274	15	n	n	CCONJ
ejpam-4918	274	16	,	,	PUNCT
ejpam-4918	274	17	1	1	NUM
ejpam-4918	274	18	≤	≤	NUM
ejpam-4918	274	19	q	q	ADJ
ejpam-4918	274	20	≤	≤	ADJ
ejpam-4918	274	21	m.	m.	NOUN
ejpam-4918	274	22	then	then	ADV
ejpam-4918	274	23	b	b	PROPN
ejpam-4918	274	24	∈	∈	PROPN
ejpam-4918	274	25	i.	i.	NOUN
ejpam-4918	274	26	since	since	SCONJ
ejpam-4918	274	27	(	(	PUNCT
ejpam-4918	274	28	a1bc1	a1bc1	NOUN
ejpam-4918	274	29	)	)	PUNCT
ejpam-4918	274	30	2	2	NUM
ejpam-4918	274	31	=	=	SYM
ejpam-4918	274	32	0	0	PUNCT
ejpam-4918	275	1	and	and	CCONJ
ejpam-4918	275	2	i	i	PRON
ejpam-4918	275	3	is	be	AUX
ejpam-4918	275	4	reduced	reduce	VERB
ejpam-4918	275	5	,	,	PUNCT
ejpam-4918	275	6	we	we	PRON
ejpam-4918	275	7	have	have	VERB
ejpam-4918	275	8	a1bc1	a1bc1	NOUN
ejpam-4918	275	9	=	=	SYM
ejpam-4918	275	10	0	0	X
ejpam-4918	275	11	.	.	PUNCT
ejpam-4918	276	1	thus	thus	ADV
ejpam-4918	276	2	,	,	PUNCT
ejpam-4918	276	3	(	(	PUNCT
ejpam-4918	276	4	a1bc2l2	a1bc2l2	PROPN
ejpam-4918	276	5	+	+	CCONJ
ejpam-4918	276	6	a1bc3l3	a1bc3l3	VERB
ejpam-4918	276	7	+	+	X
ejpam-4918	276	8	·	·	PUNCT
ejpam-4918	276	9	·	·	PUNCT
ejpam-4918	276	10	·	·	PUNCT
ejpam-4918	276	11	+	+	NUM
ejpam-4918	276	12	a1bcnln)(r	a1bcnln)(r	PROPN
ejpam-4918	276	13	∗m)(a1h1	∗m)(a1h1	X
ejpam-4918	276	14	+	+	CCONJ
ejpam-4918	276	15	a2h2	a2h2	X
ejpam-4918	276	16	+	+	X
ejpam-4918	276	17	·	·	PUNCT
ejpam-4918	276	18	·	·	PUNCT
ejpam-4918	276	19	·	·	PUNCT
ejpam-4918	277	1	+	+	NUM
ejpam-4918	277	2	amhm	amhm	NOUN
ejpam-4918	277	3	)	)	PUNCT
ejpam-4918	277	4	=	=	PUNCT
ejpam-4918	277	5	(	(	PUNCT
ejpam-4918	277	6	a1bλ)(c1l1	a1bλ)(c1l1	PROPN
ejpam-4918	277	7	+	+	NUM
ejpam-4918	277	8	c2l2	c2l2	ADJ
ejpam-4918	277	9	+	+	X
ejpam-4918	277	10	·	·	PUNCT
ejpam-4918	277	11	·	·	PUNCT
ejpam-4918	277	12	·	·	PUNCT
ejpam-4918	278	1	+	+	NUM
ejpam-4918	278	2	cnln)(r	cnln)(r	PROPN
ejpam-4918	278	3	∗m)(a1h1	∗m)(a1h1	NOUN
ejpam-4918	278	4	+	+	PROPN
ejpam-4918	278	5	a2h2	a2h2	NOUN
ejpam-4918	278	6	+	+	X
ejpam-4918	278	7	·	·	PUNCT
ejpam-4918	278	8	·	·	PUNCT
ejpam-4918	278	9	·	·	PUNCT
ejpam-4918	278	10	+	+	NUM
ejpam-4918	278	11	amhm	amhm	NOUN
ejpam-4918	278	12	)	)	PUNCT
ejpam-4918	278	13	=	=	SYM
ejpam-4918	279	1	0	0	X
ejpam-4918	279	2	.	.	PUNCT
ejpam-4918	280	1	by	by	ADP
ejpam-4918	280	2	induction	induction	NOUN
ejpam-4918	280	3	,	,	PUNCT
ejpam-4918	280	4	we	we	PRON
ejpam-4918	280	5	have	have	VERB
ejpam-4918	280	6	a1bciωl1(ωg(raj	a1bciωl1(ωg(raj	VERB
ejpam-4918	280	7	)	)	PUNCT
ejpam-4918	280	8	)	)	PUNCT
ejpam-4918	281	1	=	=	SYM
ejpam-4918	281	2	0	0	NUM
ejpam-4918	281	3	for	for	ADP
ejpam-4918	281	4	2	2	NUM
ejpam-4918	281	5	≤	≤	NUM
ejpam-4918	281	6	i	i	PRON
ejpam-4918	281	7	≤	≤	ADJ
ejpam-4918	281	8	n	n	CCONJ
ejpam-4918	281	9	and	and	CCONJ
ejpam-4918	281	10	1	1	NUM
ejpam-4918	281	11	≤	≤	NUM
ejpam-4918	281	12	j	j	PROPN
ejpam-4918	281	13	≤	≤	PROPN
ejpam-4918	281	14	m.	m.	NOUN
ejpam-4918	281	15	thus	thus	ADV
ejpam-4918	281	16	,	,	PUNCT
ejpam-4918	281	17	(	(	PUNCT
ejpam-4918	281	18	a1bcir)2	a1bcir)2	CCONJ
ejpam-4918	281	19	=	=	SYM
ejpam-4918	281	20	0	0	X
ejpam-4918	281	21	.	.	PUNCT
ejpam-4918	282	1	since	since	SCONJ
ejpam-4918	282	2	i	i	PRON
ejpam-4918	282	3	is	be	AUX
ejpam-4918	282	4	reduced	reduce	VERB
ejpam-4918	282	5	and	and	CCONJ
ejpam-4918	282	6	ω	ω	PROPN
ejpam-4918	282	7	is	be	AUX
ejpam-4918	282	8	automorphism	automorphism	NOUN
ejpam-4918	282	9	,	,	PUNCT
ejpam-4918	282	10	it	it	PRON
ejpam-4918	282	11	follows	follow	VERB
ejpam-4918	282	12	that	that	PRON
ejpam-4918	282	13	a1bciωli(r	a1bciωli(r	PUNCT
ejpam-4918	282	14	)	)	PUNCT
ejpam-4918	282	15	=	=	SYM
ejpam-4918	283	1	0	0	X
ejpam-4918	283	2	.	.	X
ejpam-4918	283	3	note	note	VERB
ejpam-4918	283	4	that	that	SCONJ
ejpam-4918	283	5	bciωli(ra1	bciωli(ra1	NOUN
ejpam-4918	283	6	)	)	PUNCT
ejpam-4918	283	7	⊆	⊆	NUM
ejpam-4918	283	8	i.	i.	NOUN
ejpam-4918	283	9	thus	thus	ADV
ejpam-4918	283	10	bciωli(ra1	bciωli(ra1	NOUN
ejpam-4918	283	11	)	)	PUNCT
ejpam-4918	283	12	=	=	SYM
ejpam-4918	283	13	0	0	NUM
ejpam-4918	284	1	for	for	ADP
ejpam-4918	284	2	any	any	DET
ejpam-4918	284	3	i.	i.	NOUN
ejpam-4918	284	4	now	now	ADV
ejpam-4918	284	5	we	we	PRON
ejpam-4918	284	6	have	have	VERB
ejpam-4918	284	7	(	(	PUNCT
ejpam-4918	284	8	bc1l1	bc1l1	NOUN
ejpam-4918	284	9	+	+	SYM
ejpam-4918	284	10	bc2l2	bc2l2	PROPN
ejpam-4918	284	11	+	+	CCONJ
ejpam-4918	284	12	·	·	PUNCT
ejpam-4918	284	13	·	·	PUNCT
ejpam-4918	284	14	·	·	PUNCT
ejpam-4918	285	1	+	+	NUM
ejpam-4918	285	2	bcnln)(r	bcnln)(r	PROPN
ejpam-4918	285	3	∗m)(a1h1+a2h2	∗m)(a1h1+a2h2	PROPN
ejpam-4918	285	4	+	+	NUM
ejpam-4918	285	5	·	·	PUNCT
ejpam-4918	285	6	·	·	PUNCT
ejpam-4918	285	7	·	·	PUNCT
ejpam-4918	285	8	+	+	NUM
ejpam-4918	285	9	amhm	amhm	NOUN
ejpam-4918	285	10	)	)	PUNCT
ejpam-4918	285	11	=	=	PUNCT
ejpam-4918	285	12	(	(	PUNCT
ejpam-4918	285	13	bλ)(c1l1	bλ)(c1l1	NOUN
ejpam-4918	285	14	+	+	CCONJ
ejpam-4918	285	15	c2l2	c2l2	X
ejpam-4918	285	16	+	+	X
ejpam-4918	285	17	·	·	PUNCT
ejpam-4918	285	18	·	·	PUNCT
ejpam-4918	285	19	·	·	PUNCT
ejpam-4918	285	20	+	+	NUM
ejpam-4918	285	21	cnln)(r	cnln)(r	NOUN
ejpam-4918	285	22	∗m)(a1h1	∗m)(a1h1	NOUN
ejpam-4918	285	23	+	+	CCONJ
ejpam-4918	285	24	a2h2	a2h2	X
ejpam-4918	285	25	+	+	X
ejpam-4918	285	26	·	·	PUNCT
ejpam-4918	285	27	·	·	PUNCT
ejpam-4918	285	28	·	·	PUNCT
ejpam-4918	285	29	+	+	NUM
ejpam-4918	285	30	amhm	amhm	NOUN
ejpam-4918	285	31	)	)	PUNCT
ejpam-4918	285	32	=	=	SYM
ejpam-4918	286	1	0	0	X
ejpam-4918	286	2	.	.	PUNCT
ejpam-4918	286	3	by	by	ADP
ejpam-4918	286	4	applying	apply	VERB
ejpam-4918	286	5	the	the	DET
ejpam-4918	286	6	induction	induction	NOUN
ejpam-4918	286	7	hypothesis	hypothesis	NOUN
ejpam-4918	286	8	,	,	PUNCT
ejpam-4918	286	9	it	it	PRON
ejpam-4918	286	10	follows	follow	VERB
ejpam-4918	286	11	that	that	SCONJ
ejpam-4918	286	12	bciωli(ωg(raj	bciωli(ωg(raj	ADJ
ejpam-4918	286	13	)	)	PUNCT
ejpam-4918	286	14	)	)	PUNCT
ejpam-4918	287	1	=	=	SYM
ejpam-4918	287	2	0	0	NUM
ejpam-4918	288	1	for	for	ADP
ejpam-4918	288	2	all	all	DET
ejpam-4918	288	3	1	1	NUM
ejpam-4918	288	4	≤	≤	NUM
ejpam-4918	288	5	i	i	PRON
ejpam-4918	288	6	≤	≤	ADJ
ejpam-4918	288	7	n	n	CCONJ
ejpam-4918	288	8	and	and	CCONJ
ejpam-4918	288	9	2	2	NUM
ejpam-4918	288	10	≤	≤	NUM
ejpam-4918	288	11	j	j	PROPN
ejpam-4918	288	12	≤	≤	PROPN
ejpam-4918	288	13	m.	m.	NOUN
ejpam-4918	288	14	thus	thus	ADV
ejpam-4918	288	15	,	,	PUNCT
ejpam-4918	288	16	we	we	PRON
ejpam-4918	288	17	have	have	VERB
ejpam-4918	288	18	ciωli(ωg(raj	ciωli(ωg(raj	NOUN
ejpam-4918	288	19	)	)	PUNCT
ejpam-4918	288	20	)	)	PUNCT
ejpam-4918	289	1	=	=	SYM
ejpam-4918	289	2	0	0	NUM
ejpam-4918	289	3	for	for	ADP
ejpam-4918	289	4	all	all	DET
ejpam-4918	289	5	i	i	PROPN
ejpam-4918	289	6	,	,	PUNCT
ejpam-4918	289	7	j	j	PROPN
ejpam-4918	289	8	and	and	CCONJ
ejpam-4918	289	9	all	all	DET
ejpam-4918	289	10	r	r	NOUN
ejpam-4918	289	11	∈	∈	PROPN
ejpam-4918	289	12	r.	r.	NOUN
ejpam-4918	289	13	particularly	particularly	ADV
ejpam-4918	289	14	,	,	PUNCT
ejpam-4918	289	15	we	we	PRON
ejpam-4918	289	16	have	have	VERB
ejpam-4918	289	17	bckωlk(ωg(raq	bckωlk(ωg(raq	PROPN
ejpam-4918	289	18	)	)	PUNCT
ejpam-4918	289	19	)	)	PUNCT
ejpam-4918	290	1	=	=	SYM
ejpam-4918	290	2	0	0	NUM
ejpam-4918	290	3	and	and	CCONJ
ejpam-4918	290	4	so	so	ADV
ejpam-4918	290	5	b2	b2	NOUN
ejpam-4918	290	6	=	=	SYM
ejpam-4918	290	7	0	0	NUM
ejpam-4918	290	8	.	.	PUNCT
ejpam-4918	291	1	thus	thus	ADV
ejpam-4918	291	2	b	b	X
ejpam-4918	291	3	=	=	SYM
ejpam-4918	291	4	0	0	PROPN
ejpam-4918	291	5	.	.	PUNCT
ejpam-4918	292	1	this	this	PRON
ejpam-4918	292	2	shows	show	VERB
ejpam-4918	292	3	that	that	SCONJ
ejpam-4918	292	4	ckωlk(ωg(raq	ckωlk(ωg(raq	NOUN
ejpam-4918	292	5	)	)	PUNCT
ejpam-4918	292	6	)	)	PUNCT
ejpam-4918	293	1	=	=	SYM
ejpam-4918	293	2	0	0	NUM
ejpam-4918	294	1	for	for	ADP
ejpam-4918	294	2	any	any	DET
ejpam-4918	294	3	1	1	NUM
ejpam-4918	294	4	≤	≤	NUM
ejpam-4918	294	5	k	k	NOUN
ejpam-4918	294	6	≤	≤	NOUN
ejpam-4918	294	7	n	n	PRON
ejpam-4918	294	8	and	and	CCONJ
ejpam-4918	294	9	1	1	NUM
ejpam-4918	294	10	≤	≤	NUM
ejpam-4918	294	11	q	q	PROPN
ejpam-4918	294	12	≤	≤	ADJ
ejpam-4918	294	13	m.	m.	NOUN
ejpam-4918	294	14	consequently	consequently	ADV
ejpam-4918	294	15	,	,	PUNCT
ejpam-4918	294	16	we	we	PRON
ejpam-4918	294	17	can	can	AUX
ejpam-4918	294	18	see	see	VERB
ejpam-4918	294	19	that	that	DET
ejpam-4918	294	20	ajωhj	ajωhj	NOUN
ejpam-4918	294	21	(	(	PUNCT
ejpam-4918	294	22	ωg(rci	ωg(rci	NOUN
ejpam-4918	294	23	)	)	PUNCT
ejpam-4918	294	24	)	)	PUNCT
ejpam-4918	295	1	=	=	SYM
ejpam-4918	295	2	0	0	NUM
ejpam-4918	296	1	for	for	ADP
ejpam-4918	296	2	all	all	PRON
ejpam-4918	296	3	g	g	PROPN
ejpam-4918	296	4	∈	∈	PROPN
ejpam-4918	296	5	m	m	NOUN
ejpam-4918	296	6	,	,	PUNCT
ejpam-4918	296	7	1	1	NUM
ejpam-4918	296	8	≤	≤	NUM
ejpam-4918	296	9	j	j	PROPN
ejpam-4918	296	10	≤	≤	NUM
ejpam-4918	296	11	m	m	PROPN
ejpam-4918	296	12	,	,	PUNCT
ejpam-4918	296	13	and	and	CCONJ
ejpam-4918	296	14	1	1	NUM
ejpam-4918	296	15	≤	≤	NUM
ejpam-4918	296	16	i	i	PRON
ejpam-4918	296	17	≤	≤	PROPN
ejpam-4918	296	18	n.	n.	NOUN
ejpam-4918	296	19	therefore	therefore	ADV
ejpam-4918	296	20	,	,	PUNCT
ejpam-4918	296	21	r	r	NOUN
ejpam-4918	296	22	is	be	AUX
ejpam-4918	296	23	strongly	strongly	ADV
ejpam-4918	296	24	cm	cm	NOUN
ejpam-4918	296	25	-reflexive	-reflexive	NOUN
ejpam-4918	296	26	.	.	PUNCT
ejpam-4918	297	1	the	the	DET
ejpam-4918	297	2	notion	notion	NOUN
ejpam-4918	297	3	of	of	ADP
ejpam-4918	297	4	complete	complete	ADJ
ejpam-4918	297	5	m	m	VERB
ejpam-4918	297	6	-compatibility	-compatibility	NOUN
ejpam-4918	297	7	is	be	AUX
ejpam-4918	297	8	important	important	ADJ
ejpam-4918	297	9	in	in	ADP
ejpam-4918	297	10	the	the	DET
ejpam-4918	297	11	following	following	ADJ
ejpam-4918	297	12	result	result	NOUN
ejpam-4918	297	13	[	[	X
ejpam-4918	297	14	18	18	NUM
ejpam-4918	297	15	]	]	PUNCT
ejpam-4918	297	16	.	.	PUNCT
ejpam-4918	298	1	corollary	corollary	ADJ
ejpam-4918	298	2	2	2	NUM
ejpam-4918	298	3	.	.	PUNCT
ejpam-4918	299	1	assuming	assume	VERB
ejpam-4918	299	2	r	r	NOUN
ejpam-4918	299	3	is	be	AUX
ejpam-4918	299	4	a	a	DET
ejpam-4918	299	5	ring	ring	NOUN
ejpam-4918	299	6	that	that	PRON
ejpam-4918	299	7	is	be	AUX
ejpam-4918	299	8	completely	completely	ADV
ejpam-4918	299	9	m	m	VERB
ejpam-4918	299	10	-compatible	-compatible	ADJ
ejpam-4918	299	11	,	,	PUNCT
ejpam-4918	299	12	where	where	SCONJ
ejpam-4918	299	13	m	m	PROPN
ejpam-4918	299	14	is	be	AUX
ejpam-4918	299	15	a	a	DET
ejpam-4918	299	16	monoid	monoid	NOUN
ejpam-4918	299	17	with	with	ADP
ejpam-4918	299	18	twisting	twist	VERB
ejpam-4918	299	19	f	f	X
ejpam-4918	299	20	:	:	PUNCT
ejpam-4918	299	21	m	m	VERB
ejpam-4918	299	22	×m	×m	NOUN
ejpam-4918	299	23	→	→	SYM
ejpam-4918	299	24	u(r	u(r	NOUN
ejpam-4918	299	25	)	)	PUNCT
ejpam-4918	299	26	and	and	CCONJ
ejpam-4918	299	27	action	action	NOUN
ejpam-4918	299	28	ω	ω	NOUN
ejpam-4918	299	29	:	:	PUNCT
ejpam-4918	299	30	m	m	PROPN
ejpam-4918	299	31	→	→	SYM
ejpam-4918	299	32	aut(r	aut(r	PROPN
ejpam-4918	299	33	)	)	PUNCT
ejpam-4918	299	34	,	,	PUNCT
ejpam-4918	299	35	and	and	CCONJ
ejpam-4918	299	36	i	i	PRON
ejpam-4918	299	37	is	be	AUX
ejpam-4918	299	38	an	an	DET
ejpam-4918	299	39	ideal	ideal	NOUN
ejpam-4918	299	40	of	of	ADP
ejpam-4918	299	41	r	r	NOUN
ejpam-4918	299	42	such	such	ADJ
ejpam-4918	299	43	that	that	SCONJ
ejpam-4918	299	44	i	i	PRON
ejpam-4918	299	45	is	be	AUX
ejpam-4918	299	46	reduced	reduce	VERB
ejpam-4918	299	47	and	and	CCONJ
ejpam-4918	299	48	r	r	AUX
ejpam-4918	299	49	/	/	SYM
ejpam-4918	299	50	i	i	PRON
ejpam-4918	299	51	is	be	AUX
ejpam-4918	299	52	cm	cm	NUM
ejpam-4918	299	53	-quasi	-quasi	NOUN
ejpam-4918	299	54	-	-	PUNCT
ejpam-4918	299	55	armendariz	armendariz	ADJ
ejpam-4918	299	56	,	,	PUNCT
ejpam-4918	299	57	then	then	ADV
ejpam-4918	299	58	r	r	NOUN
ejpam-4918	299	59	is	be	AUX
ejpam-4918	299	60	strongly	strongly	ADV
ejpam-4918	299	61	cm	cm	NOUN
ejpam-4918	299	62	-reflexive	-reflexive	NOUN
ejpam-4918	299	63	.	.	PUNCT
ejpam-4918	300	1	proof	proof	NOUN
ejpam-4918	300	2	.	.	PUNCT
ejpam-4918	301	1	as	as	SCONJ
ejpam-4918	301	2	cm	cm	PROPN
ejpam-4918	301	3	-quasi	-quasi	PROPN
ejpam-4918	301	4	-	-	PUNCT
ejpam-4918	301	5	armendariz	armendariz	ADJ
ejpam-4918	301	6	rings	ring	NOUN
ejpam-4918	301	7	are	be	AUX
ejpam-4918	301	8	strongly	strongly	ADV
ejpam-4918	301	9	cm	cm	NOUN
ejpam-4918	301	10	-reflexive	-reflexive	NOUN
ejpam-4918	301	11	,	,	PUNCT
ejpam-4918	301	12	the	the	DET
ejpam-4918	301	13	result	result	NOUN
ejpam-4918	301	14	can	can	AUX
ejpam-4918	301	15	be	be	AUX
ejpam-4918	301	16	e.	e.	PROPN
ejpam-4918	301	17	ali	ali	PROPN
ejpam-4918	301	18	/	/	SYM
ejpam-4918	301	19	eur	eur	PROPN
ejpam-4918	301	20	.	.	PUNCT
ejpam-4918	302	1	j.	j.	PROPN
ejpam-4918	302	2	pure	pure	PROPN
ejpam-4918	302	3	appl	appl	PROPN
ejpam-4918	302	4	.	.	PROPN
ejpam-4918	302	5	math	math	PROPN
ejpam-4918	302	6	,	,	PUNCT
ejpam-4918	302	7	16	16	NUM
ejpam-4918	302	8	(	(	PUNCT
ejpam-4918	302	9	4	4	NUM
ejpam-4918	302	10	)	)	PUNCT
ejpam-4918	302	11	(	(	PUNCT
ejpam-4918	302	12	2023	2023	NUM
ejpam-4918	302	13	)	)	PUNCT
ejpam-4918	302	14	,	,	PUNCT
ejpam-4918	302	15	2156	2156	NUM
ejpam-4918	302	16	-	-	SYM
ejpam-4918	302	17	2168	2168	NUM
ejpam-4918	302	18	2163	2163	NUM
ejpam-4918	302	19	obtained	obtain	VERB
ejpam-4918	302	20	from	from	ADP
ejpam-4918	302	21	theorem	theorem	ADJ
ejpam-4918	302	22	3	3	NUM
ejpam-4918	302	23	.	.	PUNCT
ejpam-4918	302	24	corollary	corollary	ADJ
ejpam-4918	302	25	3	3	NUM
ejpam-4918	302	26	.	.	PUNCT
ejpam-4918	302	27	suppose	suppose	VERB
ejpam-4918	302	28	that	that	SCONJ
ejpam-4918	302	29	r	r	NOUN
ejpam-4918	302	30	is	be	AUX
ejpam-4918	302	31	a	a	DET
ejpam-4918	302	32	completely	completely	ADV
ejpam-4918	302	33	m	m	ADJ
ejpam-4918	302	34	-compatible	-compatible	ADJ
ejpam-4918	302	35	ring	ring	NOUN
ejpam-4918	302	36	,	,	PUNCT
ejpam-4918	302	37	where	where	SCONJ
ejpam-4918	302	38	m	m	PROPN
ejpam-4918	302	39	is	be	AUX
ejpam-4918	302	40	a	a	DET
ejpam-4918	302	41	monoid	monoid	NOUN
ejpam-4918	302	42	with	with	ADP
ejpam-4918	302	43	twisting	twist	VERB
ejpam-4918	302	44	f	f	X
ejpam-4918	302	45	:	:	PUNCT
ejpam-4918	302	46	m	m	VERB
ejpam-4918	302	47	×m	×m	NOUN
ejpam-4918	302	48	→	→	SYM
ejpam-4918	302	49	u(r	u(r	NOUN
ejpam-4918	302	50	)	)	PUNCT
ejpam-4918	302	51	and	and	CCONJ
ejpam-4918	302	52	action	action	NOUN
ejpam-4918	302	53	ω	ω	NOUN
ejpam-4918	302	54	:	:	PUNCT
ejpam-4918	302	55	m	m	PROPN
ejpam-4918	302	56	→	→	SYM
ejpam-4918	302	57	aut(r	aut(r	PROPN
ejpam-4918	302	58	)	)	PUNCT
ejpam-4918	302	59	.	.	PUNCT
ejpam-4918	303	1	let	let	VERB
ejpam-4918	303	2	i	i	PRON
ejpam-4918	303	3	be	be	AUX
ejpam-4918	303	4	an	an	DET
ejpam-4918	303	5	ideal	ideal	NOUN
ejpam-4918	303	6	of	of	ADP
ejpam-4918	303	7	r	r	NOUN
ejpam-4918	303	8	such	such	ADJ
ejpam-4918	303	9	that	that	SCONJ
ejpam-4918	303	10	i	i	PRON
ejpam-4918	303	11	is	be	AUX
ejpam-4918	303	12	reduced	reduce	VERB
ejpam-4918	303	13	and	and	CCONJ
ejpam-4918	303	14	r	r	AUX
ejpam-4918	303	15	/	/	SYM
ejpam-4918	303	16	i	i	PRON
ejpam-4918	303	17	is	be	AUX
ejpam-4918	303	18	cm	cm	NOUN
ejpam-4918	303	19	-armendariz	-armendariz	NOUN
ejpam-4918	303	20	.	.	PUNCT
ejpam-4918	304	1	then	then	ADV
ejpam-4918	304	2	,	,	PUNCT
ejpam-4918	304	3	r	r	NOUN
ejpam-4918	304	4	is	be	AUX
ejpam-4918	304	5	strongly	strongly	ADV
ejpam-4918	304	6	cm	cm	NOUN
ejpam-4918	304	7	-reflexive	-reflexive	NOUN
ejpam-4918	304	8	.	.	PUNCT
ejpam-4918	305	1	proof	proof	NOUN
ejpam-4918	305	2	.	.	PUNCT
ejpam-4918	306	1	since	since	SCONJ
ejpam-4918	306	2	cm	cm	NOUN
ejpam-4918	306	3	-armendariz	-armendariz	PROPN
ejpam-4918	306	4	is	be	AUX
ejpam-4918	306	5	a	a	DET
ejpam-4918	306	6	cm	cm	NUM
ejpam-4918	306	7	-quasi	-quasi	NOUN
ejpam-4918	306	8	-	-	PUNCT
ejpam-4918	306	9	armendariz	armendariz	ADV
ejpam-4918	306	10	,	,	PUNCT
ejpam-4918	306	11	the	the	DET
ejpam-4918	306	12	result	result	NOUN
ejpam-4918	306	13	can	can	AUX
ejpam-4918	306	14	be	be	AUX
ejpam-4918	306	15	derived	derive	VERB
ejpam-4918	306	16	from	from	ADP
ejpam-4918	306	17	corollary	corollary	ADJ
ejpam-4918	306	18	2	2	NUM
ejpam-4918	306	19	.	.	PUNCT
ejpam-4918	306	20	proposition	proposition	NOUN
ejpam-4918	306	21	2	2	NUM
ejpam-4918	306	22	.	.	PUNCT
ejpam-4918	306	23	assuming	assume	VERB
ejpam-4918	306	24	r	r	NOUN
ejpam-4918	306	25	is	be	AUX
ejpam-4918	306	26	a	a	DET
ejpam-4918	306	27	ring	ring	NOUN
ejpam-4918	306	28	that	that	PRON
ejpam-4918	306	29	is	be	AUX
ejpam-4918	306	30	both	both	PRON
ejpam-4918	306	31	m	m	NOUN
ejpam-4918	306	32	-compatible	-compatible	ADJ
ejpam-4918	306	33	and	and	CCONJ
ejpam-4918	306	34	cm	cm	NOUN
ejpam-4918	306	35	-quasi	-quasi	NOUN
ejpam-4918	306	36	-	-	PUNCT
ejpam-4918	306	37	armendariz	armendariz	NOUN
ejpam-4918	306	38	,	,	PUNCT
ejpam-4918	306	39	where	where	SCONJ
ejpam-4918	306	40	m	m	NOUN
ejpam-4918	306	41	is	be	AUX
ejpam-4918	306	42	a	a	DET
ejpam-4918	306	43	monoid	monoid	NOUN
ejpam-4918	306	44	with	with	ADP
ejpam-4918	306	45	twisting	twist	VERB
ejpam-4918	306	46	f	f	X
ejpam-4918	306	47	:	:	PUNCT
ejpam-4918	306	48	m	m	VERB
ejpam-4918	306	49	×m	×m	NOUN
ejpam-4918	306	50	→	→	SYM
ejpam-4918	306	51	u(r	u(r	NOUN
ejpam-4918	306	52	)	)	PUNCT
ejpam-4918	306	53	and	and	CCONJ
ejpam-4918	307	1	action	action	NOUN
ejpam-4918	307	2	ω	ω	NOUN
ejpam-4918	307	3	:	:	PUNCT
ejpam-4918	307	4	m	m	PROPN
ejpam-4918	307	5	→	→	SYM
ejpam-4918	307	6	aut(r	aut(r	PROPN
ejpam-4918	307	7	)	)	PUNCT
ejpam-4918	307	8	,	,	PUNCT
ejpam-4918	307	9	then	then	ADV
ejpam-4918	307	10	r	r	NOUN
ejpam-4918	307	11	is	be	AUX
ejpam-4918	307	12	strongly	strongly	ADV
ejpam-4918	307	13	cm	cm	NOUN
ejpam-4918	307	14	-reflexive	-reflexive	NOUN
ejpam-4918	307	15	if	if	SCONJ
ejpam-4918	308	1	and	and	CCONJ
ejpam-4918	308	2	only	only	ADV
ejpam-4918	308	3	if	if	SCONJ
ejpam-4918	308	4	r	r	NOUN
ejpam-4918	308	5	∗m	∗m	NOUN
ejpam-4918	308	6	is	be	AUX
ejpam-4918	308	7	strongly	strongly	ADV
ejpam-4918	308	8	cm	cm	NOUN
ejpam-4918	308	9	-reflexive	-reflexive	NOUN
ejpam-4918	308	10	.	.	PUNCT
ejpam-4918	309	1	proof	proof	NOUN
ejpam-4918	309	2	.	.	PUNCT
ejpam-4918	310	1	to	to	PART
ejpam-4918	310	2	prove	prove	VERB
ejpam-4918	310	3	a	a	DET
ejpam-4918	310	4	necessary	necessary	ADJ
ejpam-4918	310	5	condition	condition	NOUN
ejpam-4918	310	6	is	be	AUX
ejpam-4918	310	7	sufficient	sufficient	ADJ
ejpam-4918	310	8	.	.	PUNCT
ejpam-4918	311	1	let	let	VERB
ejpam-4918	311	2	ϕ	ϕ	NOUN
ejpam-4918	311	3	=	=	PROPN
ejpam-4918	311	4	σni=1cili	σni=1cili	PROPN
ejpam-4918	311	5	,	,	PUNCT
ejpam-4918	311	6	ψ	ψ	X
ejpam-4918	311	7	=	=	SYM
ejpam-4918	311	8	σmj=1ajhj	σmj=1ajhj	NOUN
ejpam-4918	311	9	∈	∈	NOUN
ejpam-4918	311	10	r	r	NOUN
ejpam-4918	311	11	∗m	∗m	NOUN
ejpam-4918	311	12	satisfying	satisfy	VERB
ejpam-4918	311	13	ϕ(r	ϕ(r	PROPN
ejpam-4918	311	14	∗m)ψ	∗m)ψ	ADJ
ejpam-4918	311	15	=	=	PUNCT
ejpam-4918	312	1	0	0	X
ejpam-4918	312	2	.	.	PUNCT
ejpam-4918	313	1	since	since	SCONJ
ejpam-4918	313	2	r	r	NOUN
ejpam-4918	313	3	is	be	AUX
ejpam-4918	313	4	cm	cm	NUM
ejpam-4918	313	5	-quasi	-quasi	NOUN
ejpam-4918	313	6	-	-	PUNCT
ejpam-4918	313	7	armendariz	armendariz	ADJ
ejpam-4918	313	8	,	,	PUNCT
ejpam-4918	313	9	we	we	PRON
ejpam-4918	313	10	have	have	AUX
ejpam-4918	313	11	ciωli(ωg(raj))f(li	ciωli(ωg(raj))f(li	NOUN
ejpam-4918	313	12	,	,	PUNCT
ejpam-4918	313	13	hj)(lihj	hj)(lihj	PROPN
ejpam-4918	313	14	)	)	PUNCT
ejpam-4918	313	15	=	=	SYM
ejpam-4918	313	16	0	0	NUM
ejpam-4918	313	17	for	for	ADP
ejpam-4918	313	18	all	all	DET
ejpam-4918	313	19	i	i	PROPN
ejpam-4918	313	20	,	,	PUNCT
ejpam-4918	313	21	j.	j.	PROPN
ejpam-4918	313	22	this	this	PRON
ejpam-4918	313	23	implies	imply	VERB
ejpam-4918	313	24	that	that	SCONJ
ejpam-4918	313	25	ciωli(ωg(raj	ciωli(ωg(raj	VERB
ejpam-4918	313	26	)	)	PUNCT
ejpam-4918	313	27	)	)	PUNCT
ejpam-4918	314	1	=	=	SYM
ejpam-4918	314	2	0	0	NUM
ejpam-4918	315	1	for	for	ADP
ejpam-4918	315	2	all	all	DET
ejpam-4918	315	3	i	i	PROPN
ejpam-4918	315	4	,	,	PUNCT
ejpam-4918	315	5	j	j	PROPN
ejpam-4918	315	6	since	since	SCONJ
ejpam-4918	315	7	r	r	NOUN
ejpam-4918	315	8	is	be	AUX
ejpam-4918	315	9	m	m	PRON
ejpam-4918	315	10	-compatible	-compatible	ADJ
ejpam-4918	315	11	.	.	PUNCT
ejpam-4918	316	1	because	because	SCONJ
ejpam-4918	316	2	r	r	NOUN
ejpam-4918	316	3	is	be	AUX
ejpam-4918	316	4	a	a	DET
ejpam-4918	316	5	reflexive	reflexive	ADJ
ejpam-4918	316	6	ring	ring	NOUN
ejpam-4918	316	7	,	,	PUNCT
ejpam-4918	316	8	ajrci	ajrci	VERB
ejpam-4918	316	9	=	=	NOUN
ejpam-4918	316	10	0	0	X
ejpam-4918	316	11	.	.	PUNCT
ejpam-4918	317	1	then	then	ADV
ejpam-4918	317	2	,	,	PUNCT
ejpam-4918	317	3	ajωhj	ajωhj	PROPN
ejpam-4918	317	4	(	(	PUNCT
ejpam-4918	317	5	ωg(rci	ωg(rci	NOUN
ejpam-4918	317	6	)	)	PUNCT
ejpam-4918	317	7	)	)	PUNCT
ejpam-4918	317	8	=	=	SYM
ejpam-4918	317	9	0	0	NUM
ejpam-4918	317	10	for	for	ADP
ejpam-4918	317	11	all	all	DET
ejpam-4918	317	12	i	i	PROPN
ejpam-4918	317	13	,	,	PUNCT
ejpam-4918	317	14	j	j	PROPN
ejpam-4918	317	15	,	,	PUNCT
ejpam-4918	317	16	and	and	CCONJ
ejpam-4918	317	17	hence	hence	ADV
ejpam-4918	317	18	for	for	ADP
ejpam-4918	317	19	any	any	DET
ejpam-4918	317	20	r	r	NOUN
ejpam-4918	317	21	∈	∈	NOUN
ejpam-4918	317	22	r	r	NOUN
ejpam-4918	317	23	,	,	PUNCT
ejpam-4918	317	24	g	g	NOUN
ejpam-4918	317	25	∈m	∈m	NOUN
ejpam-4918	317	26	,	,	PUNCT
ejpam-4918	317	27	we	we	PRON
ejpam-4918	317	28	have	have	VERB
ejpam-4918	317	29	ψ(r	ψ(r	NOUN
ejpam-4918	317	30	∗m)ϕ	∗m)ϕ	NOUN
ejpam-4918	317	31	=	=	SYM
ejpam-4918	317	32	σmj=1σ	σmj=1σ	PROPN
ejpam-4918	317	33	n	n	PRON
ejpam-4918	317	34	i=1ajωhj	i=1ajωhj	NOUN
ejpam-4918	317	35	(	(	PUNCT
ejpam-4918	317	36	ωg(r	ωg(r	NUM
ejpam-4918	317	37	ci))f(hj	ci))f(hj	NOUN
ejpam-4918	317	38	,	,	PUNCT
ejpam-4918	317	39	li)(hjli	li)(hjli	PROPN
ejpam-4918	317	40	)	)	PUNCT
ejpam-4918	317	41	=	=	SYM
ejpam-4918	318	1	0	0	X
ejpam-4918	318	2	.	.	PUNCT
ejpam-4918	319	1	thus	thus	ADV
ejpam-4918	319	2	,	,	PUNCT
ejpam-4918	319	3	ajωhj	ajωhj	PROPN
ejpam-4918	319	4	(	(	PUNCT
ejpam-4918	319	5	ωg(r	ωg(r	NUM
ejpam-4918	319	6	ci	ci	NOUN
ejpam-4918	319	7	)	)	PUNCT
ejpam-4918	319	8	)	)	PUNCT
ejpam-4918	320	1	=	=	SYM
ejpam-4918	320	2	0	0	PUNCT
ejpam-4918	321	1	since	since	SCONJ
ejpam-4918	321	2	r	r	NOUN
ejpam-4918	321	3	is	be	AUX
ejpam-4918	321	4	m	m	PRON
ejpam-4918	321	5	-compatible	-compatible	ADJ
ejpam-4918	321	6	and	and	CCONJ
ejpam-4918	321	7	cm	cm	NOUN
ejpam-4918	321	8	-quasi	-quasi	NOUN
ejpam-4918	321	9	-	-	PUNCT
ejpam-4918	321	10	armendariz	armendariz	ADJ
ejpam-4918	321	11	.	.	PUNCT
ejpam-4918	322	1	therefore	therefore	ADV
ejpam-4918	322	2	,	,	PUNCT
ejpam-4918	322	3	r	r	NOUN
ejpam-4918	322	4	is	be	AUX
ejpam-4918	322	5	strongly	strongly	ADV
ejpam-4918	322	6	cm	cm	NOUN
ejpam-4918	322	7	-reflexive	-reflexive	NOUN
ejpam-4918	322	8	.	.	PUNCT
ejpam-4918	323	1	every	every	DET
ejpam-4918	323	2	left	leave	VERB
ejpam-4918	323	3	app	app	NOUN
ejpam-4918	323	4	-ring	-ring	PROPN
ejpam-4918	323	5	is	be	AUX
ejpam-4918	323	6	quasi	quasi	ADJ
ejpam-4918	323	7	-	-	ADJ
ejpam-4918	323	8	armendariz	armendariz	ADJ
ejpam-4918	323	9	,	,	PUNCT
ejpam-4918	323	10	but	but	CCONJ
ejpam-4918	323	11	not	not	PART
ejpam-4918	323	12	conversely	conversely	ADV
ejpam-4918	323	13	[	[	X
ejpam-4918	323	14	19	19	NUM
ejpam-4918	323	15	,	,	PUNCT
ejpam-4918	323	16	20	20	NUM
ejpam-4918	323	17	]	]	PUNCT
ejpam-4918	323	18	.	.	PUNCT
ejpam-4918	324	1	proposition	proposition	NOUN
ejpam-4918	324	2	3	3	X
ejpam-4918	324	3	.	.	PUNCT
ejpam-4918	325	1	let	let	VERB
ejpam-4918	325	2	m	m	PRON
ejpam-4918	325	3	be	be	AUX
ejpam-4918	325	4	a	a	DET
ejpam-4918	325	5	strictly	strictly	ADV
ejpam-4918	325	6	totally	totally	ADV
ejpam-4918	325	7	ordered	order	VERB
ejpam-4918	325	8	monoid	monoid	NOUN
ejpam-4918	325	9	with	with	ADP
ejpam-4918	325	10	twisting	twist	VERB
ejpam-4918	325	11	f	f	X
ejpam-4918	325	12	:	:	PUNCT
ejpam-4918	325	13	m	m	VERB
ejpam-4918	325	14	×m	×m	NOUN
ejpam-4918	325	15	→	→	SYM
ejpam-4918	325	16	u(r	u(r	NOUN
ejpam-4918	325	17	)	)	PUNCT
ejpam-4918	325	18	and	and	CCONJ
ejpam-4918	326	1	action	action	NOUN
ejpam-4918	326	2	ω	ω	NOUN
ejpam-4918	326	3	:	:	PUNCT
ejpam-4918	326	4	m	m	PROPN
ejpam-4918	326	5	→	→	SYM
ejpam-4918	326	6	aut(r	aut(r	PROPN
ejpam-4918	326	7	)	)	PUNCT
ejpam-4918	326	8	.	.	PUNCT
ejpam-4918	327	1	let	let	VERB
ejpam-4918	327	2	r	r	PRON
ejpam-4918	327	3	be	be	AUX
ejpam-4918	327	4	an	an	DET
ejpam-4918	327	5	m	m	PROPN
ejpam-4918	327	6	-compatible	-compatible	ADJ
ejpam-4918	327	7	left	leave	VERB
ejpam-4918	327	8	app	app	NOUN
ejpam-4918	327	9	-ring	-ring	PROPN
ejpam-4918	327	10	.	.	PUNCT
ejpam-4918	328	1	then	then	ADV
ejpam-4918	328	2	r	r	NOUN
ejpam-4918	328	3	is	be	AUX
ejpam-4918	328	4	strongly	strongly	ADV
ejpam-4918	328	5	cm	cm	NOUN
ejpam-4918	328	6	-reflexive	-reflexive	NOUN
ejpam-4918	328	7	if	if	SCONJ
ejpam-4918	329	1	and	and	CCONJ
ejpam-4918	329	2	only	only	ADV
ejpam-4918	329	3	if	if	SCONJ
ejpam-4918	329	4	r	r	NOUN
ejpam-4918	329	5	∗m	∗m	NOUN
ejpam-4918	329	6	is	be	AUX
ejpam-4918	329	7	strongly	strongly	ADV
ejpam-4918	329	8	cm	cm	NOUN
ejpam-4918	329	9	-reflexive	-reflexive	NOUN
ejpam-4918	329	10	.	.	PUNCT
ejpam-4918	330	1	proof	proof	NOUN
ejpam-4918	330	2	.	.	PUNCT
ejpam-4918	331	1	if	if	SCONJ
ejpam-4918	331	2	r	r	NOUN
ejpam-4918	331	3	is	be	AUX
ejpam-4918	331	4	a	a	DET
ejpam-4918	331	5	left	left	ADJ
ejpam-4918	331	6	app	app	NOUN
ejpam-4918	331	7	-ring	-ring	NOUN
ejpam-4918	331	8	,	,	PUNCT
ejpam-4918	331	9	then	then	ADV
ejpam-4918	331	10	it	it	PRON
ejpam-4918	331	11	is	be	AUX
ejpam-4918	331	12	m	m	PROPN
ejpam-4918	331	13	-quasi	-quasi	NOUN
ejpam-4918	331	14	-	-	ADJ
ejpam-4918	331	15	armendariz	armendariz	ADJ
ejpam-4918	332	1	[	[	X
ejpam-4918	332	2	21	21	NUM
ejpam-4918	332	3	]	]	PUNCT
ejpam-4918	332	4	.	.	PUNCT
ejpam-4918	333	1	therefore	therefore	ADV
ejpam-4918	333	2	,	,	PUNCT
ejpam-4918	333	3	the	the	DET
ejpam-4918	333	4	result	result	NOUN
ejpam-4918	333	5	follows	follow	VERB
ejpam-4918	333	6	from	from	ADP
ejpam-4918	333	7	proposition	proposition	NOUN
ejpam-4918	333	8	2	2	NUM
ejpam-4918	333	9	.	.	PUNCT
ejpam-4918	333	10	corollary	corollary	ADJ
ejpam-4918	333	11	4	4	NUM
ejpam-4918	333	12	.	.	PUNCT
ejpam-4918	334	1	let	let	VERB
ejpam-4918	334	2	r	r	PRON
ejpam-4918	334	3	be	be	AUX
ejpam-4918	334	4	a	a	DET
ejpam-4918	334	5	ring	ring	NOUN
ejpam-4918	334	6	,	,	PUNCT
ejpam-4918	334	7	m	m	VERB
ejpam-4918	334	8	be	be	VERB
ejpam-4918	334	9	a	a	DET
ejpam-4918	334	10	monoid	monoid	NOUN
ejpam-4918	334	11	with	with	ADP
ejpam-4918	334	12	twisting	twist	VERB
ejpam-4918	334	13	f	f	X
ejpam-4918	334	14	:	:	PUNCT
ejpam-4918	334	15	m	m	VERB
ejpam-4918	334	16	×m	×m	NOUN
ejpam-4918	334	17	→	→	SYM
ejpam-4918	334	18	u(r	u(r	NOUN
ejpam-4918	334	19	)	)	PUNCT
ejpam-4918	334	20	and	and	CCONJ
ejpam-4918	335	1	action	action	NOUN
ejpam-4918	335	2	ω	ω	NOUN
ejpam-4918	335	3	:	:	PUNCT
ejpam-4918	335	4	m	m	PROPN
ejpam-4918	335	5	→	→	SYM
ejpam-4918	335	6	aut(r	aut(r	PROPN
ejpam-4918	335	7	)	)	PUNCT
ejpam-4918	335	8	.	.	PUNCT
ejpam-4918	336	1	if	if	SCONJ
ejpam-4918	336	2	r	r	NOUN
ejpam-4918	336	3	is	be	AUX
ejpam-4918	336	4	a	a	DET
ejpam-4918	336	5	reduced	reduce	VERB
ejpam-4918	336	6	,	,	PUNCT
ejpam-4918	336	7	then	then	ADV
ejpam-4918	336	8	r	r	NOUN
ejpam-4918	336	9	is	be	AUX
ejpam-4918	336	10	strongly	strongly	ADV
ejpam-4918	336	11	cm	cm	NOUN
ejpam-4918	336	12	-reflexive	-reflexive	NOUN
ejpam-4918	336	13	.	.	PUNCT
ejpam-4918	337	1	proof	proof	NOUN
ejpam-4918	337	2	.	.	PUNCT
ejpam-4918	338	1	since	since	SCONJ
ejpam-4918	338	2	r	r	NOUN
ejpam-4918	338	3	is	be	AUX
ejpam-4918	338	4	reduced	reduce	VERB
ejpam-4918	338	5	,	,	PUNCT
ejpam-4918	338	6	it	it	PRON
ejpam-4918	338	7	is	be	AUX
ejpam-4918	338	8	quasi	quasi	ADJ
ejpam-4918	338	9	-	-	ADJ
ejpam-4918	338	10	armendariz	armendariz	ADJ
ejpam-4918	338	11	.	.	PUNCT
ejpam-4918	339	1	therefore	therefore	ADV
ejpam-4918	339	2	,	,	PUNCT
ejpam-4918	339	3	the	the	DET
ejpam-4918	339	4	result	result	NOUN
ejpam-4918	339	5	can	can	AUX
ejpam-4918	339	6	be	be	AUX
ejpam-4918	339	7	derived	derive	VERB
ejpam-4918	339	8	from	from	ADP
ejpam-4918	339	9	proposition	proposition	NOUN
ejpam-4918	339	10	2	2	NUM
ejpam-4918	339	11	.	.	NOUN
ejpam-4918	339	12	3	3	NUM
ejpam-4918	339	13	.	.	X
ejpam-4918	340	1	some	some	DET
ejpam-4918	340	2	results	result	NOUN
ejpam-4918	340	3	on	on	ADP
ejpam-4918	340	4	ring	ring	NOUN
ejpam-4918	340	5	extensions	extension	NOUN
ejpam-4918	340	6	of	of	ADP
ejpam-4918	340	7	crossed	cross	VERB
ejpam-4918	340	8	product	product	NOUN
ejpam-4918	340	9	type	type	NOUN
ejpam-4918	340	10	let	let	VERB
ejpam-4918	340	11	∆	∆	PROPN
ejpam-4918	340	12	be	be	AUX
ejpam-4918	340	13	a	a	DET
ejpam-4918	340	14	multiplicative	multiplicative	ADJ
ejpam-4918	340	15	monoid	monoid	NOUN
ejpam-4918	340	16	consisting	consisting	NOUN
ejpam-4918	340	17	of	of	ADP
ejpam-4918	340	18	central	central	ADJ
ejpam-4918	340	19	regular	regular	ADJ
ejpam-4918	340	20	elements	element	NOUN
ejpam-4918	340	21	of	of	ADP
ejpam-4918	340	22	r.	r.	PROPN
ejpam-4918	340	23	then	then	ADV
ejpam-4918	340	24	,	,	PUNCT
ejpam-4918	340	25	the	the	DET
ejpam-4918	340	26	set	set	NOUN
ejpam-4918	340	27	∆−1r	∆−1r	NOUN
ejpam-4918	340	28	:	:	PUNCT
ejpam-4918	340	29	=	=	SYM
ejpam-4918	340	30	{	{	PUNCT
ejpam-4918	340	31	u−1c|u	u−1c|u	PROPN
ejpam-4918	340	32	∈	∈	PROPN
ejpam-4918	340	33	∆	∆	PROPN
ejpam-4918	340	34	,	,	PUNCT
ejpam-4918	340	35	c	c	PROPN
ejpam-4918	340	36	∈	∈	PROPN
ejpam-4918	340	37	r	r	NOUN
ejpam-4918	340	38	}	}	PUNCT
ejpam-4918	340	39	forms	form	VERB
ejpam-4918	340	40	a	a	DET
ejpam-4918	340	41	ring	ring	NOUN
ejpam-4918	340	42	.	.	PUNCT
ejpam-4918	341	1	suppose	suppose	VERB
ejpam-4918	342	1	ω	ω	NOUN
ejpam-4918	342	2	:	:	PUNCT
ejpam-4918	342	3	m	m	PROPN
ejpam-4918	342	4	→	→	SYM
ejpam-4918	342	5	aut(r	aut(r	PROPN
ejpam-4918	342	6	)	)	PUNCT
ejpam-4918	342	7	is	be	AUX
ejpam-4918	342	8	a	a	DET
ejpam-4918	342	9	monoid	monoid	NOUN
ejpam-4918	342	10	homomorphism	homomorphism	NOUN
ejpam-4918	342	11	such	such	ADJ
ejpam-4918	342	12	that	that	DET
ejpam-4918	342	13	ωh(∆	ωh(∆	NOUN
ejpam-4918	342	14	)	)	PUNCT
ejpam-4918	342	15	⊆	⊆	NUM
ejpam-4918	342	16	∆	∆	PROPN
ejpam-4918	342	17	for	for	ADP
ejpam-4918	342	18	every	every	DET
ejpam-4918	342	19	h	h	NOUN
ejpam-4918	342	20	∈m	∈m	NOUN
ejpam-4918	342	21	.	.	PUNCT
ejpam-4918	343	1	then	then	ADV
ejpam-4918	343	2	,	,	PUNCT
ejpam-4918	343	3	ω	ω	PROPN
ejpam-4918	343	4	can	can	AUX
ejpam-4918	343	5	be	be	AUX
ejpam-4918	343	6	extended	extend	VERB
ejpam-4918	343	7	to	to	AUX
ejpam-4918	343	8	ω̄	ω̄	NUM
ejpam-4918	343	9	:	:	PUNCT
ejpam-4918	343	10	m	m	VERB
ejpam-4918	343	11	→	→	SYM
ejpam-4918	343	12	aut(∆−1r	aut(∆−1r	NOUN
ejpam-4918	343	13	)	)	PUNCT
ejpam-4918	343	14	defined	define	VERB
ejpam-4918	343	15	by	by	ADP
ejpam-4918	343	16	ω̄h(u−1c	ω̄h(u−1c	PROPN
ejpam-4918	343	17	)	)	PUNCT
ejpam-4918	343	18	=	=	SYM
ejpam-4918	343	19	ωh(u	ωh(u	X
ejpam-4918	343	20	)	)	PUNCT
ejpam-4918	343	21	−1ωh(c	−1ωh(c	PROPN
ejpam-4918	343	22	)	)	PUNCT
ejpam-4918	343	23	.	.	PUNCT
ejpam-4918	344	1	if	if	SCONJ
ejpam-4918	344	2	f	f	PROPN
ejpam-4918	344	3	:	:	PUNCT
ejpam-4918	344	4	m	m	VERB
ejpam-4918	344	5	×m	×m	NOUN
ejpam-4918	344	6	→	→	SYM
ejpam-4918	344	7	u(r	u(r	NOUN
ejpam-4918	344	8	)	)	PUNCT
ejpam-4918	344	9	is	be	AUX
ejpam-4918	344	10	a	a	DET
ejpam-4918	344	11	twisted	twisted	ADJ
ejpam-4918	344	12	function	function	NOUN
ejpam-4918	344	13	,	,	PUNCT
ejpam-4918	344	14	then	then	ADV
ejpam-4918	344	15	it	it	PRON
ejpam-4918	344	16	can	can	AUX
ejpam-4918	344	17	be	be	AUX
ejpam-4918	344	18	viewed	view	VERB
ejpam-4918	344	19	as	as	ADP
ejpam-4918	344	20	a	a	DET
ejpam-4918	344	21	twisted	twisted	ADJ
ejpam-4918	344	22	function	function	NOUN
ejpam-4918	344	23	from	from	ADP
ejpam-4918	344	24	m	m	PROPN
ejpam-4918	344	25	×m	×m	NOUN
ejpam-4918	344	26	to	to	ADP
ejpam-4918	344	27	u(∆−1r	u(∆−1r	PROPN
ejpam-4918	344	28	)	)	PUNCT
ejpam-4918	344	29	by	by	ADP
ejpam-4918	344	30	noting	note	VERB
ejpam-4918	344	31	that	that	SCONJ
ejpam-4918	344	32	u(r	u(r	NOUN
ejpam-4918	344	33	)	)	PUNCT
ejpam-4918	344	34	⊆	⊆	NUM
ejpam-4918	344	35	u(∆−1r	u(∆−1r	PROPN
ejpam-4918	344	36	)	)	PUNCT
ejpam-4918	344	37	.	.	PUNCT
ejpam-4918	345	1	e.	e.	PROPN
ejpam-4918	345	2	ali	ali	PROPN
ejpam-4918	345	3	/	/	SYM
ejpam-4918	345	4	eur	eur	PROPN
ejpam-4918	345	5	.	.	PUNCT
ejpam-4918	346	1	j.	j.	PROPN
ejpam-4918	346	2	pure	pure	PROPN
ejpam-4918	346	3	appl	appl	PROPN
ejpam-4918	346	4	.	.	PROPN
ejpam-4918	346	5	math	math	PROPN
ejpam-4918	346	6	,	,	PUNCT
ejpam-4918	346	7	16	16	NUM
ejpam-4918	346	8	(	(	PUNCT
ejpam-4918	346	9	4	4	NUM
ejpam-4918	346	10	)	)	PUNCT
ejpam-4918	346	11	(	(	PUNCT
ejpam-4918	346	12	2023	2023	NUM
ejpam-4918	346	13	)	)	PUNCT
ejpam-4918	346	14	,	,	PUNCT
ejpam-4918	346	15	2156	2156	NUM
ejpam-4918	346	16	-	-	SYM
ejpam-4918	346	17	2168	2168	NUM
ejpam-4918	346	18	2164	2164	NUM
ejpam-4918	346	19	theorem	theorem	NOUN
ejpam-4918	346	20	4	4	NUM
ejpam-4918	346	21	.	.	PUNCT
ejpam-4918	346	22	assuming	assume	VERB
ejpam-4918	346	23	r	r	NOUN
ejpam-4918	346	24	is	be	AUX
ejpam-4918	346	25	an	an	DET
ejpam-4918	346	26	m	m	NOUN
ejpam-4918	346	27	-compatible	-compatible	ADJ
ejpam-4918	346	28	ring	ring	NOUN
ejpam-4918	346	29	,	,	PUNCT
ejpam-4918	346	30	where	where	SCONJ
ejpam-4918	346	31	m	m	NOUN
ejpam-4918	346	32	is	be	AUX
ejpam-4918	346	33	a	a	DET
ejpam-4918	346	34	cancellative	cancellative	ADJ
ejpam-4918	346	35	monoid	monoid	NOUN
ejpam-4918	346	36	with	with	ADP
ejpam-4918	346	37	twisting	twist	VERB
ejpam-4918	346	38	f	f	X
ejpam-4918	346	39	:	:	PUNCT
ejpam-4918	346	40	m×m	m×m	ADJ
ejpam-4918	346	41	→	→	SYM
ejpam-4918	346	42	u(r	u(r	NOUN
ejpam-4918	346	43	)	)	PUNCT
ejpam-4918	346	44	and	and	CCONJ
ejpam-4918	347	1	action	action	NOUN
ejpam-4918	347	2	ω	ω	NOUN
ejpam-4918	347	3	:	:	PUNCT
ejpam-4918	347	4	m	m	PROPN
ejpam-4918	347	5	→	→	SYM
ejpam-4918	347	6	aut(r	aut(r	PROPN
ejpam-4918	347	7	)	)	PUNCT
ejpam-4918	347	8	,	,	PUNCT
ejpam-4918	347	9	then	then	ADV
ejpam-4918	347	10	r	r	NOUN
ejpam-4918	347	11	is	be	AUX
ejpam-4918	347	12	strongly	strongly	ADV
ejpam-4918	347	13	cm	cm	NOUN
ejpam-4918	347	14	-reflexive	-reflexive	NOUN
ejpam-4918	347	15	if	if	SCONJ
ejpam-4918	348	1	and	and	CCONJ
ejpam-4918	348	2	only	only	ADV
ejpam-4918	348	3	if	if	SCONJ
ejpam-4918	348	4	∆−1r	∆−1r	NOUN
ejpam-4918	348	5	is	be	AUX
ejpam-4918	348	6	strongly	strongly	ADV
ejpam-4918	348	7	cm	cm	NOUN
ejpam-4918	348	8	-reflexive	-reflexive	NOUN
ejpam-4918	348	9	,	,	PUNCT
ejpam-4918	348	10	where	where	SCONJ
ejpam-4918	348	11	∆	∆	PROPN
ejpam-4918	348	12	is	be	AUX
ejpam-4918	348	13	the	the	DET
ejpam-4918	348	14	multiplicative	multiplicative	ADJ
ejpam-4918	348	15	subset	subset	NOUN
ejpam-4918	348	16	of	of	ADP
ejpam-4918	348	17	r	r	NOUN
ejpam-4918	348	18	consisting	consist	VERB
ejpam-4918	348	19	of	of	ADP
ejpam-4918	348	20	all	all	DET
ejpam-4918	348	21	elements	element	NOUN
ejpam-4918	348	22	that	that	PRON
ejpam-4918	348	23	are	be	AUX
ejpam-4918	348	24	not	not	PART
ejpam-4918	349	1	zero	zero	NUM
ejpam-4918	349	2	divisors	divisor	NOUN
ejpam-4918	349	3	modulo	modulo	PART
ejpam-4918	349	4	m	m	VERB
ejpam-4918	349	5	.	.	PUNCT
ejpam-4918	350	1	proof	proof	NOUN
ejpam-4918	350	2	.	.	PUNCT
ejpam-4918	351	1	it	it	PRON
ejpam-4918	351	2	is	be	AUX
ejpam-4918	351	3	enough	enough	ADV
ejpam-4918	351	4	showing	show	VERB
ejpam-4918	351	5	necessary	necessary	ADJ
ejpam-4918	351	6	.	.	PUNCT
ejpam-4918	352	1	assume	assume	VERB
ejpam-4918	352	2	that	that	SCONJ
ejpam-4918	352	3	r	r	NOUN
ejpam-4918	352	4	is	be	AUX
ejpam-4918	352	5	strongly	strongly	ADV
ejpam-4918	352	6	cm	cm	NOUN
ejpam-4918	352	7	-reflexive	-reflexive	NOUN
ejpam-4918	352	8	.	.	PUNCT
ejpam-4918	353	1	let	let	VERB
ejpam-4918	353	2	ϕ	ϕ	NOUN
ejpam-4918	353	3	=	=	SYM
ejpam-4918	353	4	σni=1u	σni=1u	PROPN
ejpam-4918	353	5	−1cili	−1cili	NOUN
ejpam-4918	353	6	,	,	PUNCT
ejpam-4918	353	7	ψ	ψ	X
ejpam-4918	353	8	=	=	SYM
ejpam-4918	353	9	σmj=1v	σmj=1v	PROPN
ejpam-4918	353	10	−1ajhj	−1ajhj	PRON
ejpam-4918	353	11	be	be	AUX
ejpam-4918	353	12	elements	element	NOUN
ejpam-4918	353	13	in	in	ADP
ejpam-4918	353	14	∆−1r∗m	∆−1r∗m	PROPN
ejpam-4918	353	15	satisfying	satisfy	VERB
ejpam-4918	353	16	ϕφψ	ϕφψ	ADV
ejpam-4918	353	17	=	=	NOUN
ejpam-4918	353	18	0	0	PROPN
ejpam-4918	353	19	,	,	PUNCT
ejpam-4918	353	20	where	where	SCONJ
ejpam-4918	353	21	φ	φ	PROPN
ejpam-4918	353	22	=	=	X
ejpam-4918	354	1	σqk=1λ	σqk=1λ	PROPN
ejpam-4918	354	2	−1bkℓk	−1bkℓk	PRON
ejpam-4918	354	3	is	be	VERB
ejpam-4918	354	4	any	any	DET
ejpam-4918	354	5	nonzero	nonzero	ADJ
ejpam-4918	354	6	element	element	NOUN
ejpam-4918	354	7	in	in	ADP
ejpam-4918	354	8	∆−1r∗m	∆−1r∗m	PROPN
ejpam-4918	354	9	.	.	PUNCT
ejpam-4918	355	1	then	then	ADV
ejpam-4918	355	2	,	,	PUNCT
ejpam-4918	355	3	we	we	PRON
ejpam-4918	355	4	have	have	VERB
ejpam-4918	355	5	α	α	NOUN
ejpam-4918	355	6	=	=	SYM
ejpam-4918	355	7	(	(	PUNCT
ejpam-4918	355	8	unun−1	unun−1	PROPN
ejpam-4918	355	9	.	.	PUNCT
ejpam-4918	355	10	.	.	PUNCT
ejpam-4918	355	11	.	.	PUNCT
ejpam-4918	356	1	u1)ϕ	u1)ϕ	ADJ
ejpam-4918	356	2	,	,	PUNCT
ejpam-4918	356	3	θ	θ	X
ejpam-4918	356	4	=	=	PUNCT
ejpam-4918	356	5	(	(	PUNCT
ejpam-4918	356	6	λqλq−1	λqλq−1	PROPN
ejpam-4918	356	7	.	.	PUNCT
ejpam-4918	356	8	.	.	PUNCT
ejpam-4918	356	9	.	.	PUNCT
ejpam-4918	357	1	λ1)φ	λ1)φ	ADJ
ejpam-4918	357	2	,	,	PUNCT
ejpam-4918	357	3	β	β	X
ejpam-4918	357	4	=	=	PUNCT
ejpam-4918	357	5	(	(	PUNCT
ejpam-4918	357	6	vmvm−1	vmvm−1	PROPN
ejpam-4918	357	7	.	.	PUNCT
ejpam-4918	357	8	.	.	PUNCT
ejpam-4918	357	9	.	.	PUNCT
ejpam-4918	358	1	v1)ψ	v1)ψ	NOUN
ejpam-4918	358	2	are	be	AUX
ejpam-4918	358	3	in	in	ADP
ejpam-4918	358	4	r	r	NOUN
ejpam-4918	358	5	∗m	∗m	NOUN
ejpam-4918	358	6	.	.	PUNCT
ejpam-4918	359	1	since	since	SCONJ
ejpam-4918	359	2	r	r	NOUN
ejpam-4918	359	3	is	be	AUX
ejpam-4918	359	4	strongly	strongly	ADV
ejpam-4918	359	5	cm	cm	NOUN
ejpam-4918	359	6	-reflexive	-reflexive	NOUN
ejpam-4918	359	7	and	and	CCONJ
ejpam-4918	359	8	αθβ	αθβ	NOUN
ejpam-4918	360	1	=	=	SYM
ejpam-4918	360	2	0	0	NUM
ejpam-4918	360	3	we	we	PRON
ejpam-4918	360	4	have	have	VERB
ejpam-4918	360	5	(	(	PUNCT
ejpam-4918	360	6	unun−1	unun−1	ADJ
ejpam-4918	360	7	.	.	PUNCT
ejpam-4918	360	8	.	.	PUNCT
ejpam-4918	360	9	.	.	PUNCT
ejpam-4918	361	1	u1u	u1u	PROPN
ejpam-4918	361	2	−1	−1	NOUN
ejpam-4918	362	1	i	i	PROPN
ejpam-4918	362	2	ci)ωli(ωg(b(vmvm−1	ci)ωli(ωg(b(vmvm−1	PROPN
ejpam-4918	362	3	.	.	PUNCT
ejpam-4918	362	4	.	.	PUNCT
ejpam-4918	362	5	.	.	PUNCT
ejpam-4918	363	1	v1v	v1v	PRON
ejpam-4918	363	2	−1	−1	VERB
ejpam-4918	363	3	j	j	PROPN
ejpam-4918	363	4	)	)	PUNCT
ejpam-4918	363	5	aj))f(li	aj))f(li	PROPN
ejpam-4918	363	6	,	,	PUNCT
ejpam-4918	363	7	hj)(lihj)(vjui	hj)(lihj)(vjui	NUM
ejpam-4918	363	8	)	)	PUNCT
ejpam-4918	363	9	−1	−1	NOUN
ejpam-4918	364	1	=	=	NOUN
ejpam-4918	364	2	0	0	NUM
ejpam-4918	364	3	for	for	ADP
ejpam-4918	364	4	all	all	DET
ejpam-4918	364	5	i	i	PROPN
ejpam-4918	364	6	,	,	PUNCT
ejpam-4918	364	7	j	j	PROPN
ejpam-4918	364	8	and	and	CCONJ
ejpam-4918	364	9	b	b	PROPN
ejpam-4918	364	10	∈	∈	PROPN
ejpam-4918	364	11	r.	r.	PROPN
ejpam-4918	364	12	it	it	PRON
ejpam-4918	364	13	follows	follow	VERB
ejpam-4918	364	14	that	that	SCONJ
ejpam-4918	364	15	ciωli(ωg(raj))f(li	ciωli(ωg(raj))f(li	NOUN
ejpam-4918	364	16	,	,	PUNCT
ejpam-4918	364	17	hj)(lihj	hj)(lihj	PROPN
ejpam-4918	364	18	)	)	PUNCT
ejpam-4918	364	19	=	=	SYM
ejpam-4918	364	20	0	0	NUM
ejpam-4918	365	1	for	for	ADP
ejpam-4918	365	2	any	any	DET
ejpam-4918	365	3	g	g	PROPN
ejpam-4918	365	4	∈	∈	PROPN
ejpam-4918	365	5	m	m	NOUN
ejpam-4918	365	6	,	,	PUNCT
ejpam-4918	365	7	because	because	SCONJ
ejpam-4918	365	8	∆	∆	PROPN
ejpam-4918	365	9	is	be	AUX
ejpam-4918	365	10	a	a	DET
ejpam-4918	365	11	multiplicative	multiplicative	ADJ
ejpam-4918	365	12	monoid	monoid	NOUN
ejpam-4918	365	13	consisting	consisting	NOUN
ejpam-4918	365	14	of	of	ADP
ejpam-4918	365	15	central	central	ADJ
ejpam-4918	365	16	regular	regular	ADJ
ejpam-4918	365	17	elements	element	NOUN
ejpam-4918	365	18	of	of	ADP
ejpam-4918	365	19	r	r	NOUN
ejpam-4918	365	20	and	and	CCONJ
ejpam-4918	365	21	all	all	DET
ejpam-4918	365	22	ui	ui	PROPN
ejpam-4918	365	23	,	,	PUNCT
ejpam-4918	365	24	vj	vj	INTJ
ejpam-4918	365	25	,	,	PUNCT
ejpam-4918	365	26	λk	λk	PROPN
ejpam-4918	365	27	∈	∈	PROPN
ejpam-4918	366	1	∆.	∆.	X
ejpam-4918	366	2	hence	hence	ADV
ejpam-4918	366	3	,	,	PUNCT
ejpam-4918	366	4	(	(	PUNCT
ejpam-4918	366	5	u−1	u−1	PROPN
ejpam-4918	367	1	i	i	PRON
ejpam-4918	367	2	ci)ωli(ωg(rv	ci)ωli(ωg(rv	NUM
ejpam-4918	367	3	−1	−1	PROPN
ejpam-4918	367	4	j	j	PROPN
ejpam-4918	367	5	)	)	PUNCT
ejpam-4918	367	6	aj	aj	PROPN
ejpam-4918	367	7	)	)	PUNCT
ejpam-4918	367	8	)	)	PUNCT
ejpam-4918	368	1	=	=	SYM
ejpam-4918	368	2	ciωli(ωg(raj))(ωli(vj)ui	ciωli(ωg(raj))(ωli(vj)ui	NOUN
ejpam-4918	368	3	)	)	PUNCT
ejpam-4918	368	4	−1	−1	NOUN
ejpam-4918	369	1	=	=	NOUN
ejpam-4918	369	2	0	0	NUM
ejpam-4918	369	3	for	for	ADP
ejpam-4918	369	4	all	all	PRON
ejpam-4918	369	5	i	i	PROPN
ejpam-4918	369	6	,	,	PUNCT
ejpam-4918	369	7	j	j	PROPN
ejpam-4918	369	8	and	and	CCONJ
ejpam-4918	369	9	ω	ω	PROPN
ejpam-4918	369	10	is	be	AUX
ejpam-4918	369	11	automorphism	automorphism	NOUN
ejpam-4918	369	12	.	.	PUNCT
ejpam-4918	370	1	therefore	therefore	ADV
ejpam-4918	370	2	,	,	PUNCT
ejpam-4918	370	3	∆−1r	∆−1r	PROPN
ejpam-4918	370	4	is	be	AUX
ejpam-4918	370	5	strongly	strongly	ADV
ejpam-4918	370	6	cm	cm	NOUN
ejpam-4918	370	7	-reflexive	-reflexive	NOUN
ejpam-4918	370	8	.	.	PUNCT
ejpam-4918	371	1	the	the	DET
ejpam-4918	371	2	following	follow	VERB
ejpam-4918	371	3	statement	statement	NOUN
ejpam-4918	371	4	describes	describe	VERB
ejpam-4918	371	5	how	how	SCONJ
ejpam-4918	371	6	the	the	DET
ejpam-4918	371	7	strongly	strongly	ADV
ejpam-4918	371	8	cm	cm	NUM
ejpam-4918	371	9	-reflexive	-reflexive	ADJ
ejpam-4918	371	10	property	property	NOUN
ejpam-4918	371	11	of	of	ADP
ejpam-4918	371	12	a	a	DET
ejpam-4918	371	13	ring	ring	NOUN
ejpam-4918	371	14	r	r	NOUN
ejpam-4918	371	15	is	be	AUX
ejpam-4918	371	16	related	relate	VERB
ejpam-4918	371	17	to	to	ADP
ejpam-4918	371	18	the	the	DET
ejpam-4918	371	19	property	property	NOUN
ejpam-4918	371	20	of	of	ADP
ejpam-4918	371	21	its	its	PRON
ejpam-4918	371	22	subrings	subring	NOUN
ejpam-4918	371	23	,	,	PUNCT
ejpam-4918	371	24	which	which	PRON
ejpam-4918	371	25	are	be	AUX
ejpam-4918	371	26	created	create	VERB
ejpam-4918	371	27	by	by	ADP
ejpam-4918	371	28	a	a	DET
ejpam-4918	371	29	central	central	ADJ
ejpam-4918	371	30	idempotent	idempotent	NOUN
ejpam-4918	371	31	.	.	PUNCT
ejpam-4918	372	1	proposition	proposition	NOUN
ejpam-4918	372	2	4	4	NUM
ejpam-4918	372	3	.	.	PUNCT
ejpam-4918	373	1	the	the	DET
ejpam-4918	373	2	following	follow	VERB
ejpam-4918	373	3	conditions	condition	NOUN
ejpam-4918	373	4	are	be	AUX
ejpam-4918	373	5	equivalent	equivalent	ADJ
ejpam-4918	373	6	for	for	ADP
ejpam-4918	373	7	a	a	DET
ejpam-4918	373	8	ring	ring	NOUN
ejpam-4918	373	9	r	r	NOUN
ejpam-4918	373	10	,	,	PUNCT
ejpam-4918	373	11	a	a	DET
ejpam-4918	373	12	monoid	monoid	NOUN
ejpam-4918	373	13	m	m	VERB
ejpam-4918	373	14	with	with	ADP
ejpam-4918	373	15	twisting	twist	VERB
ejpam-4918	373	16	f	f	X
ejpam-4918	373	17	:	:	PUNCT
ejpam-4918	373	18	m	m	VERB
ejpam-4918	373	19	×m	×m	NOUN
ejpam-4918	373	20	→	→	SYM
ejpam-4918	373	21	u(r	u(r	NOUN
ejpam-4918	373	22	)	)	PUNCT
ejpam-4918	373	23	,	,	PUNCT
ejpam-4918	373	24	an	an	DET
ejpam-4918	373	25	action	action	NOUN
ejpam-4918	373	26	ω	ω	NOUN
ejpam-4918	373	27	:	:	PUNCT
ejpam-4918	373	28	m	m	PROPN
ejpam-4918	373	29	→	→	SYM
ejpam-4918	373	30	aut(r	aut(r	PROPN
ejpam-4918	373	31	)	)	PUNCT
ejpam-4918	373	32	,	,	PUNCT
ejpam-4918	373	33	and	and	CCONJ
ejpam-4918	373	34	a	a	DET
ejpam-4918	373	35	central	central	ADJ
ejpam-4918	373	36	idempotent	idempotent	ADJ
ejpam-4918	373	37	e	e	NOUN
ejpam-4918	373	38	of	of	ADP
ejpam-4918	373	39	r	r	NOUN
ejpam-4918	373	40	such	such	ADJ
ejpam-4918	373	41	that	that	DET
ejpam-4918	373	42	ωg(e	ωg(e	NUM
ejpam-4918	373	43	)	)	PUNCT
ejpam-4918	374	1	=	=	SYM
ejpam-4918	374	2	e	e	X
ejpam-4918	374	3	:	:	PUNCT
ejpam-4918	374	4	(	(	PUNCT
ejpam-4918	374	5	1	1	X
ejpam-4918	374	6	)	)	PUNCT
ejpam-4918	374	7	r	r	NOUN
ejpam-4918	374	8	is	be	AUX
ejpam-4918	374	9	strongly	strongly	ADV
ejpam-4918	374	10	cm	cm	NOUN
ejpam-4918	374	11	-reflexive	-reflexive	NOUN
ejpam-4918	374	12	.	.	PUNCT
ejpam-4918	375	1	(	(	PUNCT
ejpam-4918	375	2	2	2	X
ejpam-4918	375	3	)	)	PUNCT
ejpam-4918	375	4	er	er	INTJ
ejpam-4918	375	5	and	and	CCONJ
ejpam-4918	375	6	(	(	PUNCT
ejpam-4918	375	7	1−	1−	NUM
ejpam-4918	375	8	e)r	e)r	ADV
ejpam-4918	375	9	are	be	AUX
ejpam-4918	375	10	strongly	strongly	ADV
ejpam-4918	375	11	cm	cm	NOUN
ejpam-4918	375	12	-reflexive	-reflexive	NOUN
ejpam-4918	375	13	.	.	PUNCT
ejpam-4918	376	1	proof	proof	NOUN
ejpam-4918	376	2	.	.	PUNCT
ejpam-4918	377	1	(	(	PUNCT
ejpam-4918	377	2	1	1	X
ejpam-4918	377	3	)	)	PUNCT
ejpam-4918	377	4	⇒	⇒	NOUN
ejpam-4918	377	5	(	(	PUNCT
ejpam-4918	377	6	2	2	NUM
ejpam-4918	377	7	)	)	PUNCT
ejpam-4918	377	8	.	.	PUNCT
ejpam-4918	378	1	it	it	PRON
ejpam-4918	378	2	is	be	AUX
ejpam-4918	378	3	easy	easy	ADJ
ejpam-4918	378	4	.	.	PUNCT
ejpam-4918	379	1	(	(	PUNCT
ejpam-4918	379	2	2	2	X
ejpam-4918	379	3	)	)	PUNCT
ejpam-4918	379	4	⇒	⇒	NOUN
ejpam-4918	379	5	(	(	PUNCT
ejpam-4918	379	6	1	1	NUM
ejpam-4918	379	7	)	)	PUNCT
ejpam-4918	379	8	.	.	PUNCT
ejpam-4918	380	1	assume	assume	VERB
ejpam-4918	380	2	that	that	SCONJ
ejpam-4918	380	3	both	both	DET
ejpam-4918	380	4	er	er	INTJ
ejpam-4918	380	5	and	and	CCONJ
ejpam-4918	380	6	(	(	PUNCT
ejpam-4918	380	7	1	1	NUM
ejpam-4918	380	8	−	−	NOUN
ejpam-4918	380	9	e)r	e)r	ADV
ejpam-4918	380	10	are	be	AUX
ejpam-4918	380	11	strongly	strongly	ADV
ejpam-4918	380	12	cm	cm	NOUN
ejpam-4918	380	13	-reflexive	-reflexive	NOUN
ejpam-4918	380	14	.	.	PUNCT
ejpam-4918	381	1	let	let	VERB
ejpam-4918	381	2	ϕ	ϕ	NOUN
ejpam-4918	381	3	=	=	PROPN
ejpam-4918	381	4	σni=1cili	σni=1cili	PROPN
ejpam-4918	381	5	,	,	PUNCT
ejpam-4918	381	6	ψ	ψ	X
ejpam-4918	381	7	=	=	SYM
ejpam-4918	381	8	σmj=1ajhj	σmj=1ajhj	NOUN
ejpam-4918	381	9	∈	∈	NOUN
ejpam-4918	381	10	r	r	NOUN
ejpam-4918	381	11	∗m	∗m	NOUN
ejpam-4918	381	12	satisfying	satisfy	VERB
ejpam-4918	381	13	ϕ(r	ϕ(r	PROPN
ejpam-4918	381	14	∗m)ψ	∗m)ψ	ADJ
ejpam-4918	381	15	=	=	PUNCT
ejpam-4918	381	16	0	0	X
ejpam-4918	381	17	.	.	PUNCT
ejpam-4918	382	1	let	let	VERB
ejpam-4918	382	2	ϕ1	ϕ1	NOUN
ejpam-4918	382	3	=	=	SYM
ejpam-4918	382	4	σni=1e	σni=1e	X
ejpam-4918	382	5	cili	cili	NOUN
ejpam-4918	382	6	,	,	PUNCT
ejpam-4918	382	7	ψ1	ψ1	NOUN
ejpam-4918	382	8	=	=	SYM
ejpam-4918	382	9	σmj=1e	σmj=1e	NOUN
ejpam-4918	382	10	ajhj	ajhj	NOUN
ejpam-4918	382	11	,	,	PUNCT
ejpam-4918	382	12	ϕ2	ϕ2	ADV
ejpam-4918	382	13	=	=	SYM
ejpam-4918	382	14	σni=1(1−	σni=1(1−	PROPN
ejpam-4918	382	15	e)cili	e)cili	NOUN
ejpam-4918	382	16	,	,	PUNCT
ejpam-4918	382	17	ψ2	ψ2	NOUN
ejpam-4918	382	18	=	=	SYM
ejpam-4918	382	19	σmj=1(1−	σmj=1(1−	NOUN
ejpam-4918	382	20	e)ajhj	e)ajhj	NOUN
ejpam-4918	382	21	.	.	PUNCT
ejpam-4918	383	1	clear	clear	ADJ
ejpam-4918	383	2	that	that	SCONJ
ejpam-4918	383	3	ϕ1	ϕ1	NOUN
ejpam-4918	383	4	,	,	PUNCT
ejpam-4918	383	5	ψ1	ψ1	ADJ
ejpam-4918	383	6	∈	∈	PROPN
ejpam-4918	383	7	(	(	PUNCT
ejpam-4918	383	8	er	er	INTJ
ejpam-4918	383	9	)	)	PUNCT
ejpam-4918	383	10	∗m	∗m	NOUN
ejpam-4918	383	11	and	and	CCONJ
ejpam-4918	383	12	ϕ2	ϕ2	ADV
ejpam-4918	383	13	,	,	PUNCT
ejpam-4918	383	14	ψ2	ψ2	NOUN
ejpam-4918	383	15	∈	∈	PROPN
ejpam-4918	383	16	(	(	PUNCT
ejpam-4918	383	17	(	(	PUNCT
ejpam-4918	383	18	1−	1−	NUM
ejpam-4918	383	19	e)r	e)r	ADV
ejpam-4918	383	20	)	)	PUNCT
ejpam-4918	383	21	∗m	∗m	NOUN
ejpam-4918	383	22	.	.	PUNCT
ejpam-4918	384	1	since	since	SCONJ
ejpam-4918	384	2	e	e	PROPN
ejpam-4918	384	3	is	be	AUX
ejpam-4918	384	4	a	a	DET
ejpam-4918	384	5	central	central	ADJ
ejpam-4918	384	6	idempotent	idempotent	NOUN
ejpam-4918	384	7	of	of	ADP
ejpam-4918	384	8	r	r	NOUN
ejpam-4918	384	9	such	such	ADJ
ejpam-4918	384	10	that	that	DET
ejpam-4918	384	11	ωg(e	ωg(e	NUM
ejpam-4918	384	12	)	)	PUNCT
ejpam-4918	384	13	=	=	PUNCT
ejpam-4918	384	14	e	e	NOUN
ejpam-4918	384	15	for	for	ADP
ejpam-4918	384	16	each	each	DET
ejpam-4918	384	17	g	g	NOUN
ejpam-4918	384	18	∈m	∈m	NOUN
ejpam-4918	384	19	and	and	CCONJ
ejpam-4918	384	20	for	for	ADP
ejpam-4918	384	21	any	any	DET
ejpam-4918	384	22	r	r	NOUN
ejpam-4918	384	23	∈	∈	NOUN
ejpam-4918	384	24	r	r	NOUN
ejpam-4918	384	25	we	we	PRON
ejpam-4918	384	26	have	have	VERB
ejpam-4918	384	27	ϕ1((er	ϕ1((er	NOUN
ejpam-4918	384	28	)	)	PUNCT
ejpam-4918	384	29	∗m)ψ1	∗m)ψ1	NOUN
ejpam-4918	385	1	=	=	SYM
ejpam-4918	385	2	ec1(er)ωl1(ωg(ea1))f(l1	ec1(er)ωl1(ωg(ea1))f(l1	NOUN
ejpam-4918	385	3	,	,	PUNCT
ejpam-4918	385	4	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	385	5	+	+	PRON
ejpam-4918	385	6	·	·	PUNCT
ejpam-4918	385	7	·	·	PUNCT
ejpam-4918	385	8	·	·	PUNCT
ejpam-4918	386	1	+	+	NUM
ejpam-4918	386	2	ecn(er)ωln(ωg(eam))f(ln	ecn(er)ωln(ωg(eam))f(ln	NOUN
ejpam-4918	386	3	,	,	PUNCT
ejpam-4918	386	4	hm)lnhm	hm)lnhm	NOUN
ejpam-4918	386	5	=	=	SYM
ejpam-4918	386	6	ec1(er)ωl1(ωg(e)ωl1(ωg(a1))f(l1	ec1(er)ωl1(ωg(e)ωl1(ωg(a1))f(l1	PROPN
ejpam-4918	386	7	,	,	PUNCT
ejpam-4918	386	8	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	386	9	+	+	CCONJ
ejpam-4918	386	10	·	·	PUNCT
ejpam-4918	386	11	·	·	PUNCT
ejpam-4918	386	12	·	·	PUNCT
ejpam-4918	387	1	+	+	PUNCT
ejpam-4918	387	2	ecn(er)ωln(ωg(e))ωln(ωg(am))f(ln	ecn(er)ωln(ωg(e))ωln(ωg(am))f(ln	NOUN
ejpam-4918	387	3	,	,	PUNCT
ejpam-4918	387	4	hm)lnhm	hm)lnhm	NOUN
ejpam-4918	387	5	=	=	SYM
ejpam-4918	387	6	ec1e(er)ωl1(a1)f(l1	ec1e(er)ωl1(a1)f(l1	PROPN
ejpam-4918	387	7	,	,	PUNCT
ejpam-4918	387	8	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	387	9	+	+	CCONJ
ejpam-4918	387	10	·	·	PUNCT
ejpam-4918	387	11	·	·	PUNCT
ejpam-4918	387	12	·	·	PUNCT
ejpam-4918	388	1	+	+	PUNCT
ejpam-4918	388	2	ecne(er)ωln(am)f(ln	ecne(er)ωln(am)f(ln	NUM
ejpam-4918	388	3	,	,	PUNCT
ejpam-4918	388	4	hm)lnhm	hm)lnhm	VERB
ejpam-4918	388	5	=	=	SYM
ejpam-4918	388	6	ec1e	ec1e	PROPN
ejpam-4918	388	7	2(r)ωl1(a1)f(l1	2(r)ωl1(a1)f(l1	PROPN
ejpam-4918	388	8	,	,	PUNCT
ejpam-4918	388	9	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	388	10	+	+	CCONJ
ejpam-4918	388	11	·	·	PUNCT
ejpam-4918	388	12	·	·	PUNCT
ejpam-4918	388	13	·	·	PUNCT
ejpam-4918	389	1	+	+	PUNCT
ejpam-4918	389	2	e2cn(r)ωln(am)f(ln	e2cn(r)ωln(am)f(ln	NOUN
ejpam-4918	389	3	,	,	PUNCT
ejpam-4918	389	4	hm)lnhm	hm)lnhm	NOUN
ejpam-4918	389	5	=	=	SYM
ejpam-4918	389	6	ec1e(r)ωl1(a1)f(l1	ec1e(r)ωl1(a1)f(l1	PROPN
ejpam-4918	389	7	,	,	PUNCT
ejpam-4918	389	8	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	389	9	+	+	CCONJ
ejpam-4918	389	10	·	·	PUNCT
ejpam-4918	389	11	·	·	PUNCT
ejpam-4918	389	12	·	·	PUNCT
ejpam-4918	389	13	+	+	NUM
ejpam-4918	389	14	ecn(r)ωln(am)f(ln	ecn(r)ωln(am)f(ln	NOUN
ejpam-4918	389	15	,	,	PUNCT
ejpam-4918	389	16	hm)lnhm	hm)lnhm	NOUN
ejpam-4918	389	17	=	=	SYM
ejpam-4918	389	18	e2c1rωl1(a1)f(l1	e2c1rωl1(a1)f(l1	PROPN
ejpam-4918	389	19	,	,	PUNCT
ejpam-4918	389	20	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	389	21	+	+	CCONJ
ejpam-4918	389	22	·	·	PUNCT
ejpam-4918	389	23	·	·	PUNCT
ejpam-4918	389	24	·	·	PUNCT
ejpam-4918	389	25	+	+	NUM
ejpam-4918	389	26	e2cnrωln(am)f(ln	e2cnrωln(am)f(ln	NOUN
ejpam-4918	389	27	,	,	PUNCT
ejpam-4918	389	28	hm)lnhm	hm)lnhm	NOUN
ejpam-4918	389	29	=	=	PUNCT
ejpam-4918	389	30	ec1rωl1(a1)f(l1	ec1rωl1(a1)f(l1	NOUN
ejpam-4918	389	31	,	,	PUNCT
ejpam-4918	389	32	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	389	33	+	+	CCONJ
ejpam-4918	389	34	·	·	PUNCT
ejpam-4918	389	35	·	·	PUNCT
ejpam-4918	389	36	·	·	PUNCT
ejpam-4918	389	37	+	+	NUM
ejpam-4918	389	38	ecnrωln(am)f(ln	ecnrωln(am)f(ln	NOUN
ejpam-4918	389	39	,	,	PUNCT
ejpam-4918	389	40	hm)lnhm	hm)lnhm	NOUN
ejpam-4918	389	41	=	=	SYM
ejpam-4918	389	42	e[c1rωl1(a1)f(l1	e[c1rωl1(a1)f(l1	PROPN
ejpam-4918	389	43	,	,	PUNCT
ejpam-4918	389	44	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	389	45	+	+	CCONJ
ejpam-4918	389	46	·	·	PUNCT
ejpam-4918	389	47	·	·	PUNCT
ejpam-4918	389	48	·	·	PUNCT
ejpam-4918	389	49	+	+	NUM
ejpam-4918	389	50	cnrωln(am)f(ln	cnrωln(am)f(ln	NOUN
ejpam-4918	389	51	,	,	PUNCT
ejpam-4918	389	52	hm)lnhm	hm)lnhm	NOUN
ejpam-4918	389	53	]	]	X
ejpam-4918	389	54	=	=	SYM
ejpam-4918	389	55	eϕ(r	eϕ(r	X
ejpam-4918	389	56	∗m)ψ	∗m)ψ	X
ejpam-4918	389	57	=	=	SYM
ejpam-4918	389	58	0	0	NUM
ejpam-4918	389	59	,	,	PUNCT
ejpam-4918	389	60	e.	e.	PROPN
ejpam-4918	389	61	ali	ali	PROPN
ejpam-4918	389	62	/	/	SYM
ejpam-4918	389	63	eur	eur	PROPN
ejpam-4918	389	64	.	.	PUNCT
ejpam-4918	390	1	j.	j.	PROPN
ejpam-4918	390	2	pure	pure	PROPN
ejpam-4918	390	3	appl	appl	PROPN
ejpam-4918	390	4	.	.	PROPN
ejpam-4918	390	5	math	math	PROPN
ejpam-4918	390	6	,	,	PUNCT
ejpam-4918	390	7	16	16	NUM
ejpam-4918	390	8	(	(	PUNCT
ejpam-4918	390	9	4	4	NUM
ejpam-4918	390	10	)	)	PUNCT
ejpam-4918	390	11	(	(	PUNCT
ejpam-4918	390	12	2023	2023	NUM
ejpam-4918	390	13	)	)	PUNCT
ejpam-4918	390	14	,	,	PUNCT
ejpam-4918	390	15	2156	2156	NUM
ejpam-4918	390	16	-	-	SYM
ejpam-4918	390	17	2168	2168	NUM
ejpam-4918	390	18	2165	2165	NUM
ejpam-4918	390	19	ϕ2((1−	ϕ2((1−	X
ejpam-4918	390	20	e)r	e)r	X
ejpam-4918	390	21	∗m)ψ2	∗m)ψ2	PROPN
ejpam-4918	390	22	=	=	SYM
ejpam-4918	390	23	(	(	PUNCT
ejpam-4918	390	24	1−	1−	NUM
ejpam-4918	390	25	e)c1((1−	e)c1((1−	ADJ
ejpam-4918	390	26	e)r)ωl1(ωg((1−	e)r)ωl1(ωg((1−	PROPN
ejpam-4918	390	27	e)a1))f(l1	e)a1))f(l1	NOUN
ejpam-4918	390	28	,	,	PUNCT
ejpam-4918	390	29	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	390	30	+	+	PRON
ejpam-4918	390	31	·	·	PUNCT
ejpam-4918	390	32	·	·	PUNCT
ejpam-4918	390	33	·	·	PUNCT
ejpam-4918	391	1	+	+	CCONJ
ejpam-4918	391	2	(	(	PUNCT
ejpam-4918	391	3	1−	1−	NUM
ejpam-4918	391	4	e)cn((1−	e)cn((1−	NOUN
ejpam-4918	391	5	e)r)ωln(ωg((1−	e)r)ωln(ωg((1−	PROPN
ejpam-4918	391	6	e)(1−	e)(1−	ADJ
ejpam-4918	391	7	e)am))f(ln	e)am))f(ln	NOUN
ejpam-4918	391	8	,	,	PUNCT
ejpam-4918	391	9	hm)lnhm	hm)lnhm	NOUN
ejpam-4918	391	10	=	=	SYM
ejpam-4918	391	11	(	(	PUNCT
ejpam-4918	391	12	1−	1−	NUM
ejpam-4918	391	13	e)c1((1−	e)c1((1−	ADJ
ejpam-4918	391	14	e)r)ωl1((1−	e)r)ωl1((1−	PROPN
ejpam-4918	391	15	e)a1)f(l1	e)a1)f(l1	ADV
ejpam-4918	391	16	,	,	PUNCT
ejpam-4918	391	17	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	391	18	+	+	CCONJ
ejpam-4918	391	19	·	·	PUNCT
ejpam-4918	391	20	·	·	PUNCT
ejpam-4918	391	21	·	·	PUNCT
ejpam-4918	391	22	+	+	PUNCT
ejpam-4918	391	23	(	(	PUNCT
ejpam-4918	391	24	1−	1−	NUM
ejpam-4918	391	25	e)cn(1−	e)cn(1−	PROPN
ejpam-4918	391	26	e)rωln((1−	e)rωln((1−	PROPN
ejpam-4918	391	27	e)am)f(ln	e)am)f(ln	PROPN
ejpam-4918	391	28	,	,	PUNCT
ejpam-4918	391	29	hm)lnhm	hm)lnhm	NOUN
ejpam-4918	391	30	=	=	SYM
ejpam-4918	391	31	(	(	PUNCT
ejpam-4918	391	32	1−	1−	NUM
ejpam-4918	391	33	e)[c1rωl1(a1)f(l1	e)[c1rωl1(a1)f(l1	PROPN
ejpam-4918	391	34	,	,	PUNCT
ejpam-4918	391	35	h1)l1h1	h1)l1h1	PROPN
ejpam-4918	391	36	+	+	CCONJ
ejpam-4918	391	37	·	·	PUNCT
ejpam-4918	391	38	·	·	PUNCT
ejpam-4918	391	39	·	·	PUNCT
ejpam-4918	391	40	+	+	NUM
ejpam-4918	391	41	cnrωln(am)f(ln	cnrωln(am)f(ln	NOUN
ejpam-4918	391	42	,	,	PUNCT
ejpam-4918	391	43	hm)lnhm	hm)lnhm	PROPN
ejpam-4918	391	44	]	]	X
ejpam-4918	391	45	=	=	SYM
ejpam-4918	391	46	(	(	PUNCT
ejpam-4918	391	47	1−	1−	NUM
ejpam-4918	391	48	e)ϕ(r	e)ϕ(r	NOUN
ejpam-4918	391	49	∗m)ψ	∗m)ψ	ADV
ejpam-4918	391	50	=	=	PUNCT
ejpam-4918	392	1	0	0	X
ejpam-4918	392	2	.	.	PUNCT
ejpam-4918	393	1	because	because	SCONJ
ejpam-4918	393	2	er	er	INTJ
ejpam-4918	393	3	and	and	CCONJ
ejpam-4918	393	4	(	(	PUNCT
ejpam-4918	393	5	1	1	NUM
ejpam-4918	393	6	−	−	NOUN
ejpam-4918	393	7	e)r	e)r	ADV
ejpam-4918	393	8	are	be	AUX
ejpam-4918	393	9	strongly	strongly	ADV
ejpam-4918	393	10	cm	cm	NUM
ejpam-4918	393	11	-reflexive	-reflexive	ADJ
ejpam-4918	393	12	subrings	subring	NOUN
ejpam-4918	393	13	of	of	ADP
ejpam-4918	393	14	r	r	NOUN
ejpam-4918	393	15	,	,	PUNCT
ejpam-4918	393	16	we	we	PRON
ejpam-4918	393	17	conclude	conclude	VERB
ejpam-4918	393	18	that	that	SCONJ
ejpam-4918	393	19	ψ1((er	ψ1((er	NOUN
ejpam-4918	393	20	)	)	PUNCT
ejpam-4918	393	21	∗m)ϕ1	∗m)ϕ1	NOUN
ejpam-4918	394	1	=	=	SYM
ejpam-4918	394	2	0	0	NUM
ejpam-4918	394	3	,	,	PUNCT
ejpam-4918	394	4	ψ2(((1−	ψ2(((1−	X
ejpam-4918	394	5	e)r	e)r	ADV
ejpam-4918	394	6	)	)	PUNCT
ejpam-4918	394	7	∗m)ϕ2	∗m)ϕ2	PUNCT
ejpam-4918	395	1	=	=	SYM
ejpam-4918	395	2	0	0	X
ejpam-4918	395	3	.	.	PUNCT
ejpam-4918	396	1	therefore	therefore	ADV
ejpam-4918	396	2	,	,	PUNCT
ejpam-4918	396	3	we	we	PRON
ejpam-4918	396	4	have	have	VERB
ejpam-4918	396	5	ψ(r	ψ(r	NOUN
ejpam-4918	396	6	∗m)ϕ	∗m)ϕ	NOUN
ejpam-4918	396	7	=	=	SYM
ejpam-4918	396	8	ψ1((er	ψ1((er	PROPN
ejpam-4918	396	9	)	)	PUNCT
ejpam-4918	396	10	∗m)ϕ1	∗m)ϕ1	NOUN
ejpam-4918	397	1	+	+	CCONJ
ejpam-4918	397	2	ψ2(((1−	ψ2(((1−	ADJ
ejpam-4918	397	3	e)r	e)r	ADV
ejpam-4918	397	4	)	)	PUNCT
ejpam-4918	397	5	∗m)ϕ2	∗m)ϕ2	ADP
ejpam-4918	397	6	=	=	SYM
ejpam-4918	397	7	eψ(r	eψ(r	NUM
ejpam-4918	397	8	∗m)ϕ+	∗m)ϕ+	NOUN
ejpam-4918	397	9	(	(	PUNCT
ejpam-4918	397	10	1−	1−	NUM
ejpam-4918	397	11	e)ψ(r	e)ψ(r	NOUN
ejpam-4918	397	12	∗m)ϕ	∗m)ϕ	NOUN
ejpam-4918	397	13	=	=	SYM
ejpam-4918	397	14	0	0	X
ejpam-4918	397	15	.	.	PUNCT
ejpam-4918	398	1	therefore	therefore	ADV
ejpam-4918	398	2	,	,	PUNCT
ejpam-4918	398	3	r	r	NOUN
ejpam-4918	398	4	is	be	AUX
ejpam-4918	398	5	strongly	strongly	ADV
ejpam-4918	398	6	cm	cm	NOUN
ejpam-4918	398	7	-reflexive	-reflexive	NOUN
ejpam-4918	398	8	,	,	PUNCT
ejpam-4918	398	9	which	which	PRON
ejpam-4918	398	10	concludes	conclude	VERB
ejpam-4918	398	11	the	the	DET
ejpam-4918	398	12	proof	proof	NOUN
ejpam-4918	398	13	.	.	PUNCT
ejpam-4918	399	1	proposition	proposition	NOUN
ejpam-4918	399	2	5	5	NUM
ejpam-4918	399	3	.	.	PUNCT
ejpam-4918	400	1	let	let	VERB
ejpam-4918	400	2	r	r	PRON
ejpam-4918	400	3	be	be	AUX
ejpam-4918	400	4	a	a	DET
ejpam-4918	400	5	ring	ring	NOUN
ejpam-4918	400	6	and	and	CCONJ
ejpam-4918	400	7	m	m	NOUN
ejpam-4918	400	8	is	be	AUX
ejpam-4918	400	9	a	a	DET
ejpam-4918	400	10	strictly	strictly	ADV
ejpam-4918	400	11	ordered	order	VERB
ejpam-4918	400	12	monoid	monoid	NOUN
ejpam-4918	400	13	with	with	ADP
ejpam-4918	400	14	a	a	DET
ejpam-4918	400	15	twisting	twisting	NOUN
ejpam-4918	400	16	f	f	NOUN
ejpam-4918	400	17	:	:	PUNCT
ejpam-4918	400	18	m×m	m×m	ADJ
ejpam-4918	400	19	→	→	SYM
ejpam-4918	400	20	u(r	u(r	NOUN
ejpam-4918	400	21	)	)	PUNCT
ejpam-4918	400	22	and	and	CCONJ
ejpam-4918	400	23	an	an	DET
ejpam-4918	400	24	action	action	NOUN
ejpam-4918	400	25	ω	ω	NOUN
ejpam-4918	400	26	:	:	PUNCT
ejpam-4918	400	27	m	m	PROPN
ejpam-4918	400	28	→	→	SYM
ejpam-4918	400	29	aut(r	aut(r	PROPN
ejpam-4918	400	30	)	)	PUNCT
ejpam-4918	400	31	.	.	PUNCT
ejpam-4918	401	1	assume	assume	VERB
ejpam-4918	401	2	that	that	SCONJ
ejpam-4918	401	3	r	r	NOUN
ejpam-4918	401	4	is	be	AUX
ejpam-4918	401	5	cm	cm	NUM
ejpam-4918	401	6	-quasi	-quasi	NOUN
ejpam-4918	401	7	-	-	PUNCT
ejpam-4918	401	8	armendariz	armendariz	ADV
ejpam-4918	401	9	.	.	PUNCT
ejpam-4918	402	1	let	let	VERB
ejpam-4918	402	2	e	e	PRON
ejpam-4918	402	3	be	be	AUX
ejpam-4918	402	4	a	a	DET
ejpam-4918	402	5	nonzero	nonzero	NOUN
ejpam-4918	402	6	idempotent	idempotent	NOUN
ejpam-4918	402	7	in	in	ADP
ejpam-4918	402	8	r	r	NOUN
ejpam-4918	402	9	such	such	ADJ
ejpam-4918	402	10	that	that	DET
ejpam-4918	402	11	ωg(e	ωg(e	NUM
ejpam-4918	402	12	)	)	PUNCT
ejpam-4918	403	1	=	=	PUNCT
ejpam-4918	403	2	e	e	X
ejpam-4918	403	3	for	for	ADP
ejpam-4918	403	4	all	all	DET
ejpam-4918	403	5	g	g	PROPN
ejpam-4918	403	6	∈	∈	NOUN
ejpam-4918	403	7	m	m	VERB
ejpam-4918	403	8	.	.	PUNCT
ejpam-4918	404	1	then	then	ADV
ejpam-4918	404	2	,	,	PUNCT
ejpam-4918	404	3	the	the	DET
ejpam-4918	404	4	subring	subring	NOUN
ejpam-4918	404	5	ere	ere	NOUN
ejpam-4918	404	6	is	be	AUX
ejpam-4918	404	7	strongly	strongly	ADV
ejpam-4918	404	8	cm	cm	NOUN
ejpam-4918	404	9	-reflexive	-reflexive	NOUN
ejpam-4918	404	10	.	.	PUNCT
ejpam-4918	405	1	proof	proof	NOUN
ejpam-4918	405	2	.	.	PUNCT
ejpam-4918	406	1	the	the	DET
ejpam-4918	406	2	proof	proof	NOUN
ejpam-4918	406	3	is	be	AUX
ejpam-4918	406	4	a	a	DET
ejpam-4918	406	5	variant	variant	NOUN
ejpam-4918	406	6	of	of	ADP
ejpam-4918	406	7	the	the	DET
ejpam-4918	406	8	proof	proof	NOUN
ejpam-4918	406	9	given	give	VERB
ejpam-4918	406	10	in	in	ADP
ejpam-4918	406	11	proposition	proposition	NOUN
ejpam-4918	406	12	2.9	2.9	NUM
ejpam-4918	406	13	[	[	X
ejpam-4918	406	14	17	17	NUM
ejpam-4918	406	15	]	]	PUNCT
ejpam-4918	406	16	.	.	PUNCT
ejpam-4918	407	1	let	let	VERB
ejpam-4918	407	2	ϕ	ϕ	NOUN
ejpam-4918	407	3	=	=	PUNCT
ejpam-4918	407	4	c1l1	c1l1	X
ejpam-4918	408	1	+	+	CCONJ
ejpam-4918	408	2	c2l2	c2l2	X
ejpam-4918	408	3	+	+	NUM
ejpam-4918	408	4	·	·	PUNCT
ejpam-4918	408	5	·	·	PUNCT
ejpam-4918	408	6	·	·	PUNCT
ejpam-4918	408	7	+	+	NUM
ejpam-4918	408	8	cnln	cnln	NOUN
ejpam-4918	408	9	and	and	CCONJ
ejpam-4918	408	10	ψ	ψ	X
ejpam-4918	408	11	=	=	X
ejpam-4918	408	12	a1h1	a1h1	X
ejpam-4918	408	13	+	+	NOUN
ejpam-4918	408	14	a2h2	a2h2	X
ejpam-4918	408	15	+	+	X
ejpam-4918	408	16	·	·	PUNCT
ejpam-4918	408	17	·	·	PUNCT
ejpam-4918	408	18	·	·	PUNCT
ejpam-4918	408	19	+	+	NUM
ejpam-4918	408	20	amhm	amhm	NOUN
ejpam-4918	408	21	∈	∈	PROPN
ejpam-4918	408	22	(	(	PUNCT
ejpam-4918	408	23	ere	ere	NOUN
ejpam-4918	408	24	)	)	PUNCT
ejpam-4918	408	25	∗m	∗m	NOUN
ejpam-4918	408	26	satisfy	satisfy	VERB
ejpam-4918	408	27	ϕ((ere	ϕ((ere	ADV
ejpam-4918	408	28	)	)	PUNCT
ejpam-4918	408	29	∗	∗	NOUN
ejpam-4918	408	30	m)ψ	m)ψ	X
ejpam-4918	409	1	=	=	NOUN
ejpam-4918	409	2	0	0	X
ejpam-4918	409	3	.	.	PUNCT
ejpam-4918	410	1	since	since	SCONJ
ejpam-4918	410	2	m	m	PROPN
ejpam-4918	410	3	is	be	AUX
ejpam-4918	410	4	a	a	DET
ejpam-4918	410	5	strictly	strictly	ADV
ejpam-4918	410	6	totally	totally	ADV
ejpam-4918	410	7	ordered	order	VERB
ejpam-4918	410	8	monoid	monoid	NOUN
ejpam-4918	410	9	,	,	PUNCT
ejpam-4918	410	10	we	we	PRON
ejpam-4918	410	11	can	can	AUX
ejpam-4918	410	12	assume	assume	VERB
ejpam-4918	410	13	that	that	SCONJ
ejpam-4918	410	14	li	li	PROPN
ejpam-4918	410	15	⪯	⪯	PROPN
ejpam-4918	410	16	lj	lj	PROPN
ejpam-4918	410	17	and	and	CCONJ
ejpam-4918	410	18	hi	hi	INTJ
ejpam-4918	410	19	⪯	⪯	NOUN
ejpam-4918	410	20	hj	hj	INTJ
ejpam-4918	410	21	whenever	whenever	SCONJ
ejpam-4918	410	22	i	i	PRON
ejpam-4918	410	23	<	<	X
ejpam-4918	410	24	j.	j.	PROPN
ejpam-4918	410	25	since	since	SCONJ
ejpam-4918	410	26	r	r	NOUN
ejpam-4918	410	27	is	be	AUX
ejpam-4918	410	28	cm	cm	NUM
ejpam-4918	410	29	-quasi	-quasi	NOUN
ejpam-4918	410	30	-	-	PUNCT
ejpam-4918	410	31	armendariz	armendariz	ADJ
ejpam-4918	410	32	,	,	PUNCT
ejpam-4918	410	33	then	then	ADV
ejpam-4918	410	34	so	so	ADV
ejpam-4918	410	35	is	be	AUX
ejpam-4918	410	36	ere	ere	PROPN
ejpam-4918	410	37	.	.	PUNCT
ejpam-4918	411	1	thus	thus	ADV
ejpam-4918	411	2	,	,	PUNCT
ejpam-4918	411	3	we	we	PRON
ejpam-4918	411	4	have	have	VERB
ejpam-4918	411	5	ciωli(ωg((ere)aj))f(li	ciωli(ωg((ere)aj))f(li	NUM
ejpam-4918	411	6	,	,	PUNCT
ejpam-4918	411	7	hj)(lihj	hj)(lihj	PROPN
ejpam-4918	411	8	)	)	PUNCT
ejpam-4918	411	9	=	=	SYM
ejpam-4918	411	10	0	0	NUM
ejpam-4918	411	11	for	for	ADP
ejpam-4918	411	12	all	all	DET
ejpam-4918	411	13	i	i	PROPN
ejpam-4918	411	14	,	,	PUNCT
ejpam-4918	411	15	j.	j.	PROPN
ejpam-4918	411	16	this	this	PRON
ejpam-4918	411	17	implies	imply	VERB
ejpam-4918	411	18	that	that	SCONJ
ejpam-4918	411	19	ciωli(ωg((ere)aj	ciωli(ωg((ere)aj	NUM
ejpam-4918	411	20	)	)	PUNCT
ejpam-4918	411	21	)	)	PUNCT
ejpam-4918	412	1	=	=	SYM
ejpam-4918	412	2	0	0	NUM
ejpam-4918	413	1	for	for	ADP
ejpam-4918	413	2	all	all	DET
ejpam-4918	413	3	i	i	PROPN
ejpam-4918	413	4	,	,	PUNCT
ejpam-4918	413	5	j	j	PROPN
ejpam-4918	413	6	since	since	SCONJ
ejpam-4918	413	7	r	r	NOUN
ejpam-4918	413	8	is	be	AUX
ejpam-4918	413	9	m	m	PRON
ejpam-4918	413	10	-compatible	-compatible	ADJ
ejpam-4918	413	11	and	and	CCONJ
ejpam-4918	413	12	ω	ω	NOUN
ejpam-4918	413	13	is	be	AUX
ejpam-4918	413	14	an	an	DET
ejpam-4918	413	15	automorphism	automorphism	NOUN
ejpam-4918	413	16	.	.	PUNCT
ejpam-4918	414	1	therefore	therefore	ADV
ejpam-4918	414	2	,	,	PUNCT
ejpam-4918	414	3	by	by	ADP
ejpam-4918	414	4	proposition	proposition	NOUN
ejpam-4918	414	5	2	2	NUM
ejpam-4918	414	6	,	,	PUNCT
ejpam-4918	414	7	ere	ere	PROPN
ejpam-4918	414	8	is	be	AUX
ejpam-4918	414	9	strongly	strongly	ADV
ejpam-4918	414	10	cm	cm	NOUN
ejpam-4918	414	11	-reflexive	-reflexive	NOUN
ejpam-4918	414	12	.	.	PUNCT
ejpam-4918	415	1	corollary	corollary	ADJ
ejpam-4918	415	2	5	5	NUM
ejpam-4918	415	3	.	.	PUNCT
ejpam-4918	416	1	[	[	X
ejpam-4918	416	2	20	20	NUM
ejpam-4918	416	3	,	,	PUNCT
ejpam-4918	416	4	proposition	proposition	NOUN
ejpam-4918	416	5	3.7	3.7	NUM
ejpam-4918	416	6	]	]	PUNCT
ejpam-4918	416	7	let	let	VERB
ejpam-4918	416	8	e	e	X
ejpam-4918	416	9	∈	∈	NOUN
ejpam-4918	416	10	r	r	NOUN
ejpam-4918	416	11	be	be	AUX
ejpam-4918	416	12	an	an	DET
ejpam-4918	416	13	idempotent	idempotent	NOUN
ejpam-4918	416	14	.	.	PUNCT
ejpam-4918	417	1	if	if	SCONJ
ejpam-4918	417	2	r	r	NOUN
ejpam-4918	417	3	is	be	AUX
ejpam-4918	417	4	a	a	DET
ejpam-4918	417	5	left	left	ADJ
ejpam-4918	417	6	app	app	NOUN
ejpam-4918	417	7	,	,	PUNCT
ejpam-4918	417	8	then	then	ADV
ejpam-4918	417	9	ere	ere	PROPN
ejpam-4918	417	10	is	be	AUX
ejpam-4918	417	11	a	a	DET
ejpam-4918	417	12	left	left	ADJ
ejpam-4918	417	13	app	app	NOUN
ejpam-4918	417	14	-ring	-ring	NOUN
ejpam-4918	417	15	.	.	PUNCT
ejpam-4918	418	1	corollary	corollary	ADJ
ejpam-4918	418	2	6	6	NUM
ejpam-4918	418	3	.	.	PUNCT
ejpam-4918	419	1	[	[	X
ejpam-4918	419	2	22	22	NUM
ejpam-4918	419	3	,	,	PUNCT
ejpam-4918	419	4	corollary	corollary	NOUN
ejpam-4918	419	5	3.19	3.19	NUM
ejpam-4918	419	6	]	]	PUNCT
ejpam-4918	419	7	let	let	VERB
ejpam-4918	419	8	m	m	PRON
ejpam-4918	419	9	be	be	AUX
ejpam-4918	419	10	a	a	DET
ejpam-4918	419	11	strictly	strictly	ADV
ejpam-4918	419	12	totally	totally	ADV
ejpam-4918	419	13	ordered	order	VERB
ejpam-4918	419	14	monoid	monoid	NOUN
ejpam-4918	419	15	and	and	CCONJ
ejpam-4918	419	16	ω	ω	NUM
ejpam-4918	419	17	:	:	PUNCT
ejpam-4918	419	18	m	m	VERB
ejpam-4918	419	19	→	→	SYM
ejpam-4918	419	20	end(r	end(r	NOUN
ejpam-4918	419	21	)	)	PUNCT
ejpam-4918	419	22	a	a	DET
ejpam-4918	419	23	monoid	monoid	NOUN
ejpam-4918	419	24	homomorphism	homomorphism	NOUN
ejpam-4918	419	25	.	.	PUNCT
ejpam-4918	420	1	assume	assume	VERB
ejpam-4918	420	2	that	that	SCONJ
ejpam-4918	420	3	e	e	PRON
ejpam-4918	420	4	be	be	AUX
ejpam-4918	420	5	an	an	DET
ejpam-4918	420	6	idempotent	idempotent	NOUN
ejpam-4918	420	7	.	.	PUNCT
ejpam-4918	421	1	if	if	SCONJ
ejpam-4918	421	2	r	r	NOUN
ejpam-4918	421	3	is	be	AUX
ejpam-4918	421	4	left	leave	VERB
ejpam-4918	421	5	app	app	NOUN
ejpam-4918	421	6	,	,	PUNCT
ejpam-4918	421	7	then	then	ADV
ejpam-4918	421	8	ere	ere	PROPN
ejpam-4918	421	9	is	be	AUX
ejpam-4918	421	10	(	(	PUNCT
ejpam-4918	421	11	m	m	PROPN
ejpam-4918	421	12	,	,	PUNCT
ejpam-4918	421	13	ω)-quasi	ω)-quasi	NOUN
ejpam-4918	421	14	-	-	ADJ
ejpam-4918	421	15	armendariz	armendariz	ADV
ejpam-4918	421	16	.	.	PUNCT
ejpam-4918	422	1	proposition	proposition	NOUN
ejpam-4918	422	2	6	6	NUM
ejpam-4918	422	3	.	.	PUNCT
ejpam-4918	423	1	let	let	VERB
ejpam-4918	423	2	m	m	PRON
ejpam-4918	423	3	be	be	AUX
ejpam-4918	423	4	a	a	DET
ejpam-4918	423	5	strictly	strictly	ADV
ejpam-4918	423	6	totally	totally	ADV
ejpam-4918	423	7	ordered	order	VERB
ejpam-4918	423	8	monoid	monoid	NOUN
ejpam-4918	423	9	with	with	ADP
ejpam-4918	423	10	twisting	twist	VERB
ejpam-4918	423	11	f	f	X
ejpam-4918	423	12	:	:	PUNCT
ejpam-4918	423	13	m	m	VERB
ejpam-4918	423	14	×m	×m	NOUN
ejpam-4918	423	15	→	→	SYM
ejpam-4918	423	16	u(r	u(r	NOUN
ejpam-4918	423	17	)	)	PUNCT
ejpam-4918	423	18	and	and	CCONJ
ejpam-4918	424	1	action	action	NOUN
ejpam-4918	424	2	ω	ω	NOUN
ejpam-4918	424	3	:	:	PUNCT
ejpam-4918	424	4	m	m	PROPN
ejpam-4918	424	5	→	→	SYM
ejpam-4918	424	6	aut(r	aut(r	PROPN
ejpam-4918	424	7	)	)	PUNCT
ejpam-4918	424	8	.	.	PUNCT
ejpam-4918	425	1	assume	assume	VERB
ejpam-4918	425	2	that	that	SCONJ
ejpam-4918	425	3	e	e	PRON
ejpam-4918	425	4	be	be	AUX
ejpam-4918	425	5	an	an	DET
ejpam-4918	425	6	idempotent	idempotent	NOUN
ejpam-4918	425	7	.	.	PUNCT
ejpam-4918	426	1	if	if	SCONJ
ejpam-4918	426	2	r	r	NOUN
ejpam-4918	426	3	is	be	AUX
ejpam-4918	426	4	a	a	DET
ejpam-4918	426	5	left	left	ADJ
ejpam-4918	426	6	app	app	NOUN
ejpam-4918	426	7	,	,	PUNCT
ejpam-4918	426	8	then	then	ADV
ejpam-4918	426	9	ere	ere	PROPN
ejpam-4918	426	10	is	be	AUX
ejpam-4918	426	11	strongly	strongly	ADV
ejpam-4918	426	12	cm	cm	NOUN
ejpam-4918	426	13	-reflexive	-reflexive	NOUN
ejpam-4918	426	14	.	.	PUNCT
ejpam-4918	427	1	proof	proof	NOUN
ejpam-4918	427	2	.	.	PUNCT
ejpam-4918	428	1	by	by	ADP
ejpam-4918	428	2	corollary	corollary	ADJ
ejpam-4918	428	3	5	5	NUM
ejpam-4918	428	4	,	,	PUNCT
ejpam-4918	428	5	ere	ere	PROPN
ejpam-4918	428	6	is	be	AUX
ejpam-4918	428	7	a	a	DET
ejpam-4918	428	8	left	left	ADJ
ejpam-4918	428	9	app	app	NOUN
ejpam-4918	428	10	.	.	PUNCT
ejpam-4918	429	1	so	so	ADV
ejpam-4918	429	2	,	,	PUNCT
ejpam-4918	429	3	ere	ere	PROPN
ejpam-4918	429	4	is	be	AUX
ejpam-4918	429	5	(	(	PUNCT
ejpam-4918	429	6	m	m	PROPN
ejpam-4918	429	7	,	,	PUNCT
ejpam-4918	429	8	ω)-quasi	ω)-quasi	NOUN
ejpam-4918	429	9	-	-	PUNCT
ejpam-4918	429	10	armendariz	armendariz	ADV
ejpam-4918	429	11	by	by	ADP
ejpam-4918	429	12	corollary	corollary	ADJ
ejpam-4918	429	13	6	6	NUM
ejpam-4918	429	14	.	.	PUNCT
ejpam-4918	430	1	thus	thus	ADV
ejpam-4918	430	2	,	,	PUNCT
ejpam-4918	430	3	the	the	DET
ejpam-4918	430	4	result	result	NOUN
ejpam-4918	430	5	follows	follow	VERB
ejpam-4918	430	6	from	from	ADP
ejpam-4918	430	7	proposition	proposition	NOUN
ejpam-4918	430	8	5	5	NUM
ejpam-4918	430	9	.	.	PUNCT
ejpam-4918	431	1	let	let	VERB
ejpam-4918	431	2	i	i	PRON
ejpam-4918	431	3	be	be	AUX
ejpam-4918	431	4	an	an	DET
ejpam-4918	431	5	index	index	NOUN
ejpam-4918	431	6	set	set	VERB
ejpam-4918	431	7	and	and	CCONJ
ejpam-4918	431	8	ri	ri	PROPN
ejpam-4918	431	9	be	be	AUX
ejpam-4918	431	10	a	a	DET
ejpam-4918	431	11	ring	ring	NOUN
ejpam-4918	431	12	for	for	ADP
ejpam-4918	431	13	each	each	DET
ejpam-4918	431	14	i	i	PRON
ejpam-4918	431	15	∈	∈	PROPN
ejpam-4918	431	16	i.	i.	NOUN
ejpam-4918	431	17	let	let	VERB
ejpam-4918	431	18	m	m	PRON
ejpam-4918	431	19	be	be	AUX
ejpam-4918	431	20	a	a	DET
ejpam-4918	431	21	strictly	strictly	ADV
ejpam-4918	431	22	ordered	order	VERB
ejpam-4918	431	23	monoid	monoid	NOUN
ejpam-4918	431	24	and	and	CCONJ
ejpam-4918	431	25	ωi	ωi	INTJ
ejpam-4918	431	26	:	:	PUNCT
ejpam-4918	431	27	m	m	VERB
ejpam-4918	431	28	→	→	SYM
ejpam-4918	431	29	end(ri	end(ri	NUM
ejpam-4918	431	30	)	)	PUNCT
ejpam-4918	431	31	a	a	DET
ejpam-4918	431	32	monoid	monoid	NOUN
ejpam-4918	431	33	homomorphism	homomorphism	NOUN
ejpam-4918	431	34	.	.	PUNCT
ejpam-4918	432	1	then	then	ADV
ejpam-4918	432	2	the	the	DET
ejpam-4918	432	3	mapping	mapping	NOUN
ejpam-4918	432	4	ω	ω	PROPN
ejpam-4918	432	5	:	:	PUNCT
ejpam-4918	432	6	m	m	VERB
ejpam-4918	432	7	→	→	SYM
ejpam-4918	432	8	end	end	NOUN
ejpam-4918	432	9	(	(	PUNCT
ejpam-4918	432	10	∏	∏	PROPN
ejpam-4918	432	11	i∈i	i∈i	PROPN
ejpam-4918	432	12	ri	ri	PROPN
ejpam-4918	432	13	)	)	PUNCT
ejpam-4918	432	14	is	be	AUX
ejpam-4918	432	15	a	a	DET
ejpam-4918	432	16	monoid	monoid	NOUN
ejpam-4918	432	17	homomorphism	homomorphism	NOUN
ejpam-4918	432	18	given	give	VERB
ejpam-4918	432	19	by	by	ADP
ejpam-4918	432	20	ωg({ri}i∈i	ωg({ri}i∈i	NOUN
ejpam-4918	432	21	)	)	PUNCT
ejpam-4918	432	22	=	=	PRON
ejpam-4918	432	23	{	{	PUNCT
ejpam-4918	432	24	(	(	PUNCT
ejpam-4918	432	25	ωi)g(ri)}i∈i	ωi)g(ri)}i∈i	NOUN
ejpam-4918	432	26	}	}	PUNCT
ejpam-4918	432	27	for	for	ADP
ejpam-4918	432	28	all	all	DET
ejpam-4918	432	29	g	g	PROPN
ejpam-4918	432	30	∈m	∈m	NOUN
ejpam-4918	432	31	.	.	PUNCT
ejpam-4918	433	1	e.	e.	PROPN
ejpam-4918	433	2	ali	ali	PROPN
ejpam-4918	433	3	/	/	SYM
ejpam-4918	433	4	eur	eur	PROPN
ejpam-4918	433	5	.	.	PUNCT
ejpam-4918	434	1	j.	j.	PROPN
ejpam-4918	434	2	pure	pure	PROPN
ejpam-4918	434	3	appl	appl	PROPN
ejpam-4918	434	4	.	.	PROPN
ejpam-4918	434	5	math	math	PROPN
ejpam-4918	434	6	,	,	PUNCT
ejpam-4918	434	7	16	16	NUM
ejpam-4918	434	8	(	(	PUNCT
ejpam-4918	434	9	4	4	NUM
ejpam-4918	434	10	)	)	PUNCT
ejpam-4918	434	11	(	(	PUNCT
ejpam-4918	434	12	2023	2023	NUM
ejpam-4918	434	13	)	)	PUNCT
ejpam-4918	434	14	,	,	PUNCT
ejpam-4918	434	15	2156	2156	NUM
ejpam-4918	434	16	-	-	SYM
ejpam-4918	434	17	2168	2168	NUM
ejpam-4918	434	18	2166	2166	NUM
ejpam-4918	434	19	proposition	proposition	NOUN
ejpam-4918	434	20	7	7	NUM
ejpam-4918	434	21	.	.	PUNCT
ejpam-4918	435	1	let	let	VERB
ejpam-4918	435	2	ri	ri	PRON
ejpam-4918	435	3	be	be	AUX
ejpam-4918	435	4	a	a	DET
ejpam-4918	435	5	ring	ring	NOUN
ejpam-4918	435	6	for	for	ADP
ejpam-4918	435	7	each	each	DET
ejpam-4918	435	8	i	i	PRON
ejpam-4918	435	9	in	in	ADP
ejpam-4918	435	10	a	a	DET
ejpam-4918	435	11	finite	finite	ADJ
ejpam-4918	435	12	index	index	NOUN
ejpam-4918	435	13	set	set	VERB
ejpam-4918	435	14	i	i	PRON
ejpam-4918	435	15	,	,	PUNCT
ejpam-4918	435	16	and	and	CCONJ
ejpam-4918	435	17	let	let	VERB
ejpam-4918	435	18	m	m	PRON
ejpam-4918	435	19	be	be	AUX
ejpam-4918	435	20	a	a	DET
ejpam-4918	435	21	monoid	monoid	NOUN
ejpam-4918	435	22	with	with	ADP
ejpam-4918	435	23	a	a	DET
ejpam-4918	435	24	twisting	twisting	NOUN
ejpam-4918	435	25	f	f	X
ejpam-4918	435	26	:	:	PUNCT
ejpam-4918	435	27	m	m	VERB
ejpam-4918	435	28	×m	×m	NOUN
ejpam-4918	435	29	→	→	SYM
ejpam-4918	435	30	⋃	⋃	NOUN
ejpam-4918	435	31	i∈i	i∈i	NOUN
ejpam-4918	435	32	u(ri	u(ri	PROPN
ejpam-4918	435	33	)	)	PUNCT
ejpam-4918	435	34	and	and	CCONJ
ejpam-4918	436	1	an	an	DET
ejpam-4918	436	2	action	action	NOUN
ejpam-4918	436	3	ωi	ωi	X
ejpam-4918	436	4	:	:	PUNCT
ejpam-4918	436	5	m	m	VERB
ejpam-4918	436	6	→	→	SYM
ejpam-4918	436	7	aut(ri	aut(ri	NUM
ejpam-4918	436	8	)	)	PUNCT
ejpam-4918	436	9	on	on	ADP
ejpam-4918	436	10	each	each	DET
ejpam-4918	436	11	ri	ri	PROPN
ejpam-4918	436	12	.	.	PROPN
ejpam-4918	436	13	suppose	suppose	VERB
ejpam-4918	436	14	that	that	SCONJ
ejpam-4918	436	15	each	each	DET
ejpam-4918	436	16	ri	ri	NOUN
ejpam-4918	436	17	is	be	AUX
ejpam-4918	436	18	strongly	strongly	ADV
ejpam-4918	436	19	cm	cm	NOUN
ejpam-4918	436	20	-reflexive	-reflexive	NOUN
ejpam-4918	436	21	.	.	PUNCT
ejpam-4918	437	1	then	then	ADV
ejpam-4918	437	2	,	,	PUNCT
ejpam-4918	437	3	the	the	DET
ejpam-4918	437	4	direct	direct	ADJ
ejpam-4918	437	5	product	product	NOUN
ejpam-4918	437	6	r	r	NOUN
ejpam-4918	437	7	=	=	SYM
ejpam-4918	437	8	∏	∏	PROPN
ejpam-4918	437	9	i∈i	i∈i	PROPN
ejpam-4918	437	10	ri	ri	PROPN
ejpam-4918	437	11	,	,	PUNCT
ejpam-4918	437	12	equipped	equip	VERB
ejpam-4918	437	13	with	with	ADP
ejpam-4918	437	14	the	the	DET
ejpam-4918	437	15	product	product	NOUN
ejpam-4918	437	16	action	action	NOUN
ejpam-4918	437	17	ω	ω	NOUN
ejpam-4918	437	18	=	=	SYM
ejpam-4918	437	19	∏	∏	PROPN
ejpam-4918	437	20	i∈i	i∈i	PROPN
ejpam-4918	437	21	ω	ω	PROPN
ejpam-4918	437	22	i	i	PROPN
ejpam-4918	437	23	,	,	PUNCT
ejpam-4918	437	24	is	be	AUX
ejpam-4918	437	25	strongly	strongly	ADV
ejpam-4918	437	26	cm	cm	NOUN
ejpam-4918	437	27	-reflexive	-reflexive	NOUN
ejpam-4918	437	28	.	.	PUNCT
ejpam-4918	438	1	proof	proof	NOUN
ejpam-4918	438	2	.	.	PUNCT
ejpam-4918	439	1	let	let	VERB
ejpam-4918	439	2	r	r	NOUN
ejpam-4918	439	3	=	=	SYM
ejpam-4918	439	4	∏	∏	PROPN
ejpam-4918	439	5	i∈i	i∈i	NOUN
ejpam-4918	439	6	ri	ri	PROPN
ejpam-4918	439	7	be	be	AUX
ejpam-4918	439	8	the	the	DET
ejpam-4918	439	9	direct	direct	ADJ
ejpam-4918	439	10	product	product	NOUN
ejpam-4918	439	11	of	of	ADP
ejpam-4918	439	12	rings	ring	NOUN
ejpam-4918	439	13	(	(	PUNCT
ejpam-4918	439	14	ri)i∈i	ri)i∈i	NUM
ejpam-4918	439	15	and	and	CCONJ
ejpam-4918	439	16	ri	ri	PROPN
ejpam-4918	439	17	is	be	AUX
ejpam-4918	439	18	is	be	AUX
ejpam-4918	439	19	strongly	strongly	ADV
ejpam-4918	439	20	cm	cm	NOUN
ejpam-4918	439	21	reflexive	reflexive	ADJ
ejpam-4918	439	22	for	for	SCONJ
ejpam-4918	439	23	each	each	DET
ejpam-4918	439	24	i	i	PROPN
ejpam-4918	439	25	∈	∈	PROPN
ejpam-4918	439	26	i.	i.	NOUN
ejpam-4918	439	27	denote	denote	VERB
ejpam-4918	439	28	the	the	DET
ejpam-4918	439	29	projection	projection	NOUN
ejpam-4918	439	30	r→	r→	PROPN
ejpam-4918	439	31	ri	ri	PROPN
ejpam-4918	439	32	as	as	ADP
ejpam-4918	439	33	πi	πi	ADV
ejpam-4918	439	34	.	.	PUNCT
ejpam-4918	439	35	suppose	suppose	VERB
ejpam-4918	439	36	that	that	SCONJ
ejpam-4918	439	37	ϕ	ϕ	NOUN
ejpam-4918	439	38	,	,	PUNCT
ejpam-4918	439	39	ψ	ψ	ADP
ejpam-4918	439	40	∈	∈	NOUN
ejpam-4918	439	41	r∗m	r∗m	NOUN
ejpam-4918	439	42	are	be	AUX
ejpam-4918	439	43	such	such	ADJ
ejpam-4918	439	44	that	that	SCONJ
ejpam-4918	439	45	ϕ(r	ϕ(r	PROPN
ejpam-4918	439	46	∗m)ψ	∗m)ψ	VERB
ejpam-4918	439	47	=	=	PUNCT
ejpam-4918	440	1	0	0	X
ejpam-4918	440	2	.	.	PUNCT
ejpam-4918	440	3	set	set	VERB
ejpam-4918	440	4	ϕi	ϕi	ADP
ejpam-4918	440	5	=	=	SYM
ejpam-4918	440	6	∏	∏	PROPN
ejpam-4918	441	1	i	i	NOUN
ejpam-4918	441	2	ϕ	ϕ	NOUN
ejpam-4918	441	3	,	,	PUNCT
ejpam-4918	441	4	ψi	ψi	ADP
ejpam-4918	441	5	=	=	SYM
ejpam-4918	441	6	∏	∏	X
ejpam-4918	441	7	i	i	NOUN
ejpam-4918	441	8	ψ	ψ	NOUN
ejpam-4918	441	9	and	and	CCONJ
ejpam-4918	441	10	φi	φi	ADP
ejpam-4918	441	11	=	=	PUNCT
ejpam-4918	441	12	∏	∏	PROPN
ejpam-4918	442	1	i	i	PROPN
ejpam-4918	442	2	φ	φ	NOUN
ejpam-4918	442	3	.	.	PUNCT
ejpam-4918	443	1	then	then	ADV
ejpam-4918	443	2	ϕi	ϕi	ADP
ejpam-4918	443	3	,	,	PUNCT
ejpam-4918	443	4	ψi	ψi	ADP
ejpam-4918	443	5	∈	∈	PROPN
ejpam-4918	443	6	ri	ri	X
ejpam-4918	443	7	∗m	∗m	NOUN
ejpam-4918	443	8	.	.	PUNCT
ejpam-4918	444	1	for	for	ADP
ejpam-4918	444	2	any	any	DET
ejpam-4918	444	3	u	u	NOUN
ejpam-4918	444	4	,	,	PUNCT
ejpam-4918	444	5	v	v	NOUN
ejpam-4918	444	6	∈	∈	PROPN
ejpam-4918	444	7	m	m	X
ejpam-4918	444	8	,	,	PUNCT
ejpam-4918	444	9	assume	assume	VERB
ejpam-4918	444	10	ϕ(u	ϕ(u	X
ejpam-4918	444	11	)	)	PUNCT
ejpam-4918	445	1	=	=	PUNCT
ejpam-4918	445	2	(	(	PUNCT
ejpam-4918	445	3	cui	cui	NOUN
ejpam-4918	445	4	)	)	PUNCT
ejpam-4918	446	1	i∈i	i∈i	ADJ
ejpam-4918	446	2	,	,	PUNCT
ejpam-4918	446	3	ψ(v	ψ(v	PROPN
ejpam-4918	446	4	)	)	PUNCT
ejpam-4918	446	5	=	=	PUNCT
ejpam-4918	446	6	(	(	PUNCT
ejpam-4918	446	7	avi	avi	NOUN
ejpam-4918	446	8	)	)	PUNCT
ejpam-4918	447	1	i∈i	i∈i	ADJ
ejpam-4918	447	2	.	.	PUNCT
ejpam-4918	448	1	now	now	ADV
ejpam-4918	448	2	,	,	PUNCT
ejpam-4918	448	3	for	for	ADP
ejpam-4918	448	4	any	any	DET
ejpam-4918	448	5	r	r	NOUN
ejpam-4918	448	6	∈	∈	NOUN
ejpam-4918	448	7	r	r	NOUN
ejpam-4918	448	8	and	and	CCONJ
ejpam-4918	448	9	any	any	DET
ejpam-4918	448	10	g	g	NOUN
ejpam-4918	448	11	∈m	∈m	NOUN
ejpam-4918	448	12	,	,	PUNCT
ejpam-4918	448	13	ϕ(r	ϕ(r	PROPN
ejpam-4918	448	14	∗m)ψ	∗m)ψ	VERB
ejpam-4918	448	15	=	=	PUNCT
ejpam-4918	448	16	∑	∑	PUNCT
ejpam-4918	448	17	(	(	PUNCT
ejpam-4918	448	18	u	u	NOUN
ejpam-4918	448	19	,	,	PUNCT
ejpam-4918	448	20	v)∈xs(ϕ,crψ	v)∈xs(ϕ,crψ	PROPN
ejpam-4918	448	21	)	)	PUNCT
ejpam-4918	449	1	ϕ(u)ωu(ωg(rψ(v)))f(um	ϕ(u)ωu(ωg(rψ(v)))f(um	NOUN
ejpam-4918	449	2	,	,	PUNCT
ejpam-4918	449	3	vn)umvn	vn)umvn	VERB
ejpam-4918	449	4	=	=	SYM
ejpam-4918	449	5	∑	∑	PUNCT
ejpam-4918	449	6	(	(	PUNCT
ejpam-4918	449	7	u	u	NOUN
ejpam-4918	449	8	,	,	PUNCT
ejpam-4918	449	9	v)∈xs(ϕ,crψ	v)∈xs(ϕ,crψ	NUM
ejpam-4918	449	10	)	)	PUNCT
ejpam-4918	449	11	(	(	PUNCT
ejpam-4918	449	12	cui	cui	NOUN
ejpam-4918	449	13	)	)	PUNCT
ejpam-4918	449	14	i∈i	i∈i	NOUN
ejpam-4918	449	15	(	(	PUNCT
ejpam-4918	449	16	(	(	PUNCT
ejpam-4918	449	17	∏	∏	X
ejpam-4918	449	18	i∈i	i∈i	ADJ
ejpam-4918	449	19	ωi)u(ωg(ria	ωi)u(ωg(ria	NOUN
ejpam-4918	449	20	v	v	NOUN
ejpam-4918	449	21	i	i	NOUN
ejpam-4918	449	22	)	)	PUNCT
ejpam-4918	449	23	)	)	PUNCT
ejpam-4918	450	1	f(u	f(u	PROPN
ejpam-4918	450	2	i	i	PRON
ejpam-4918	450	3	m	m	PROPN
ejpam-4918	450	4	,	,	PUNCT
ejpam-4918	450	5	v	v	ADP
ejpam-4918	450	6	i	i	PRON
ejpam-4918	450	7	n)u	n)u	ADJ
ejpam-4918	451	1	i	i	PRON
ejpam-4918	451	2	mv	mv	VERB
ejpam-4918	451	3	i	i	NOUN
ejpam-4918	451	4	n)i∈i	n)i∈i	PROPN
ejpam-4918	451	5	=	=	SYM
ejpam-4918	451	6	∑	∑	PUNCT
ejpam-4918	451	7	(	(	PUNCT
ejpam-4918	451	8	u	u	NOUN
ejpam-4918	451	9	,	,	PUNCT
ejpam-4918	451	10	v)∈xs(ϕ,crψ	v)∈xs(ϕ,crψ	NUM
ejpam-4918	451	11	)	)	PUNCT
ejpam-4918	451	12	(	(	PUNCT
ejpam-4918	451	13	cui	cui	NOUN
ejpam-4918	451	14	)	)	PUNCT
ejpam-4918	451	15	i∈i	i∈i	PROPN
ejpam-4918	451	16	(	(	PUNCT
ejpam-4918	451	17	∏	∏	PROPN
ejpam-4918	451	18	i∈i	i∈i	ADJ
ejpam-4918	451	19	ωiu)(ωg(ria	ωiu)(ωg(ria	NUM
ejpam-4918	451	20	v	v	ADP
ejpam-4918	451	21	i	i	PRON
ejpam-4918	451	22	)	)	PUNCT
ejpam-4918	452	1	f(u	f(u	PROPN
ejpam-4918	452	2	i	i	PRON
ejpam-4918	452	3	m	m	PROPN
ejpam-4918	452	4	,	,	PUNCT
ejpam-4918	452	5	v	v	ADP
ejpam-4918	452	6	i	i	PRON
ejpam-4918	452	7	n)u	n)u	ADJ
ejpam-4918	453	1	i	i	PRON
ejpam-4918	453	2	mv	mv	VERB
ejpam-4918	453	3	i	i	NOUN
ejpam-4918	453	4	n)i∈i	n)i∈i	PROPN
ejpam-4918	453	5	=	=	SYM
ejpam-4918	453	6	∑	∑	PUNCT
ejpam-4918	453	7	(	(	PUNCT
ejpam-4918	453	8	u	u	NOUN
ejpam-4918	453	9	,	,	PUNCT
ejpam-4918	453	10	v)∈xs(ϕ,crψ	v)∈xs(ϕ,crψ	NUM
ejpam-4918	453	11	)	)	PUNCT
ejpam-4918	453	12	(	(	PUNCT
ejpam-4918	453	13	cui	cui	NOUN
ejpam-4918	453	14	ω	ω	PROPN
ejpam-4918	453	15	i	i	PRON
ejpam-4918	453	16	u(ωg(ria	u(ωg(ria	PROPN
ejpam-4918	453	17	v	v	ADP
ejpam-4918	453	18	i	i	PROPN
ejpam-4918	453	19	)	)	PUNCT
ejpam-4918	453	20	)	)	PUNCT
ejpam-4918	453	21	f(ϕi	f(ϕi	NUM
ejpam-4918	453	22	,	,	PUNCT
ejpam-4918	453	23	ψi)u	ψi)u	PROPN
ejpam-4918	454	1	i	i	PRON
ejpam-4918	454	2	mv	mv	VERB
ejpam-4918	454	3	i	i	NOUN
ejpam-4918	454	4	n)i∈i	n)i∈i	PROPN
ejpam-4918	454	5	=	=	SYM
ejpam-4918	454	6	∑	∑	PUNCT
ejpam-4918	454	7	(	(	PUNCT
ejpam-4918	454	8	u	u	NOUN
ejpam-4918	454	9	,	,	PUNCT
ejpam-4918	454	10	v)∈xs(ϕ,crψ	v)∈xs(ϕ,crψ	NUM
ejpam-4918	454	11	)	)	PUNCT
ejpam-4918	454	12	(	(	PUNCT
ejpam-4918	454	13	ϕi(u)ω	ϕi(u)ω	VERB
ejpam-4918	454	14	i	i	PRON
ejpam-4918	454	15	u(riψi(v)))f(ϕi	u(riψi(v)))f(ϕi	VERB
ejpam-4918	454	16	,	,	PUNCT
ejpam-4918	454	17	ψi)u	ψi)u	PROPN
ejpam-4918	455	1	i	i	PRON
ejpam-4918	455	2	mv	mv	VERB
ejpam-4918	455	3	i	i	NOUN
ejpam-4918	455	4	n)i∈i	n)i∈i	PROPN
ejpam-4918	455	5	=	=	PUNCT
ejpam-4918	455	6	(	(	PUNCT
ejpam-4918	455	7	∑	∑	PROPN
ejpam-4918	455	8	(	(	PUNCT
ejpam-4918	455	9	u	u	NOUN
ejpam-4918	455	10	,	,	PUNCT
ejpam-4918	455	11	v)∈xs(ϕ,crψ	v)∈xs(ϕ,crψ	NUM
ejpam-4918	455	12	)	)	PUNCT
ejpam-4918	456	1	ϕi(u)ω	ϕi(u)ω	VERB
ejpam-4918	456	2	i	i	PRON
ejpam-4918	456	3	u(ωg(riψi(v	u(ωg(riψi(v	ADJ
ejpam-4918	456	4	)	)	PUNCT
ejpam-4918	456	5	)	)	PUNCT
ejpam-4918	456	6	)	)	PUNCT
ejpam-4918	457	1	f(ϕi	f(ϕi	NUM
ejpam-4918	457	2	,	,	PUNCT
ejpam-4918	457	3	ψi)u	ψi)u	PROPN
ejpam-4918	458	1	i	i	PRON
ejpam-4918	458	2	mv	mv	VERB
ejpam-4918	458	3	i	i	NOUN
ejpam-4918	458	4	n)i∈i	n)i∈i	PROPN
ejpam-4918	458	5	=	=	PUNCT
ejpam-4918	458	6	(	(	PUNCT
ejpam-4918	458	7	∑	∑	PROPN
ejpam-4918	458	8	(	(	PUNCT
ejpam-4918	458	9	u	u	NOUN
ejpam-4918	458	10	,	,	PUNCT
ejpam-4918	458	11	v)∈xs(ϕi	v)∈xs(ϕi	NUM
ejpam-4918	458	12	,	,	PUNCT
ejpam-4918	458	13	criψi	criψi	NUM
ejpam-4918	458	14	)	)	PUNCT
ejpam-4918	459	1	ϕi(u)ω	ϕi(u)ω	PROPN
ejpam-4918	459	2	i	i	PRON
ejpam-4918	459	3	u(ωg(riψi(v	u(ωg(riψi(v	ADJ
ejpam-4918	459	4	)	)	PUNCT
ejpam-4918	459	5	)	)	PUNCT
ejpam-4918	459	6	)	)	PUNCT
ejpam-4918	459	7	)	)	PUNCT
ejpam-4918	460	1	f(ϕi	f(ϕi	NUM
ejpam-4918	460	2	,	,	PUNCT
ejpam-4918	460	3	ψi)u	ψi)u	PROPN
ejpam-4918	461	1	i	i	PRON
ejpam-4918	461	2	mv	mv	VERB
ejpam-4918	461	3	i	i	NOUN
ejpam-4918	461	4	n)i∈i	n)i∈i	PROPN
ejpam-4918	461	5	=	=	SYM
ejpam-4918	461	6	(	(	PUNCT
ejpam-4918	461	7	ϕi(ri	ϕi(ri	PROPN
ejpam-4918	461	8	∗m)ψi)i∈i	∗m)ψi)i∈i	NOUN
ejpam-4918	461	9	.	.	PUNCT
ejpam-4918	462	1	since	since	SCONJ
ejpam-4918	462	2	ϕ(r	ϕ(r	PROPN
ejpam-4918	462	3	∗m)ψ	∗m)ψ	VERB
ejpam-4918	462	4	=	=	SYM
ejpam-4918	462	5	0	0	NUM
ejpam-4918	462	6	,	,	PUNCT
ejpam-4918	462	7	we	we	PRON
ejpam-4918	462	8	have	have	VERB
ejpam-4918	462	9	ϕi(ri	ϕi(ri	NOUN
ejpam-4918	463	1	∗m)ψi	∗m)ψi	CCONJ
ejpam-4918	463	2	=	=	SYM
ejpam-4918	463	3	0	0	X
ejpam-4918	463	4	.	.	PUNCT
ejpam-4918	464	1	now	now	ADV
ejpam-4918	464	2	it	it	PRON
ejpam-4918	464	3	follows	follow	VERB
ejpam-4918	464	4	ϕi(u)ωiu(ωg(riψi(v	ϕi(u)ωiu(ωg(riψi(v	PROPN
ejpam-4918	464	5	)	)	PUNCT
ejpam-4918	464	6	)	)	PUNCT
ejpam-4918	464	7	)	)	PUNCT
ejpam-4918	465	1	=	=	SYM
ejpam-4918	465	2	0	0	NUM
ejpam-4918	466	1	for	for	ADP
ejpam-4918	466	2	any	any	DET
ejpam-4918	466	3	r	r	NOUN
ejpam-4918	466	4	∈	∈	NOUN
ejpam-4918	466	5	r	r	NOUN
ejpam-4918	466	6	,	,	PUNCT
ejpam-4918	466	7	any	any	DET
ejpam-4918	466	8	u	u	NOUN
ejpam-4918	466	9	,	,	PUNCT
ejpam-4918	466	10	v	v	NOUN
ejpam-4918	466	11	,	,	PUNCT
ejpam-4918	466	12	g	g	NOUN
ejpam-4918	466	13	∈m	∈m	NOUN
ejpam-4918	466	14	and	and	CCONJ
ejpam-4918	466	15	any	any	DET
ejpam-4918	466	16	i	i	PRON
ejpam-4918	466	17	∈	∈	PROPN
ejpam-4918	466	18	i	i	PRON
ejpam-4918	466	19	,	,	PUNCT
ejpam-4918	466	20	since	since	SCONJ
ejpam-4918	466	21	ri	ri	PROPN
ejpam-4918	466	22	is	be	AUX
ejpam-4918	466	23	strongly	strongly	ADV
ejpam-4918	466	24	cm	cm	NOUN
ejpam-4918	466	25	-reflexive	-reflexive	NOUN
ejpam-4918	466	26	.	.	PUNCT
ejpam-4918	467	1	hence	hence	ADV
ejpam-4918	467	2	,	,	PUNCT
ejpam-4918	467	3	for	for	ADP
ejpam-4918	467	4	any	any	DET
ejpam-4918	467	5	u	u	NOUN
ejpam-4918	467	6	,	,	PUNCT
ejpam-4918	467	7	v	v	NOUN
ejpam-4918	467	8	∈m	∈m	NOUN
ejpam-4918	467	9	,	,	PUNCT
ejpam-4918	467	10	ψ(v)ωv(ωg(rϕ(u	ψ(v)ωv(ωg(rϕ(u	NOUN
ejpam-4918	467	11	)	)	PUNCT
ejpam-4918	467	12	)	)	PUNCT
ejpam-4918	467	13	)	)	PUNCT
ejpam-4918	468	1	=	=	PUNCT
ejpam-4918	468	2	(	(	PUNCT
ejpam-4918	468	3	ψi(v)ω	ψi(v)ω	PUNCT
ejpam-4918	468	4	i	i	PRON
ejpam-4918	468	5	v(ωg(riϕi(u))))i∈i	v(ωg(riϕi(u))))i∈i	NOUN
ejpam-4918	468	6	=	=	SYM
ejpam-4918	468	7	0	0	PUNCT
ejpam-4918	468	8	since	since	SCONJ
ejpam-4918	468	9	i	i	PRON
ejpam-4918	468	10	is	be	AUX
ejpam-4918	468	11	finite	finite	ADJ
ejpam-4918	468	12	.	.	PUNCT
ejpam-4918	469	1	thus	thus	ADV
ejpam-4918	469	2	,	,	PUNCT
ejpam-4918	469	3	ψ(v)ωv(ωg(rϕ(u	ψ(v)ωv(ωg(rϕ(u	NOUN
ejpam-4918	469	4	)	)	PUNCT
ejpam-4918	469	5	)	)	PUNCT
ejpam-4918	469	6	)	)	PUNCT
ejpam-4918	470	1	=	=	PUNCT
ejpam-4918	470	2	0	0	NUM
ejpam-4918	470	3	by	by	ADP
ejpam-4918	470	4	the	the	DET
ejpam-4918	470	5	compatibility	compatibility	NOUN
ejpam-4918	470	6	of	of	ADP
ejpam-4918	470	7	ω	ω	PROPN
ejpam-4918	470	8	.	.	PUNCT
ejpam-4918	471	1	therefore	therefore	ADV
ejpam-4918	471	2	,	,	PUNCT
ejpam-4918	471	3	ψ(r	ψ(r	PROPN
ejpam-4918	471	4	∗m)ϕ	∗m)ϕ	NOUN
ejpam-4918	471	5	=	=	SYM
ejpam-4918	471	6	0	0	X
ejpam-4918	471	7	.	.	PUNCT
ejpam-4918	472	1	this	this	PRON
ejpam-4918	472	2	means	mean	VERB
ejpam-4918	472	3	that	that	SCONJ
ejpam-4918	472	4	r	r	NOUN
ejpam-4918	472	5	is	be	AUX
ejpam-4918	472	6	strongly	strongly	ADV
ejpam-4918	472	7	cm	cm	NOUN
ejpam-4918	472	8	-reflexive	-reflexive	NOUN
ejpam-4918	472	9	.	.	PUNCT
ejpam-4918	473	1	theorem	theorem	NOUN
ejpam-4918	473	2	5	5	NUM
ejpam-4918	473	3	.	.	PUNCT
ejpam-4918	473	4	assuming	assume	VERB
ejpam-4918	473	5	that	that	SCONJ
ejpam-4918	473	6	r	r	NOUN
ejpam-4918	473	7	is	be	AUX
ejpam-4918	473	8	an	an	DET
ejpam-4918	473	9	m	m	NOUN
ejpam-4918	473	10	-compatible	-compatible	ADJ
ejpam-4918	473	11	ring	ring	NOUN
ejpam-4918	473	12	and	and	CCONJ
ejpam-4918	473	13	m	m	NOUN
ejpam-4918	473	14	is	be	AUX
ejpam-4918	473	15	a	a	DET
ejpam-4918	473	16	cancellative	cancellative	ADJ
ejpam-4918	473	17	monoid	monoid	NOUN
ejpam-4918	473	18	with	with	ADP
ejpam-4918	473	19	a	a	DET
ejpam-4918	473	20	twisting	twisting	NOUN
ejpam-4918	473	21	map	map	NOUN
ejpam-4918	474	1	f	f	X
ejpam-4918	474	2	:	:	PUNCT
ejpam-4918	474	3	m	m	VERB
ejpam-4918	474	4	×	×	NOUN
ejpam-4918	474	5	m	m	INTJ
ejpam-4918	474	6	→	→	SYM
ejpam-4918	474	7	u(r	u(r	NOUN
ejpam-4918	474	8	)	)	PUNCT
ejpam-4918	474	9	and	and	CCONJ
ejpam-4918	474	10	an	an	DET
ejpam-4918	474	11	action	action	NOUN
ejpam-4918	474	12	map	map	NOUN
ejpam-4918	474	13	ω	ω	NOUN
ejpam-4918	474	14	:	:	PUNCT
ejpam-4918	474	15	m	m	PROPN
ejpam-4918	474	16	→	→	SYM
ejpam-4918	474	17	aut(r	aut(r	PROPN
ejpam-4918	474	18	)	)	PUNCT
ejpam-4918	474	19	,	,	PUNCT
ejpam-4918	474	20	and	and	CCONJ
ejpam-4918	474	21	considering	consider	VERB
ejpam-4918	474	22	r	r	NOUN
ejpam-4918	474	23	as	as	ADP
ejpam-4918	474	24	a	a	DET
ejpam-4918	474	25	right	right	ADJ
ejpam-4918	474	26	ore	ore	NOUN
ejpam-4918	474	27	ring	ring	NOUN
ejpam-4918	474	28	with	with	ADP
ejpam-4918	474	29	the	the	DET
ejpam-4918	474	30	classical	classical	ADJ
ejpam-4918	474	31	right	right	ADJ
ejpam-4918	474	32	quotient	quotient	NOUN
ejpam-4918	474	33	ring	ring	PROPN
ejpam-4918	474	34	q	q	PROPN
ejpam-4918	474	35	,	,	PUNCT
ejpam-4918	474	36	the	the	DET
ejpam-4918	474	37	r	r	NOUN
ejpam-4918	474	38	is	be	AUX
ejpam-4918	474	39	strongly	strongly	ADV
ejpam-4918	474	40	cm	cm	NOUN
ejpam-4918	474	41	-reflexive	-reflexive	NOUN
ejpam-4918	474	42	if	if	SCONJ
ejpam-4918	474	43	and	and	CCONJ
ejpam-4918	474	44	only	only	ADV
ejpam-4918	474	45	if	if	SCONJ
ejpam-4918	474	46	q	q	NOUN
ejpam-4918	474	47	is	be	AUX
ejpam-4918	474	48	strongly	strongly	ADV
ejpam-4918	474	49	cm	cm	NOUN
ejpam-4918	474	50	-reflexive	-reflexive	NOUN
ejpam-4918	474	51	.	.	PUNCT
ejpam-4918	475	1	proof	proof	NOUN
ejpam-4918	475	2	.	.	PUNCT
ejpam-4918	476	1	it	it	PRON
ejpam-4918	476	2	is	be	AUX
ejpam-4918	476	3	enough	enough	ADV
ejpam-4918	476	4	showing	show	VERB
ejpam-4918	476	5	necessary	necessary	ADJ
ejpam-4918	476	6	.	.	PUNCT
ejpam-4918	477	1	assume	assume	VERB
ejpam-4918	477	2	that	that	SCONJ
ejpam-4918	477	3	r	r	NOUN
ejpam-4918	477	4	is	be	AUX
ejpam-4918	477	5	strongly	strongly	ADV
ejpam-4918	477	6	cm	cm	NOUN
ejpam-4918	477	7	-reflexive	-reflexive	NOUN
ejpam-4918	477	8	.	.	PUNCT
ejpam-4918	478	1	let	let	VERB
ejpam-4918	478	2	ϕ	ϕ	NOUN
ejpam-4918	478	3	=	=	X
ejpam-4918	478	4	σmi=1αili	σmi=1αili	PROPN
ejpam-4918	478	5	,	,	PUNCT
ejpam-4918	478	6	ψ	ψ	X
ejpam-4918	478	7	=	=	NOUN
ejpam-4918	478	8	σpk=1γkhk	σpk=1γkhk	NOUN
ejpam-4918	478	9	be	be	VERB
ejpam-4918	478	10	elements	element	NOUN
ejpam-4918	478	11	in	in	ADP
ejpam-4918	478	12	q∗m	q∗m	DET
ejpam-4918	478	13	satisfying	satisfying	NOUN
ejpam-4918	478	14	ϕφψ	ϕφψ	NOUN
ejpam-4918	478	15	=	=	NOUN
ejpam-4918	478	16	0	0	PROPN
ejpam-4918	478	17	,	,	PUNCT
ejpam-4918	478	18	where	where	SCONJ
ejpam-4918	478	19	φ	φ	PROPN
ejpam-4918	478	20	=	=	SYM
ejpam-4918	478	21	σnj=1βjgj	σnj=1βjgj	PROPN
ejpam-4918	478	22	is	be	AUX
ejpam-4918	478	23	any	any	DET
ejpam-4918	478	24	nonzero	nonzero	ADJ
ejpam-4918	478	25	element	element	NOUN
ejpam-4918	478	26	in	in	ADP
ejpam-4918	478	27	q	q	PROPN
ejpam-4918	478	28	∗m	∗m	NOUN
ejpam-4918	478	29	.	.	PUNCT
ejpam-4918	479	1	by	by	ADP
ejpam-4918	479	2	proposition	proposition	NOUN
ejpam-4918	479	3	2.1.16	2.1.16	NUM
ejpam-4918	480	1	[	[	X
ejpam-4918	480	2	23	23	NUM
ejpam-4918	480	3	]	]	PUNCT
ejpam-4918	480	4	,	,	PUNCT
ejpam-4918	480	5	we	we	PRON
ejpam-4918	480	6	may	may	AUX
ejpam-4918	480	7	assume	assume	VERB
ejpam-4918	480	8	that	that	SCONJ
ejpam-4918	480	9	αi	αi	ADV
ejpam-4918	480	10	=	=	PUNCT
ejpam-4918	481	1	aiu	aiu	PROPN
ejpam-4918	481	2	−1	−1	ADP
ejpam-4918	481	3	,	,	PUNCT
ejpam-4918	481	4	βj	βj	PRON
ejpam-4918	481	5	=	=	PUNCT
ejpam-4918	482	1	bjv	bjv	CCONJ
ejpam-4918	482	2	−1	−1	NOUN
ejpam-4918	482	3	and	and	CCONJ
ejpam-4918	482	4	γk	γk	X
ejpam-4918	482	5	=	=	PUNCT
ejpam-4918	482	6	ckw	ckw	VERB
ejpam-4918	482	7	−1	−1	NOUN
ejpam-4918	482	8	with	with	ADP
ejpam-4918	482	9	regular	regular	ADJ
ejpam-4918	482	10	u	u	PROPN
ejpam-4918	482	11	,	,	PUNCT
ejpam-4918	482	12	v	v	NOUN
ejpam-4918	482	13	,	,	PUNCT
ejpam-4918	482	14	w	w	PROPN
ejpam-4918	482	15	∈	∈	PROPN
ejpam-4918	482	16	r.	r.	PROPN
ejpam-4918	482	17	also	also	ADV
ejpam-4918	482	18	,	,	PUNCT
ejpam-4918	482	19	proposition	proposition	NOUN
ejpam-4918	482	20	2.1.16	2.1.16	NUM
ejpam-4918	483	1	[	[	X
ejpam-4918	483	2	23	23	NUM
ejpam-4918	483	3	]	]	PUNCT
ejpam-4918	483	4	,	,	PUNCT
ejpam-4918	483	5	for	for	ADP
ejpam-4918	483	6	each	each	DET
ejpam-4918	483	7	j	j	PROPN
ejpam-4918	483	8	and	and	CCONJ
ejpam-4918	483	9	k	k	PROPN
ejpam-4918	483	10	,	,	PUNCT
ejpam-4918	483	11	there	there	PRON
ejpam-4918	483	12	exist	exist	VERB
ejpam-4918	483	13	dj	dj	NOUN
ejpam-4918	483	14	,	,	PUNCT
ejpam-4918	483	15	ek	ek	PROPN
ejpam-4918	483	16	∈	∈	PROPN
ejpam-4918	483	17	r	r	NOUN
ejpam-4918	483	18	and	and	CCONJ
ejpam-4918	483	19	regular	regular	ADJ
ejpam-4918	483	20	s	s	PROPN
ejpam-4918	483	21	,	,	PUNCT
ejpam-4918	483	22	t	t	PROPN
ejpam-4918	483	23	∈	∈	PROPN
ejpam-4918	483	24	r	r	NOUN
ejpam-4918	483	25	such	such	DET
ejpam-4918	483	26	that	that	DET
ejpam-4918	483	27	u−1bj	u−1bj	NOUN
ejpam-4918	483	28	=	=	SYM
ejpam-4918	483	29	djs	djs	NOUN
ejpam-4918	483	30	−1	−1	NOUN
ejpam-4918	483	31	references	reference	NOUN
ejpam-4918	483	32	2167	2167	NUM
ejpam-4918	483	33	and	and	CCONJ
ejpam-4918	483	34	(	(	PUNCT
ejpam-4918	483	35	vs)−1ck	vs)−1ck	NOUN
ejpam-4918	483	36	=	=	PUNCT
ejpam-4918	483	37	ekt	ekt	PROPN
ejpam-4918	483	38	−1	−1	NOUN
ejpam-4918	483	39	.	.	PUNCT
ejpam-4918	484	1	suppose	suppose	VERB
ejpam-4918	484	2	ϕ1	ϕ1	NOUN
ejpam-4918	484	3	=	=	SYM
ejpam-4918	484	4	σmi=1aili	σmi=1aili	PROPN
ejpam-4918	484	5	,	,	PUNCT
ejpam-4918	484	6	φ1	φ1	NOUN
ejpam-4918	484	7	=	=	SYM
ejpam-4918	484	8	σnj=1bjgj	σnj=1bjgj	PROPN
ejpam-4918	484	9	,	,	PUNCT
ejpam-4918	484	10	φ2	φ2	NOUN
ejpam-4918	484	11	=	=	SYM
ejpam-4918	484	12	σnj=1djgj	σnj=1djgj	PROPN
ejpam-4918	484	13	,	,	PUNCT
ejpam-4918	484	14	ψ1	ψ1	ADJ
ejpam-4918	484	15	=	=	SYM
ejpam-4918	484	16	σpk=1ckhk	σpk=1ckhk	NOUN
ejpam-4918	484	17	,	,	PUNCT
ejpam-4918	484	18	ψ2	ψ2	NOUN
ejpam-4918	484	19	=	=	SYM
ejpam-4918	484	20	σpk=1ekhk	σpk=1ekhk	PROPN
ejpam-4918	484	21	∈	∈	PROPN
ejpam-4918	484	22	r	r	NOUN
ejpam-4918	484	23	∗m	∗m	NOUN
ejpam-4918	484	24	.	.	PUNCT
ejpam-4918	485	1	since	since	SCONJ
ejpam-4918	485	2	m	m	PROPN
ejpam-4918	485	3	is	be	AUX
ejpam-4918	485	4	a	a	DET
ejpam-4918	485	5	cancellative	cancellative	ADJ
ejpam-4918	485	6	monoid	monoid	NOUN
ejpam-4918	485	7	by	by	ADP
ejpam-4918	485	8	lemma	lemma	PROPN
ejpam-4918	485	9	1	1	NUM
ejpam-4918	485	10	.	.	PUNCT
ejpam-4918	486	1	thus	thus	ADV
ejpam-4918	486	2	,	,	PUNCT
ejpam-4918	486	3	gish1	gish1	PROPN
ejpam-4918	486	4	̸=	̸=	PROPN
ejpam-4918	486	5	gjsh1	gjsh1	NOUN
ejpam-4918	486	6	for	for	ADP
ejpam-4918	486	7	gi	gi	NOUN
ejpam-4918	486	8	̸=	̸=	PROPN
ejpam-4918	486	9	gj	gj	NOUN
ejpam-4918	486	10	.	.	PUNCT
ejpam-4918	487	1	then	then	ADV
ejpam-4918	487	2	,	,	PUNCT
ejpam-4918	487	3	we	we	PRON
ejpam-4918	487	4	have	have	VERB
ejpam-4918	487	5	0	0	NUM
ejpam-4918	487	6	=	=	SYM
ejpam-4918	487	7	ϕφψ	ϕφψ	NOUN
ejpam-4918	487	8	=	=	PUNCT
ejpam-4918	488	1	σmi=1σ	σmi=1σ	PUNCT
ejpam-4918	489	1	p	p	NOUN
ejpam-4918	489	2	k=1(aiu	k=1(aiu	PROPN
ejpam-4918	489	3	−1)ωli(ωg(rckw	−1)ωli(ωg(rckw	PROPN
ejpam-4918	489	4	−1	−1	NOUN
ejpam-4918	489	5	)	)	PUNCT
ejpam-4918	489	6	)	)	PUNCT
ejpam-4918	490	1	f(li	f(li	PROPN
ejpam-4918	490	2	,	,	PUNCT
ejpam-4918	490	3	hk)(lihk	hk)(lihk	NOUN
ejpam-4918	490	4	)	)	PUNCT
ejpam-4918	490	5	=	=	PUNCT
ejpam-4918	491	1	σmi=1σ	σmi=1σ	PROPN
ejpam-4918	491	2	p	p	X
ejpam-4918	491	3	k=1aiωli(ωg(rek))f(li	k=1aiωli(ωg(rek))f(li	NOUN
ejpam-4918	491	4	,	,	PUNCT
ejpam-4918	491	5	hk)(lihk)(ωli(t)w	hk)(lihk)(ωli(t)w	NOUN
ejpam-4918	491	6	)	)	PUNCT
ejpam-4918	491	7	−1	−1	NOUN
ejpam-4918	491	8	=	=	SYM
ejpam-4918	491	9	0	0	NUM
ejpam-4918	491	10	=	=	SYM
ejpam-4918	491	11	ϕ1φ2ψ2(wt	ϕ1φ2ψ2(wt	PROPN
ejpam-4918	491	12	)	)	PUNCT
ejpam-4918	491	13	−1	−1	NOUN
ejpam-4918	491	14	.	.	PUNCT
ejpam-4918	492	1	therefore	therefore	ADV
ejpam-4918	492	2	,	,	PUNCT
ejpam-4918	492	3	ϕ1φ2ψ2	ϕ1φ2ψ2	PROPN
ejpam-4918	492	4	=	=	NOUN
ejpam-4918	492	5	0	0	X
ejpam-4918	492	6	.	.	PUNCT
ejpam-4918	493	1	since	since	SCONJ
ejpam-4918	493	2	r	r	NOUN
ejpam-4918	493	3	is	be	AUX
ejpam-4918	493	4	strongly	strongly	ADV
ejpam-4918	493	5	cm	cm	NOUN
ejpam-4918	493	6	-reflexive	-reflexive	NOUN
ejpam-4918	493	7	,	,	PUNCT
ejpam-4918	493	8	then	then	ADV
ejpam-4918	493	9	ψ2φ2ϕ1	ψ2φ2ϕ1	NOUN
ejpam-4918	493	10	=	=	NOUN
ejpam-4918	493	11	0	0	NUM
ejpam-4918	493	12	.	.	PUNCT
ejpam-4918	494	1	this	this	PRON
ejpam-4918	494	2	implies	imply	VERB
ejpam-4918	494	3	that	that	SCONJ
ejpam-4918	494	4	ϕ1uφ2ψ2	ϕ1uφ2ψ2	PRON
ejpam-4918	494	5	=	=	PUNCT
ejpam-4918	495	1	ϕ1φ1sψ2	ϕ1φ1sψ2	PUNCT
ejpam-4918	496	1	=	=	SYM
ejpam-4918	496	2	0	0	PROPN
ejpam-4918	496	3	since	since	SCONJ
ejpam-4918	496	4	u−1bj	u−1bj	NOUN
ejpam-4918	496	5	=	=	SYM
ejpam-4918	496	6	djs	djs	PROPN
ejpam-4918	496	7	−1	−1	NOUN
ejpam-4918	496	8	,	,	PUNCT
ejpam-4918	496	9	then	then	ADV
ejpam-4918	496	10	sψ2φ1ϕ1	sψ2φ1ϕ1	PROPN
ejpam-4918	496	11	=	=	SYM
ejpam-4918	496	12	0	0	NUM
ejpam-4918	496	13	and	and	CCONJ
ejpam-4918	496	14	(	(	PUNCT
ejpam-4918	496	15	vs)ψ2φ1ϕ1	vs)ψ2φ1ϕ1	NOUN
ejpam-4918	496	16	=	=	SYM
ejpam-4918	496	17	0	0	NUM
ejpam-4918	496	18	,	,	PUNCT
ejpam-4918	496	19	so	so	ADV
ejpam-4918	496	20	ψ1φ1ϕ1	ψ1φ1ϕ1	NOUN
ejpam-4918	496	21	=	=	SYM
ejpam-4918	496	22	0	0	PUNCT
ejpam-4918	496	23	since	since	SCONJ
ejpam-4918	496	24	(	(	PUNCT
ejpam-4918	496	25	vs)−1ck	vs)−1ck	NOUN
ejpam-4918	496	26	=	=	PUNCT
ejpam-4918	496	27	ekt	ekt	PROPN
ejpam-4918	496	28	−1	−1	NOUN
ejpam-4918	496	29	.	.	PUNCT
ejpam-4918	497	1	using	use	VERB
ejpam-4918	497	2	proposition	proposition	NOUN
ejpam-4918	497	3	2.1.16	2.1.16	NUM
ejpam-4918	498	1	[	[	X
ejpam-4918	498	2	23	23	NUM
ejpam-4918	498	3	]	]	PUNCT
ejpam-4918	498	4	again	again	ADV
ejpam-4918	498	5	,	,	PUNCT
ejpam-4918	498	6	for	for	ADP
ejpam-4918	498	7	each	each	DET
ejpam-4918	498	8	i	i	PRON
ejpam-4918	498	9	,	,	PUNCT
ejpam-4918	498	10	j	j	PROPN
ejpam-4918	498	11	there	there	PRON
ejpam-4918	498	12	exist	exist	VERB
ejpam-4918	498	13	φi	φi	ADP
ejpam-4918	498	14	,	,	PUNCT
ejpam-4918	498	15	ϕj	ϕj	ADP
ejpam-4918	498	16	∈	∈	PROPN
ejpam-4918	498	17	r	r	NOUN
ejpam-4918	498	18	∗m	∗m	NOUN
ejpam-4918	498	19	and	and	CCONJ
ejpam-4918	498	20	regular	regular	ADJ
ejpam-4918	498	21	element	element	NOUN
ejpam-4918	498	22	q	q	NOUN
ejpam-4918	498	23	,	,	PUNCT
ejpam-4918	498	24	p	p	NOUN
ejpam-4918	498	25	∈	∈	PROPN
ejpam-4918	498	26	r	r	NOUN
ejpam-4918	498	27	such	such	ADJ
ejpam-4918	498	28	that	that	SCONJ
ejpam-4918	498	29	w−1bj	w−1bj	ADV
ejpam-4918	499	1	=	=	PUNCT
ejpam-4918	499	2	ϕjq	ϕjq	NOUN
ejpam-4918	499	3	−1	−1	NOUN
ejpam-4918	499	4	and	and	CCONJ
ejpam-4918	499	5	(	(	PUNCT
ejpam-4918	499	6	vq)−1ai	vq)−1ai	NOUN
ejpam-4918	499	7	=	=	SYM
ejpam-4918	499	8	φip	φip	NOUN
ejpam-4918	499	9	−1	−1	NOUN
ejpam-4918	499	10	.	.	PUNCT
ejpam-4918	500	1	let	let	VERB
ejpam-4918	500	2	ϕ2	ϕ2	ADV
ejpam-4918	500	3	=	=	SYM
ejpam-4918	500	4	σmi=1φili	σmi=1φili	NOUN
ejpam-4918	500	5	,	,	PUNCT
ejpam-4918	500	6	φ3	φ3	NOUN
ejpam-4918	500	7	=	=	PUNCT
ejpam-4918	500	8	σnj=1ϕjgj	σnj=1ϕjgj	NOUN
ejpam-4918	500	9	.	.	PUNCT
ejpam-4918	501	1	then	then	ADV
ejpam-4918	501	2	,	,	PUNCT
ejpam-4918	501	3	qψ1φ1ϕ1	qψ1φ1ϕ1	PROPN
ejpam-4918	501	4	=	=	PUNCT
ejpam-4918	501	5	σmi=1σ	σmi=1σ	PROPN
ejpam-4918	502	1	p	p	NOUN
ejpam-4918	502	2	k=1q(ckw	k=1q(ckw	ADJ
ejpam-4918	502	3	−1)ωhk(ωg(raiu	−1)ωhk(ωg(raiu	PROPN
ejpam-4918	502	4	−1))f(hk	−1))f(hk	PROPN
ejpam-4918	502	5	,	,	PUNCT
ejpam-4918	502	6	li)(hkli	li)(hkli	PROPN
ejpam-4918	502	7	)	)	PUNCT
ejpam-4918	502	8	=	=	PUNCT
ejpam-4918	503	1	σmi=1σ	σmi=1σ	PUNCT
ejpam-4918	503	2	p	p	X
ejpam-4918	503	3	k=1q(ck)ωhk(ωg(rai))×	k=1q(ck)ωhk(ωg(rai))×	PROPN
ejpam-4918	503	4	(	(	PUNCT
ejpam-4918	503	5	ωhk(u)w	ωhk(u)w	ADJ
ejpam-4918	503	6	)	)	PUNCT
ejpam-4918	503	7	−1	−1	NOUN
ejpam-4918	504	1	=	=	NOUN
ejpam-4918	504	2	0	0	NUM
ejpam-4918	504	3	since	since	SCONJ
ejpam-4918	504	4	ψ1φ1ϕ1	ψ1φ1ϕ1	NOUN
ejpam-4918	504	5	=	=	SYM
ejpam-4918	504	6	0	0	NUM
ejpam-4918	504	7	.	.	PUNCT
ejpam-4918	505	1	thus	thus	ADV
ejpam-4918	505	2	,	,	PUNCT
ejpam-4918	505	3	for	for	ADP
ejpam-4918	505	4	all	all	DET
ejpam-4918	505	5	k	k	PROPN
ejpam-4918	505	6	,	,	PUNCT
ejpam-4918	505	7	i	i	PRON
ejpam-4918	505	8	we	we	PRON
ejpam-4918	505	9	have	have	VERB
ejpam-4918	505	10	ckωhk(ωg(rai	ckωhk(ωg(rai	NOUN
ejpam-4918	505	11	)	)	PUNCT
ejpam-4918	505	12	)	)	PUNCT
ejpam-4918	506	1	=	=	PUNCT
ejpam-4918	506	2	0	0	NUM
ejpam-4918	506	3	,	,	PUNCT
ejpam-4918	506	4	and	and	CCONJ
ejpam-4918	506	5	it	it	PRON
ejpam-4918	506	6	follows	follow	VERB
ejpam-4918	506	7	that	that	PRON
ejpam-4918	506	8	ϕ1wφ3qψ2	ϕ1wφ3qψ2	PRON
ejpam-4918	507	1	=	=	PUNCT
ejpam-4918	507	2	σmi=1σ	σmi=1σ	PROPN
ejpam-4918	508	1	p	p	NOUN
ejpam-4918	508	2	k=1waiωli(ωg(rek	k=1waiωli(ωg(rek	PROPN
ejpam-4918	508	3	)	)	PUNCT
ejpam-4918	508	4	)	)	PUNCT
ejpam-4918	509	1	=	=	PUNCT
ejpam-4918	509	2	0	0	PUNCT
ejpam-4918	509	3	since	since	SCONJ
ejpam-4918	509	4	w−1bj	w−1bj	ADV
ejpam-4918	509	5	=	=	PUNCT
ejpam-4918	509	6	ϕjq	ϕjq	VERB
ejpam-4918	509	7	−1	−1	NOUN
ejpam-4918	509	8	.	.	PUNCT
ejpam-4918	510	1	then	then	ADV
ejpam-4918	510	2	,	,	PUNCT
ejpam-4918	510	3	ψ1φ3ϕ1w	ψ1φ3ϕ1w	PUNCT
ejpam-4918	510	4	=	=	PUNCT
ejpam-4918	510	5	0	0	PUNCT
ejpam-4918	510	6	since	since	SCONJ
ejpam-4918	510	7	r	r	NOUN
ejpam-4918	510	8	is	be	AUX
ejpam-4918	510	9	strongly	strongly	ADV
ejpam-4918	510	10	cm	cm	NOUN
ejpam-4918	510	11	-reflexive	-reflexive	NOUN
ejpam-4918	510	12	,	,	PUNCT
ejpam-4918	510	13	and	and	CCONJ
ejpam-4918	510	14	so	so	ADV
ejpam-4918	510	15	ψ1φ3ϕ1	ψ1φ3ϕ1	ADJ
ejpam-4918	510	16	=	=	SYM
ejpam-4918	510	17	0	0	X
ejpam-4918	510	18	.	.	PUNCT
ejpam-4918	511	1	therefore	therefore	ADV
ejpam-4918	511	2	,	,	PUNCT
ejpam-4918	511	3	ψ1φ3ϕ1p	ψ1φ3ϕ1p	PUNCT
ejpam-4918	511	4	=	=	PUNCT
ejpam-4918	511	5	σpk=1σ	σpk=1σ	PROPN
ejpam-4918	511	6	m	m	NOUN
ejpam-4918	511	7	i=1ckωhk(ωg(rai))f(hk	i=1ckωhk(ωg(rai))f(hk	NOUN
ejpam-4918	511	8	,	,	PUNCT
ejpam-4918	511	9	li)(hkli)p	li)(hkli)p	PROPN
ejpam-4918	511	10	=	=	SYM
ejpam-4918	511	11	ψ1φ3ϕ2(vq	ψ1φ3ϕ2(vq	PROPN
ejpam-4918	511	12	)	)	PUNCT
ejpam-4918	511	13	=	=	PUNCT
ejpam-4918	511	14	σpk=1σ	σpk=1σ	PROPN
ejpam-4918	511	15	n	n	NUM
ejpam-4918	511	16	j=1ckωhk(ωg(rdj))(pv	j=1ckωhk(ωg(rdj))(pv	NOUN
ejpam-4918	511	17	)	)	PUNCT
ejpam-4918	511	18	=	=	SYM
ejpam-4918	511	19	0	0	NUM
ejpam-4918	511	20	,	,	PUNCT
ejpam-4918	511	21	and	and	CCONJ
ejpam-4918	511	22	thus	thus	ADV
ejpam-4918	511	23	ψ1φ3ϕ2	ψ1φ3ϕ2	ADJ
ejpam-4918	511	24	=	=	SYM
ejpam-4918	511	25	0	0	X
ejpam-4918	511	26	.	.	PUNCT
ejpam-4918	512	1	therefore	therefore	ADV
ejpam-4918	512	2	,	,	PUNCT
ejpam-4918	512	3	ψφϕ	ψφϕ	X
ejpam-4918	512	4	=	=	SYM
ejpam-4918	512	5	σpk=1σ	σpk=1σ	PROPN
ejpam-4918	512	6	m	m	VERB
ejpam-4918	512	7	i=1(ckw	i=1(ckw	ADJ
ejpam-4918	512	8	−1)ωhk(ωg(raiu	−1)ωhk(ωg(raiu	PROPN
ejpam-4918	512	9	−1	−1	NOUN
ejpam-4918	512	10	)	)	PUNCT
ejpam-4918	512	11	)	)	PUNCT
ejpam-4918	513	1	=	=	PUNCT
ejpam-4918	514	1	σpk=1σ	σpk=1σ	PROPN
ejpam-4918	514	2	m	m	NOUN
ejpam-4918	514	3	i=1ckωhk(ωg(rai))(ωhk(u)w	i=1ckωhk(ωg(rai))(ωhk(u)w	NOUN
ejpam-4918	514	4	)	)	PUNCT
ejpam-4918	514	5	−1	−1	NOUN
ejpam-4918	514	6	=	=	SYM
ejpam-4918	514	7	0	0	NUM
ejpam-4918	514	8	.	.	PUNCT
ejpam-4918	515	1	thus	thus	ADV
ejpam-4918	515	2	,	,	PUNCT
ejpam-4918	515	3	ckωhk(ωg(rai))(up	ckωhk(ωg(rai))(up	NOUN
ejpam-4918	515	4	)	)	PUNCT
ejpam-4918	515	5	−1	−1	NOUN
ejpam-4918	515	6	=	=	SYM
ejpam-4918	515	7	0.therefore	0.therefore	NUM
ejpam-4918	515	8	,	,	PUNCT
ejpam-4918	515	9	qisstronglycm−reflexive	qisstronglycm−reflexive	NUM
ejpam-4918	515	10	.	.	PUNCT
ejpam-4918	516	1	references	reference	NOUN
ejpam-4918	516	2	[	[	X
ejpam-4918	516	3	1	1	NUM
ejpam-4918	516	4	]	]	PUNCT
ejpam-4918	516	5	l.	l.	PROPN
ejpam-4918	516	6	zhao	zhao	PROPN
ejpam-4918	516	7	,	,	PUNCT
ejpam-4918	516	8	x.	x.	PROPN
ejpam-4918	516	9	zhu	zhu	PROPN
ejpam-4918	516	10	,	,	PUNCT
ejpam-4918	516	11	and	and	CCONJ
ejpam-4918	516	12	q.	q.	PROPN
ejpam-4918	516	13	gu	gu	PROPN
ejpam-4918	516	14	.	.	PROPN
ejpam-4918	516	15	reflexive	reflexive	ADJ
ejpam-4918	516	16	rings	ring	NOUN
ejpam-4918	516	17	and	and	CCONJ
ejpam-4918	516	18	their	their	PRON
ejpam-4918	516	19	extensions	extension	NOUN
ejpam-4918	516	20	.	.	PUNCT
ejpam-4918	517	1	math	math	NOUN
ejpam-4918	517	2	.	.	PUNCT
ejpam-4918	518	1	slovaca	slovaca	PROPN
ejpam-4918	518	2	,	,	PUNCT
ejpam-4918	518	3	63(3):417–430	63(3):417–430	PROPN
ejpam-4918	518	4	,	,	PUNCT
ejpam-4918	518	5	2013	2013	NUM
ejpam-4918	518	6	.	.	PUNCT
ejpam-4918	519	1	[	[	X
ejpam-4918	519	2	2	2	X
ejpam-4918	519	3	]	]	PUNCT
ejpam-4918	519	4	e.	e.	PROPN
ejpam-4918	519	5	ali	ali	PROPN
ejpam-4918	519	6	.	.	PUNCT
ejpam-4918	520	1	the	the	DET
ejpam-4918	520	2	reflexive	reflexive	ADJ
ejpam-4918	520	3	condition	condition	NOUN
ejpam-4918	520	4	on	on	ADP
ejpam-4918	520	5	skew	skew	ADJ
ejpam-4918	520	6	monoid	monoid	PROPN
ejpam-4918	520	7	rings	ring	NOUN
ejpam-4918	520	8	.	.	PUNCT
ejpam-4918	521	1	eur	eur	PROPN
ejpam-4918	521	2	.	.	PUNCT
ejpam-4918	522	1	j.	j.	PROPN
ejpam-4918	522	2	pure	pure	PROPN
ejpam-4918	522	3	appl	appl	PROPN
ejpam-4918	522	4	.	.	PUNCT
ejpam-4918	522	5	math	math	PROPN
ejpam-4918	522	6	,	,	PUNCT
ejpam-4918	522	7	16(3):1878–1893	16(3):1878–1893	NUM
ejpam-4918	522	8	,	,	PUNCT
ejpam-4918	522	9	2023	2023	NUM
ejpam-4918	522	10	.	.	PUNCT
ejpam-4918	523	1	[	[	X
ejpam-4918	523	2	3	3	X
ejpam-4918	523	3	]	]	X
ejpam-4918	523	4	e.	e.	PROPN
ejpam-4918	523	5	ali	ali	PROPN
ejpam-4918	523	6	,	,	PUNCT
ejpam-4918	523	7	a.	a.	NOUN
ejpam-4918	523	8	elshokry	elshokry	NOUN
ejpam-4918	523	9	,	,	PUNCT
ejpam-4918	523	10	and	and	CCONJ
ejpam-4918	523	11	z.	z.	PROPN
ejpam-4918	523	12	kui	kui	PROPN
ejpam-4918	523	13	.	.	PUNCT
ejpam-4918	524	1	strongly	strongly	ADV
ejpam-4918	524	2	α	α	NUM
ejpam-4918	524	3	-	-	ADJ
ejpam-4918	524	4	reversible	reversible	ADJ
ejpam-4918	524	5	rings	ring	NOUN
ejpam-4918	524	6	relative	relative	ADJ
ejpam-4918	524	7	to	to	ADP
ejpam-4918	524	8	a	a	DET
ejpam-4918	524	9	monoid	monoid	NOUN
ejpam-4918	524	10	.	.	PUNCT
ejpam-4918	524	11	int	int	PROPN
ejpam-4918	524	12	.	.	PUNCT
ejpam-4918	525	1	j.	j.	PROPN
ejpam-4918	525	2	of	of	ADP
ejpam-4918	525	3	algebra	algebra	PROPN
ejpam-4918	525	4	,	,	PUNCT
ejpam-4918	525	5	8:375–387	8:375–387	NOUN
ejpam-4918	525	6	,	,	PUNCT
ejpam-4918	525	7	2014	2014	NUM
ejpam-4918	525	8	.	.	PUNCT
ejpam-4918	526	1	[	[	X
ejpam-4918	526	2	4	4	X
ejpam-4918	526	3	]	]	PUNCT
ejpam-4918	526	4	z.	z.	PROPN
ejpam-4918	526	5	peng	peng	PROPN
ejpam-4918	526	6	,	,	PUNCT
ejpam-4918	526	7	q.	q.	PROPN
ejpam-4918	526	8	gu	gu	PROPN
ejpam-4918	526	9	,	,	PUNCT
ejpam-4918	526	10	and	and	CCONJ
ejpam-4918	526	11	l.	l.	PROPN
ejpam-4918	526	12	zhao	zhao	PROPN
ejpam-4918	526	13	.	.	PUNCT
ejpam-4918	527	1	extensions	extension	NOUN
ejpam-4918	527	2	of	of	ADP
ejpam-4918	527	3	strongly	strongly	ADV
ejpam-4918	527	4	reflexive	reflexive	ADJ
ejpam-4918	527	5	rings	ring	NOUN
ejpam-4918	527	6	.	.	PUNCT
ejpam-4918	528	1	asian	asian	ADJ
ejpam-4918	528	2	-	-	PUNCT
ejpam-4918	528	3	european	european	ADJ
ejpam-4918	528	4	journal	journal	NOUN
ejpam-4918	528	5	of	of	ADP
ejpam-4918	528	6	mathematics	mathematic	NOUN
ejpam-4918	528	7	,	,	PUNCT
ejpam-4918	528	8	8(4):1550078	8(4):1550078	NUM
ejpam-4918	528	9	,	,	PUNCT
ejpam-4918	528	10	2015	2015	NUM
ejpam-4918	528	11	.	.	PUNCT
ejpam-4918	529	1	[	[	X
ejpam-4918	529	2	5	5	X
ejpam-4918	529	3	]	]	PUNCT
ejpam-4918	529	4	z.	z.	PROPN
ejpam-4918	529	5	k.	k.	PROPN
ejpam-4918	529	6	liu	liu	PROPN
ejpam-4918	529	7	.	.	PUNCT
ejpam-4918	530	1	armendariz	armendariz	PROPN
ejpam-4918	530	2	rings	ring	NOUN
ejpam-4918	530	3	relative	relative	ADJ
ejpam-4918	530	4	to	to	ADP
ejpam-4918	530	5	a	a	DET
ejpam-4918	530	6	monoid	monoid	NOUN
ejpam-4918	530	7	.	.	PUNCT
ejpam-4918	530	8	comm	comm	NOUN
ejpam-4918	530	9	.	.	PUNCT
ejpam-4918	531	1	algebra	algebra	NOUN
ejpam-4918	531	2	,	,	PUNCT
ejpam-4918	531	3	33(3):649–661	33(3):649–661	PROPN
ejpam-4918	531	4	,	,	PUNCT
ejpam-4918	531	5	2005	2005	NUM
ejpam-4918	531	6	.	.	PUNCT
ejpam-4918	532	1	[	[	X
ejpam-4918	532	2	6	6	NUM
ejpam-4918	532	3	]	]	PUNCT
ejpam-4918	532	4	l.	l.	PROPN
ejpam-4918	532	5	zhao	zhao	PROPN
ejpam-4918	532	6	and	and	CCONJ
ejpam-4918	532	7	y.	y.	PROPN
ejpam-4918	532	8	q.	q.	PROPN
ejpam-4918	532	9	zhou	zhou	PROPN
ejpam-4918	532	10	.	.	PUNCT
ejpam-4918	533	1	generalized	generalize	VERB
ejpam-4918	533	2	armendariz	armendariz	ADJ
ejpam-4918	533	3	properties	property	NOUN
ejpam-4918	533	4	of	of	ADP
ejpam-4918	533	5	crossed	cross	VERB
ejpam-4918	533	6	product	product	NOUN
ejpam-4918	533	7	type	type	NOUN
ejpam-4918	533	8	.	.	PUNCT
ejpam-4918	534	1	glasgow	glasgow	PROPN
ejpam-4918	534	2	math	math	PROPN
ejpam-4918	534	3	.	.	PUNCT
ejpam-4918	535	1	j.	j.	PROPN
ejpam-4918	535	2	,	,	PUNCT
ejpam-4918	535	3	58:313–323	58:313–323	PROPN
ejpam-4918	535	4	,	,	PUNCT
ejpam-4918	535	5	2016	2016	NUM
ejpam-4918	535	6	.	.	PUNCT
ejpam-4918	536	1	[	[	X
ejpam-4918	536	2	7	7	NUM
ejpam-4918	536	3	]	]	X
ejpam-4918	536	4	a.	a.	NOUN
ejpam-4918	536	5	v.	v.	PROPN
ejpam-4918	536	6	kelarev	kelarev	PROPN
ejpam-4918	536	7	.	.	PUNCT
ejpam-4918	537	1	ring	ring	NOUN
ejpam-4918	537	2	constructions	construction	NOUN
ejpam-4918	537	3	and	and	CCONJ
ejpam-4918	537	4	applications	application	NOUN
ejpam-4918	537	5	.	.	PUNCT
ejpam-4918	538	1	world	world	NOUN
ejpam-4918	538	2	scientific	scientific	PROPN
ejpam-4918	538	3	publishing	publishing	PROPN
ejpam-4918	538	4	co.	co.	PROPN
ejpam-4918	538	5	pte	pte	PROPN
ejpam-4918	538	6	.	.	PROPN
ejpam-4918	538	7	ltd	ltd	PROPN
ejpam-4918	538	8	.	.	PROPN
ejpam-4918	538	9	,	,	PUNCT
ejpam-4918	538	10	singapore	singapore	PROPN
ejpam-4918	538	11	,	,	PUNCT
ejpam-4918	538	12	2002	2002	NUM
ejpam-4918	538	13	.	.	PUNCT
ejpam-4918	539	1	[	[	X
ejpam-4918	539	2	8	8	NUM
ejpam-4918	539	3	]	]	X
ejpam-4918	539	4	d.	d.	PROPN
ejpam-4918	539	5	s.	s.	PROPN
ejpam-4918	539	6	passman	passman	PROPN
ejpam-4918	539	7	.	.	PUNCT
ejpam-4918	540	1	the	the	DET
ejpam-4918	540	2	algebraic	algebraic	ADJ
ejpam-4918	540	3	structure	structure	NOUN
ejpam-4918	540	4	of	of	ADP
ejpam-4918	540	5	group	group	NOUN
ejpam-4918	540	6	rings	ring	NOUN
ejpam-4918	540	7	.	.	PUNCT
ejpam-4918	541	1	john	john	PROPN
ejpam-4918	541	2	wiley	wiley	PROPN
ejpam-4918	541	3	&	&	CCONJ
ejpam-4918	541	4	sons	sons	PROPN
ejpam-4918	541	5	ltd	ltd	PROPN
ejpam-4918	541	6	.	.	PROPN
ejpam-4918	541	7	,	,	PUNCT
ejpam-4918	541	8	new	new	PROPN
ejpam-4918	541	9	york	york	PROPN
ejpam-4918	541	10	,	,	PUNCT
ejpam-4918	541	11	1977	1977	NUM
ejpam-4918	541	12	.	.	PUNCT
ejpam-4918	542	1	references	reference	NOUN
ejpam-4918	542	2	2168	2168	NUM
ejpam-4918	542	3	[	[	X
ejpam-4918	542	4	9	9	NUM
ejpam-4918	542	5	]	]	PUNCT
ejpam-4918	542	6	a.	a.	PROPN
ejpam-4918	542	7	r.	r.	PROPN
ejpam-4918	542	8	nasr	nasr	PROPN
ejpam-4918	542	9	-	-	PUNCT
ejpam-4918	542	10	isfahani	isfahani	PROPN
ejpam-4918	542	11	and	and	CCONJ
ejpam-4918	542	12	a.	a.	NOUN
ejpam-4918	542	13	moussavi	moussavi	PROPN
ejpam-4918	542	14	.	.	PUNCT
ejpam-4918	543	1	on	on	ADP
ejpam-4918	543	2	weakly	weakly	ADJ
ejpam-4918	543	3	rigid	rigid	ADJ
ejpam-4918	543	4	rings	ring	NOUN
ejpam-4918	543	5	.	.	PUNCT
ejpam-4918	543	6	glasg	glasg	PROPN
ejpam-4918	543	7	.	.	PUNCT
ejpam-4918	544	1	math	math	NOUN
ejpam-4918	544	2	.	.	PUNCT
ejpam-4918	545	1	j.	j.	PROPN
ejpam-4918	545	2	,	,	PUNCT
ejpam-4918	545	3	51(3):425–440	51(3):425–440	PROPN
ejpam-4918	545	4	,	,	PUNCT
ejpam-4918	545	5	2009	2009	NUM
ejpam-4918	545	6	.	.	PUNCT
ejpam-4918	546	1	[	[	X
ejpam-4918	546	2	10	10	NUM
ejpam-4918	546	3	]	]	X
ejpam-4918	546	4	e.	e.	PROPN
ejpam-4918	546	5	hashemi	hashemi	PROPN
ejpam-4918	546	6	and	and	CCONJ
ejpam-4918	546	7	a.	a.	NOUN
ejpam-4918	546	8	moussavi	moussavi	PROPN
ejpam-4918	546	9	.	.	PUNCT
ejpam-4918	547	1	polynomial	polynomial	ADJ
ejpam-4918	547	2	extensions	extension	NOUN
ejpam-4918	547	3	of	of	ADP
ejpam-4918	547	4	quasi	quasi	ADJ
ejpam-4918	547	5	-	-	PROPN
ejpam-4918	547	6	baer	baer	PROPN
ejpam-4918	547	7	rings	ring	NOUN
ejpam-4918	547	8	.	.	PUNCT
ejpam-4918	548	1	acta	acta	PROPN
ejpam-4918	548	2	math	math	PROPN
ejpam-4918	548	3	.	.	PUNCT
ejpam-4918	549	1	hungar	hungar	NOUN
ejpam-4918	549	2	,	,	PUNCT
ejpam-4918	549	3	107(3):207–224	107(3):207–224	NUM
ejpam-4918	549	4	,	,	PUNCT
ejpam-4918	549	5	2005	2005	NUM
ejpam-4918	549	6	.	.	PUNCT
ejpam-4918	550	1	[	[	X
ejpam-4918	550	2	11	11	NUM
ejpam-4918	550	3	]	]	X
ejpam-4918	550	4	g.	g.	PROPN
ejpam-4918	550	5	f.	f.	PROPN
ejpam-4918	550	6	birkenmeier	birkenmeier	PROPN
ejpam-4918	550	7	and	and	CCONJ
ejpam-4918	550	8	j.	j.	PROPN
ejpam-4918	550	9	k.	k.	PROPN
ejpam-4918	550	10	park	park	PROPN
ejpam-4918	550	11	.	.	PUNCT
ejpam-4918	551	1	triangular	triangular	NOUN
ejpam-4918	551	2	matrix	matrix	NOUN
ejpam-4918	551	3	representation	representation	NOUN
ejpam-4918	551	4	of	of	ADP
ejpam-4918	551	5	ring	ring	NOUN
ejpam-4918	551	6	extensions	extension	NOUN
ejpam-4918	551	7	.	.	PUNCT
ejpam-4918	552	1	j.	j.	PROPN
ejpam-4918	552	2	algebra	algebra	PROPN
ejpam-4918	552	3	,	,	PUNCT
ejpam-4918	552	4	265:457–477	265:457–477	NUM
ejpam-4918	552	5	,	,	PUNCT
ejpam-4918	552	6	2003	2003	NUM
ejpam-4918	552	7	.	.	PUNCT
ejpam-4918	553	1	[	[	X
ejpam-4918	553	2	12	12	NUM
ejpam-4918	553	3	]	]	X
ejpam-4918	553	4	c.	c.	PROPN
ejpam-4918	553	5	y.	y.	PROPN
ejpam-4918	553	6	hong	hong	PROPN
ejpam-4918	553	7	,	,	PUNCT
ejpam-4918	553	8	n.	n.	PROPN
ejpam-4918	553	9	k.	k.	PROPN
ejpam-4918	553	10	kim	kim	PROPN
ejpam-4918	553	11	,	,	PUNCT
ejpam-4918	553	12	and	and	CCONJ
ejpam-4918	553	13	t.	t.	PROPN
ejpam-4918	553	14	k.	k.	PROPN
ejpam-4918	553	15	kwak	kwak	PROPN
ejpam-4918	553	16	.	.	PUNCT
ejpam-4918	554	1	ore	ore	NOUN
ejpam-4918	554	2	extensions	extension	NOUN
ejpam-4918	554	3	of	of	ADP
ejpam-4918	554	4	baer	baer	PROPN
ejpam-4918	554	5	and	and	CCONJ
ejpam-4918	554	6	p.p.-rings	p.p.-ring	NOUN
ejpam-4918	554	7	.	.	PUNCT
ejpam-4918	555	1	j.	j.	PROPN
ejpam-4918	555	2	pure	pure	PROPN
ejpam-4918	555	3	appl	appl	PROPN
ejpam-4918	555	4	.	.	PUNCT
ejpam-4918	556	1	algebra	algebra	PROPN
ejpam-4918	556	2	,	,	PUNCT
ejpam-4918	556	3	151(3):215–226	151(3):215–226	NUM
ejpam-4918	556	4	,	,	PUNCT
ejpam-4918	556	5	2000	2000	NUM
ejpam-4918	556	6	.	.	PUNCT
ejpam-4918	557	1	[	[	X
ejpam-4918	557	2	13	13	NUM
ejpam-4918	557	3	]	]	X
ejpam-4918	557	4	g.	g.	PROPN
ejpam-4918	557	5	birkenmeier	birkenmeier	PROPN
ejpam-4918	557	6	,	,	PUNCT
ejpam-4918	557	7	f.	f.	PROPN
ejpam-4918	557	8	kim	kim	PROPN
ejpam-4918	557	9	,	,	PUNCT
ejpam-4918	557	10	j.	j.	PROPN
ejpam-4918	557	11	y	y	PROPN
ejpam-4918	557	12	,	,	PUNCT
ejpam-4918	557	13	and	and	CCONJ
ejpam-4918	557	14	j.	j.	PROPN
ejpam-4918	557	15	k	k	PROPN
ejpam-4918	557	16	park	park	PROPN
ejpam-4918	557	17	.	.	PUNCT
ejpam-4918	558	1	principally	principally	ADV
ejpam-4918	558	2	quasi	quasi	ADJ
ejpam-4918	558	3	-	-	PROPN
ejpam-4918	558	4	baer	baer	PROPN
ejpam-4918	558	5	rings	ring	NOUN
ejpam-4918	558	6	.	.	PUNCT
ejpam-4918	559	1	comm	comm	NOUN
ejpam-4918	559	2	.	.	PUNCT
ejpam-4918	560	1	algebra	algebra	NOUN
ejpam-4918	560	2	,	,	PUNCT
ejpam-4918	560	3	29:639–660	29:639–660	PROPN
ejpam-4918	560	4	,	,	PUNCT
ejpam-4918	560	5	2001	2001	NUM
ejpam-4918	560	6	.	.	PUNCT
ejpam-4918	561	1	[	[	X
ejpam-4918	561	2	14	14	NUM
ejpam-4918	561	3	]	]	X
ejpam-4918	561	4	g.	g.	PROPN
ejpam-4918	561	5	birkenmeier	birkenmeier	PROPN
ejpam-4918	561	6	,	,	PUNCT
ejpam-4918	561	7	f.	f.	PROPN
ejpam-4918	561	8	kim	kim	PROPN
ejpam-4918	561	9	,	,	PUNCT
ejpam-4918	561	10	j.	j.	PROPN
ejpam-4918	561	11	y	y	PROPN
ejpam-4918	561	12	,	,	PUNCT
ejpam-4918	561	13	and	and	CCONJ
ejpam-4918	561	14	j.	j.	PROPN
ejpam-4918	561	15	k	k	PROPN
ejpam-4918	561	16	park	park	PROPN
ejpam-4918	561	17	.	.	PUNCT
ejpam-4918	562	1	on	on	ADP
ejpam-4918	562	2	polynomial	polynomial	ADJ
ejpam-4918	562	3	extensions	extension	NOUN
ejpam-4918	562	4	of	of	ADP
ejpam-4918	562	5	principally	principally	ADV
ejpam-4918	562	6	quasi	quasi	ADJ
ejpam-4918	562	7	-	-	PROPN
ejpam-4918	562	8	baer	baer	PROPN
ejpam-4918	562	9	rings	ring	NOUN
ejpam-4918	562	10	.	.	PUNCT
ejpam-4918	563	1	kyungpook	kyungpook	PROPN
ejpam-4918	563	2	mathematical	mathematical	PROPN
ejpam-4918	563	3	j.	j.	PROPN
ejpam-4918	563	4	,	,	PUNCT
ejpam-4918	563	5	40:247–254	40:247–254	PROPN
ejpam-4918	563	6	,	,	PUNCT
ejpam-4918	563	7	2000	2000	NUM
ejpam-4918	563	8	.	.	PUNCT
ejpam-4918	564	1	[	[	X
ejpam-4918	564	2	15	15	NUM
ejpam-4918	564	3	]	]	PUNCT
ejpam-4918	564	4	z.	z.	PROPN
ejpam-4918	564	5	k.	k.	PROPN
ejpam-4918	564	6	liu	liu	PROPN
ejpam-4918	564	7	.	.	PUNCT
ejpam-4918	565	1	a	a	DET
ejpam-4918	565	2	note	note	NOUN
ejpam-4918	565	3	on	on	ADP
ejpam-4918	565	4	principally	principally	ADV
ejpam-4918	565	5	quasi	quasi	ADJ
ejpam-4918	565	6	-	-	PROPN
ejpam-4918	565	7	baer	baer	PROPN
ejpam-4918	565	8	rings	ring	NOUN
ejpam-4918	565	9	.	.	PUNCT
ejpam-4918	566	1	comm	comm	NOUN
ejpam-4918	566	2	.	.	PUNCT
ejpam-4918	567	1	algebra	algebra	NOUN
ejpam-4918	567	2	,	,	PUNCT
ejpam-4918	567	3	30:3885–3890	30:3885–3890	NUM
ejpam-4918	567	4	,	,	PUNCT
ejpam-4918	567	5	2002	2002	NUM
ejpam-4918	567	6	.	.	PUNCT
ejpam-4918	568	1	[	[	X
ejpam-4918	568	2	16	16	NUM
ejpam-4918	568	3	]	]	PUNCT
ejpam-4918	568	4	z.	z.	PROPN
ejpam-4918	568	5	k.	k.	PROPN
ejpam-4918	568	6	liu	liu	PROPN
ejpam-4918	568	7	and	and	CCONJ
ejpam-4918	568	8	r.	r.	PROPN
ejpam-4918	568	9	y.	y.	PROPN
ejpam-4918	568	10	zhao	zhao	PROPN
ejpam-4918	568	11	.	.	PUNCT
ejpam-4918	569	1	a	a	DET
ejpam-4918	569	2	generalization	generalization	NOUN
ejpam-4918	569	3	of	of	ADP
ejpam-4918	569	4	pp	pp	NOUN
ejpam-4918	569	5	-	-	PUNCT
ejpam-4918	569	6	rings	ring	NOUN
ejpam-4918	569	7	and	and	CCONJ
ejpam-4918	569	8	p.q.-baer	p.q.-baer	PROPN
ejpam-4918	569	9	rings	ring	NOUN
ejpam-4918	569	10	.	.	PUNCT
ejpam-4918	570	1	glasgow	glasgow	PROPN
ejpam-4918	570	2	j.	j.	PROPN
ejpam-4918	570	3	math	math	PROPN
ejpam-4918	570	4	.	.	PUNCT
ejpam-4918	570	5	,	,	PUNCT
ejpam-4918	570	6	48:217–229	48:217–229	NUM
ejpam-4918	570	7	,	,	PUNCT
ejpam-4918	570	8	2006	2006	NUM
ejpam-4918	570	9	.	.	PUNCT
ejpam-4918	571	1	[	[	X
ejpam-4918	571	2	17	17	NUM
ejpam-4918	571	3	]	]	X
ejpam-4918	571	4	e.	e.	PROPN
ejpam-4918	571	5	ali	ali	PROPN
ejpam-4918	571	6	and	and	CCONJ
ejpam-4918	571	7	a.	a.	NOUN
ejpam-4918	571	8	elshokry	elshokry	PROPN
ejpam-4918	571	9	.	.	PUNCT
ejpam-4918	572	1	some	some	DET
ejpam-4918	572	2	properties	property	NOUN
ejpam-4918	572	3	of	of	ADP
ejpam-4918	572	4	quasi	quasi	ADJ
ejpam-4918	572	5	-	-	ADJ
ejpam-4918	572	6	armendariz	armendariz	ADJ
ejpam-4918	572	7	rings	ring	NOUN
ejpam-4918	572	8	and	and	CCONJ
ejpam-4918	572	9	their	their	PRON
ejpam-4918	572	10	generalization	generalization	NOUN
ejpam-4918	572	11	.	.	PUNCT
ejpam-4918	573	1	asia	asia	PROPN
ejpam-4918	573	2	p.	p.	PROPN
ejpam-4918	573	3	j.	j.	PROPN
ejpam-4918	573	4	math	math	PROPN
ejpam-4918	573	5	,	,	PUNCT
ejpam-4918	573	6	5(1):14–26	5(1):14–26	NUM
ejpam-4918	573	7	,	,	PUNCT
ejpam-4918	573	8	2018	2018	NUM
ejpam-4918	573	9	.	.	PUNCT
ejpam-4918	574	1	[	[	X
ejpam-4918	574	2	18	18	NUM
ejpam-4918	574	3	]	]	X
ejpam-4918	574	4	e.	e.	PROPN
ejpam-4918	574	5	ali	ali	PROPN
ejpam-4918	574	6	and	and	CCONJ
ejpam-4918	574	7	a.	a.	NOUN
ejpam-4918	574	8	elshokry	elshokry	PROPN
ejpam-4918	574	9	.	.	PUNCT
ejpam-4918	575	1	some	some	DET
ejpam-4918	575	2	results	result	NOUN
ejpam-4918	575	3	on	on	ADP
ejpam-4918	575	4	a	a	DET
ejpam-4918	575	5	generalization	generalization	NOUN
ejpam-4918	575	6	of	of	ADP
ejpam-4918	575	7	armendariz	armendariz	ADJ
ejpam-4918	575	8	rings	ring	NOUN
ejpam-4918	575	9	.	.	PUNCT
ejpam-4918	576	1	asia	asia	PROPN
ejpam-4918	576	2	p.	p.	PROPN
ejpam-4918	576	3	j.	j.	PROPN
ejpam-4918	576	4	math	math	PROPN
ejpam-4918	576	5	,	,	PUNCT
ejpam-4918	576	6	6(1):1–17	6(1):1–17	PROPN
ejpam-4918	576	7	,	,	PUNCT
ejpam-4918	576	8	2019	2019	NUM
ejpam-4918	576	9	.	.	PUNCT
ejpam-4918	577	1	[	[	X
ejpam-4918	577	2	19	19	NUM
ejpam-4918	577	3	]	]	X
ejpam-4918	577	4	n.	n.	PROPN
ejpam-4918	577	5	k.	k.	PROPN
ejpam-4918	577	6	kim	kim	PROPN
ejpam-4918	577	7	and	and	CCONJ
ejpam-4918	577	8	y.	y.	PROPN
ejpam-4918	577	9	lee	lee	PROPN
ejpam-4918	577	10	.	.	PUNCT
ejpam-4918	578	1	extensions	extension	NOUN
ejpam-4918	578	2	of	of	ADP
ejpam-4918	578	3	reversible	reversible	ADJ
ejpam-4918	578	4	rings	ring	NOUN
ejpam-4918	578	5	.	.	PUNCT
ejpam-4918	579	1	j.	j.	PROPN
ejpam-4918	579	2	pure	pure	PROPN
ejpam-4918	579	3	appl	appl	PROPN
ejpam-4918	579	4	.	.	PUNCT
ejpam-4918	580	1	algebra	algebra	PROPN
ejpam-4918	580	2	,	,	PUNCT
ejpam-4918	580	3	185:207	185:207	NOUN
ejpam-4918	580	4	–	–	PUNCT
ejpam-4918	580	5	223	223	NUM
ejpam-4918	580	6	,	,	PUNCT
ejpam-4918	580	7	2003	2003	NUM
ejpam-4918	580	8	.	.	PUNCT
ejpam-4918	581	1	[	[	X
ejpam-4918	581	2	20	20	NUM
ejpam-4918	581	3	]	]	PUNCT
ejpam-4918	581	4	z.	z.	PROPN
ejpam-4918	581	5	k.	k.	PROPN
ejpam-4918	581	6	liu	liu	PROPN
ejpam-4918	581	7	and	and	CCONJ
ejpam-4918	581	8	z.	z.	PROPN
ejpam-4918	581	9	renyu	renyu	PROPN
ejpam-4918	581	10	.	.	PUNCT
ejpam-4918	582	1	a	a	DET
ejpam-4918	582	2	generalization	generalization	NOUN
ejpam-4918	582	3	of	of	ADP
ejpam-4918	582	4	pp	pp	NOUN
ejpam-4918	582	5	-	-	PUNCT
ejpam-4918	582	6	rings	ring	NOUN
ejpam-4918	582	7	and	and	CCONJ
ejpam-4918	582	8	p.q.-baer	p.q.-baer	PROPN
ejpam-4918	582	9	rings	rings	PROPN
ejpam-4918	582	10	.	.	PUNCT
ejpam-4918	582	11	glasg	glasg	PROPN
ejpam-4918	582	12	.	.	PUNCT
ejpam-4918	583	1	math	math	NOUN
ejpam-4918	583	2	.	.	PUNCT
ejpam-4918	584	1	j.	j.	PROPN
ejpam-4918	584	2	,	,	PUNCT
ejpam-4918	584	3	48(2):217–229	48(2):217–229	PROPN
ejpam-4918	584	4	,	,	PUNCT
ejpam-4918	584	5	2006	2006	NUM
ejpam-4918	584	6	.	.	PUNCT
ejpam-4918	585	1	[	[	X
ejpam-4918	585	2	21	21	NUM
ejpam-4918	585	3	]	]	PUNCT
ejpam-4918	585	4	z.	z.	PROPN
ejpam-4918	585	5	k.	k.	PROPN
ejpam-4918	585	6	liu	liu	PROPN
ejpam-4918	585	7	and	and	CCONJ
ejpam-4918	585	8	z.	z.	PROPN
ejpam-4918	585	9	wenhui	wenhui	PROPN
ejpam-4918	585	10	.	.	PUNCT
ejpam-4918	586	1	quasi	quasi	ADJ
ejpam-4918	586	2	-	-	ADJ
ejpam-4918	586	3	armendariz	armendariz	ADJ
ejpam-4918	586	4	rings	ring	NOUN
ejpam-4918	586	5	relative	relative	ADJ
ejpam-4918	586	6	to	to	ADP
ejpam-4918	586	7	a	a	DET
ejpam-4918	586	8	monoid	monoid	NOUN
ejpam-4918	586	9	.	.	PUNCT
ejpam-4918	586	10	comm	comm	NOUN
ejpam-4918	586	11	.	.	PUNCT
ejpam-4918	587	1	algebra	algebra	PROPN
ejpam-4918	587	2	,	,	PUNCT
ejpam-4918	587	3	36:928–947	36:928–947	PROPN
ejpam-4918	587	4	,	,	PUNCT
ejpam-4918	587	5	2008	2008	NUM
ejpam-4918	587	6	.	.	PUNCT
ejpam-4918	588	1	[	[	X
ejpam-4918	588	2	22	22	NUM
ejpam-4918	588	3	]	]	X
ejpam-4918	588	4	e.	e.	PROPN
ejpam-4918	588	5	ali	ali	PROPN
ejpam-4918	588	6	and	and	CCONJ
ejpam-4918	588	7	a.	a.	NOUN
ejpam-4918	588	8	elshokry	elshokry	PROPN
ejpam-4918	588	9	.	.	PUNCT
ejpam-4918	589	1	a	a	DET
ejpam-4918	589	2	note	note	NOUN
ejpam-4918	589	3	on	on	ADP
ejpam-4918	589	4	(	(	PUNCT
ejpam-4918	589	5	s	s	NOUN
ejpam-4918	589	6	,	,	PUNCT
ejpam-4918	589	7	ω)-quasi	ω)-quasi	NOUN
ejpam-4918	589	8	-	-	SYM
ejpam-4918	589	9	armendariz	armendariz	ADJ
ejpam-4918	589	10	rings	ring	NOUN
ejpam-4918	589	11	.	.	PUNCT
ejpam-4918	589	12	accepted	accept	VERB
ejpam-4918	589	13	.	.	PUNCT
ejpam-4918	590	1	[	[	X
ejpam-4918	590	2	23	23	NUM
ejpam-4918	590	3	]	]	PUNCT
ejpam-4918	590	4	j.	j.	PROPN
ejpam-4918	590	5	c.	c.	PROPN
ejpam-4918	590	6	mcconnell	mcconnell	PROPN
ejpam-4918	590	7	and	and	CCONJ
ejpam-4918	590	8	j.	j.	PROPN
ejpam-4918	590	9	c.	c.	PROPN
ejpam-4918	590	10	robson	robson	PROPN
ejpam-4918	590	11	.	.	PUNCT
ejpam-4918	591	1	noncommutative	noncommutative	ADJ
ejpam-4918	591	2	noetherian	noetherian	ADJ
ejpam-4918	591	3	rings	ring	NOUN
ejpam-4918	591	4	.	.	PUNCT
ejpam-4918	592	1	john	john	PROPN
ejpam-4918	592	2	wiley	wiley	PROPN
ejpam-4918	592	3	and	and	CCONJ
ejpam-4918	592	4	sons	son	NOUN
ejpam-4918	592	5	,	,	PUNCT
ejpam-4918	592	6	chichester	chichester	PROPN
ejpam-4918	592	7	,	,	PUNCT
ejpam-4918	592	8	1987	1987	NUM
ejpam-4918	592	9	.	.	PUNCT