id	sid	tid	token	lemma	pos
ejpam-5915	1	1	european	european	PROPN
ejpam-5915	1	2	journal	journal	PROPN
ejpam-5915	1	3	of	of	ADP
ejpam-5915	1	4	pure	pure	ADJ
ejpam-5915	1	5	and	and	CCONJ
ejpam-5915	1	6	applied	applied	ADJ
ejpam-5915	1	7	mathematics	mathematic	NOUN
ejpam-5915	1	8	2025	2025	NUM
ejpam-5915	1	9	,	,	PUNCT
ejpam-5915	1	10	vol	vol	NOUN
ejpam-5915	1	11	.	.	PROPN
ejpam-5915	1	12	18	18	NUM
ejpam-5915	1	13	,	,	PUNCT
ejpam-5915	1	14	issue	issue	NOUN
ejpam-5915	1	15	2	2	NUM
ejpam-5915	1	16	,	,	PUNCT
ejpam-5915	1	17	article	article	NOUN
ejpam-5915	1	18	number	number	NOUN
ejpam-5915	1	19	5915	5915	NUM
ejpam-5915	1	20	issn	issn	PROPN
ejpam-5915	1	21	1307	1307	NUM
ejpam-5915	1	22	-	-	SYM
ejpam-5915	1	23	5543	5543	NUM
ejpam-5915	1	24	–	–	PUNCT
ejpam-5915	1	25	ejpam.com	ejpam.com	X
ejpam-5915	1	26	published	publish	VERB
ejpam-5915	1	27	by	by	ADP
ejpam-5915	1	28	new	new	PROPN
ejpam-5915	1	29	york	york	PROPN
ejpam-5915	1	30	business	business	PROPN
ejpam-5915	1	31	global	global	ADJ
ejpam-5915	1	32	fuzzy	fuzzy	ADJ
ejpam-5915	1	33	hom	hom	NOUN
ejpam-5915	1	34	-	-	PUNCT
ejpam-5915	1	35	groups	group	NOUN
ejpam-5915	1	36	:	:	PUNCT
ejpam-5915	1	37	a	a	DET
ejpam-5915	1	38	new	new	ADJ
ejpam-5915	1	39	perspective	perspective	NOUN
ejpam-5915	1	40	on	on	ADP
ejpam-5915	1	41	algebraic	algebraic	ADJ
ejpam-5915	1	42	generalization	generalization	NOUN
ejpam-5915	1	43	shadi	shadi	PROPN
ejpam-5915	1	44	m.	m.	PROPN
ejpam-5915	1	45	shaqaqha	shaqaqha	PROPN
ejpam-5915	1	46	department	department	PROPN
ejpam-5915	1	47	of	of	ADP
ejpam-5915	1	48	mathematics	mathematic	NOUN
ejpam-5915	1	49	,	,	PUNCT
ejpam-5915	1	50	yarmouk	yarmouk	CCONJ
ejpam-5915	1	51	university	university	NOUN
ejpam-5915	1	52	,	,	PUNCT
ejpam-5915	1	53	shafiq	shafiq	PROPN
ejpam-5915	1	54	irshidat	irshidat	PROPN
ejpam-5915	1	55	street	street	PROPN
ejpam-5915	1	56	,	,	PUNCT
ejpam-5915	1	57	irbid	irbid	VERB
ejpam-5915	1	58	21163	21163	NUM
ejpam-5915	1	59	,	,	PUNCT
ejpam-5915	1	60	jordan	jordan	PROPN
ejpam-5915	1	61	abstract	abstract	PROPN
ejpam-5915	1	62	.	.	PUNCT
ejpam-5915	2	1	fuzzy	fuzzy	ADJ
ejpam-5915	2	2	algebraic	algebraic	ADJ
ejpam-5915	2	3	structures	structure	NOUN
ejpam-5915	2	4	extend	extend	VERB
ejpam-5915	2	5	classical	classical	ADJ
ejpam-5915	2	6	algebra	algebra	NOUN
ejpam-5915	2	7	to	to	ADP
ejpam-5915	2	8	model	model	NOUN
ejpam-5915	2	9	uncertainty	uncertainty	NOUN
ejpam-5915	2	10	,	,	PUNCT
ejpam-5915	2	11	while	while	SCONJ
ejpam-5915	2	12	homgroups	homgroup	NOUN
ejpam-5915	2	13	introduce	introduce	VERB
ejpam-5915	2	14	a	a	DET
ejpam-5915	2	15	twisting	twisting	NOUN
ejpam-5915	2	16	map	map	NOUN
ejpam-5915	2	17	α	α	NOUN
ejpam-5915	2	18	that	that	PRON
ejpam-5915	2	19	modifies	modify	VERB
ejpam-5915	2	20	associativity	associativity	NOUN
ejpam-5915	2	21	and	and	CCONJ
ejpam-5915	2	22	identity	identity	NOUN
ejpam-5915	2	23	conditions	condition	NOUN
ejpam-5915	2	24	.	.	PUNCT
ejpam-5915	3	1	this	this	DET
ejpam-5915	3	2	paper	paper	NOUN
ejpam-5915	3	3	unifies	unify	VERB
ejpam-5915	3	4	these	these	DET
ejpam-5915	3	5	concepts	concept	NOUN
ejpam-5915	3	6	by	by	ADP
ejpam-5915	3	7	introducing	introduce	VERB
ejpam-5915	3	8	fuzzy	fuzzy	ADJ
ejpam-5915	3	9	hom	hom	NOUN
ejpam-5915	3	10	-	-	PUNCT
ejpam-5915	3	11	groups	group	NOUN
ejpam-5915	3	12	,	,	PUNCT
ejpam-5915	3	13	a	a	DET
ejpam-5915	3	14	generalization	generalization	NOUN
ejpam-5915	3	15	of	of	ADP
ejpam-5915	3	16	fuzzy	fuzzy	ADJ
ejpam-5915	3	17	groups	group	NOUN
ejpam-5915	3	18	within	within	ADP
ejpam-5915	3	19	the	the	DET
ejpam-5915	3	20	hom	hom	NOUN
ejpam-5915	3	21	-	-	PUNCT
ejpam-5915	3	22	group	group	NOUN
ejpam-5915	3	23	framework	framework	NOUN
ejpam-5915	3	24	.	.	PUNCT
ejpam-5915	4	1	we	we	PRON
ejpam-5915	4	2	define	define	VERB
ejpam-5915	4	3	fuzzy	fuzzy	ADJ
ejpam-5915	4	4	hom	hom	NOUN
ejpam-5915	4	5	-	-	PUNCT
ejpam-5915	4	6	subgroups	subgroup	NOUN
ejpam-5915	4	7	and	and	CCONJ
ejpam-5915	4	8	fuzzy	fuzzy	ADJ
ejpam-5915	4	9	hom	hom	NOUN
ejpam-5915	4	10	-	-	PUNCT
ejpam-5915	4	11	normal	normal	ADJ
ejpam-5915	4	12	subgroups	subgroup	NOUN
ejpam-5915	4	13	,	,	PUNCT
ejpam-5915	4	14	establishing	establish	VERB
ejpam-5915	4	15	their	their	PRON
ejpam-5915	4	16	fundamental	fundamental	ADJ
ejpam-5915	4	17	properties	property	NOUN
ejpam-5915	4	18	.	.	PUNCT
ejpam-5915	5	1	a	a	DET
ejpam-5915	5	2	key	key	ADJ
ejpam-5915	5	3	result	result	NOUN
ejpam-5915	5	4	shows	show	VERB
ejpam-5915	5	5	that	that	SCONJ
ejpam-5915	5	6	each	each	DET
ejpam-5915	5	7	fuzzy	fuzzy	ADJ
ejpam-5915	5	8	hom	hom	NOUN
ejpam-5915	5	9	-	-	PUNCT
ejpam-5915	5	10	subgroup	subgroup	NOUN
ejpam-5915	5	11	induces	induce	VERB
ejpam-5915	5	12	an	an	DET
ejpam-5915	5	13	upper	upper	ADJ
ejpam-5915	5	14	-	-	PUNCT
ejpam-5915	5	15	level	level	NOUN
ejpam-5915	5	16	set	set	NOUN
ejpam-5915	5	17	forming	form	VERB
ejpam-5915	5	18	a	a	DET
ejpam-5915	5	19	classical	classical	ADJ
ejpam-5915	5	20	hom	hom	NOUN
ejpam-5915	5	21	-	-	PUNCT
ejpam-5915	5	22	subgroup	subgroup	NOUN
ejpam-5915	5	23	,	,	PUNCT
ejpam-5915	5	24	bridging	bridge	VERB
ejpam-5915	5	25	fuzzy	fuzzy	ADJ
ejpam-5915	5	26	group	group	NOUN
ejpam-5915	5	27	theory	theory	NOUN
ejpam-5915	5	28	and	and	CCONJ
ejpam-5915	5	29	hom	hom	NOUN
ejpam-5915	5	30	-	-	PUNCT
ejpam-5915	5	31	algebra	algebra	NOUN
ejpam-5915	5	32	.	.	PUNCT
ejpam-5915	6	1	we	we	PRON
ejpam-5915	6	2	further	far	ADV
ejpam-5915	6	3	analyze	analyze	VERB
ejpam-5915	6	4	the	the	DET
ejpam-5915	6	5	structural	structural	ADJ
ejpam-5915	6	6	relationships	relationship	NOUN
ejpam-5915	6	7	between	between	ADP
ejpam-5915	6	8	fuzzy	fuzzy	ADJ
ejpam-5915	6	9	hom	hom	NOUN
ejpam-5915	6	10	-	-	PUNCT
ejpam-5915	6	11	subgroups	subgroup	NOUN
ejpam-5915	6	12	and	and	CCONJ
ejpam-5915	6	13	hom	hom	NOUN
ejpam-5915	6	14	-	-	PUNCT
ejpam-5915	6	15	subgroups	subgroup	NOUN
ejpam-5915	6	16	.	.	PUNCT
ejpam-5915	7	1	illustrative	illustrative	ADJ
ejpam-5915	7	2	examples	example	NOUN
ejpam-5915	7	3	highlight	highlight	VERB
ejpam-5915	7	4	how	how	SCONJ
ejpam-5915	7	5	the	the	DET
ejpam-5915	7	6	twisting	twisting	NOUN
ejpam-5915	7	7	map	map	NOUN
ejpam-5915	7	8	influences	influence	VERB
ejpam-5915	7	9	fuzzy	fuzzy	ADJ
ejpam-5915	7	10	hom	hom	NOUN
ejpam-5915	7	11	-	-	PUNCT
ejpam-5915	7	12	structures	structure	NOUN
ejpam-5915	7	13	.	.	PUNCT
ejpam-5915	8	1	this	this	DET
ejpam-5915	8	2	study	study	NOUN
ejpam-5915	8	3	extends	extend	VERB
ejpam-5915	8	4	fuzzy	fuzzy	ADJ
ejpam-5915	8	5	algebra	algebra	NOUN
ejpam-5915	8	6	and	and	CCONJ
ejpam-5915	8	7	hom	hom	NOUN
ejpam-5915	8	8	-	-	PUNCT
ejpam-5915	8	9	group	group	NOUN
ejpam-5915	8	10	theory	theory	NOUN
ejpam-5915	8	11	,	,	PUNCT
ejpam-5915	8	12	with	with	ADP
ejpam-5915	8	13	potential	potential	ADJ
ejpam-5915	8	14	applications	application	NOUN
ejpam-5915	8	15	in	in	ADP
ejpam-5915	8	16	decision	decision	NOUN
ejpam-5915	8	17	-	-	PUNCT
ejpam-5915	8	18	making	making	NOUN
ejpam-5915	8	19	,	,	PUNCT
ejpam-5915	8	20	fuzzy	fuzzy	ADJ
ejpam-5915	8	21	control	control	NOUN
ejpam-5915	8	22	,	,	PUNCT
ejpam-5915	8	23	and	and	CCONJ
ejpam-5915	8	24	uncertainty	uncertainty	NOUN
ejpam-5915	8	25	modeling	modeling	NOUN
ejpam-5915	8	26	.	.	PUNCT
ejpam-5915	9	1	2020	2020	NUM
ejpam-5915	9	2	mathematics	mathematic	NOUN
ejpam-5915	9	3	subject	subject	NOUN
ejpam-5915	9	4	classifications	classification	NOUN
ejpam-5915	9	5	:	:	PUNCT
ejpam-5915	9	6	20n25	20n25	NUM
ejpam-5915	9	7	,	,	PUNCT
ejpam-5915	9	8	03e72	03e72	NUM
ejpam-5915	9	9	,	,	PUNCT
ejpam-5915	9	10	20m99	20m99	NUM
ejpam-5915	9	11	,	,	PUNCT
ejpam-5915	9	12	08a72	08a72	NUM
ejpam-5915	9	13	,	,	PUNCT
ejpam-5915	9	14	20d99	20d99	NUM
ejpam-5915	9	15	key	key	ADJ
ejpam-5915	9	16	words	word	NOUN
ejpam-5915	9	17	and	and	CCONJ
ejpam-5915	9	18	phrases	phrase	NOUN
ejpam-5915	9	19	:	:	PUNCT
ejpam-5915	9	20	fuzzy	fuzzy	ADJ
ejpam-5915	9	21	hom	hom	NOUN
ejpam-5915	9	22	-	-	PUNCT
ejpam-5915	9	23	groups	group	NOUN
ejpam-5915	9	24	,	,	PUNCT
ejpam-5915	9	25	fuzzy	fuzzy	ADJ
ejpam-5915	9	26	hom	hom	NOUN
ejpam-5915	9	27	-	-	PUNCT
ejpam-5915	9	28	subgroups	subgroup	NOUN
ejpam-5915	9	29	,	,	PUNCT
ejpam-5915	9	30	hom	hom	NOUN
ejpam-5915	9	31	-	-	PUNCT
ejpam-5915	9	32	groups	group	NOUN
ejpam-5915	9	33	,	,	PUNCT
ejpam-5915	9	34	fuzzy	fuzzy	ADJ
ejpam-5915	9	35	algebraic	algebraic	ADJ
ejpam-5915	9	36	structures	structure	NOUN
ejpam-5915	9	37	,	,	PUNCT
ejpam-5915	9	38	hom	hom	ADV
ejpam-5915	9	39	-	-	PUNCT
ejpam-5915	9	40	algebraic	algebraic	ADJ
ejpam-5915	9	41	systems	system	NOUN
ejpam-5915	9	42	,	,	PUNCT
ejpam-5915	9	43	generalized	generalize	VERB
ejpam-5915	9	44	fuzzy	fuzzy	ADJ
ejpam-5915	9	45	groups	group	NOUN
ejpam-5915	9	46	,	,	PUNCT
ejpam-5915	9	47	fuzzy	fuzzy	ADJ
ejpam-5915	9	48	normal	normal	ADJ
ejpam-5915	9	49	subgroups	subgroup	NOUN
ejpam-5915	9	50	,	,	PUNCT
ejpam-5915	9	51	hom	hom	NOUN
ejpam-5915	9	52	-	-	PUNCT
ejpam-5915	9	53	normal	normal	ADJ
ejpam-5915	9	54	subgroups	subgroup	NOUN
ejpam-5915	9	55	1	1	NUM
ejpam-5915	9	56	.	.	X
ejpam-5915	10	1	introduction	introduction	NOUN
ejpam-5915	10	2	fuzzy	fuzzy	ADJ
ejpam-5915	10	3	set	set	NOUN
ejpam-5915	10	4	theory	theory	NOUN
ejpam-5915	10	5	and	and	CCONJ
ejpam-5915	10	6	algebraic	algebraic	ADJ
ejpam-5915	10	7	structures	structure	NOUN
ejpam-5915	10	8	have	have	AUX
ejpam-5915	10	9	been	be	AUX
ejpam-5915	10	10	widely	widely	ADV
ejpam-5915	10	11	studied	study	VERB
ejpam-5915	10	12	because	because	SCONJ
ejpam-5915	10	13	they	they	PRON
ejpam-5915	10	14	help	help	VERB
ejpam-5915	10	15	model	model	VERB
ejpam-5915	10	16	uncertainty	uncertainty	NOUN
ejpam-5915	10	17	in	in	ADP
ejpam-5915	10	18	a	a	DET
ejpam-5915	10	19	precise	precise	ADJ
ejpam-5915	10	20	mathematical	mathematical	ADJ
ejpam-5915	10	21	way	way	NOUN
ejpam-5915	10	22	.	.	PUNCT
ejpam-5915	11	1	introduced	introduce	VERB
ejpam-5915	11	2	by	by	ADP
ejpam-5915	11	3	lotfi	lotfi	PROPN
ejpam-5915	11	4	zadeh	zadeh	PROPN
ejpam-5915	11	5	in	in	ADP
ejpam-5915	11	6	1965	1965	NUM
ejpam-5915	11	7	,	,	PUNCT
ejpam-5915	11	8	fuzzy	fuzzy	ADJ
ejpam-5915	11	9	sets	set	NOUN
ejpam-5915	11	10	provide	provide	VERB
ejpam-5915	11	11	a	a	DET
ejpam-5915	11	12	mathematical	mathematical	ADJ
ejpam-5915	11	13	framework	framework	NOUN
ejpam-5915	11	14	for	for	ADP
ejpam-5915	11	15	representing	represent	VERB
ejpam-5915	11	16	and	and	CCONJ
ejpam-5915	11	17	handling	handle	VERB
ejpam-5915	11	18	imprecise	imprecise	NOUN
ejpam-5915	11	19	and	and	CCONJ
ejpam-5915	11	20	uncertain	uncertain	ADJ
ejpam-5915	11	21	information	information	NOUN
ejpam-5915	12	1	[	[	X
ejpam-5915	12	2	1	1	NUM
ejpam-5915	12	3	]	]	PUNCT
ejpam-5915	12	4	.	.	PUNCT
ejpam-5915	13	1	since	since	SCONJ
ejpam-5915	13	2	rosenfeld	rosenfeld	PROPN
ejpam-5915	13	3	introduced	introduce	VERB
ejpam-5915	13	4	fuzzy	fuzzy	ADJ
ejpam-5915	13	5	groups	group	NOUN
ejpam-5915	13	6	[	[	X
ejpam-5915	13	7	2	2	NUM
ejpam-5915	13	8	]	]	PUNCT
ejpam-5915	13	9	,	,	PUNCT
ejpam-5915	13	10	researchers	researcher	NOUN
ejpam-5915	13	11	have	have	AUX
ejpam-5915	13	12	extended	extend	VERB
ejpam-5915	13	13	this	this	DET
ejpam-5915	13	14	idea	idea	NOUN
ejpam-5915	13	15	to	to	ADP
ejpam-5915	13	16	fuzzy	fuzzy	ADJ
ejpam-5915	13	17	rings	ring	NOUN
ejpam-5915	13	18	,	,	PUNCT
ejpam-5915	13	19	fuzzy	fuzzy	ADJ
ejpam-5915	13	20	lie	lie	NOUN
ejpam-5915	13	21	algebras	algebra	NOUN
ejpam-5915	13	22	,	,	PUNCT
ejpam-5915	13	23	and	and	CCONJ
ejpam-5915	13	24	fuzzy	fuzzy	ADJ
ejpam-5915	13	25	modules	module	NOUN
ejpam-5915	13	26	,	,	PUNCT
ejpam-5915	13	27	which	which	PRON
ejpam-5915	13	28	have	have	VERB
ejpam-5915	13	29	applications	application	NOUN
ejpam-5915	13	30	in	in	ADP
ejpam-5915	13	31	decision	decision	NOUN
ejpam-5915	13	32	-	-	PUNCT
ejpam-5915	13	33	making	making	NOUN
ejpam-5915	13	34	,	,	PUNCT
ejpam-5915	13	35	control	control	NOUN
ejpam-5915	13	36	systems	system	NOUN
ejpam-5915	13	37	,	,	PUNCT
ejpam-5915	13	38	and	and	CCONJ
ejpam-5915	13	39	artificial	artificial	ADJ
ejpam-5915	13	40	intelligence	intelligence	NOUN
ejpam-5915	14	1	[	[	X
ejpam-5915	14	2	3–6	3–6	NUM
ejpam-5915	14	3	]	]	X
ejpam-5915	14	4	.	.	PUNCT
ejpam-5915	15	1	studies	study	NOUN
ejpam-5915	15	2	on	on	ADP
ejpam-5915	15	3	equivalence	equivalence	NOUN
ejpam-5915	15	4	relations	relation	NOUN
ejpam-5915	15	5	in	in	ADP
ejpam-5915	15	6	fuzzy	fuzzy	ADJ
ejpam-5915	15	7	subgroups	subgroup	NOUN
ejpam-5915	15	8	[	[	X
ejpam-5915	15	9	7	7	X
ejpam-5915	15	10	]	]	PUNCT
ejpam-5915	15	11	and	and	CCONJ
ejpam-5915	15	12	the	the	DET
ejpam-5915	15	13	classification	classification	NOUN
ejpam-5915	15	14	of	of	ADP
ejpam-5915	15	15	fuzzy	fuzzy	ADJ
ejpam-5915	15	16	normal	normal	ADJ
ejpam-5915	15	17	subgroups	subgroup	NOUN
ejpam-5915	15	18	in	in	ADP
ejpam-5915	15	19	finite	finite	ADJ
ejpam-5915	15	20	groups	group	NOUN
ejpam-5915	15	21	[	[	X
ejpam-5915	15	22	8	8	NUM
ejpam-5915	15	23	]	]	PUNCT
ejpam-5915	15	24	have	have	AUX
ejpam-5915	15	25	further	far	ADV
ejpam-5915	15	26	developed	develop	VERB
ejpam-5915	15	27	the	the	DET
ejpam-5915	15	28	theory	theory	NOUN
ejpam-5915	15	29	,	,	PUNCT
ejpam-5915	15	30	offering	offer	VERB
ejpam-5915	15	31	insights	insight	NOUN
ejpam-5915	15	32	into	into	ADP
ejpam-5915	15	33	their	their	PRON
ejpam-5915	15	34	structure	structure	NOUN
ejpam-5915	15	35	and	and	CCONJ
ejpam-5915	15	36	properties	property	NOUN
ejpam-5915	15	37	.	.	PUNCT
ejpam-5915	16	1	these	these	DET
ejpam-5915	16	2	contributions	contribution	NOUN
ejpam-5915	16	3	provide	provide	VERB
ejpam-5915	16	4	a	a	DET
ejpam-5915	16	5	useful	useful	ADJ
ejpam-5915	16	6	mathematical	mathematical	ADJ
ejpam-5915	16	7	framework	framework	NOUN
ejpam-5915	16	8	for	for	ADP
ejpam-5915	16	9	both	both	CCONJ
ejpam-5915	16	10	theoretical	theoretical	ADJ
ejpam-5915	16	11	and	and	CCONJ
ejpam-5915	16	12	practical	practical	ADJ
ejpam-5915	16	13	problems	problem	NOUN
ejpam-5915	16	14	.	.	PUNCT
ejpam-5915	17	1	doi	doi	NOUN
ejpam-5915	17	2	:	:	PUNCT
ejpam-5915	17	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5915	https://doi.org/10.29020/nybg.ejpam.v18i2.5915	NUM
ejpam-5915	17	4	email	email	NOUN
ejpam-5915	17	5	address	address	NOUN
ejpam-5915	17	6	:	:	PUNCT
ejpam-5915	17	7	shadi.s@yu.edu.jo	shadi.s@yu.edu.jo	ADJ
ejpam-5915	17	8	(	(	PUNCT
ejpam-5915	17	9	s.	s.	PROPN
ejpam-5915	17	10	shaqaqha	shaqaqha	PROPN
ejpam-5915	17	11	)	)	PUNCT
ejpam-5915	17	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5915	18	1	1	1	NUM
ejpam-5915	18	2	copyright	copyright	NOUN
ejpam-5915	18	3	:	:	PUNCT
ejpam-5915	18	4	©	©	PROPN
ejpam-5915	18	5	2025	2025	NUM
ejpam-5915	18	6	the	the	DET
ejpam-5915	18	7	author(s	author(s	NOUN
ejpam-5915	18	8	)	)	PUNCT
ejpam-5915	18	9	.	.	PUNCT
ejpam-5915	19	1	(	(	PUNCT
ejpam-5915	19	2	cc	cc	NOUN
ejpam-5915	19	3	by	by	ADP
ejpam-5915	19	4	-	-	PUNCT
ejpam-5915	19	5	nc	nc	PROPN
ejpam-5915	19	6	4.0	4.0	NUM
ejpam-5915	19	7	)	)	PUNCT
ejpam-5915	19	8	s.	s.	PROPN
ejpam-5915	19	9	shaqaqha	shaqaqha	PROPN
ejpam-5915	19	10	/	/	SYM
ejpam-5915	19	11	eur	eur	PROPN
ejpam-5915	19	12	.	.	PUNCT
ejpam-5915	20	1	j.	j.	PROPN
ejpam-5915	20	2	pure	pure	PROPN
ejpam-5915	20	3	appl	appl	PROPN
ejpam-5915	20	4	.	.	PROPN
ejpam-5915	20	5	math	math	PROPN
ejpam-5915	20	6	,	,	PUNCT
ejpam-5915	20	7	18	18	NUM
ejpam-5915	20	8	(	(	PUNCT
ejpam-5915	20	9	2	2	NUM
ejpam-5915	20	10	)	)	PUNCT
ejpam-5915	20	11	(	(	PUNCT
ejpam-5915	20	12	2025	2025	NUM
ejpam-5915	20	13	)	)	PUNCT
ejpam-5915	20	14	,	,	PUNCT
ejpam-5915	20	15	5915	5915	NUM
ejpam-5915	20	16	2	2	NUM
ejpam-5915	20	17	of	of	ADP
ejpam-5915	20	18	16	16	NUM
ejpam-5915	20	19	at	at	ADP
ejpam-5915	20	20	the	the	DET
ejpam-5915	20	21	same	same	ADJ
ejpam-5915	20	22	time	time	NOUN
ejpam-5915	20	23	,	,	PUNCT
ejpam-5915	20	24	algebraic	algebraic	ADJ
ejpam-5915	20	25	structures	structure	NOUN
ejpam-5915	20	26	have	have	AUX
ejpam-5915	20	27	been	be	AUX
ejpam-5915	20	28	generalized	generalize	VERB
ejpam-5915	20	29	using	use	VERB
ejpam-5915	20	30	hom	hom	NOUN
ejpam-5915	20	31	-	-	PUNCT
ejpam-5915	20	32	algebra	algebra	NOUN
ejpam-5915	20	33	theory	theory	NOUN
ejpam-5915	20	34	,	,	PUNCT
ejpam-5915	20	35	which	which	PRON
ejpam-5915	20	36	introduces	introduce	VERB
ejpam-5915	20	37	a	a	DET
ejpam-5915	20	38	twisting	twisting	NOUN
ejpam-5915	20	39	map	map	NOUN
ejpam-5915	20	40	α	α	NOUN
ejpam-5915	20	41	that	that	PRON
ejpam-5915	20	42	modifies	modify	VERB
ejpam-5915	20	43	traditional	traditional	ADJ
ejpam-5915	20	44	algebraic	algebraic	ADJ
ejpam-5915	20	45	properties	property	NOUN
ejpam-5915	20	46	.	.	PUNCT
ejpam-5915	21	1	this	this	DET
ejpam-5915	21	2	approach	approach	NOUN
ejpam-5915	21	3	has	have	AUX
ejpam-5915	21	4	led	lead	VERB
ejpam-5915	21	5	to	to	ADP
ejpam-5915	21	6	new	new	ADJ
ejpam-5915	21	7	generalizations	generalization	NOUN
ejpam-5915	21	8	of	of	ADP
ejpam-5915	21	9	groups	group	NOUN
ejpam-5915	21	10	,	,	PUNCT
ejpam-5915	21	11	lie	lie	NOUN
ejpam-5915	21	12	algebras	algebra	NOUN
ejpam-5915	21	13	,	,	PUNCT
ejpam-5915	21	14	and	and	CCONJ
ejpam-5915	21	15	related	related	ADJ
ejpam-5915	21	16	structures	structure	NOUN
ejpam-5915	21	17	[	[	X
ejpam-5915	21	18	9	9	NUM
ejpam-5915	21	19	–	–	SYM
ejpam-5915	21	20	13	13	NUM
ejpam-5915	21	21	]	]	PUNCT
ejpam-5915	21	22	.	.	PUNCT
ejpam-5915	22	1	hom	hom	X
ejpam-5915	22	2	-	-	PUNCT
ejpam-5915	22	3	algebraic	algebraic	ADJ
ejpam-5915	22	4	structures	structure	NOUN
ejpam-5915	22	5	have	have	AUX
ejpam-5915	22	6	been	be	AUX
ejpam-5915	22	7	particularly	particularly	ADV
ejpam-5915	22	8	useful	useful	ADJ
ejpam-5915	22	9	in	in	ADP
ejpam-5915	22	10	the	the	DET
ejpam-5915	22	11	study	study	NOUN
ejpam-5915	22	12	of	of	ADP
ejpam-5915	22	13	deformations	deformation	NOUN
ejpam-5915	22	14	,	,	PUNCT
ejpam-5915	22	15	quantum	quantum	ADJ
ejpam-5915	22	16	symmetries	symmetry	NOUN
ejpam-5915	22	17	,	,	PUNCT
ejpam-5915	22	18	and	and	CCONJ
ejpam-5915	22	19	non	non	ADJ
ejpam-5915	22	20	-	-	ADJ
ejpam-5915	22	21	associative	associative	ADJ
ejpam-5915	22	22	algebraic	algebraic	ADJ
ejpam-5915	22	23	systems	system	NOUN
ejpam-5915	22	24	.	.	PUNCT
ejpam-5915	23	1	the	the	DET
ejpam-5915	23	2	development	development	NOUN
ejpam-5915	23	3	of	of	ADP
ejpam-5915	23	4	homgroup	homgroup	PROPN
ejpam-5915	23	5	theory	theory	PROPN
ejpam-5915	23	6	by	by	ADP
ejpam-5915	23	7	chen	chen	PROPN
ejpam-5915	23	8	et	et	PROPN
ejpam-5915	23	9	al	al	PROPN
ejpam-5915	23	10	.	.	PUNCT
ejpam-5915	24	1	[	[	X
ejpam-5915	24	2	14	14	NUM
ejpam-5915	24	3	]	]	PUNCT
ejpam-5915	24	4	and	and	CCONJ
ejpam-5915	24	5	further	further	ADJ
ejpam-5915	24	6	extensions	extension	NOUN
ejpam-5915	24	7	such	such	ADJ
ejpam-5915	24	8	as	as	ADP
ejpam-5915	24	9	ordered	order	VERB
ejpam-5915	24	10	hom	hom	NOUN
ejpam-5915	24	11	-	-	PUNCT
ejpam-5915	24	12	groups	group	NOUN
ejpam-5915	25	1	[	[	X
ejpam-5915	25	2	15	15	NUM
ejpam-5915	25	3	]	]	PUNCT
ejpam-5915	25	4	have	have	AUX
ejpam-5915	25	5	expanded	expand	VERB
ejpam-5915	25	6	algebraic	algebraic	ADJ
ejpam-5915	25	7	research	research	NOUN
ejpam-5915	25	8	by	by	ADP
ejpam-5915	25	9	adding	add	VERB
ejpam-5915	25	10	more	more	ADJ
ejpam-5915	25	11	flexibility	flexibility	NOUN
ejpam-5915	25	12	to	to	ADP
ejpam-5915	25	13	classical	classical	ADJ
ejpam-5915	25	14	group	group	NOUN
ejpam-5915	25	15	theory	theory	NOUN
ejpam-5915	25	16	through	through	ADP
ejpam-5915	25	17	the	the	DET
ejpam-5915	25	18	twisting	twisting	NOUN
ejpam-5915	25	19	map	map	NOUN
ejpam-5915	25	20	α	α	NOUN
ejpam-5915	25	21	.	.	PUNCT
ejpam-5915	26	1	despite	despite	SCONJ
ejpam-5915	26	2	the	the	DET
ejpam-5915	26	3	progress	progress	NOUN
ejpam-5915	26	4	in	in	ADP
ejpam-5915	26	5	fuzzy	fuzzy	ADJ
ejpam-5915	26	6	algebra	algebra	NOUN
ejpam-5915	26	7	and	and	CCONJ
ejpam-5915	26	8	hom	hom	NOUN
ejpam-5915	26	9	-	-	PUNCT
ejpam-5915	26	10	group	group	NOUN
ejpam-5915	26	11	theory	theory	NOUN
ejpam-5915	26	12	,	,	PUNCT
ejpam-5915	26	13	the	the	DET
ejpam-5915	26	14	study	study	NOUN
ejpam-5915	26	15	of	of	ADP
ejpam-5915	26	16	fuzzy	fuzzy	ADJ
ejpam-5915	26	17	homgroups	homgroup	NOUN
ejpam-5915	26	18	and	and	CCONJ
ejpam-5915	26	19	their	their	PRON
ejpam-5915	26	20	substructures	substructure	NOUN
ejpam-5915	26	21	remains	remain	VERB
ejpam-5915	26	22	largely	largely	ADV
ejpam-5915	26	23	unexplored	unexplored	ADJ
ejpam-5915	26	24	.	.	PUNCT
ejpam-5915	27	1	fuzzy	fuzzy	ADJ
ejpam-5915	27	2	hom	hom	NOUN
ejpam-5915	27	3	-	-	PUNCT
ejpam-5915	27	4	groups	group	NOUN
ejpam-5915	27	5	extend	extend	VERB
ejpam-5915	27	6	both	both	DET
ejpam-5915	27	7	fuzzy	fuzzy	ADJ
ejpam-5915	27	8	groups	group	NOUN
ejpam-5915	27	9	[	[	X
ejpam-5915	27	10	2	2	NUM
ejpam-5915	27	11	]	]	PUNCT
ejpam-5915	27	12	and	and	CCONJ
ejpam-5915	27	13	hom	hom	NOUN
ejpam-5915	27	14	-	-	PUNCT
ejpam-5915	27	15	groups	group	NOUN
ejpam-5915	27	16	[	[	X
ejpam-5915	27	17	14	14	NUM
ejpam-5915	27	18	]	]	PUNCT
ejpam-5915	27	19	by	by	ADP
ejpam-5915	27	20	incorporating	incorporate	VERB
ejpam-5915	27	21	a	a	DET
ejpam-5915	27	22	twisting	twisting	NOUN
ejpam-5915	27	23	map	map	NOUN
ejpam-5915	27	24	α	α	NOUN
ejpam-5915	27	25	,	,	PUNCT
ejpam-5915	27	26	which	which	PRON
ejpam-5915	27	27	adds	add	VERB
ejpam-5915	27	28	more	more	ADJ
ejpam-5915	27	29	structural	structural	ADJ
ejpam-5915	27	30	flexibility	flexibility	NOUN
ejpam-5915	27	31	while	while	SCONJ
ejpam-5915	27	32	maintaining	maintain	VERB
ejpam-5915	27	33	key	key	ADJ
ejpam-5915	27	34	fuzzy	fuzzy	ADJ
ejpam-5915	27	35	properties	property	NOUN
ejpam-5915	27	36	.	.	PUNCT
ejpam-5915	28	1	this	this	DET
ejpam-5915	28	2	work	work	NOUN
ejpam-5915	28	3	systematically	systematically	ADV
ejpam-5915	28	4	develops	develop	VERB
ejpam-5915	28	5	the	the	DET
ejpam-5915	28	6	theory	theory	NOUN
ejpam-5915	28	7	of	of	ADP
ejpam-5915	28	8	fuzzy	fuzzy	ADJ
ejpam-5915	28	9	hom	hom	NOUN
ejpam-5915	28	10	-	-	PUNCT
ejpam-5915	28	11	subgroups	subgroup	NOUN
ejpam-5915	28	12	and	and	CCONJ
ejpam-5915	28	13	fuzzy	fuzzy	ADJ
ejpam-5915	28	14	hom	hom	NOUN
ejpam-5915	28	15	-	-	PUNCT
ejpam-5915	28	16	normal	normal	ADJ
ejpam-5915	28	17	subgroups	subgroup	NOUN
ejpam-5915	28	18	,	,	PUNCT
ejpam-5915	28	19	linking	link	VERB
ejpam-5915	28	20	fuzzy	fuzzy	ADJ
ejpam-5915	28	21	group	group	NOUN
ejpam-5915	28	22	theory	theory	NOUN
ejpam-5915	28	23	with	with	ADP
ejpam-5915	28	24	hom	hom	ADV
ejpam-5915	28	25	-	-	PUNCT
ejpam-5915	28	26	algebraic	algebraic	ADJ
ejpam-5915	28	27	structures	structure	NOUN
ejpam-5915	28	28	.	.	PUNCT
ejpam-5915	29	1	previous	previous	ADJ
ejpam-5915	29	2	research	research	NOUN
ejpam-5915	29	3	on	on	ADP
ejpam-5915	29	4	fuzzy	fuzzy	ADJ
ejpam-5915	29	5	cosets	coset	NOUN
ejpam-5915	29	6	,	,	PUNCT
ejpam-5915	29	7	fuzzy	fuzzy	ADJ
ejpam-5915	29	8	normal	normal	ADJ
ejpam-5915	29	9	subgroups	subgroup	NOUN
ejpam-5915	29	10	,	,	PUNCT
ejpam-5915	29	11	and	and	CCONJ
ejpam-5915	29	12	their	their	PRON
ejpam-5915	29	13	homomorphic	homomorphic	ADJ
ejpam-5915	29	14	images	image	NOUN
ejpam-5915	29	15	[	[	X
ejpam-5915	29	16	8	8	NUM
ejpam-5915	29	17	,	,	PUNCT
ejpam-5915	29	18	16	16	NUM
ejpam-5915	29	19	]	]	PUNCT
ejpam-5915	29	20	suggests	suggest	VERB
ejpam-5915	29	21	that	that	SCONJ
ejpam-5915	29	22	combining	combine	VERB
ejpam-5915	29	23	fuzzy	fuzzy	ADJ
ejpam-5915	29	24	set	set	NOUN
ejpam-5915	29	25	theory	theory	NOUN
ejpam-5915	29	26	with	with	ADP
ejpam-5915	29	27	hom	hom	NOUN
ejpam-5915	29	28	-	-	PUNCT
ejpam-5915	29	29	groups	group	NOUN
ejpam-5915	29	30	could	could	AUX
ejpam-5915	29	31	lead	lead	VERB
ejpam-5915	29	32	to	to	ADP
ejpam-5915	29	33	significant	significant	ADJ
ejpam-5915	29	34	new	new	ADJ
ejpam-5915	29	35	results	result	NOUN
ejpam-5915	29	36	.	.	PUNCT
ejpam-5915	30	1	additionally	additionally	ADV
ejpam-5915	30	2	,	,	PUNCT
ejpam-5915	30	3	intuitionistic	intuitionistic	ADJ
ejpam-5915	30	4	fuzzy	fuzzy	ADJ
ejpam-5915	30	5	structures	structure	NOUN
ejpam-5915	30	6	provide	provide	VERB
ejpam-5915	30	7	a	a	DET
ejpam-5915	30	8	foundation	foundation	NOUN
ejpam-5915	30	9	for	for	ADP
ejpam-5915	30	10	extended	extended	ADJ
ejpam-5915	30	11	logic	logic	NOUN
ejpam-5915	30	12	systems	system	NOUN
ejpam-5915	30	13	[	[	X
ejpam-5915	30	14	17	17	NUM
ejpam-5915	30	15	]	]	PUNCT
ejpam-5915	30	16	,	,	PUNCT
ejpam-5915	30	17	which	which	PRON
ejpam-5915	30	18	may	may	AUX
ejpam-5915	30	19	inspire	inspire	VERB
ejpam-5915	30	20	further	further	ADJ
ejpam-5915	30	21	generalizations	generalization	NOUN
ejpam-5915	30	22	of	of	ADP
ejpam-5915	30	23	fuzzy	fuzzy	ADJ
ejpam-5915	30	24	hom	hom	ADV
ejpam-5915	30	25	-	-	PUNCT
ejpam-5915	30	26	algebraic	algebraic	ADJ
ejpam-5915	30	27	systems	system	NOUN
ejpam-5915	30	28	.	.	PUNCT
ejpam-5915	31	1	by	by	ADP
ejpam-5915	31	2	combining	combine	VERB
ejpam-5915	31	3	fuzzy	fuzzy	ADJ
ejpam-5915	31	4	set	set	NOUN
ejpam-5915	31	5	theory	theory	NOUN
ejpam-5915	31	6	with	with	ADP
ejpam-5915	31	7	hom	hom	ADV
ejpam-5915	31	8	-	-	PUNCT
ejpam-5915	31	9	algebraic	algebraic	ADJ
ejpam-5915	31	10	structures	structure	NOUN
ejpam-5915	31	11	,	,	PUNCT
ejpam-5915	31	12	we	we	PRON
ejpam-5915	31	13	define	define	VERB
ejpam-5915	31	14	fuzzy	fuzzy	ADJ
ejpam-5915	31	15	homsubgroups	homsubgroup	NOUN
ejpam-5915	31	16	and	and	CCONJ
ejpam-5915	31	17	fuzzy	fuzzy	ADJ
ejpam-5915	31	18	hom	hom	NOUN
ejpam-5915	31	19	-	-	PUNCT
ejpam-5915	31	20	normal	normal	ADJ
ejpam-5915	31	21	subgroups	subgroup	NOUN
ejpam-5915	31	22	,	,	PUNCT
ejpam-5915	31	23	establishing	establish	VERB
ejpam-5915	31	24	their	their	PRON
ejpam-5915	31	25	fundamental	fundamental	ADJ
ejpam-5915	31	26	properties	property	NOUN
ejpam-5915	31	27	and	and	CCONJ
ejpam-5915	31	28	examining	examine	VERB
ejpam-5915	31	29	their	their	PRON
ejpam-5915	31	30	behavior	behavior	NOUN
ejpam-5915	31	31	under	under	ADP
ejpam-5915	31	32	the	the	DET
ejpam-5915	31	33	structural	structural	ADJ
ejpam-5915	31	34	constraints	constraint	NOUN
ejpam-5915	31	35	imposed	impose	VERB
ejpam-5915	31	36	by	by	ADP
ejpam-5915	31	37	the	the	DET
ejpam-5915	31	38	twisting	twisting	NOUN
ejpam-5915	31	39	map	map	NOUN
ejpam-5915	31	40	α	α	NOUN
ejpam-5915	31	41	.	.	PUNCT
ejpam-5915	32	1	this	this	DET
ejpam-5915	32	2	study	study	NOUN
ejpam-5915	32	3	builds	build	VERB
ejpam-5915	32	4	on	on	ADP
ejpam-5915	32	5	recent	recent	ADJ
ejpam-5915	32	6	developments	development	NOUN
ejpam-5915	32	7	in	in	ADP
ejpam-5915	32	8	fuzzy	fuzzy	ADJ
ejpam-5915	32	9	algebraic	algebraic	ADJ
ejpam-5915	32	10	systems	system	NOUN
ejpam-5915	32	11	,	,	PUNCT
ejpam-5915	32	12	such	such	ADJ
ejpam-5915	32	13	as	as	ADP
ejpam-5915	32	14	fuzzy	fuzzy	ADJ
ejpam-5915	32	15	(	(	PUNCT
ejpam-5915	32	16	n-)lie	n-)lie	PROPN
ejpam-5915	32	17	algebras	algebras	X
ejpam-5915	33	1	[	[	X
ejpam-5915	33	2	18	18	NUM
ejpam-5915	33	3	,	,	PUNCT
ejpam-5915	33	4	19	19	NUM
ejpam-5915	33	5	]	]	PUNCT
ejpam-5915	33	6	,	,	PUNCT
ejpam-5915	33	7	fuzzy	fuzzy	ADJ
ejpam-5915	33	8	hom	hom	NOUN
ejpam-5915	33	9	-	-	PUNCT
ejpam-5915	33	10	lie	lie	NOUN
ejpam-5915	33	11	ideals	ideal	NOUN
ejpam-5915	33	12	[	[	X
ejpam-5915	33	13	20	20	NUM
ejpam-5915	33	14	]	]	PUNCT
ejpam-5915	33	15	,	,	PUNCT
ejpam-5915	33	16	and	and	CCONJ
ejpam-5915	33	17	complex	complex	ADJ
ejpam-5915	33	18	fuzzy	fuzzy	ADJ
ejpam-5915	33	19	gamma	gamma	NOUN
ejpam-5915	33	20	rings	ring	NOUN
ejpam-5915	34	1	[	[	X
ejpam-5915	34	2	21	21	NUM
ejpam-5915	34	3	,	,	PUNCT
ejpam-5915	34	4	22	22	NUM
ejpam-5915	34	5	]	]	PUNCT
ejpam-5915	34	6	.	.	PUNCT
ejpam-5915	35	1	it	it	PRON
ejpam-5915	35	2	also	also	ADV
ejpam-5915	35	3	aligns	align	VERB
ejpam-5915	35	4	with	with	ADP
ejpam-5915	35	5	structural	structural	ADJ
ejpam-5915	35	6	studies	study	NOUN
ejpam-5915	35	7	of	of	ADP
ejpam-5915	35	8	hom	hom	NOUN
ejpam-5915	35	9	-	-	PUNCT
ejpam-5915	35	10	algebras	algebras	PROPN
ejpam-5915	35	11	and	and	CCONJ
ejpam-5915	35	12	their	their	PRON
ejpam-5915	35	13	fuzzy	fuzzy	ADJ
ejpam-5915	35	14	extensions	extension	NOUN
ejpam-5915	35	15	[	[	X
ejpam-5915	35	16	9	9	NUM
ejpam-5915	35	17	,	,	PUNCT
ejpam-5915	35	18	10	10	NUM
ejpam-5915	35	19	,	,	PUNCT
ejpam-5915	35	20	15	15	NUM
ejpam-5915	35	21	]	]	PUNCT
ejpam-5915	35	22	.	.	PUNCT
ejpam-5915	36	1	the	the	DET
ejpam-5915	36	2	relationships	relationship	NOUN
ejpam-5915	36	3	between	between	ADP
ejpam-5915	36	4	fuzzy	fuzzy	ADJ
ejpam-5915	36	5	hom	hom	NOUN
ejpam-5915	36	6	-	-	PUNCT
ejpam-5915	36	7	subgroups	subgroup	NOUN
ejpam-5915	36	8	,	,	PUNCT
ejpam-5915	36	9	classical	classical	ADJ
ejpam-5915	36	10	fuzzy	fuzzy	ADJ
ejpam-5915	36	11	groups	group	NOUN
ejpam-5915	36	12	,	,	PUNCT
ejpam-5915	36	13	and	and	CCONJ
ejpam-5915	36	14	upperlevel	upperlevel	NOUN
ejpam-5915	36	15	sets	set	NOUN
ejpam-5915	36	16	of	of	ADP
ejpam-5915	36	17	hom	hom	NOUN
ejpam-5915	36	18	-	-	PUNCT
ejpam-5915	36	19	subgroups	subgroup	NOUN
ejpam-5915	36	20	are	be	AUX
ejpam-5915	36	21	explored	explore	VERB
ejpam-5915	36	22	,	,	PUNCT
ejpam-5915	36	23	extending	extend	VERB
ejpam-5915	36	24	the	the	DET
ejpam-5915	36	25	scope	scope	NOUN
ejpam-5915	36	26	of	of	ADP
ejpam-5915	36	27	generalized	generalized	ADJ
ejpam-5915	36	28	algebraic	algebraic	ADJ
ejpam-5915	36	29	systems	system	NOUN
ejpam-5915	36	30	.	.	PUNCT
ejpam-5915	37	1	this	this	DET
ejpam-5915	37	2	work	work	NOUN
ejpam-5915	37	3	lays	lay	VERB
ejpam-5915	37	4	the	the	DET
ejpam-5915	37	5	groundwork	groundwork	NOUN
ejpam-5915	37	6	for	for	ADP
ejpam-5915	37	7	further	further	ADJ
ejpam-5915	37	8	studies	study	NOUN
ejpam-5915	37	9	on	on	ADP
ejpam-5915	37	10	fuzzy	fuzzy	ADJ
ejpam-5915	37	11	hom	hom	NOUN
ejpam-5915	37	12	-	-	PUNCT
ejpam-5915	37	13	structures	structure	NOUN
ejpam-5915	37	14	,	,	PUNCT
ejpam-5915	37	15	including	include	VERB
ejpam-5915	37	16	their	their	PRON
ejpam-5915	37	17	links	link	NOUN
ejpam-5915	37	18	to	to	ADP
ejpam-5915	37	19	intuitionistic	intuitionistic	ADJ
ejpam-5915	37	20	fuzzy	fuzzy	ADJ
ejpam-5915	37	21	logic	logic	NOUN
ejpam-5915	37	22	and	and	CCONJ
ejpam-5915	37	23	algebraic	algebraic	ADJ
ejpam-5915	37	24	models	model	NOUN
ejpam-5915	37	25	in	in	ADP
ejpam-5915	37	26	uncertain	uncertain	ADJ
ejpam-5915	37	27	environments	environment	NOUN
ejpam-5915	37	28	.	.	PUNCT
ejpam-5915	38	1	this	this	DET
ejpam-5915	38	2	paper	paper	NOUN
ejpam-5915	38	3	is	be	AUX
ejpam-5915	38	4	organized	organize	VERB
ejpam-5915	38	5	as	as	SCONJ
ejpam-5915	38	6	follows	follow	VERB
ejpam-5915	38	7	.	.	PUNCT
ejpam-5915	39	1	section	section	NOUN
ejpam-5915	39	2	2	2	NUM
ejpam-5915	39	3	provides	provide	VERB
ejpam-5915	39	4	a	a	DET
ejpam-5915	39	5	brief	brief	ADJ
ejpam-5915	39	6	review	review	NOUN
ejpam-5915	39	7	of	of	ADP
ejpam-5915	39	8	hom	hom	NOUN
ejpam-5915	39	9	-	-	PUNCT
ejpam-5915	39	10	groups	group	NOUN
ejpam-5915	39	11	,	,	PUNCT
ejpam-5915	39	12	including	include	VERB
ejpam-5915	39	13	their	their	PRON
ejpam-5915	39	14	essential	essential	ADJ
ejpam-5915	39	15	properties	property	NOUN
ejpam-5915	39	16	and	and	CCONJ
ejpam-5915	39	17	examples	example	NOUN
ejpam-5915	39	18	.	.	PUNCT
ejpam-5915	40	1	section	section	NOUN
ejpam-5915	40	2	3	3	NUM
ejpam-5915	40	3	introduces	introduce	NOUN
ejpam-5915	40	4	hom	hom	NOUN
ejpam-5915	40	5	-	-	PUNCT
ejpam-5915	40	6	subgroups	subgroup	NOUN
ejpam-5915	40	7	and	and	CCONJ
ejpam-5915	40	8	hom	hom	ADV
ejpam-5915	40	9	-	-	PUNCT
ejpam-5915	40	10	normal	normal	ADJ
ejpam-5915	40	11	subgroups	subgroup	NOUN
ejpam-5915	40	12	,	,	PUNCT
ejpam-5915	40	13	establishing	establish	VERB
ejpam-5915	40	14	their	their	PRON
ejpam-5915	40	15	foundational	foundational	ADJ
ejpam-5915	40	16	properties	property	NOUN
ejpam-5915	40	17	.	.	PUNCT
ejpam-5915	41	1	section	section	NOUN
ejpam-5915	41	2	4	4	NUM
ejpam-5915	41	3	focuses	focus	VERB
ejpam-5915	41	4	on	on	ADP
ejpam-5915	41	5	fuzzy	fuzzy	ADJ
ejpam-5915	41	6	hom	hom	NOUN
ejpam-5915	41	7	-	-	PUNCT
ejpam-5915	41	8	subgroups	subgroup	NOUN
ejpam-5915	41	9	,	,	PUNCT
ejpam-5915	41	10	while	while	SCONJ
ejpam-5915	41	11	section	section	NOUN
ejpam-5915	41	12	5	5	NUM
ejpam-5915	41	13	develops	develop	VERB
ejpam-5915	41	14	the	the	DET
ejpam-5915	41	15	concept	concept	NOUN
ejpam-5915	41	16	of	of	ADP
ejpam-5915	41	17	fuzzy	fuzzy	ADJ
ejpam-5915	41	18	hom	hom	NOUN
ejpam-5915	41	19	-	-	PUNCT
ejpam-5915	41	20	normal	normal	ADJ
ejpam-5915	41	21	subgroups	subgroup	NOUN
ejpam-5915	41	22	.	.	PUNCT
ejpam-5915	42	1	in	in	ADP
ejpam-5915	42	2	section	section	NOUN
ejpam-5915	42	3	6	6	NUM
ejpam-5915	42	4	,	,	PUNCT
ejpam-5915	42	5	we	we	PRON
ejpam-5915	42	6	explore	explore	VERB
ejpam-5915	42	7	the	the	DET
ejpam-5915	42	8	relationships	relationship	NOUN
ejpam-5915	42	9	between	between	ADP
ejpam-5915	42	10	fuzzy	fuzzy	ADJ
ejpam-5915	42	11	hom	hom	NOUN
ejpam-5915	42	12	-	-	PUNCT
ejpam-5915	42	13	subgroups	subgroup	NOUN
ejpam-5915	42	14	and	and	CCONJ
ejpam-5915	42	15	classical	classical	ADJ
ejpam-5915	42	16	hom	hom	NOUN
ejpam-5915	42	17	-	-	PUNCT
ejpam-5915	42	18	subgroups	subgroup	NOUN
ejpam-5915	42	19	.	.	PUNCT
ejpam-5915	43	1	section	section	NOUN
ejpam-5915	43	2	7	7	NUM
ejpam-5915	43	3	investigates	investigate	VERB
ejpam-5915	43	4	the	the	DET
ejpam-5915	43	5	connections	connection	NOUN
ejpam-5915	43	6	between	between	ADP
ejpam-5915	43	7	strong	strong	ADJ
ejpam-5915	43	8	fuzzy	fuzzy	ADJ
ejpam-5915	43	9	hom	hom	NOUN
ejpam-5915	43	10	-	-	PUNCT
ejpam-5915	43	11	subgroups	subgroup	NOUN
ejpam-5915	43	12	and	and	CCONJ
ejpam-5915	43	13	hom	hom	NOUN
ejpam-5915	43	14	-	-	PUNCT
ejpam-5915	43	15	subgroups	subgroup	NOUN
ejpam-5915	43	16	.	.	PUNCT
ejpam-5915	44	1	section	section	NOUN
ejpam-5915	44	2	8	8	NUM
ejpam-5915	44	3	examines	examine	VERB
ejpam-5915	44	4	the	the	DET
ejpam-5915	44	5	relationships	relationship	NOUN
ejpam-5915	44	6	between	between	ADP
ejpam-5915	44	7	fuzzy	fuzzy	ADJ
ejpam-5915	44	8	hom	hom	NOUN
ejpam-5915	44	9	-	-	PUNCT
ejpam-5915	44	10	normal	normal	ADJ
ejpam-5915	44	11	subgroups	subgroup	NOUN
ejpam-5915	44	12	and	and	CCONJ
ejpam-5915	44	13	classical	classical	ADJ
ejpam-5915	44	14	hom	hom	ADJ
ejpam-5915	44	15	-	-	PUNCT
ejpam-5915	44	16	normal	normal	ADJ
ejpam-5915	44	17	subgroups	subgroup	NOUN
ejpam-5915	44	18	,	,	PUNCT
ejpam-5915	44	19	while	while	SCONJ
ejpam-5915	44	20	section	section	NOUN
ejpam-5915	44	21	9	9	NUM
ejpam-5915	44	22	extends	extend	VERB
ejpam-5915	44	23	this	this	DET
ejpam-5915	44	24	discussion	discussion	NOUN
ejpam-5915	44	25	to	to	ADP
ejpam-5915	44	26	strong	strong	ADJ
ejpam-5915	44	27	fuzzy	fuzzy	ADJ
ejpam-5915	44	28	hom	hom	NOUN
ejpam-5915	44	29	-	-	PUNCT
ejpam-5915	44	30	normal	normal	ADJ
ejpam-5915	44	31	subgroups	subgroup	NOUN
ejpam-5915	44	32	.	.	PUNCT
ejpam-5915	45	1	finally	finally	ADV
ejpam-5915	45	2	,	,	PUNCT
ejpam-5915	45	3	section	section	NOUN
ejpam-5915	45	4	10	10	NUM
ejpam-5915	45	5	concludes	conclude	VERB
ejpam-5915	45	6	the	the	DET
ejpam-5915	45	7	paper	paper	NOUN
ejpam-5915	45	8	and	and	CCONJ
ejpam-5915	45	9	highlights	highlight	NOUN
ejpam-5915	45	10	directions	direction	NOUN
ejpam-5915	45	11	for	for	ADP
ejpam-5915	45	12	future	future	ADJ
ejpam-5915	45	13	research	research	NOUN
ejpam-5915	45	14	,	,	PUNCT
ejpam-5915	45	15	including	include	VERB
ejpam-5915	45	16	extensions	extension	NOUN
ejpam-5915	45	17	to	to	ADP
ejpam-5915	45	18	intuitionistic	intuitionistic	ADJ
ejpam-5915	45	19	fuzzy	fuzzy	ADJ
ejpam-5915	45	20	,	,	PUNCT
ejpam-5915	45	21	bipolar	bipolar	ADJ
ejpam-5915	45	22	fuzzy	fuzzy	ADJ
ejpam-5915	45	23	,	,	PUNCT
ejpam-5915	45	24	and	and	CCONJ
ejpam-5915	45	25	generalized	generalize	VERB
ejpam-5915	45	26	fuzzy	fuzzy	ADJ
ejpam-5915	45	27	hom	hom	NOUN
ejpam-5915	45	28	-	-	PUNCT
ejpam-5915	45	29	structures	structure	NOUN
ejpam-5915	45	30	,	,	PUNCT
ejpam-5915	45	31	as	as	ADV
ejpam-5915	45	32	well	well	ADV
ejpam-5915	45	33	as	as	ADP
ejpam-5915	45	34	applications	application	NOUN
ejpam-5915	45	35	in	in	ADP
ejpam-5915	45	36	uncertain	uncertain	ADJ
ejpam-5915	45	37	environments	environment	NOUN
ejpam-5915	45	38	.	.	PUNCT
ejpam-5915	46	1	s.	s.	PROPN
ejpam-5915	46	2	shaqaqha	shaqaqha	PROPN
ejpam-5915	46	3	/	/	SYM
ejpam-5915	46	4	eur	eur	PROPN
ejpam-5915	46	5	.	.	PUNCT
ejpam-5915	47	1	j.	j.	PROPN
ejpam-5915	47	2	pure	pure	PROPN
ejpam-5915	47	3	appl	appl	PROPN
ejpam-5915	47	4	.	.	PROPN
ejpam-5915	47	5	math	math	PROPN
ejpam-5915	47	6	,	,	PUNCT
ejpam-5915	47	7	18	18	NUM
ejpam-5915	47	8	(	(	PUNCT
ejpam-5915	47	9	2	2	NUM
ejpam-5915	47	10	)	)	PUNCT
ejpam-5915	47	11	(	(	PUNCT
ejpam-5915	47	12	2025	2025	NUM
ejpam-5915	47	13	)	)	PUNCT
ejpam-5915	47	14	,	,	PUNCT
ejpam-5915	47	15	5915	5915	NUM
ejpam-5915	47	16	3	3	NUM
ejpam-5915	47	17	of	of	ADP
ejpam-5915	47	18	16	16	NUM
ejpam-5915	47	19	2	2	NUM
ejpam-5915	47	20	.	.	PUNCT
ejpam-5915	47	21	hom	hom	NOUN
ejpam-5915	47	22	-	-	PUNCT
ejpam-5915	47	23	groups	group	NOUN
ejpam-5915	47	24	this	this	DET
ejpam-5915	47	25	section	section	NOUN
ejpam-5915	47	26	provides	provide	VERB
ejpam-5915	47	27	an	an	DET
ejpam-5915	47	28	overview	overview	NOUN
ejpam-5915	47	29	of	of	ADP
ejpam-5915	47	30	hom	hom	NOUN
ejpam-5915	47	31	-	-	PUNCT
ejpam-5915	47	32	groups	group	NOUN
ejpam-5915	47	33	and	and	CCONJ
ejpam-5915	47	34	hom	hom	NOUN
ejpam-5915	47	35	-	-	PUNCT
ejpam-5915	47	36	subgroups	subgroup	NOUN
ejpam-5915	47	37	,	,	PUNCT
ejpam-5915	47	38	recalling	recall	VERB
ejpam-5915	47	39	key	key	ADJ
ejpam-5915	47	40	definitions	definition	NOUN
ejpam-5915	47	41	and	and	CCONJ
ejpam-5915	47	42	properties	property	NOUN
ejpam-5915	47	43	while	while	SCONJ
ejpam-5915	47	44	incorporating	incorporate	VERB
ejpam-5915	47	45	refinements	refinement	NOUN
ejpam-5915	47	46	and	and	CCONJ
ejpam-5915	47	47	generalizations	generalization	NOUN
ejpam-5915	47	48	.	.	PUNCT
ejpam-5915	48	1	some	some	DET
ejpam-5915	48	2	definitions	definition	NOUN
ejpam-5915	48	3	are	be	AUX
ejpam-5915	48	4	adapted	adapt	VERB
ejpam-5915	48	5	from	from	ADP
ejpam-5915	48	6	[	[	X
ejpam-5915	48	7	14	14	NUM
ejpam-5915	48	8	]	]	PUNCT
ejpam-5915	48	9	,	,	PUNCT
ejpam-5915	48	10	but	but	CCONJ
ejpam-5915	48	11	with	with	ADP
ejpam-5915	48	12	modifications	modification	NOUN
ejpam-5915	48	13	for	for	ADP
ejpam-5915	48	14	consistency	consistency	NOUN
ejpam-5915	48	15	.	.	PUNCT
ejpam-5915	49	1	additionally	additionally	ADV
ejpam-5915	49	2	,	,	PUNCT
ejpam-5915	49	3	certain	certain	ADJ
ejpam-5915	49	4	results	result	NOUN
ejpam-5915	49	5	originally	originally	ADV
ejpam-5915	49	6	stated	state	VERB
ejpam-5915	49	7	as	as	SCONJ
ejpam-5915	49	8	theorems	theorem	NOUN
ejpam-5915	49	9	are	be	AUX
ejpam-5915	49	10	reformulated	reformulate	VERB
ejpam-5915	49	11	as	as	ADP
ejpam-5915	49	12	definitions	definition	NOUN
ejpam-5915	49	13	to	to	PART
ejpam-5915	49	14	better	well	ADV
ejpam-5915	49	15	reflect	reflect	VERB
ejpam-5915	49	16	their	their	PRON
ejpam-5915	49	17	foundational	foundational	ADJ
ejpam-5915	49	18	role	role	NOUN
ejpam-5915	49	19	.	.	PUNCT
ejpam-5915	50	1	new	new	ADJ
ejpam-5915	50	2	examples	example	NOUN
ejpam-5915	50	3	are	be	AUX
ejpam-5915	50	4	introduced	introduce	VERB
ejpam-5915	50	5	,	,	PUNCT
ejpam-5915	50	6	including	include	VERB
ejpam-5915	50	7	a	a	DET
ejpam-5915	50	8	generalization	generalization	NOUN
ejpam-5915	50	9	of	of	ADP
ejpam-5915	50	10	a	a	DET
ejpam-5915	50	11	previously	previously	ADV
ejpam-5915	50	12	studied	study	VERB
ejpam-5915	50	13	case	case	NOUN
ejpam-5915	50	14	,	,	PUNCT
ejpam-5915	50	15	to	to	PART
ejpam-5915	50	16	illustrate	illustrate	VERB
ejpam-5915	50	17	structural	structural	ADJ
ejpam-5915	50	18	flexibility	flexibility	NOUN
ejpam-5915	50	19	within	within	ADP
ejpam-5915	50	20	the	the	DET
ejpam-5915	50	21	hom	hom	NOUN
ejpam-5915	50	22	-	-	PUNCT
ejpam-5915	50	23	group	group	NOUN
ejpam-5915	50	24	framework	framework	NOUN
ejpam-5915	50	25	.	.	PUNCT
ejpam-5915	51	1	these	these	DET
ejpam-5915	51	2	refinements	refinement	NOUN
ejpam-5915	51	3	and	and	CCONJ
ejpam-5915	51	4	additions	addition	NOUN
ejpam-5915	51	5	serve	serve	VERB
ejpam-5915	51	6	as	as	ADP
ejpam-5915	51	7	the	the	DET
ejpam-5915	51	8	basis	basis	NOUN
ejpam-5915	51	9	for	for	ADP
ejpam-5915	51	10	developing	develop	VERB
ejpam-5915	51	11	fuzzy	fuzzy	ADJ
ejpam-5915	51	12	hom	hom	NOUN
ejpam-5915	51	13	-	-	PUNCT
ejpam-5915	51	14	groups	group	NOUN
ejpam-5915	51	15	in	in	ADP
ejpam-5915	51	16	subsequent	subsequent	ADJ
ejpam-5915	51	17	sections	section	NOUN
ejpam-5915	51	18	.	.	PUNCT
ejpam-5915	52	1	in	in	ADP
ejpam-5915	52	2	algebraic	algebraic	ADJ
ejpam-5915	52	3	structures	structure	NOUN
ejpam-5915	52	4	,	,	PUNCT
ejpam-5915	52	5	hom	hom	NOUN
ejpam-5915	52	6	-	-	PUNCT
ejpam-5915	52	7	groups	group	NOUN
ejpam-5915	52	8	are	be	AUX
ejpam-5915	52	9	generalizations	generalization	NOUN
ejpam-5915	52	10	of	of	ADP
ejpam-5915	52	11	classical	classical	ADJ
ejpam-5915	52	12	groups	group	NOUN
ejpam-5915	52	13	,	,	PUNCT
ejpam-5915	52	14	incorporating	incorporate	VERB
ejpam-5915	52	15	a	a	DET
ejpam-5915	52	16	twisting	twisting	NOUN
ejpam-5915	52	17	map	map	NOUN
ejpam-5915	52	18	that	that	PRON
ejpam-5915	52	19	modifies	modify	VERB
ejpam-5915	52	20	the	the	DET
ejpam-5915	52	21	usual	usual	ADJ
ejpam-5915	52	22	associativity	associativity	NOUN
ejpam-5915	52	23	and	and	CCONJ
ejpam-5915	52	24	identity	identity	NOUN
ejpam-5915	52	25	properties	property	NOUN
ejpam-5915	52	26	.	.	PUNCT
ejpam-5915	53	1	the	the	DET
ejpam-5915	53	2	following	follow	VERB
ejpam-5915	53	3	formal	formal	ADJ
ejpam-5915	53	4	definition	definition	NOUN
ejpam-5915	53	5	outlines	outline	VERB
ejpam-5915	53	6	the	the	DET
ejpam-5915	53	7	essential	essential	ADJ
ejpam-5915	53	8	components	component	NOUN
ejpam-5915	53	9	and	and	CCONJ
ejpam-5915	53	10	properties	property	NOUN
ejpam-5915	53	11	of	of	ADP
ejpam-5915	53	12	a	a	DET
ejpam-5915	53	13	homgroup	homgroup	NOUN
ejpam-5915	53	14	:	:	PUNCT
ejpam-5915	53	15	definition	definition	NOUN
ejpam-5915	53	16	1	1	NUM
ejpam-5915	53	17	(	(	PUNCT
ejpam-5915	53	18	[	[	X
ejpam-5915	53	19	14	14	NUM
ejpam-5915	53	20	]	]	NUM
ejpam-5915	53	21	)	)	PUNCT
ejpam-5915	53	22	.	.	PUNCT
ejpam-5915	54	1	a	a	DET
ejpam-5915	54	2	hom	hom	NOUN
ejpam-5915	54	3	-	-	PUNCT
ejpam-5915	54	4	group	group	NOUN
ejpam-5915	54	5	is	be	AUX
ejpam-5915	54	6	a	a	DET
ejpam-5915	54	7	triple	triple	ADJ
ejpam-5915	54	8	(	(	PUNCT
ejpam-5915	54	9	g	g	NOUN
ejpam-5915	54	10	,	,	PUNCT
ejpam-5915	54	11	·	·	PUNCT
ejpam-5915	54	12	,	,	PUNCT
ejpam-5915	54	13	α	α	NOUN
ejpam-5915	54	14	)	)	PUNCT
ejpam-5915	54	15	,	,	PUNCT
ejpam-5915	54	16	where	where	SCONJ
ejpam-5915	54	17	:	:	PUNCT
ejpam-5915	54	18	•	•	NUM
ejpam-5915	54	19	g	g	NOUN
ejpam-5915	54	20	is	be	AUX
ejpam-5915	54	21	a	a	DET
ejpam-5915	54	22	set	set	NOUN
ejpam-5915	54	23	,	,	PUNCT
ejpam-5915	54	24	•	•	NOUN
ejpam-5915	54	25	·	·	PUNCT
ejpam-5915	54	26	:	:	PUNCT
ejpam-5915	55	1	g×g	g×g	PROPN
ejpam-5915	55	2	→	→	SYM
ejpam-5915	55	3	g	g	PROPN
ejpam-5915	55	4	is	be	AUX
ejpam-5915	55	5	a	a	DET
ejpam-5915	55	6	binary	binary	ADJ
ejpam-5915	55	7	operation	operation	NOUN
ejpam-5915	55	8	,	,	PUNCT
ejpam-5915	55	9	•	•	NUM
ejpam-5915	55	10	α	α	NOUN
ejpam-5915	55	11	:	:	PUNCT
ejpam-5915	55	12	g	g	NOUN
ejpam-5915	55	13	→	→	SYM
ejpam-5915	55	14	g	g	PROPN
ejpam-5915	55	15	is	be	AUX
ejpam-5915	55	16	a	a	DET
ejpam-5915	55	17	bijective	bijective	ADJ
ejpam-5915	55	18	map	map	NOUN
ejpam-5915	55	19	,	,	PUNCT
ejpam-5915	55	20	satisfying	satisfy	VERB
ejpam-5915	55	21	the	the	DET
ejpam-5915	55	22	following	follow	VERB
ejpam-5915	55	23	identities	identity	NOUN
ejpam-5915	55	24	for	for	ADP
ejpam-5915	55	25	all	all	PRON
ejpam-5915	55	26	g	g	PROPN
ejpam-5915	55	27	,	,	PUNCT
ejpam-5915	55	28	h	h	NOUN
ejpam-5915	55	29	,	,	PUNCT
ejpam-5915	55	30	k	k	PROPN
ejpam-5915	55	31	∈	∈	PROPN
ejpam-5915	55	32	g	g	NOUN
ejpam-5915	55	33	:	:	PUNCT
ejpam-5915	55	34	(	(	PUNCT
ejpam-5915	55	35	hg1	hg1	PROPN
ejpam-5915	55	36	)	)	PUNCT
ejpam-5915	55	37	hom	hom	NOUN
ejpam-5915	55	38	-	-	PUNCT
ejpam-5915	55	39	associativity	associativity	NOUN
ejpam-5915	55	40	:	:	PUNCT
ejpam-5915	55	41	α(g	α(g	NUM
ejpam-5915	55	42	)	)	PUNCT
ejpam-5915	55	43	·	·	PUNCT
ejpam-5915	55	44	(	(	PUNCT
ejpam-5915	55	45	h	h	NOUN
ejpam-5915	55	46	·	·	PUNCT
ejpam-5915	55	47	k	k	X
ejpam-5915	55	48	)	)	PUNCT
ejpam-5915	55	49	=	=	SYM
ejpam-5915	55	50	(	(	PUNCT
ejpam-5915	55	51	g	g	NOUN
ejpam-5915	55	52	·	·	SYM
ejpam-5915	55	53	h	h	NOUN
ejpam-5915	55	54	)	)	PUNCT
ejpam-5915	55	55	·	·	PUNCT
ejpam-5915	55	56	α(k	α(k	NOUN
ejpam-5915	55	57	)	)	PUNCT
ejpam-5915	55	58	,	,	PUNCT
ejpam-5915	55	59	(	(	PUNCT
ejpam-5915	55	60	hg2	hg2	NOUN
ejpam-5915	55	61	)	)	PUNCT
ejpam-5915	55	62	hom	hom	NOUN
ejpam-5915	55	63	-	-	PUNCT
ejpam-5915	55	64	unitarity	unitarity	NOUN
ejpam-5915	55	65	:	:	PUNCT
ejpam-5915	55	66	there	there	PRON
ejpam-5915	55	67	exists	exist	VERB
ejpam-5915	55	68	an	an	DET
ejpam-5915	55	69	element	element	NOUN
ejpam-5915	55	70	e	e	PROPN
ejpam-5915	55	71	∈	∈	PROPN
ejpam-5915	55	72	g	g	PROPN
ejpam-5915	55	73	such	such	ADJ
ejpam-5915	55	74	that	that	PRON
ejpam-5915	55	75	for	for	ADP
ejpam-5915	55	76	all	all	DET
ejpam-5915	55	77	g	g	PROPN
ejpam-5915	55	78	∈	∈	PROPN
ejpam-5915	55	79	g	g	NOUN
ejpam-5915	55	80	:	:	PUNCT
ejpam-5915	55	81	g	g	PROPN
ejpam-5915	55	82	·	·	PUNCT
ejpam-5915	55	83	e	e	X
ejpam-5915	55	84	=	=	SYM
ejpam-5915	55	85	α(g	α(g	PROPN
ejpam-5915	55	86	)	)	PUNCT
ejpam-5915	55	87	and	and	CCONJ
ejpam-5915	55	88	e	e	X
ejpam-5915	55	89	·	·	PUNCT
ejpam-5915	55	90	g	g	X
ejpam-5915	55	91	=	=	SYM
ejpam-5915	55	92	α(g	α(g	NUM
ejpam-5915	55	93	)	)	PUNCT
ejpam-5915	55	94	,	,	PUNCT
ejpam-5915	55	95	and	and	CCONJ
ejpam-5915	55	96	in	in	ADP
ejpam-5915	55	97	addition	addition	NOUN
ejpam-5915	55	98	,	,	PUNCT
ejpam-5915	55	99	the	the	DET
ejpam-5915	55	100	condition	condition	NOUN
ejpam-5915	55	101	α(e	α(e	NOUN
ejpam-5915	55	102	)	)	PUNCT
ejpam-5915	56	1	=	=	PUNCT
ejpam-5915	56	2	e	e	NOUN
ejpam-5915	56	3	holds	hold	VERB
ejpam-5915	56	4	.	.	PUNCT
ejpam-5915	57	1	(	(	PUNCT
ejpam-5915	57	2	hg3	hg3	X
ejpam-5915	57	3	)	)	PUNCT
ejpam-5915	57	4	hom	hom	NOUN
ejpam-5915	57	5	-	-	PUNCT
ejpam-5915	57	6	inverses	inverse	VERB
ejpam-5915	57	7	:	:	PUNCT
ejpam-5915	57	8	for	for	ADP
ejpam-5915	57	9	every	every	DET
ejpam-5915	57	10	g	g	PROPN
ejpam-5915	57	11	∈	∈	PROPN
ejpam-5915	57	12	g	g	NOUN
ejpam-5915	57	13	,	,	PUNCT
ejpam-5915	57	14	there	there	PRON
ejpam-5915	57	15	exists	exist	VERB
ejpam-5915	57	16	an	an	DET
ejpam-5915	57	17	inverse	inverse	NOUN
ejpam-5915	57	18	element	element	NOUN
ejpam-5915	57	19	g−1	g−1	PROPN
ejpam-5915	57	20	∈	∈	PROPN
ejpam-5915	57	21	g	g	NOUN
ejpam-5915	57	22	such	such	ADJ
ejpam-5915	57	23	that	that	SCONJ
ejpam-5915	57	24	:	:	PUNCT
ejpam-5915	58	1	g	g	X
ejpam-5915	58	2	·	·	PUNCT
ejpam-5915	58	3	g−1	g−1	PROPN
ejpam-5915	58	4	=	=	SYM
ejpam-5915	58	5	e	e	NOUN
ejpam-5915	58	6	and	and	CCONJ
ejpam-5915	58	7	g−1	g−1	PROPN
ejpam-5915	58	8	·	·	PUNCT
ejpam-5915	58	9	g	g	NOUN
ejpam-5915	58	10	=	=	SYM
ejpam-5915	58	11	e	e	NOUN
ejpam-5915	58	12	,	,	PUNCT
ejpam-5915	58	13	where	where	SCONJ
ejpam-5915	58	14	e	e	NOUN
ejpam-5915	58	15	is	be	AUX
ejpam-5915	58	16	the	the	DET
ejpam-5915	58	17	hom	hom	NOUN
ejpam-5915	58	18	-	-	PUNCT
ejpam-5915	58	19	identity	identity	NOUN
ejpam-5915	58	20	element	element	NOUN
ejpam-5915	58	21	.	.	PUNCT
ejpam-5915	59	1	(	(	PUNCT
ejpam-5915	59	2	hg4	hg4	NOUN
ejpam-5915	59	3	)	)	PUNCT
ejpam-5915	59	4	multiplicativity	multiplicativity	NOUN
ejpam-5915	59	5	of	of	ADP
ejpam-5915	59	6	α	α	NOUN
ejpam-5915	59	7	:	:	PUNCT
ejpam-5915	59	8	α(g	α(g	PROPN
ejpam-5915	59	9	·	·	PUNCT
ejpam-5915	59	10	h	h	NOUN
ejpam-5915	59	11	)	)	PUNCT
ejpam-5915	59	12	=	=	SYM
ejpam-5915	59	13	α(g	α(g	NUM
ejpam-5915	59	14	)	)	PUNCT
ejpam-5915	59	15	·	·	PUNCT
ejpam-5915	59	16	α(h	α(h	NOUN
ejpam-5915	59	17	)	)	PUNCT
ejpam-5915	59	18	for	for	ADP
ejpam-5915	59	19	all	all	DET
ejpam-5915	59	20	g	g	NOUN
ejpam-5915	59	21	,	,	PUNCT
ejpam-5915	59	22	h	h	PROPN
ejpam-5915	59	23	∈	∈	PROPN
ejpam-5915	59	24	g.	g.	PROPN
ejpam-5915	59	25	remark	remark	NOUN
ejpam-5915	59	26	1	1	NUM
ejpam-5915	59	27	.	.	PUNCT
ejpam-5915	60	1	in	in	ADP
ejpam-5915	60	2	earlier	early	ADJ
ejpam-5915	60	3	literature	literature	NOUN
ejpam-5915	60	4	,	,	PUNCT
ejpam-5915	60	5	such	such	ADJ
ejpam-5915	60	6	as	as	ADP
ejpam-5915	60	7	in	in	ADP
ejpam-5915	60	8	[	[	X
ejpam-5915	60	9	14	14	NUM
ejpam-5915	60	10	]	]	PUNCT
ejpam-5915	60	11	,	,	PUNCT
ejpam-5915	60	12	the	the	DET
ejpam-5915	60	13	condition	condition	NOUN
ejpam-5915	60	14	α(e	α(e	NOUN
ejpam-5915	60	15	)	)	PUNCT
ejpam-5915	61	1	=	=	PUNCT
ejpam-5915	61	2	e	e	NOUN
ejpam-5915	61	3	was	be	AUX
ejpam-5915	61	4	not	not	PART
ejpam-5915	61	5	always	always	ADV
ejpam-5915	61	6	required	require	VERB
ejpam-5915	61	7	in	in	ADP
ejpam-5915	61	8	the	the	DET
ejpam-5915	61	9	definition	definition	NOUN
ejpam-5915	61	10	of	of	ADP
ejpam-5915	61	11	hom	hom	NOUN
ejpam-5915	61	12	-	-	PUNCT
ejpam-5915	61	13	groups	group	NOUN
ejpam-5915	61	14	.	.	PUNCT
ejpam-5915	62	1	this	this	PRON
ejpam-5915	62	2	led	lead	VERB
ejpam-5915	62	3	to	to	ADP
ejpam-5915	62	4	cases	case	NOUN
ejpam-5915	62	5	where	where	SCONJ
ejpam-5915	62	6	the	the	DET
ejpam-5915	62	7	behavior	behavior	NOUN
ejpam-5915	62	8	of	of	ADP
ejpam-5915	62	9	the	the	DET
ejpam-5915	62	10	identity	identity	NOUN
ejpam-5915	62	11	element	element	NOUN
ejpam-5915	62	12	under	under	ADP
ejpam-5915	62	13	the	the	DET
ejpam-5915	62	14	twisting	twisting	NOUN
ejpam-5915	62	15	map	map	NOUN
ejpam-5915	62	16	needed	need	VERB
ejpam-5915	62	17	to	to	PART
ejpam-5915	62	18	be	be	AUX
ejpam-5915	62	19	verified	verify	VERB
ejpam-5915	62	20	separately	separately	ADV
ejpam-5915	62	21	.	.	PUNCT
ejpam-5915	63	1	however	however	ADV
ejpam-5915	63	2	,	,	PUNCT
ejpam-5915	63	3	in	in	ADP
ejpam-5915	63	4	this	this	DET
ejpam-5915	63	5	paper	paper	NOUN
ejpam-5915	63	6	,	,	PUNCT
ejpam-5915	63	7	we	we	PRON
ejpam-5915	63	8	adopt	adopt	VERB
ejpam-5915	63	9	a	a	DET
ejpam-5915	63	10	more	more	ADV
ejpam-5915	63	11	structured	structured	ADJ
ejpam-5915	63	12	definition	definition	NOUN
ejpam-5915	63	13	by	by	ADP
ejpam-5915	63	14	explicitly	explicitly	ADV
ejpam-5915	63	15	requiring	require	VERB
ejpam-5915	63	16	that	that	SCONJ
ejpam-5915	63	17	α(e	α(e	NOUN
ejpam-5915	63	18	)	)	PUNCT
ejpam-5915	64	1	=	=	VERB
ejpam-5915	64	2	e.	e.	PROPN
ejpam-5915	65	1	this	this	DET
ejpam-5915	65	2	condition	condition	NOUN
ejpam-5915	65	3	simplifies	simplify	VERB
ejpam-5915	65	4	many	many	ADJ
ejpam-5915	65	5	proofs	proof	NOUN
ejpam-5915	65	6	and	and	CCONJ
ejpam-5915	65	7	provides	provide	VERB
ejpam-5915	65	8	a	a	DET
ejpam-5915	65	9	consistent	consistent	ADJ
ejpam-5915	65	10	framework	framework	NOUN
ejpam-5915	65	11	for	for	ADP
ejpam-5915	65	12	studying	study	VERB
ejpam-5915	65	13	hom	hom	NOUN
ejpam-5915	65	14	-	-	PUNCT
ejpam-5915	65	15	subgroups	subgroup	NOUN
ejpam-5915	65	16	and	and	CCONJ
ejpam-5915	65	17	other	other	ADJ
ejpam-5915	65	18	related	related	ADJ
ejpam-5915	65	19	structures	structure	NOUN
ejpam-5915	65	20	,	,	PUNCT
ejpam-5915	65	21	ensuring	ensure	VERB
ejpam-5915	65	22	that	that	SCONJ
ejpam-5915	65	23	the	the	DET
ejpam-5915	65	24	identity	identity	NOUN
ejpam-5915	65	25	element	element	NOUN
ejpam-5915	65	26	behaves	behave	VERB
ejpam-5915	65	27	uniformly	uniformly	ADV
ejpam-5915	65	28	within	within	ADP
ejpam-5915	65	29	the	the	DET
ejpam-5915	65	30	hom	hom	NOUN
ejpam-5915	65	31	-	-	PUNCT
ejpam-5915	65	32	group	group	NOUN
ejpam-5915	65	33	.	.	PUNCT
ejpam-5915	66	1	s.	s.	PROPN
ejpam-5915	66	2	shaqaqha	shaqaqha	PROPN
ejpam-5915	66	3	/	/	SYM
ejpam-5915	66	4	eur	eur	PROPN
ejpam-5915	66	5	.	.	PUNCT
ejpam-5915	67	1	j.	j.	PROPN
ejpam-5915	67	2	pure	pure	PROPN
ejpam-5915	67	3	appl	appl	PROPN
ejpam-5915	67	4	.	.	PROPN
ejpam-5915	67	5	math	math	PROPN
ejpam-5915	67	6	,	,	PUNCT
ejpam-5915	67	7	18	18	NUM
ejpam-5915	67	8	(	(	PUNCT
ejpam-5915	67	9	2	2	NUM
ejpam-5915	67	10	)	)	PUNCT
ejpam-5915	67	11	(	(	PUNCT
ejpam-5915	67	12	2025	2025	NUM
ejpam-5915	67	13	)	)	PUNCT
ejpam-5915	67	14	,	,	PUNCT
ejpam-5915	67	15	5915	5915	NUM
ejpam-5915	67	16	4	4	NUM
ejpam-5915	67	17	of	of	ADP
ejpam-5915	67	18	16	16	NUM
ejpam-5915	67	19	example	example	NOUN
ejpam-5915	67	20	1	1	NUM
ejpam-5915	67	21	(	(	PUNCT
ejpam-5915	67	22	[	[	X
ejpam-5915	67	23	14	14	NUM
ejpam-5915	67	24	]	]	NUM
ejpam-5915	67	25	)	)	PUNCT
ejpam-5915	67	26	.	.	PUNCT
ejpam-5915	68	1	any	any	DET
ejpam-5915	68	2	classical	classical	ADJ
ejpam-5915	68	3	group	group	NOUN
ejpam-5915	68	4	(	(	PUNCT
ejpam-5915	68	5	g	g	NOUN
ejpam-5915	68	6	,	,	PUNCT
ejpam-5915	68	7	·	·	PUNCT
ejpam-5915	68	8	)	)	PUNCT
ejpam-5915	68	9	can	can	AUX
ejpam-5915	68	10	be	be	AUX
ejpam-5915	68	11	viewed	view	VERB
ejpam-5915	68	12	as	as	ADP
ejpam-5915	68	13	a	a	DET
ejpam-5915	68	14	hom	hom	NOUN
ejpam-5915	68	15	-	-	PUNCT
ejpam-5915	68	16	group	group	NOUN
ejpam-5915	68	17	by	by	ADP
ejpam-5915	68	18	defining	define	VERB
ejpam-5915	68	19	the	the	DET
ejpam-5915	68	20	twisting	twisting	NOUN
ejpam-5915	68	21	map	map	NOUN
ejpam-5915	68	22	α	α	NOUN
ejpam-5915	68	23	as	as	ADP
ejpam-5915	68	24	the	the	DET
ejpam-5915	68	25	identity	identity	NOUN
ejpam-5915	68	26	map	map	NOUN
ejpam-5915	68	27	,	,	PUNCT
ejpam-5915	68	28	i.e.	i.e.	X
ejpam-5915	68	29	,	,	PUNCT
ejpam-5915	68	30	α(g	α(g	NUM
ejpam-5915	68	31	)	)	PUNCT
ejpam-5915	68	32	=	=	SYM
ejpam-5915	68	33	ig(g	ig(g	X
ejpam-5915	68	34	)	)	PUNCT
ejpam-5915	68	35	=	=	SYM
ejpam-5915	68	36	g	g	NOUN
ejpam-5915	68	37	for	for	ADP
ejpam-5915	68	38	all	all	DET
ejpam-5915	68	39	g	g	PROPN
ejpam-5915	68	40	∈	∈	PROPN
ejpam-5915	68	41	g.	g.	NOUN
ejpam-5915	68	42	example	example	NOUN
ejpam-5915	68	43	2	2	X
ejpam-5915	68	44	.	.	PUNCT
ejpam-5915	69	1	let	let	VERB
ejpam-5915	69	2	g	g	NOUN
ejpam-5915	69	3	=	=	SYM
ejpam-5915	69	4	r	r	NOUN
ejpam-5915	69	5	,	,	PUNCT
ejpam-5915	69	6	and	and	CCONJ
ejpam-5915	69	7	define	define	VERB
ejpam-5915	69	8	the	the	DET
ejpam-5915	69	9	binary	binary	ADJ
ejpam-5915	69	10	operation	operation	NOUN
ejpam-5915	69	11	⊕k	⊕k	NOUN
ejpam-5915	69	12	and	and	CCONJ
ejpam-5915	69	13	the	the	DET
ejpam-5915	69	14	twisting	twisting	NOUN
ejpam-5915	69	15	map	map	NOUN
ejpam-5915	69	16	αk	αk	NOUN
ejpam-5915	69	17	for	for	ADP
ejpam-5915	69	18	a	a	DET
ejpam-5915	69	19	fixed	fix	VERB
ejpam-5915	69	20	constant	constant	ADJ
ejpam-5915	69	21	k	k	PROPN
ejpam-5915	69	22	̸=	̸=	PROPN
ejpam-5915	69	23	0	0	PUNCT
ejpam-5915	69	24	as	as	SCONJ
ejpam-5915	69	25	follows	follow	VERB
ejpam-5915	69	26	:	:	PUNCT
ejpam-5915	69	27	a⊕k	a⊕k	PROPN
ejpam-5915	69	28	b	b	X
ejpam-5915	69	29	=	=	PRON
ejpam-5915	69	30	a+	a+	PUNCT
ejpam-5915	69	31	b	b	PROPN
ejpam-5915	69	32	k	k	PROPN
ejpam-5915	69	33	and	and	CCONJ
ejpam-5915	69	34	αk(a	αk(a	NUM
ejpam-5915	69	35	)	)	PUNCT
ejpam-5915	70	1	=	=	SYM
ejpam-5915	70	2	a	a	DET
ejpam-5915	70	3	k	k	X
ejpam-5915	70	4	,	,	PUNCT
ejpam-5915	70	5	for	for	ADP
ejpam-5915	70	6	all	all	DET
ejpam-5915	70	7	a	a	DET
ejpam-5915	70	8	,	,	PUNCT
ejpam-5915	70	9	b	b	X
ejpam-5915	70	10	∈	∈	PROPN
ejpam-5915	70	11	r.	r.	NOUN
ejpam-5915	70	12	we	we	PRON
ejpam-5915	70	13	verify	verify	VERB
ejpam-5915	70	14	that	that	SCONJ
ejpam-5915	70	15	(	(	PUNCT
ejpam-5915	70	16	g	g	NOUN
ejpam-5915	70	17	=	=	SYM
ejpam-5915	70	18	r,⊕k	r,⊕k	X
ejpam-5915	70	19	,	,	PUNCT
ejpam-5915	70	20	αk	αk	NOUN
ejpam-5915	70	21	)	)	PUNCT
ejpam-5915	70	22	forms	form	VERB
ejpam-5915	70	23	a	a	DET
ejpam-5915	70	24	hom	hom	NOUN
ejpam-5915	70	25	-	-	PUNCT
ejpam-5915	70	26	group	group	NOUN
ejpam-5915	70	27	by	by	ADP
ejpam-5915	70	28	checking	check	VERB
ejpam-5915	70	29	the	the	DET
ejpam-5915	70	30	required	require	VERB
ejpam-5915	70	31	properties	property	NOUN
ejpam-5915	70	32	:	:	PUNCT
ejpam-5915	70	33	(	(	PUNCT
ejpam-5915	70	34	hg1	hg1	PROPN
ejpam-5915	70	35	)	)	PUNCT
ejpam-5915	70	36	hom	hom	NOUN
ejpam-5915	70	37	-	-	PUNCT
ejpam-5915	70	38	associativity	associativity	NOUN
ejpam-5915	70	39	:	:	PUNCT
ejpam-5915	70	40	for	for	ADP
ejpam-5915	70	41	all	all	DET
ejpam-5915	70	42	a	a	DET
ejpam-5915	70	43	,	,	PUNCT
ejpam-5915	70	44	b	b	NOUN
ejpam-5915	70	45	,	,	PUNCT
ejpam-5915	70	46	c	c	PROPN
ejpam-5915	70	47	∈	∈	PROPN
ejpam-5915	70	48	r	r	NOUN
ejpam-5915	70	49	,	,	PUNCT
ejpam-5915	70	50	we	we	PRON
ejpam-5915	70	51	verify	verify	VERB
ejpam-5915	70	52	:	:	PUNCT
ejpam-5915	70	53	αk(a)⊕k	αk(a)⊕k	PROPN
ejpam-5915	70	54	(	(	PUNCT
ejpam-5915	70	55	b⊕k	b⊕k	PROPN
ejpam-5915	70	56	c	c	NOUN
ejpam-5915	70	57	)	)	PUNCT
ejpam-5915	70	58	=	=	SYM
ejpam-5915	70	59	(	(	PUNCT
ejpam-5915	70	60	a⊕k	a⊕k	NOUN
ejpam-5915	70	61	b)⊕k	b)⊕k	NOUN
ejpam-5915	70	62	αk(c	αk(c	NUM
ejpam-5915	70	63	)	)	PUNCT
ejpam-5915	70	64	.	.	PUNCT
ejpam-5915	71	1	the	the	DET
ejpam-5915	71	2	left	left	ADJ
ejpam-5915	71	3	-	-	PUNCT
ejpam-5915	71	4	hand	hand	NOUN
ejpam-5915	71	5	side	side	NOUN
ejpam-5915	71	6	(	(	PUNCT
ejpam-5915	71	7	lhs	lhs	PROPN
ejpam-5915	71	8	)	)	PUNCT
ejpam-5915	71	9	is	be	AUX
ejpam-5915	71	10	:	:	PUNCT
ejpam-5915	71	11	αk(a)⊕k	αk(a)⊕k	PROPN
ejpam-5915	71	12	(	(	PUNCT
ejpam-5915	71	13	b⊕k	b⊕k	PROPN
ejpam-5915	71	14	c	c	NOUN
ejpam-5915	71	15	)	)	PUNCT
ejpam-5915	71	16	=	=	PUNCT
ejpam-5915	72	1	a	a	PRON
ejpam-5915	72	2	k	k	PROPN
ejpam-5915	73	1	+	+	CCONJ
ejpam-5915	73	2	b+c	b+c	PROPN
ejpam-5915	74	1	k	k	X
ejpam-5915	74	2	k	k	PROPN
ejpam-5915	74	3	=	=	X
ejpam-5915	74	4	a+	a+	PUNCT
ejpam-5915	74	5	b+	b+	X
ejpam-5915	74	6	c	c	PROPN
ejpam-5915	74	7	k2	k2	PROPN
ejpam-5915	74	8	.	.	PUNCT
ejpam-5915	75	1	the	the	DET
ejpam-5915	75	2	right	right	ADJ
ejpam-5915	75	3	-	-	PUNCT
ejpam-5915	75	4	hand	hand	NOUN
ejpam-5915	75	5	side	side	NOUN
ejpam-5915	75	6	(	(	PUNCT
ejpam-5915	75	7	rhs	rhs	PROPN
ejpam-5915	75	8	)	)	PUNCT
ejpam-5915	75	9	is	be	AUX
ejpam-5915	75	10	:	:	PUNCT
ejpam-5915	75	11	(	(	PUNCT
ejpam-5915	75	12	a⊕k	a⊕k	NOUN
ejpam-5915	75	13	b)⊕k	b)⊕k	NOUN
ejpam-5915	75	14	αk(c	αk(c	NUM
ejpam-5915	75	15	)	)	PUNCT
ejpam-5915	75	16	=	=	SYM
ejpam-5915	76	1	a+b	a+b	NUM
ejpam-5915	76	2	k	k	X
ejpam-5915	77	1	+	+	CCONJ
ejpam-5915	77	2	c	c	NOUN
ejpam-5915	77	3	k	k	PROPN
ejpam-5915	77	4	k	k	PROPN
ejpam-5915	77	5	=	=	X
ejpam-5915	77	6	a+	a+	PUNCT
ejpam-5915	77	7	b+	b+	X
ejpam-5915	77	8	c	c	PROPN
ejpam-5915	77	9	k2	k2	PROPN
ejpam-5915	77	10	.	.	PUNCT
ejpam-5915	78	1	since	since	SCONJ
ejpam-5915	78	2	the	the	DET
ejpam-5915	78	3	lhs	lhs	PROPN
ejpam-5915	78	4	equals	equal	VERB
ejpam-5915	78	5	the	the	DET
ejpam-5915	78	6	rhs	rhs	PROPN
ejpam-5915	78	7	,	,	PUNCT
ejpam-5915	78	8	the	the	DET
ejpam-5915	78	9	hom	hom	NOUN
ejpam-5915	78	10	-	-	PUNCT
ejpam-5915	78	11	associativity	associativity	NOUN
ejpam-5915	78	12	condition	condition	NOUN
ejpam-5915	78	13	is	be	AUX
ejpam-5915	78	14	satisfied	satisfied	ADJ
ejpam-5915	78	15	.	.	PUNCT
ejpam-5915	79	1	(	(	PUNCT
ejpam-5915	79	2	hg2	hg2	NOUN
ejpam-5915	79	3	)	)	PUNCT
ejpam-5915	79	4	hom	hom	NOUN
ejpam-5915	79	5	-	-	PUNCT
ejpam-5915	79	6	unitarity	unitarity	NOUN
ejpam-5915	79	7	:	:	PUNCT
ejpam-5915	79	8	we	we	PRON
ejpam-5915	79	9	need	need	VERB
ejpam-5915	79	10	to	to	PART
ejpam-5915	79	11	find	find	VERB
ejpam-5915	79	12	an	an	DET
ejpam-5915	79	13	element	element	NOUN
ejpam-5915	79	14	e	e	NOUN
ejpam-5915	79	15	∈	∈	PROPN
ejpam-5915	79	16	g	g	PROPN
ejpam-5915	79	17	such	such	ADJ
ejpam-5915	79	18	that	that	PRON
ejpam-5915	79	19	for	for	ADP
ejpam-5915	79	20	all	all	DET
ejpam-5915	79	21	a	a	DET
ejpam-5915	79	22	∈	∈	PROPN
ejpam-5915	79	23	g	g	NOUN
ejpam-5915	79	24	,	,	PUNCT
ejpam-5915	79	25	we	we	PRON
ejpam-5915	79	26	have	have	VERB
ejpam-5915	79	27	:	:	PUNCT
ejpam-5915	79	28	a⊕k	a⊕k	NOUN
ejpam-5915	79	29	e	e	NOUN
ejpam-5915	79	30	=	=	PUNCT
ejpam-5915	79	31	αk(a	αk(a	X
ejpam-5915	79	32	)	)	PUNCT
ejpam-5915	79	33	and	and	CCONJ
ejpam-5915	79	34	e⊕k	e⊕k	NOUN
ejpam-5915	79	35	a	a	PRON
ejpam-5915	79	36	=	=	PUNCT
ejpam-5915	79	37	αk(a	αk(a	NOUN
ejpam-5915	79	38	)	)	PUNCT
ejpam-5915	79	39	,	,	PUNCT
ejpam-5915	79	40	and	and	CCONJ
ejpam-5915	79	41	αk(e	αk(e	NUM
ejpam-5915	79	42	)	)	PUNCT
ejpam-5915	80	1	=	=	SYM
ejpam-5915	80	2	e.	e.	PROPN
ejpam-5915	80	3	let	let	VERB
ejpam-5915	80	4	e	e	NOUN
ejpam-5915	80	5	=	=	NOUN
ejpam-5915	80	6	0	0	X
ejpam-5915	80	7	.	.	PUNCT
ejpam-5915	81	1	we	we	PRON
ejpam-5915	81	2	check	check	VERB
ejpam-5915	81	3	:	:	PUNCT
ejpam-5915	81	4	a⊕k	a⊕k	NOUN
ejpam-5915	81	5	0	0	PUNCT
ejpam-5915	81	6	=	=	SYM
ejpam-5915	81	7	a+	a+	PUNCT
ejpam-5915	81	8	0	0	NUM
ejpam-5915	82	1	k	k	X
ejpam-5915	82	2	=	=	PUNCT
ejpam-5915	82	3	a	a	PRON
ejpam-5915	82	4	k	k	X
ejpam-5915	82	5	=	=	PUNCT
ejpam-5915	82	6	αk(a	αk(a	NOUN
ejpam-5915	82	7	)	)	PUNCT
ejpam-5915	82	8	,	,	PUNCT
ejpam-5915	82	9	and	and	CCONJ
ejpam-5915	82	10	0⊕k	0⊕k	NUM
ejpam-5915	82	11	a	a	DET
ejpam-5915	82	12	=	=	SYM
ejpam-5915	82	13	0	0	PUNCT
ejpam-5915	83	1	+	+	CCONJ
ejpam-5915	83	2	a	a	DET
ejpam-5915	83	3	k	k	X
ejpam-5915	83	4	=	=	PUNCT
ejpam-5915	83	5	a	a	DET
ejpam-5915	83	6	k	k	X
ejpam-5915	83	7	=	=	PUNCT
ejpam-5915	83	8	αk(a	αk(a	NOUN
ejpam-5915	83	9	)	)	PUNCT
ejpam-5915	83	10	.	.	PUNCT
ejpam-5915	84	1	also	also	ADV
ejpam-5915	84	2	,	,	PUNCT
ejpam-5915	84	3	αk(0	αk(0	PROPN
ejpam-5915	84	4	)	)	PUNCT
ejpam-5915	84	5	=	=	SYM
ejpam-5915	84	6	0	0	PUNCT
ejpam-5915	85	1	k	k	X
ejpam-5915	85	2	=	=	PUNCT
ejpam-5915	85	3	0	0	PROPN
ejpam-5915	85	4	.	.	PUNCT
ejpam-5915	86	1	thus	thus	ADV
ejpam-5915	86	2	,	,	PUNCT
ejpam-5915	86	3	e	e	X
ejpam-5915	86	4	=	=	SYM
ejpam-5915	86	5	0	0	NUM
ejpam-5915	86	6	is	be	AUX
ejpam-5915	86	7	the	the	DET
ejpam-5915	86	8	hom	hom	NOUN
ejpam-5915	86	9	-	-	PUNCT
ejpam-5915	86	10	identity	identity	NOUN
ejpam-5915	86	11	element	element	NOUN
ejpam-5915	86	12	,	,	PUNCT
ejpam-5915	86	13	and	and	CCONJ
ejpam-5915	86	14	αk(e	αk(e	NUM
ejpam-5915	86	15	)	)	PUNCT
ejpam-5915	87	1	=	=	SYM
ejpam-5915	87	2	e.	e.	PROPN
ejpam-5915	87	3	(	(	PUNCT
ejpam-5915	87	4	hg3	hg3	PROPN
ejpam-5915	87	5	)	)	PUNCT
ejpam-5915	87	6	hom	hom	NOUN
ejpam-5915	87	7	-	-	PUNCT
ejpam-5915	87	8	inverses	inverse	VERB
ejpam-5915	87	9	:	:	PUNCT
ejpam-5915	87	10	for	for	ADP
ejpam-5915	87	11	each	each	PRON
ejpam-5915	87	12	a	a	DET
ejpam-5915	87	13	∈	∈	PROPN
ejpam-5915	87	14	g	g	NOUN
ejpam-5915	87	15	,	,	PUNCT
ejpam-5915	87	16	we	we	PRON
ejpam-5915	87	17	need	need	VERB
ejpam-5915	87	18	to	to	PART
ejpam-5915	87	19	find	find	VERB
ejpam-5915	87	20	a∗	a∗	PROPN
ejpam-5915	87	21	∈	∈	PROPN
ejpam-5915	87	22	g	g	ADP
ejpam-5915	87	23	such	such	ADJ
ejpam-5915	87	24	that	that	PRON
ejpam-5915	87	25	:	:	PUNCT
ejpam-5915	87	26	a⊕k	a⊕k	NOUN
ejpam-5915	87	27	a	a	DET
ejpam-5915	87	28	∗	∗	NOUN
ejpam-5915	87	29	=	=	SYM
ejpam-5915	87	30	e	e	NOUN
ejpam-5915	87	31	and	and	CCONJ
ejpam-5915	87	32	a∗	a∗	PROPN
ejpam-5915	87	33	⊕k	⊕k	PROPN
ejpam-5915	87	34	a	a	DET
ejpam-5915	87	35	=	=	SYM
ejpam-5915	87	36	e	e	NOUN
ejpam-5915	87	37	,	,	PUNCT
ejpam-5915	87	38	where	where	SCONJ
ejpam-5915	87	39	e	e	NOUN
ejpam-5915	87	40	=	=	NOUN
ejpam-5915	87	41	0	0	X
ejpam-5915	87	42	.	.	PUNCT
ejpam-5915	88	1	solving	solve	VERB
ejpam-5915	88	2	for	for	ADP
ejpam-5915	88	3	a∗	a∗	NOUN
ejpam-5915	88	4	,	,	PUNCT
ejpam-5915	88	5	we	we	PRON
ejpam-5915	88	6	have	have	VERB
ejpam-5915	88	7	:	:	PUNCT
ejpam-5915	88	8	a+	a+	PUNCT
ejpam-5915	88	9	a∗	a∗	PROPN
ejpam-5915	88	10	k	k	PROPN
ejpam-5915	88	11	=	=	SYM
ejpam-5915	88	12	0	0	PUNCT
ejpam-5915	89	1	=	=	NOUN
ejpam-5915	89	2	⇒	⇒	NOUN
ejpam-5915	89	3	a+	a+	PUNCT
ejpam-5915	89	4	a∗	a∗	NOUN
ejpam-5915	89	5	=	=	SYM
ejpam-5915	89	6	0	0	PUNCT
ejpam-5915	89	7	=	=	NOUN
ejpam-5915	89	8	⇒	⇒	X
ejpam-5915	89	9	a∗	a∗	NOUN
ejpam-5915	89	10	=	=	SYM
ejpam-5915	89	11	−a	−a	NOUN
ejpam-5915	89	12	.	.	PUNCT
ejpam-5915	90	1	thus	thus	ADV
ejpam-5915	90	2	,	,	PUNCT
ejpam-5915	90	3	the	the	DET
ejpam-5915	90	4	inverse	inverse	NOUN
ejpam-5915	90	5	of	of	ADP
ejpam-5915	90	6	a	a	DET
ejpam-5915	90	7	∈	∈	PROPN
ejpam-5915	90	8	g	g	NOUN
ejpam-5915	90	9	under	under	ADP
ejpam-5915	90	10	⊕k	⊕k	NOUN
ejpam-5915	90	11	is	be	AUX
ejpam-5915	90	12	denoted	denote	VERB
ejpam-5915	90	13	by	by	ADP
ejpam-5915	90	14	a∗	a∗	PROPN
ejpam-5915	90	15	,	,	PUNCT
ejpam-5915	90	16	where	where	SCONJ
ejpam-5915	90	17	a∗	a∗	NOUN
ejpam-5915	90	18	=	=	SYM
ejpam-5915	90	19	−a	−a	NOUN
ejpam-5915	90	20	.	.	PUNCT
ejpam-5915	91	1	s.	s.	PROPN
ejpam-5915	91	2	shaqaqha	shaqaqha	PROPN
ejpam-5915	91	3	/	/	SYM
ejpam-5915	91	4	eur	eur	PROPN
ejpam-5915	91	5	.	.	PUNCT
ejpam-5915	92	1	j.	j.	PROPN
ejpam-5915	92	2	pure	pure	PROPN
ejpam-5915	92	3	appl	appl	PROPN
ejpam-5915	92	4	.	.	PROPN
ejpam-5915	92	5	math	math	PROPN
ejpam-5915	92	6	,	,	PUNCT
ejpam-5915	92	7	18	18	NUM
ejpam-5915	92	8	(	(	PUNCT
ejpam-5915	92	9	2	2	NUM
ejpam-5915	92	10	)	)	PUNCT
ejpam-5915	92	11	(	(	PUNCT
ejpam-5915	92	12	2025	2025	NUM
ejpam-5915	92	13	)	)	PUNCT
ejpam-5915	92	14	,	,	PUNCT
ejpam-5915	92	15	5915	5915	NUM
ejpam-5915	92	16	5	5	NUM
ejpam-5915	92	17	of	of	ADP
ejpam-5915	92	18	16	16	NUM
ejpam-5915	92	19	(	(	PUNCT
ejpam-5915	92	20	hg4	hg4	NOUN
ejpam-5915	92	21	)	)	PUNCT
ejpam-5915	92	22	multiplicativity	multiplicativity	NOUN
ejpam-5915	92	23	of	of	ADP
ejpam-5915	92	24	αk	αk	NOUN
ejpam-5915	92	25	:	:	PUNCT
ejpam-5915	92	26	we	we	PRON
ejpam-5915	92	27	verify	verify	VERB
ejpam-5915	92	28	that	that	SCONJ
ejpam-5915	92	29	:	:	PUNCT
ejpam-5915	92	30	αk(a⊕k	αk(a⊕k	PROPN
ejpam-5915	92	31	b	b	X
ejpam-5915	92	32	)	)	PUNCT
ejpam-5915	92	33	=	=	SYM
ejpam-5915	92	34	αk(a)⊕k	αk(a)⊕k	PROPN
ejpam-5915	92	35	αk(b	αk(b	NUM
ejpam-5915	92	36	)	)	PUNCT
ejpam-5915	92	37	.	.	PUNCT
ejpam-5915	93	1	lhs	lhs	PROPN
ejpam-5915	93	2	:	:	PUNCT
ejpam-5915	93	3	αk(a⊕k	αk(a⊕k	PROPN
ejpam-5915	93	4	b	b	X
ejpam-5915	93	5	)	)	PUNCT
ejpam-5915	93	6	=	=	SYM
ejpam-5915	94	1	αk	αk	INTJ
ejpam-5915	94	2	(	(	PUNCT
ejpam-5915	94	3	a+	a+	PRON
ejpam-5915	94	4	b	b	X
ejpam-5915	94	5	k	k	NOUN
ejpam-5915	94	6	)	)	PUNCT
ejpam-5915	94	7	=	=	PUNCT
ejpam-5915	95	1	a+b	a+b	NUM
ejpam-5915	95	2	k	k	X
ejpam-5915	95	3	k	k	X
ejpam-5915	95	4	=	=	PUNCT
ejpam-5915	95	5	a+	a+	PUNCT
ejpam-5915	95	6	b	b	PROPN
ejpam-5915	95	7	k2	k2	PROPN
ejpam-5915	95	8	.	.	PUNCT
ejpam-5915	96	1	rhs	rhs	PROPN
ejpam-5915	96	2	:	:	PUNCT
ejpam-5915	96	3	αk(a)⊕k	αk(a)⊕k	NUM
ejpam-5915	96	4	αk(b	αk(b	NUM
ejpam-5915	96	5	)	)	PUNCT
ejpam-5915	96	6	=	=	PUNCT
ejpam-5915	97	1	a	a	PRON
ejpam-5915	97	2	k	k	PROPN
ejpam-5915	98	1	+	+	PROPN
ejpam-5915	98	2	b	b	PROPN
ejpam-5915	98	3	k	k	PROPN
ejpam-5915	98	4	k	k	PROPN
ejpam-5915	98	5	=	=	PUNCT
ejpam-5915	98	6	a+	a+	PUNCT
ejpam-5915	98	7	b	b	PROPN
ejpam-5915	98	8	k2	k2	PROPN
ejpam-5915	98	9	.	.	PUNCT
ejpam-5915	99	1	since	since	SCONJ
ejpam-5915	99	2	lhs	lhs	PROPN
ejpam-5915	99	3	=	=	SYM
ejpam-5915	99	4	rhs	rhs	PROPN
ejpam-5915	99	5	,	,	PUNCT
ejpam-5915	99	6	the	the	DET
ejpam-5915	99	7	multiplicativity	multiplicativity	NOUN
ejpam-5915	99	8	condition	condition	NOUN
ejpam-5915	99	9	is	be	AUX
ejpam-5915	99	10	satisfied	satisfied	ADJ
ejpam-5915	99	11	.	.	PUNCT
ejpam-5915	100	1	therefore	therefore	ADV
ejpam-5915	100	2	,	,	PUNCT
ejpam-5915	100	3	the	the	DET
ejpam-5915	100	4	set	set	NOUN
ejpam-5915	100	5	g	g	NOUN
ejpam-5915	100	6	=	=	NOUN
ejpam-5915	100	7	r	r	NOUN
ejpam-5915	100	8	with	with	ADP
ejpam-5915	100	9	the	the	DET
ejpam-5915	100	10	binary	binary	PROPN
ejpam-5915	100	11	operation	operation	NOUN
ejpam-5915	100	12	⊕k	⊕k	PROPN
ejpam-5915	100	13	and	and	CCONJ
ejpam-5915	100	14	the	the	DET
ejpam-5915	100	15	twisting	twisting	NOUN
ejpam-5915	100	16	map	map	NOUN
ejpam-5915	100	17	αk	αk	SCONJ
ejpam-5915	100	18	forms	form	VERB
ejpam-5915	100	19	a	a	DET
ejpam-5915	100	20	hom	hom	NOUN
ejpam-5915	100	21	-	-	PUNCT
ejpam-5915	100	22	group	group	NOUN
ejpam-5915	100	23	for	for	ADP
ejpam-5915	100	24	any	any	DET
ejpam-5915	100	25	k	k	PROPN
ejpam-5915	100	26	̸=	̸=	PROPN
ejpam-5915	100	27	0	0	NUM
ejpam-5915	100	28	.	.	PUNCT
ejpam-5915	100	29	remark	remark	NOUN
ejpam-5915	100	30	2	2	NUM
ejpam-5915	100	31	.	.	PUNCT
ejpam-5915	101	1	in	in	ADP
ejpam-5915	101	2	[	[	X
ejpam-5915	101	3	14	14	NUM
ejpam-5915	101	4	]	]	PUNCT
ejpam-5915	101	5	,	,	PUNCT
ejpam-5915	101	6	the	the	DET
ejpam-5915	101	7	case	case	NOUN
ejpam-5915	101	8	k	k	NOUN
ejpam-5915	101	9	=	=	SYM
ejpam-5915	101	10	2	2	NUM
ejpam-5915	101	11	was	be	AUX
ejpam-5915	101	12	specifically	specifically	ADV
ejpam-5915	101	13	considered	consider	VERB
ejpam-5915	101	14	in	in	ADP
ejpam-5915	101	15	the	the	DET
ejpam-5915	101	16	construction	construction	NOUN
ejpam-5915	101	17	of	of	ADP
ejpam-5915	101	18	a	a	DET
ejpam-5915	101	19	hom	hom	NOUN
ejpam-5915	101	20	-	-	PUNCT
ejpam-5915	101	21	group	group	NOUN
ejpam-5915	101	22	.	.	PUNCT
ejpam-5915	102	1	however	however	ADV
ejpam-5915	102	2	,	,	PUNCT
ejpam-5915	102	3	in	in	ADP
ejpam-5915	102	4	example	example	NOUN
ejpam-5915	102	5	2	2	NUM
ejpam-5915	102	6	,	,	PUNCT
ejpam-5915	102	7	we	we	PRON
ejpam-5915	102	8	generalize	generalize	VERB
ejpam-5915	102	9	this	this	DET
ejpam-5915	102	10	concept	concept	NOUN
ejpam-5915	102	11	by	by	ADP
ejpam-5915	102	12	defining	define	VERB
ejpam-5915	102	13	the	the	DET
ejpam-5915	102	14	operation	operation	NOUN
ejpam-5915	102	15	and	and	CCONJ
ejpam-5915	102	16	twisting	twisting	NOUN
ejpam-5915	102	17	map	map	NOUN
ejpam-5915	102	18	for	for	ADP
ejpam-5915	102	19	an	an	DET
ejpam-5915	102	20	arbitrary	arbitrary	ADJ
ejpam-5915	102	21	nonzero	nonzero	NOUN
ejpam-5915	102	22	parameter	parameter	PROPN
ejpam-5915	102	23	k.	k.	PROPN
ejpam-5915	103	1	this	this	DET
ejpam-5915	103	2	generalization	generalization	NOUN
ejpam-5915	103	3	provides	provide	VERB
ejpam-5915	103	4	a	a	DET
ejpam-5915	103	5	broader	broad	ADJ
ejpam-5915	103	6	class	class	NOUN
ejpam-5915	103	7	of	of	ADP
ejpam-5915	103	8	hom	hom	NOUN
ejpam-5915	103	9	-	-	PUNCT
ejpam-5915	103	10	groups	group	NOUN
ejpam-5915	103	11	,	,	PUNCT
ejpam-5915	103	12	demonstrating	demonstrate	VERB
ejpam-5915	103	13	how	how	SCONJ
ejpam-5915	103	14	different	different	ADJ
ejpam-5915	103	15	choices	choice	NOUN
ejpam-5915	103	16	of	of	ADP
ejpam-5915	103	17	k	k	PROPN
ejpam-5915	103	18	influence	influence	NOUN
ejpam-5915	103	19	the	the	DET
ejpam-5915	103	20	underlying	underlie	VERB
ejpam-5915	103	21	structure	structure	NOUN
ejpam-5915	103	22	.	.	PUNCT
ejpam-5915	104	1	definition	definition	NOUN
ejpam-5915	104	2	2	2	NUM
ejpam-5915	104	3	.	.	PUNCT
ejpam-5915	105	1	let	let	VERB
ejpam-5915	105	2	(	(	PUNCT
ejpam-5915	105	3	g	g	NOUN
ejpam-5915	105	4	,	,	PUNCT
ejpam-5915	105	5	·	·	PUNCT
ejpam-5915	105	6	,	,	PUNCT
ejpam-5915	105	7	α	α	X
ejpam-5915	105	8	)	)	PUNCT
ejpam-5915	105	9	be	be	VERB
ejpam-5915	105	10	a	a	DET
ejpam-5915	105	11	hom	hom	NOUN
ejpam-5915	105	12	-	-	PUNCT
ejpam-5915	105	13	group	group	NOUN
ejpam-5915	105	14	.	.	PUNCT
ejpam-5915	106	1	a	a	DET
ejpam-5915	106	2	subset	subset	NOUN
ejpam-5915	106	3	h	h	NOUN
ejpam-5915	106	4	⊆	⊆	NUM
ejpam-5915	106	5	g	g	NOUN
ejpam-5915	106	6	is	be	AUX
ejpam-5915	106	7	called	call	VERB
ejpam-5915	106	8	a	a	DET
ejpam-5915	106	9	hom	hom	NOUN
ejpam-5915	106	10	-	-	PUNCT
ejpam-5915	106	11	subgroup	subgroup	NOUN
ejpam-5915	106	12	if	if	SCONJ
ejpam-5915	106	13	(	(	PUNCT
ejpam-5915	106	14	h	h	NOUN
ejpam-5915	106	15	,	,	PUNCT
ejpam-5915	106	16	·	·	PUNCT
ejpam-5915	106	17	,	,	PUNCT
ejpam-5915	106	18	α|h	α|h	PROPN
ejpam-5915	106	19	)	)	PUNCT
ejpam-5915	106	20	is	be	AUX
ejpam-5915	106	21	a	a	DET
ejpam-5915	106	22	hom	hom	NOUN
ejpam-5915	106	23	-	-	PUNCT
ejpam-5915	106	24	group	group	NOUN
ejpam-5915	106	25	.	.	PUNCT
ejpam-5915	107	1	theorem	theorem	NOUN
ejpam-5915	107	2	1	1	NUM
ejpam-5915	107	3	.	.	PUNCT
ejpam-5915	108	1	let	let	AUX
ejpam-5915	108	2	(	(	PUNCT
ejpam-5915	108	3	g	g	NOUN
ejpam-5915	108	4	,	,	PUNCT
ejpam-5915	108	5	·	·	PUNCT
ejpam-5915	108	6	,	,	PUNCT
ejpam-5915	108	7	α	α	X
ejpam-5915	108	8	)	)	PUNCT
ejpam-5915	108	9	be	be	VERB
ejpam-5915	108	10	a	a	DET
ejpam-5915	108	11	hom	hom	NOUN
ejpam-5915	108	12	-	-	PUNCT
ejpam-5915	108	13	group	group	NOUN
ejpam-5915	108	14	,	,	PUNCT
ejpam-5915	108	15	and	and	CCONJ
ejpam-5915	108	16	let	let	VERB
ejpam-5915	108	17	h	h	PRON
ejpam-5915	108	18	⊆	⊆	NUM
ejpam-5915	108	19	g	g	NOUN
ejpam-5915	108	20	be	be	AUX
ejpam-5915	108	21	a	a	DET
ejpam-5915	108	22	non	non	ADJ
ejpam-5915	108	23	-	-	ADJ
ejpam-5915	108	24	empty	empty	ADJ
ejpam-5915	108	25	subset	subset	NOUN
ejpam-5915	108	26	.	.	PUNCT
ejpam-5915	109	1	then	then	ADV
ejpam-5915	109	2	h	h	PROPN
ejpam-5915	109	3	is	be	AUX
ejpam-5915	109	4	a	a	DET
ejpam-5915	109	5	hom	hom	NOUN
ejpam-5915	109	6	-	-	PUNCT
ejpam-5915	109	7	subgroup	subgroup	NOUN
ejpam-5915	109	8	of	of	ADP
ejpam-5915	109	9	g	g	PROPN
ejpam-5915	109	10	if	if	SCONJ
ejpam-5915	110	1	and	and	CCONJ
ejpam-5915	110	2	only	only	ADV
ejpam-5915	110	3	if	if	SCONJ
ejpam-5915	110	4	the	the	DET
ejpam-5915	110	5	following	follow	VERB
ejpam-5915	110	6	conditions	condition	NOUN
ejpam-5915	110	7	hold	hold	VERB
ejpam-5915	110	8	:	:	PUNCT
ejpam-5915	110	9	(	(	PUNCT
ejpam-5915	110	10	hs1	hs1	X
ejpam-5915	110	11	)	)	PUNCT
ejpam-5915	110	12	h	h	PROPN
ejpam-5915	110	13	is	be	AUX
ejpam-5915	110	14	closed	close	VERB
ejpam-5915	110	15	under	under	ADP
ejpam-5915	110	16	the	the	DET
ejpam-5915	110	17	hom	hom	NOUN
ejpam-5915	110	18	-	-	PUNCT
ejpam-5915	110	19	group	group	NOUN
ejpam-5915	110	20	operation	operation	NOUN
ejpam-5915	110	21	,	,	PUNCT
ejpam-5915	110	22	i.e.	i.e.	X
ejpam-5915	110	23	,	,	PUNCT
ejpam-5915	110	24	for	for	ADP
ejpam-5915	110	25	all	all	DET
ejpam-5915	110	26	h1	h1	PROPN
ejpam-5915	110	27	,	,	PUNCT
ejpam-5915	110	28	h2	h2	PROPN
ejpam-5915	110	29	∈	∈	PROPN
ejpam-5915	110	30	h	h	NOUN
ejpam-5915	110	31	,	,	PUNCT
ejpam-5915	110	32	h1	h1	PROPN
ejpam-5915	110	33	·	·	PUNCT
ejpam-5915	110	34	h2	h2	PROPN
ejpam-5915	110	35	∈	∈	PROPN
ejpam-5915	110	36	h	h	NOUN
ejpam-5915	110	37	,	,	PUNCT
ejpam-5915	110	38	(	(	PUNCT
ejpam-5915	110	39	hs2	hs2	NOUN
ejpam-5915	110	40	)	)	PUNCT
ejpam-5915	110	41	h	h	NOUN
ejpam-5915	110	42	is	be	AUX
ejpam-5915	110	43	closed	close	VERB
ejpam-5915	110	44	under	under	ADP
ejpam-5915	110	45	the	the	DET
ejpam-5915	110	46	twisting	twisting	NOUN
ejpam-5915	110	47	map	map	NOUN
ejpam-5915	110	48	α	α	NOUN
ejpam-5915	110	49	,	,	PUNCT
ejpam-5915	110	50	i.e.	i.e.	X
ejpam-5915	110	51	,	,	PUNCT
ejpam-5915	110	52	for	for	ADP
ejpam-5915	110	53	all	all	DET
ejpam-5915	110	54	h	h	NOUN
ejpam-5915	110	55	∈	∈	PROPN
ejpam-5915	110	56	h	h	NOUN
ejpam-5915	110	57	,	,	PUNCT
ejpam-5915	110	58	α(h	α(h	NOUN
ejpam-5915	110	59	)	)	PUNCT
ejpam-5915	110	60	∈	∈	PROPN
ejpam-5915	110	61	h	h	NOUN
ejpam-5915	110	62	,	,	PUNCT
ejpam-5915	110	63	(	(	PUNCT
ejpam-5915	110	64	hs3	hs3	NOUN
ejpam-5915	110	65	)	)	PUNCT
ejpam-5915	110	66	h	h	NOUN
ejpam-5915	110	67	is	be	AUX
ejpam-5915	110	68	closed	close	VERB
ejpam-5915	110	69	under	under	ADP
ejpam-5915	110	70	inverses	inverse	NOUN
ejpam-5915	110	71	,	,	PUNCT
ejpam-5915	110	72	i.e.	i.e.	X
ejpam-5915	110	73	,	,	PUNCT
ejpam-5915	110	74	for	for	ADP
ejpam-5915	110	75	all	all	DET
ejpam-5915	110	76	h	h	NOUN
ejpam-5915	110	77	∈	∈	PROPN
ejpam-5915	110	78	h	h	NOUN
ejpam-5915	110	79	,	,	PUNCT
ejpam-5915	111	1	h−1	h−1	PROPN
ejpam-5915	111	2	∈	∈	PROPN
ejpam-5915	111	3	h	h	NOUN
ejpam-5915	111	4	,	,	PUNCT
ejpam-5915	111	5	where	where	SCONJ
ejpam-5915	111	6	h−1	h−1	PROPN
ejpam-5915	111	7	denotes	denote	VERB
ejpam-5915	111	8	the	the	DET
ejpam-5915	111	9	inverse	inverse	NOUN
ejpam-5915	111	10	of	of	ADP
ejpam-5915	111	11	h	h	NOUN
ejpam-5915	111	12	under	under	ADP
ejpam-5915	111	13	the	the	DET
ejpam-5915	111	14	hom	hom	NOUN
ejpam-5915	111	15	-	-	PUNCT
ejpam-5915	111	16	group	group	NOUN
ejpam-5915	111	17	operation	operation	NOUN
ejpam-5915	111	18	·	·	PUNCT
ejpam-5915	111	19	.	.	PUNCT
ejpam-5915	112	1	proof	proof	NOUN
ejpam-5915	112	2	.	.	PUNCT
ejpam-5915	113	1	suppose	suppose	VERB
ejpam-5915	113	2	h	h	PRON
ejpam-5915	113	3	⊆	⊆	NUM
ejpam-5915	113	4	g	g	NOUN
ejpam-5915	113	5	satisfies	satisfy	VERB
ejpam-5915	113	6	the	the	DET
ejpam-5915	113	7	conditions	condition	NOUN
ejpam-5915	113	8	(	(	PUNCT
ejpam-5915	113	9	hs1	hs1	PROPN
ejpam-5915	113	10	)	)	PUNCT
ejpam-5915	113	11	,	,	PUNCT
ejpam-5915	113	12	(	(	PUNCT
ejpam-5915	113	13	hs2	hs2	NOUN
ejpam-5915	113	14	)	)	PUNCT
ejpam-5915	113	15	,	,	PUNCT
ejpam-5915	113	16	and	and	CCONJ
ejpam-5915	113	17	(	(	PUNCT
ejpam-5915	113	18	hs3	hs3	NOUN
ejpam-5915	113	19	)	)	PUNCT
ejpam-5915	113	20	.	.	PUNCT
ejpam-5915	114	1	we	we	PRON
ejpam-5915	114	2	will	will	AUX
ejpam-5915	114	3	prove	prove	VERB
ejpam-5915	114	4	that	that	SCONJ
ejpam-5915	114	5	h	h	PROPN
ejpam-5915	114	6	forms	form	VERB
ejpam-5915	114	7	a	a	DET
ejpam-5915	114	8	hom	hom	NOUN
ejpam-5915	114	9	-	-	PUNCT
ejpam-5915	114	10	subgroup	subgroup	NOUN
ejpam-5915	114	11	by	by	ADP
ejpam-5915	114	12	showing	show	VERB
ejpam-5915	114	13	that	that	SCONJ
ejpam-5915	114	14	it	it	PRON
ejpam-5915	114	15	satisfies	satisfy	VERB
ejpam-5915	114	16	both	both	DET
ejpam-5915	114	17	*	*	PROPN
ejpam-5915	114	18	*	*	PUNCT
ejpam-5915	114	19	hom	hom	NOUN
ejpam-5915	114	20	-	-	PUNCT
ejpam-5915	114	21	associativity	associativity	NOUN
ejpam-5915	114	22	*	*	NOUN
ejpam-5915	114	23	*	*	PUNCT
ejpam-5915	114	24	(	(	PUNCT
ejpam-5915	114	25	hg1	hg1	PROPN
ejpam-5915	114	26	)	)	PUNCT
ejpam-5915	114	27	and	and	CCONJ
ejpam-5915	114	28	*	*	PUNCT
ejpam-5915	114	29	*	*	PUNCT
ejpam-5915	114	30	hom	hom	NOUN
ejpam-5915	114	31	-	-	PUNCT
ejpam-5915	114	32	unitarity	unitarity	NOUN
ejpam-5915	114	33	*	*	NOUN
ejpam-5915	114	34	*	*	NOUN
ejpam-5915	114	35	(	(	PUNCT
ejpam-5915	114	36	hg2	hg2	NOUN
ejpam-5915	114	37	)	)	PUNCT
ejpam-5915	114	38	as	as	SCONJ
ejpam-5915	114	39	inherited	inherit	VERB
ejpam-5915	114	40	from	from	ADP
ejpam-5915	114	41	g.	g.	PROPN
ejpam-5915	114	42	step	step	NOUN
ejpam-5915	114	43	1	1	NUM
ejpam-5915	114	44	:	:	PUNCT
ejpam-5915	114	45	hom	hom	NOUN
ejpam-5915	114	46	-	-	PUNCT
ejpam-5915	114	47	associativity	associativity	NOUN
ejpam-5915	114	48	(	(	PUNCT
ejpam-5915	114	49	hg1	hg1	PROPN
ejpam-5915	114	50	):	):	PUNCT
ejpam-5915	114	51	since	since	SCONJ
ejpam-5915	114	52	h	h	NOUN
ejpam-5915	114	53	is	be	AUX
ejpam-5915	114	54	closed	close	VERB
ejpam-5915	114	55	under	under	ADP
ejpam-5915	114	56	the	the	DET
ejpam-5915	114	57	operation	operation	NOUN
ejpam-5915	114	58	·	·	PUNCT
ejpam-5915	114	59	and	and	CCONJ
ejpam-5915	114	60	g	g	NOUN
ejpam-5915	114	61	satisfies	satisfie	NOUN
ejpam-5915	114	62	hom	hom	NOUN
ejpam-5915	114	63	-	-	PUNCT
ejpam-5915	114	64	associativity	associativity	NOUN
ejpam-5915	114	65	,	,	PUNCT
ejpam-5915	114	66	for	for	ADP
ejpam-5915	114	67	all	all	DET
ejpam-5915	114	68	h1	h1	PROPN
ejpam-5915	114	69	,	,	PUNCT
ejpam-5915	114	70	h2	h2	PROPN
ejpam-5915	114	71	,	,	PUNCT
ejpam-5915	114	72	h3	h3	NOUN
ejpam-5915	114	73	∈	∈	PROPN
ejpam-5915	114	74	h	h	NOUN
ejpam-5915	114	75	,	,	PUNCT
ejpam-5915	114	76	we	we	PRON
ejpam-5915	114	77	have	have	VERB
ejpam-5915	114	78	:	:	PUNCT
ejpam-5915	114	79	α(h1	α(h1	NOUN
ejpam-5915	114	80	)	)	PUNCT
ejpam-5915	114	81	·	·	PUNCT
ejpam-5915	114	82	(	(	PUNCT
ejpam-5915	114	83	h2	h2	NOUN
ejpam-5915	114	84	·	·	PUNCT
ejpam-5915	114	85	h3	h3	NOUN
ejpam-5915	114	86	)	)	PUNCT
ejpam-5915	114	87	=	=	PUNCT
ejpam-5915	114	88	(	(	PUNCT
ejpam-5915	114	89	h1	h1	PROPN
ejpam-5915	114	90	·	·	SYM
ejpam-5915	114	91	h2	h2	NOUN
ejpam-5915	114	92	)	)	PUNCT
ejpam-5915	114	93	·	·	PUNCT
ejpam-5915	114	94	α(h3	α(h3	NOUN
ejpam-5915	114	95	)	)	PUNCT
ejpam-5915	114	96	.	.	PUNCT
ejpam-5915	115	1	s.	s.	PROPN
ejpam-5915	115	2	shaqaqha	shaqaqha	PROPN
ejpam-5915	115	3	/	/	SYM
ejpam-5915	115	4	eur	eur	PROPN
ejpam-5915	115	5	.	.	PUNCT
ejpam-5915	116	1	j.	j.	PROPN
ejpam-5915	116	2	pure	pure	PROPN
ejpam-5915	116	3	appl	appl	PROPN
ejpam-5915	116	4	.	.	PROPN
ejpam-5915	116	5	math	math	PROPN
ejpam-5915	116	6	,	,	PUNCT
ejpam-5915	116	7	18	18	NUM
ejpam-5915	116	8	(	(	PUNCT
ejpam-5915	116	9	2	2	NUM
ejpam-5915	116	10	)	)	PUNCT
ejpam-5915	116	11	(	(	PUNCT
ejpam-5915	116	12	2025	2025	NUM
ejpam-5915	116	13	)	)	PUNCT
ejpam-5915	116	14	,	,	PUNCT
ejpam-5915	116	15	5915	5915	NUM
ejpam-5915	116	16	6	6	NUM
ejpam-5915	116	17	of	of	ADP
ejpam-5915	116	18	16	16	NUM
ejpam-5915	116	19	therefore	therefore	ADV
ejpam-5915	116	20	,	,	PUNCT
ejpam-5915	116	21	h	h	NOUN
ejpam-5915	116	22	inherits	inherit	VERB
ejpam-5915	116	23	hom	hom	ADV
ejpam-5915	116	24	-	-	PUNCT
ejpam-5915	116	25	associativity	associativity	NOUN
ejpam-5915	116	26	from	from	ADP
ejpam-5915	116	27	g	g	PROPN
ejpam-5915	116	28	,	,	PUNCT
ejpam-5915	116	29	and	and	CCONJ
ejpam-5915	116	30	(	(	PUNCT
ejpam-5915	116	31	hg1	hg1	PROPN
ejpam-5915	116	32	)	)	PUNCT
ejpam-5915	116	33	is	be	AUX
ejpam-5915	116	34	satisfied	satisfied	ADJ
ejpam-5915	116	35	within	within	ADP
ejpam-5915	116	36	h.	h.	PROPN
ejpam-5915	116	37	step	step	NOUN
ejpam-5915	116	38	2	2	NUM
ejpam-5915	116	39	:	:	PUNCT
ejpam-5915	116	40	hom	hom	NOUN
ejpam-5915	116	41	-	-	PUNCT
ejpam-5915	116	42	unitarity	unitarity	NOUN
ejpam-5915	116	43	(	(	PUNCT
ejpam-5915	116	44	hg2	hg2	NOUN
ejpam-5915	116	45	):	):	PUNCT
ejpam-5915	116	46	we	we	PRON
ejpam-5915	116	47	need	need	VERB
ejpam-5915	116	48	to	to	PART
ejpam-5915	116	49	check	check	VERB
ejpam-5915	116	50	whether	whether	SCONJ
ejpam-5915	116	51	the	the	DET
ejpam-5915	116	52	hom	hom	NOUN
ejpam-5915	116	53	-	-	PUNCT
ejpam-5915	116	54	identity	identity	NOUN
ejpam-5915	116	55	element	element	NOUN
ejpam-5915	116	56	e	e	PROPN
ejpam-5915	116	57	∈	∈	PROPN
ejpam-5915	116	58	g	g	PROPN
ejpam-5915	116	59	belongs	belong	VERB
ejpam-5915	116	60	to	to	ADP
ejpam-5915	116	61	h.	h.	PROPN
ejpam-5915	116	62	since	since	SCONJ
ejpam-5915	116	63	h	h	PROPN
ejpam-5915	116	64	is	be	AUX
ejpam-5915	116	65	non	non	ADJ
ejpam-5915	116	66	-	-	ADJ
ejpam-5915	116	67	empty	empty	ADJ
ejpam-5915	116	68	,	,	PUNCT
ejpam-5915	116	69	take	take	VERB
ejpam-5915	116	70	any	any	DET
ejpam-5915	116	71	h	h	NOUN
ejpam-5915	116	72	∈	∈	PROPN
ejpam-5915	116	73	h.	h.	NOUN
ejpam-5915	116	74	by	by	ADP
ejpam-5915	116	75	(	(	PUNCT
ejpam-5915	116	76	hs3	hs3	NOUN
ejpam-5915	116	77	)	)	PUNCT
ejpam-5915	116	78	,	,	PUNCT
ejpam-5915	116	79	we	we	PRON
ejpam-5915	116	80	know	know	VERB
ejpam-5915	116	81	that	that	SCONJ
ejpam-5915	116	82	the	the	DET
ejpam-5915	116	83	inverse	inverse	NOUN
ejpam-5915	116	84	h−1	h−1	PROPN
ejpam-5915	116	85	∈	∈	PROPN
ejpam-5915	116	86	h.	h.	NOUN
ejpam-5915	116	87	now	now	ADV
ejpam-5915	116	88	,	,	PUNCT
ejpam-5915	116	89	by	by	ADP
ejpam-5915	116	90	(	(	PUNCT
ejpam-5915	116	91	hs1	hs1	PROPN
ejpam-5915	116	92	)	)	PUNCT
ejpam-5915	116	93	,	,	PUNCT
ejpam-5915	116	94	the	the	DET
ejpam-5915	116	95	operation	operation	NOUN
ejpam-5915	116	96	h	h	NOUN
ejpam-5915	116	97	·	·	PUNCT
ejpam-5915	117	1	h−1	h−1	NOUN
ejpam-5915	117	2	=	=	PUNCT
ejpam-5915	117	3	e	e	NOUN
ejpam-5915	117	4	must	must	AUX
ejpam-5915	117	5	also	also	ADV
ejpam-5915	117	6	belong	belong	VERB
ejpam-5915	117	7	to	to	ADP
ejpam-5915	117	8	h	h	NOUN
ejpam-5915	117	9	because	because	SCONJ
ejpam-5915	117	10	h	h	NOUN
ejpam-5915	117	11	is	be	AUX
ejpam-5915	117	12	closed	close	VERB
ejpam-5915	117	13	under	under	ADP
ejpam-5915	117	14	the	the	DET
ejpam-5915	117	15	operation	operation	NOUN
ejpam-5915	117	16	·	·	PUNCT
ejpam-5915	117	17	.	.	PUNCT
ejpam-5915	118	1	therefore	therefore	ADV
ejpam-5915	118	2	,	,	PUNCT
ejpam-5915	118	3	the	the	DET
ejpam-5915	118	4	hom	hom	NOUN
ejpam-5915	118	5	-	-	PUNCT
ejpam-5915	118	6	identity	identity	NOUN
ejpam-5915	118	7	element	element	NOUN
ejpam-5915	118	8	e	e	PROPN
ejpam-5915	118	9	∈	∈	PROPN
ejpam-5915	118	10	h.	h.	PROPN
ejpam-5915	118	11	since	since	SCONJ
ejpam-5915	118	12	e	e	PROPN
ejpam-5915	118	13	∈	∈	PROPN
ejpam-5915	118	14	h	h	NOUN
ejpam-5915	118	15	and	and	CCONJ
ejpam-5915	118	16	α(e	α(e	NOUN
ejpam-5915	118	17	)	)	PUNCT
ejpam-5915	119	1	=	=	SYM
ejpam-5915	119	2	e	e	X
ejpam-5915	119	3	(	(	PUNCT
ejpam-5915	119	4	inherited	inherit	VERB
ejpam-5915	119	5	from	from	ADP
ejpam-5915	119	6	g	g	PROPN
ejpam-5915	119	7	)	)	PUNCT
ejpam-5915	119	8	,	,	PUNCT
ejpam-5915	119	9	h	h	PROPN
ejpam-5915	119	10	satisfies	satisfy	VERB
ejpam-5915	119	11	the	the	DET
ejpam-5915	119	12	hom	hom	NOUN
ejpam-5915	119	13	-	-	PUNCT
ejpam-5915	119	14	unitarity	unitarity	NOUN
ejpam-5915	119	15	condition	condition	NOUN
ejpam-5915	119	16	(	(	PUNCT
ejpam-5915	119	17	hg2	hg2	NOUN
ejpam-5915	119	18	)	)	PUNCT
ejpam-5915	119	19	.	.	PUNCT
ejpam-5915	120	1	example	example	NOUN
ejpam-5915	121	1	3	3	X
ejpam-5915	121	2	.	.	X
ejpam-5915	121	3	consider	consider	VERB
ejpam-5915	121	4	the	the	DET
ejpam-5915	121	5	hom	hom	NOUN
ejpam-5915	121	6	-	-	PUNCT
ejpam-5915	121	7	group	group	NOUN
ejpam-5915	121	8	(	(	PUNCT
ejpam-5915	121	9	z,⊕	z,⊕	PROPN
ejpam-5915	121	10	,	,	PUNCT
ejpam-5915	121	11	α	α	NOUN
ejpam-5915	121	12	)	)	PUNCT
ejpam-5915	121	13	,	,	PUNCT
ejpam-5915	121	14	where	where	SCONJ
ejpam-5915	121	15	the	the	DET
ejpam-5915	121	16	operation	operation	NOUN
ejpam-5915	121	17	⊕	⊕	PROPN
ejpam-5915	121	18	and	and	CCONJ
ejpam-5915	121	19	twisting	twisting	NOUN
ejpam-5915	121	20	map	map	NOUN
ejpam-5915	121	21	α	α	NOUN
ejpam-5915	121	22	are	be	AUX
ejpam-5915	121	23	defined	define	VERB
ejpam-5915	121	24	as	as	SCONJ
ejpam-5915	121	25	follows	follow	VERB
ejpam-5915	121	26	:	:	PUNCT
ejpam-5915	121	27	a⊕	a⊕	PROPN
ejpam-5915	121	28	b	b	X
ejpam-5915	121	29	=	=	NOUN
ejpam-5915	121	30	−(a+	−(a+	NUM
ejpam-5915	121	31	b	b	NOUN
ejpam-5915	121	32	)	)	PUNCT
ejpam-5915	121	33	for	for	ADP
ejpam-5915	121	34	all	all	DET
ejpam-5915	121	35	a	a	PRON
ejpam-5915	121	36	,	,	PUNCT
ejpam-5915	121	37	b	b	PROPN
ejpam-5915	121	38	∈	∈	PROPN
ejpam-5915	121	39	z	z	PROPN
ejpam-5915	121	40	,	,	PUNCT
ejpam-5915	121	41	α(a	α(a	NOUN
ejpam-5915	121	42	)	)	PUNCT
ejpam-5915	121	43	=	=	VERB
ejpam-5915	122	1	−a	−a	VERB
ejpam-5915	122	2	for	for	ADP
ejpam-5915	122	3	all	all	DET
ejpam-5915	122	4	a	a	DET
ejpam-5915	122	5	∈	∈	PROPN
ejpam-5915	122	6	z.	z.	NOUN
ejpam-5915	122	7	this	this	DET
ejpam-5915	122	8	structure	structure	NOUN
ejpam-5915	122	9	satisfies	satisfy	VERB
ejpam-5915	122	10	the	the	DET
ejpam-5915	122	11	conditions	condition	NOUN
ejpam-5915	122	12	for	for	ADP
ejpam-5915	122	13	a	a	DET
ejpam-5915	122	14	hom	hom	NOUN
ejpam-5915	122	15	-	-	PUNCT
ejpam-5915	122	16	group	group	NOUN
ejpam-5915	122	17	,	,	PUNCT
ejpam-5915	122	18	as	as	SCONJ
ejpam-5915	122	19	described	describe	VERB
ejpam-5915	122	20	in	in	ADP
ejpam-5915	122	21	[	[	X
ejpam-5915	122	22	14	14	NUM
ejpam-5915	122	23	]	]	PUNCT
ejpam-5915	122	24	.	.	PUNCT
ejpam-5915	123	1	now	now	ADV
ejpam-5915	123	2	,	,	PUNCT
ejpam-5915	123	3	consider	consider	VERB
ejpam-5915	123	4	the	the	DET
ejpam-5915	123	5	subset	subset	NOUN
ejpam-5915	123	6	h	h	NOUN
ejpam-5915	124	1	=	=	PUNCT
ejpam-5915	124	2	2z	2z	NUM
ejpam-5915	124	3	,	,	PUNCT
ejpam-5915	124	4	the	the	DET
ejpam-5915	124	5	set	set	NOUN
ejpam-5915	124	6	of	of	ADP
ejpam-5915	124	7	all	all	DET
ejpam-5915	124	8	even	even	ADV
ejpam-5915	124	9	integers	integer	NOUN
ejpam-5915	124	10	.	.	PUNCT
ejpam-5915	125	1	we	we	PRON
ejpam-5915	125	2	claim	claim	VERB
ejpam-5915	125	3	that	that	SCONJ
ejpam-5915	125	4	h	h	NOUN
ejpam-5915	125	5	is	be	AUX
ejpam-5915	125	6	a	a	DET
ejpam-5915	125	7	hom	hom	NOUN
ejpam-5915	125	8	-	-	PUNCT
ejpam-5915	125	9	subgroup	subgroup	NOUN
ejpam-5915	125	10	of	of	ADP
ejpam-5915	125	11	(	(	PUNCT
ejpam-5915	125	12	z,⊕	z,⊕	PROPN
ejpam-5915	125	13	,	,	PUNCT
ejpam-5915	125	14	α	α	NOUN
ejpam-5915	125	15	)	)	PUNCT
ejpam-5915	125	16	.	.	PUNCT
ejpam-5915	126	1	this	this	PRON
ejpam-5915	126	2	is	be	AUX
ejpam-5915	126	3	verified	verify	VERB
ejpam-5915	126	4	as	as	SCONJ
ejpam-5915	126	5	follows	follow	VERB
ejpam-5915	126	6	:	:	PUNCT
ejpam-5915	126	7	(	(	PUNCT
ejpam-5915	126	8	hs1	hs1	X
ejpam-5915	126	9	)	)	PUNCT
ejpam-5915	126	10	closure	closure	NOUN
ejpam-5915	126	11	under	under	ADP
ejpam-5915	126	12	the	the	DET
ejpam-5915	126	13	operation	operation	NOUN
ejpam-5915	126	14	⊕	⊕	PROPN
ejpam-5915	126	15	:	:	PUNCT
ejpam-5915	126	16	for	for	ADP
ejpam-5915	126	17	any	any	DET
ejpam-5915	126	18	h1	h1	NOUN
ejpam-5915	126	19	,	,	PUNCT
ejpam-5915	126	20	h2	h2	PROPN
ejpam-5915	126	21	∈	∈	PROPN
ejpam-5915	126	22	2z	2z	NUM
ejpam-5915	126	23	,	,	PUNCT
ejpam-5915	126	24	we	we	PRON
ejpam-5915	126	25	have	have	AUX
ejpam-5915	126	26	:	:	PUNCT
ejpam-5915	126	27	h1	h1	VERB
ejpam-5915	126	28	⊕	⊕	PROPN
ejpam-5915	126	29	h2	h2	PROPN
ejpam-5915	126	30	=	=	PUNCT
ejpam-5915	126	31	−(h1	−(h1	NUM
ejpam-5915	126	32	+	+	NUM
ejpam-5915	126	33	h2	h2	NOUN
ejpam-5915	126	34	)	)	PUNCT
ejpam-5915	126	35	.	.	PUNCT
ejpam-5915	127	1	since	since	SCONJ
ejpam-5915	127	2	h1+h2	h1+h2	PROPN
ejpam-5915	127	3	∈	∈	PROPN
ejpam-5915	127	4	2z	2z	NUM
ejpam-5915	127	5	(	(	PUNCT
ejpam-5915	127	6	the	the	DET
ejpam-5915	127	7	sum	sum	NOUN
ejpam-5915	127	8	of	of	ADP
ejpam-5915	127	9	two	two	NUM
ejpam-5915	127	10	even	even	ADV
ejpam-5915	127	11	integers	integer	NOUN
ejpam-5915	127	12	is	be	AUX
ejpam-5915	127	13	even	even	ADV
ejpam-5915	127	14	)	)	PUNCT
ejpam-5915	127	15	,	,	PUNCT
ejpam-5915	127	16	we	we	PRON
ejpam-5915	127	17	know	know	VERB
ejpam-5915	127	18	that	that	SCONJ
ejpam-5915	127	19	−(h1+h2	−(h1+h2	X
ejpam-5915	127	20	)	)	PUNCT
ejpam-5915	127	21	∈	∈	PROPN
ejpam-5915	127	22	2z	2z	NUM
ejpam-5915	127	23	.	.	PUNCT
ejpam-5915	128	1	therefore	therefore	ADV
ejpam-5915	128	2	,	,	PUNCT
ejpam-5915	128	3	h	h	NOUN
ejpam-5915	128	4	is	be	AUX
ejpam-5915	128	5	closed	close	VERB
ejpam-5915	128	6	under	under	ADP
ejpam-5915	128	7	the	the	DET
ejpam-5915	128	8	operation	operation	NOUN
ejpam-5915	128	9	⊕.	⊕.	NOUN
ejpam-5915	128	10	(	(	PUNCT
ejpam-5915	128	11	hs2	hs2	NOUN
ejpam-5915	128	12	)	)	PUNCT
ejpam-5915	128	13	closure	closure	NOUN
ejpam-5915	128	14	under	under	ADP
ejpam-5915	128	15	the	the	DET
ejpam-5915	128	16	twisting	twisting	NOUN
ejpam-5915	128	17	map	map	NOUN
ejpam-5915	128	18	α	α	NOUN
ejpam-5915	128	19	:	:	PUNCT
ejpam-5915	128	20	for	for	ADP
ejpam-5915	128	21	any	any	DET
ejpam-5915	128	22	h	h	NOUN
ejpam-5915	128	23	∈	∈	NOUN
ejpam-5915	128	24	2z	2z	NUM
ejpam-5915	128	25	,	,	PUNCT
ejpam-5915	128	26	we	we	PRON
ejpam-5915	128	27	have	have	VERB
ejpam-5915	128	28	:	:	PUNCT
ejpam-5915	128	29	α(h	α(h	VERB
ejpam-5915	128	30	)	)	PUNCT
ejpam-5915	128	31	=	=	PUNCT
ejpam-5915	129	1	−h	−h	ADJ
ejpam-5915	129	2	.	.	PUNCT
ejpam-5915	130	1	since	since	SCONJ
ejpam-5915	130	2	the	the	DET
ejpam-5915	130	3	negative	negative	NOUN
ejpam-5915	130	4	of	of	ADP
ejpam-5915	130	5	an	an	DET
ejpam-5915	130	6	even	even	ADV
ejpam-5915	130	7	integer	integer	NOUN
ejpam-5915	130	8	is	be	AUX
ejpam-5915	130	9	still	still	ADV
ejpam-5915	130	10	an	an	DET
ejpam-5915	130	11	even	even	ADV
ejpam-5915	130	12	integer	integer	NOUN
ejpam-5915	130	13	,	,	PUNCT
ejpam-5915	130	14	α(h	α(h	NOUN
ejpam-5915	130	15	)	)	PUNCT
ejpam-5915	130	16	∈	∈	NOUN
ejpam-5915	130	17	2z	2z	NUM
ejpam-5915	130	18	.	.	PUNCT
ejpam-5915	131	1	thus	thus	ADV
ejpam-5915	131	2	,	,	PUNCT
ejpam-5915	131	3	h	h	NOUN
ejpam-5915	131	4	is	be	AUX
ejpam-5915	131	5	closed	close	VERB
ejpam-5915	131	6	under	under	ADP
ejpam-5915	131	7	the	the	DET
ejpam-5915	131	8	twisting	twisting	NOUN
ejpam-5915	131	9	map	map	NOUN
ejpam-5915	131	10	.	.	PUNCT
ejpam-5915	132	1	(	(	PUNCT
ejpam-5915	132	2	hs3	hs3	NOUN
ejpam-5915	132	3	)	)	PUNCT
ejpam-5915	132	4	closure	closure	NOUN
ejpam-5915	132	5	under	under	ADP
ejpam-5915	132	6	inverses	inverse	NOUN
ejpam-5915	132	7	:	:	PUNCT
ejpam-5915	132	8	for	for	ADP
ejpam-5915	132	9	any	any	DET
ejpam-5915	132	10	h	h	NOUN
ejpam-5915	132	11	∈	∈	PROPN
ejpam-5915	132	12	2z	2z	NUM
ejpam-5915	132	13	,	,	PUNCT
ejpam-5915	132	14	the	the	DET
ejpam-5915	132	15	inverse	inverse	NOUN
ejpam-5915	132	16	under	under	ADP
ejpam-5915	132	17	the	the	DET
ejpam-5915	132	18	operation	operation	NOUN
ejpam-5915	132	19	⊕	⊕	PROPN
ejpam-5915	132	20	is	be	AUX
ejpam-5915	132	21	h∗	h∗	NOUN
ejpam-5915	132	22	=	=	PUNCT
ejpam-5915	132	23	−h	−h	ADJ
ejpam-5915	132	24	.	.	PUNCT
ejpam-5915	133	1	since	since	SCONJ
ejpam-5915	133	2	−h	−h	ADJ
ejpam-5915	133	3	∈	∈	PROPN
ejpam-5915	133	4	2z	2z	NOUN
ejpam-5915	133	5	(	(	PUNCT
ejpam-5915	133	6	the	the	DET
ejpam-5915	133	7	set	set	NOUN
ejpam-5915	133	8	of	of	ADP
ejpam-5915	133	9	even	even	ADV
ejpam-5915	133	10	integers	integer	NOUN
ejpam-5915	133	11	is	be	AUX
ejpam-5915	133	12	closed	close	VERB
ejpam-5915	133	13	under	under	ADP
ejpam-5915	133	14	negation	negation	NOUN
ejpam-5915	133	15	)	)	PUNCT
ejpam-5915	133	16	,	,	PUNCT
ejpam-5915	133	17	h	h	NOUN
ejpam-5915	133	18	is	be	AUX
ejpam-5915	133	19	closed	close	VERB
ejpam-5915	133	20	under	under	ADP
ejpam-5915	133	21	inverses	inverse	NOUN
ejpam-5915	133	22	.	.	PUNCT
ejpam-5915	134	1	therefore	therefore	ADV
ejpam-5915	134	2	,	,	PUNCT
ejpam-5915	134	3	h	h	NOUN
ejpam-5915	134	4	=	=	NOUN
ejpam-5915	134	5	2z	2z	NUM
ejpam-5915	134	6	is	be	AUX
ejpam-5915	134	7	a	a	DET
ejpam-5915	134	8	nontrivial	nontrivial	ADJ
ejpam-5915	134	9	hom	hom	NOUN
ejpam-5915	134	10	-	-	PUNCT
ejpam-5915	134	11	subgroup	subgroup	NOUN
ejpam-5915	134	12	of	of	ADP
ejpam-5915	134	13	(	(	PUNCT
ejpam-5915	134	14	z,⊕	z,⊕	PROPN
ejpam-5915	134	15	,	,	PUNCT
ejpam-5915	134	16	α	α	NOUN
ejpam-5915	134	17	)	)	PUNCT
ejpam-5915	134	18	.	.	PUNCT
ejpam-5915	135	1	definition	definition	NOUN
ejpam-5915	135	2	3	3	NUM
ejpam-5915	135	3	.	.	PUNCT
ejpam-5915	136	1	[	[	X
ejpam-5915	136	2	14	14	NUM
ejpam-5915	136	3	]	]	X
ejpam-5915	136	4	let	let	VERB
ejpam-5915	136	5	(	(	PUNCT
ejpam-5915	136	6	g	g	NOUN
ejpam-5915	136	7	,	,	PUNCT
ejpam-5915	136	8	·	·	PUNCT
ejpam-5915	136	9	,	,	PUNCT
ejpam-5915	136	10	α	α	X
ejpam-5915	136	11	)	)	PUNCT
ejpam-5915	136	12	be	be	VERB
ejpam-5915	136	13	a	a	DET
ejpam-5915	136	14	hom	hom	NOUN
ejpam-5915	136	15	-	-	PUNCT
ejpam-5915	136	16	group	group	NOUN
ejpam-5915	136	17	.	.	PUNCT
ejpam-5915	137	1	a	a	DET
ejpam-5915	137	2	subset	subset	NOUN
ejpam-5915	137	3	n	n	PRON
ejpam-5915	137	4	⊆	⊆	NUM
ejpam-5915	137	5	g	g	NOUN
ejpam-5915	137	6	is	be	AUX
ejpam-5915	137	7	called	call	VERB
ejpam-5915	137	8	a	a	DET
ejpam-5915	137	9	homnormal	homnormal	ADJ
ejpam-5915	137	10	subgroup	subgroup	NOUN
ejpam-5915	137	11	if	if	SCONJ
ejpam-5915	137	12	:	:	PUNCT
ejpam-5915	137	13	(	(	PUNCT
ejpam-5915	137	14	hn1	hn1	NOUN
ejpam-5915	137	15	)	)	PUNCT
ejpam-5915	137	16	n	n	PRON
ejpam-5915	137	17	is	be	AUX
ejpam-5915	137	18	a	a	DET
ejpam-5915	137	19	hom	hom	NOUN
ejpam-5915	137	20	-	-	PUNCT
ejpam-5915	137	21	subgroup	subgroup	NOUN
ejpam-5915	137	22	of	of	ADP
ejpam-5915	137	23	g	g	PROPN
ejpam-5915	137	24	,	,	PUNCT
ejpam-5915	137	25	(	(	PUNCT
ejpam-5915	137	26	hn2	hn2	PROPN
ejpam-5915	137	27	)	)	PUNCT
ejpam-5915	138	1	n	n	PRON
ejpam-5915	138	2	is	be	AUX
ejpam-5915	138	3	closed	close	VERB
ejpam-5915	138	4	under	under	ADP
ejpam-5915	138	5	conjugation	conjugation	NOUN
ejpam-5915	138	6	,	,	PUNCT
ejpam-5915	138	7	i.e.	i.e.	X
ejpam-5915	138	8	,	,	PUNCT
ejpam-5915	138	9	for	for	ADP
ejpam-5915	138	10	all	all	DET
ejpam-5915	138	11	g	g	PROPN
ejpam-5915	138	12	∈	∈	PROPN
ejpam-5915	138	13	g	g	NOUN
ejpam-5915	138	14	and	and	CCONJ
ejpam-5915	138	15	n	n	CCONJ
ejpam-5915	138	16	∈	∈	PROPN
ejpam-5915	138	17	n	n	NOUN
ejpam-5915	138	18	,	,	PUNCT
ejpam-5915	138	19	g	g	PROPN
ejpam-5915	138	20	·	·	PUNCT
ejpam-5915	138	21	n	n	X
ejpam-5915	138	22	·	·	PUNCT
ejpam-5915	138	23	g−1	g−1	PROPN
ejpam-5915	138	24	∈	∈	PROPN
ejpam-5915	138	25	n.	n.	NOUN
ejpam-5915	138	26	example	example	NOUN
ejpam-5915	139	1	4	4	X
ejpam-5915	139	2	.	.	PUNCT
ejpam-5915	139	3	consider	consider	VERB
ejpam-5915	139	4	the	the	DET
ejpam-5915	139	5	hom	hom	NOUN
ejpam-5915	139	6	-	-	PUNCT
ejpam-5915	139	7	group	group	NOUN
ejpam-5915	139	8	(	(	PUNCT
ejpam-5915	139	9	g	g	NOUN
ejpam-5915	139	10	=	=	SYM
ejpam-5915	139	11	z,⊕	z,⊕	PROPN
ejpam-5915	139	12	,	,	PUNCT
ejpam-5915	139	13	α	α	NOUN
ejpam-5915	139	14	)	)	PUNCT
ejpam-5915	139	15	,	,	PUNCT
ejpam-5915	139	16	where	where	SCONJ
ejpam-5915	139	17	:	:	PUNCT
ejpam-5915	139	18	s.	s.	PROPN
ejpam-5915	139	19	shaqaqha	shaqaqha	PROPN
ejpam-5915	139	20	/	/	SYM
ejpam-5915	139	21	eur	eur	PROPN
ejpam-5915	139	22	.	.	PUNCT
ejpam-5915	140	1	j.	j.	PROPN
ejpam-5915	140	2	pure	pure	PROPN
ejpam-5915	140	3	appl	appl	PROPN
ejpam-5915	140	4	.	.	PROPN
ejpam-5915	140	5	math	math	PROPN
ejpam-5915	140	6	,	,	PUNCT
ejpam-5915	140	7	18	18	NUM
ejpam-5915	140	8	(	(	PUNCT
ejpam-5915	140	9	2	2	NUM
ejpam-5915	140	10	)	)	PUNCT
ejpam-5915	140	11	(	(	PUNCT
ejpam-5915	140	12	2025	2025	NUM
ejpam-5915	140	13	)	)	PUNCT
ejpam-5915	140	14	,	,	PUNCT
ejpam-5915	140	15	5915	5915	NUM
ejpam-5915	140	16	7	7	NUM
ejpam-5915	140	17	of	of	ADP
ejpam-5915	140	18	16	16	NUM
ejpam-5915	140	19	•	•	NOUN
ejpam-5915	140	20	the	the	DET
ejpam-5915	140	21	operation	operation	NOUN
ejpam-5915	140	22	is	be	AUX
ejpam-5915	140	23	defined	define	VERB
ejpam-5915	140	24	as	as	ADP
ejpam-5915	140	25	a⊕	a⊕	PROPN
ejpam-5915	140	26	b	b	PROPN
ejpam-5915	140	27	=	=	NOUN
ejpam-5915	140	28	−(a+	−(a+	NUM
ejpam-5915	140	29	b	b	NOUN
ejpam-5915	140	30	)	)	PUNCT
ejpam-5915	140	31	for	for	ADP
ejpam-5915	140	32	all	all	DET
ejpam-5915	140	33	a	a	DET
ejpam-5915	140	34	,	,	PUNCT
ejpam-5915	140	35	b	b	PROPN
ejpam-5915	140	36	∈	∈	PROPN
ejpam-5915	140	37	z	z	PROPN
ejpam-5915	140	38	,	,	PUNCT
ejpam-5915	140	39	•	•	ADP
ejpam-5915	140	40	the	the	DET
ejpam-5915	140	41	twisting	twisting	NOUN
ejpam-5915	140	42	map	map	NOUN
ejpam-5915	140	43	is	be	AUX
ejpam-5915	140	44	α(a	α(a	NOUN
ejpam-5915	140	45	)	)	PUNCT
ejpam-5915	141	1	=	=	VERB
ejpam-5915	141	2	−a	−a	VERB
ejpam-5915	141	3	for	for	SCONJ
ejpam-5915	141	4	all	all	DET
ejpam-5915	141	5	a	a	DET
ejpam-5915	141	6	∈	∈	NOUN
ejpam-5915	141	7	z.	z.	NOUN
ejpam-5915	141	8	let	let	VERB
ejpam-5915	141	9	n	n	NOUN
ejpam-5915	141	10	=	=	VERB
ejpam-5915	141	11	2z	2z	NUM
ejpam-5915	141	12	,	,	PUNCT
ejpam-5915	141	13	the	the	DET
ejpam-5915	141	14	set	set	NOUN
ejpam-5915	141	15	of	of	ADP
ejpam-5915	141	16	all	all	DET
ejpam-5915	141	17	even	even	ADV
ejpam-5915	141	18	integers	integer	NOUN
ejpam-5915	141	19	.	.	PUNCT
ejpam-5915	142	1	we	we	PRON
ejpam-5915	142	2	previously	previously	ADV
ejpam-5915	142	3	showed	show	VERB
ejpam-5915	142	4	that	that	SCONJ
ejpam-5915	142	5	n	n	PRON
ejpam-5915	142	6	is	be	AUX
ejpam-5915	142	7	a	a	DET
ejpam-5915	142	8	homsubgroup	homsubgroup	NOUN
ejpam-5915	142	9	of	of	ADP
ejpam-5915	142	10	(	(	PUNCT
ejpam-5915	142	11	z,⊕	z,⊕	PROPN
ejpam-5915	142	12	,	,	PUNCT
ejpam-5915	142	13	α	α	NOUN
ejpam-5915	142	14	)	)	PUNCT
ejpam-5915	142	15	(	(	PUNCT
ejpam-5915	142	16	see	see	VERB
ejpam-5915	142	17	example	example	NOUN
ejpam-5915	142	18	3	3	NUM
ejpam-5915	142	19	)	)	PUNCT
ejpam-5915	142	20	.	.	PUNCT
ejpam-5915	143	1	now	now	ADV
ejpam-5915	143	2	,	,	PUNCT
ejpam-5915	143	3	we	we	PRON
ejpam-5915	143	4	will	will	AUX
ejpam-5915	143	5	verify	verify	VERB
ejpam-5915	143	6	that	that	SCONJ
ejpam-5915	143	7	n	n	PRON
ejpam-5915	143	8	is	be	AUX
ejpam-5915	143	9	also	also	ADV
ejpam-5915	143	10	a	a	DET
ejpam-5915	143	11	hom	hom	ADJ
ejpam-5915	143	12	-	-	PUNCT
ejpam-5915	143	13	normal	normal	ADJ
ejpam-5915	143	14	subgroup	subgroup	NOUN
ejpam-5915	143	15	.	.	PUNCT
ejpam-5915	144	1	hom	hom	X
ejpam-5915	144	2	-	-	PUNCT
ejpam-5915	144	3	normal	normal	ADJ
ejpam-5915	144	4	subgroup	subgroup	NOUN
ejpam-5915	144	5	property	property	NOUN
ejpam-5915	144	6	(	(	PUNCT
ejpam-5915	144	7	hn2	hn2	PROPN
ejpam-5915	144	8	):	):	PUNCT
ejpam-5915	144	9	to	to	PART
ejpam-5915	144	10	check	check	VERB
ejpam-5915	144	11	that	that	PRON
ejpam-5915	144	12	n	n	PRON
ejpam-5915	144	13	is	be	AUX
ejpam-5915	144	14	closed	close	VERB
ejpam-5915	144	15	under	under	ADP
ejpam-5915	144	16	conjugation	conjugation	NOUN
ejpam-5915	144	17	,	,	PUNCT
ejpam-5915	144	18	take	take	VERB
ejpam-5915	144	19	any	any	DET
ejpam-5915	144	20	g	g	PROPN
ejpam-5915	144	21	∈	∈	PROPN
ejpam-5915	144	22	z	z	NOUN
ejpam-5915	144	23	and	and	CCONJ
ejpam-5915	144	24	n	n	CCONJ
ejpam-5915	144	25	∈	∈	PROPN
ejpam-5915	144	26	n	n	NOUN
ejpam-5915	144	27	.	.	PUNCT
ejpam-5915	145	1	we	we	PRON
ejpam-5915	145	2	compute	compute	VERB
ejpam-5915	145	3	the	the	DET
ejpam-5915	145	4	conjugation	conjugation	NOUN
ejpam-5915	145	5	:	:	PUNCT
ejpam-5915	145	6	g	g	PROPN
ejpam-5915	145	7	⊕	⊕	PROPN
ejpam-5915	145	8	n⊕	n⊕	PROPN
ejpam-5915	145	9	g∗	g∗	PROPN
ejpam-5915	145	10	=	=	PUNCT
ejpam-5915	145	11	−(g	−(g	NOUN
ejpam-5915	145	12	+	+	CCONJ
ejpam-5915	145	13	n)⊕	n)⊕	ADJ
ejpam-5915	145	14	(	(	PUNCT
ejpam-5915	145	15	−g	−g	NOUN
ejpam-5915	145	16	)	)	PUNCT
ejpam-5915	145	17	.	.	PUNCT
ejpam-5915	146	1	using	use	VERB
ejpam-5915	146	2	the	the	DET
ejpam-5915	146	3	operation	operation	NOUN
ejpam-5915	146	4	⊕	⊕	PROPN
ejpam-5915	146	5	,	,	PUNCT
ejpam-5915	146	6	this	this	DET
ejpam-5915	146	7	simplifies	simplifie	NOUN
ejpam-5915	146	8	to	to	PART
ejpam-5915	146	9	:	:	PUNCT
ejpam-5915	146	10	−(g	−(g	VERB
ejpam-5915	146	11	+	+	CCONJ
ejpam-5915	146	12	n)⊕	n)⊕	ADJ
ejpam-5915	146	13	(	(	PUNCT
ejpam-5915	146	14	−g	−g	NOUN
ejpam-5915	146	15	)	)	PUNCT
ejpam-5915	146	16	=	=	PRON
ejpam-5915	146	17	−(−(g	−(−(g	NOUN
ejpam-5915	146	18	+	+	CCONJ
ejpam-5915	146	19	n	n	CCONJ
ejpam-5915	146	20	)	)	PUNCT
ejpam-5915	146	21	+	+	CCONJ
ejpam-5915	146	22	g	g	NOUN
ejpam-5915	146	23	)	)	PUNCT
ejpam-5915	146	24	=	=	PUNCT
ejpam-5915	147	1	−n	−n	ADJ
ejpam-5915	147	2	.	.	PUNCT
ejpam-5915	148	1	since	since	SCONJ
ejpam-5915	148	2	n	n	NOUN
ejpam-5915	148	3	∈	∈	PROPN
ejpam-5915	148	4	2z	2z	NUM
ejpam-5915	148	5	,	,	PUNCT
ejpam-5915	148	6	we	we	PRON
ejpam-5915	148	7	know	know	VERB
ejpam-5915	148	8	that	that	SCONJ
ejpam-5915	148	9	−n	−n	NOUN
ejpam-5915	148	10	∈	∈	PROPN
ejpam-5915	148	11	2z	2z	NUM
ejpam-5915	148	12	,	,	PUNCT
ejpam-5915	148	13	so	so	CCONJ
ejpam-5915	148	14	the	the	DET
ejpam-5915	148	15	conjugation	conjugation	NOUN
ejpam-5915	148	16	result	result	VERB
ejpam-5915	148	17	g	g	PROPN
ejpam-5915	148	18	⊕	⊕	PROPN
ejpam-5915	148	19	n⊕	n⊕	PROPN
ejpam-5915	148	20	g∗	g∗	PROPN
ejpam-5915	148	21	∈	∈	PROPN
ejpam-5915	148	22	2z	2z	NUM
ejpam-5915	148	23	.	.	PUNCT
ejpam-5915	149	1	therefore	therefore	ADV
ejpam-5915	149	2	,	,	PUNCT
ejpam-5915	149	3	n	n	PROPN
ejpam-5915	149	4	=	=	PRON
ejpam-5915	149	5	2z	2z	NUM
ejpam-5915	149	6	is	be	AUX
ejpam-5915	149	7	a	a	DET
ejpam-5915	149	8	hom	hom	ADV
ejpam-5915	149	9	-	-	PUNCT
ejpam-5915	149	10	normal	normal	ADJ
ejpam-5915	149	11	subgroup	subgroup	NOUN
ejpam-5915	149	12	of	of	ADP
ejpam-5915	149	13	g.	g.	PROPN
ejpam-5915	149	14	3	3	NUM
ejpam-5915	149	15	.	.	PUNCT
ejpam-5915	149	16	fuzzy	fuzzy	ADJ
ejpam-5915	149	17	hom	hom	NOUN
ejpam-5915	149	18	-	-	PUNCT
ejpam-5915	149	19	subgroups	subgroup	NOUN
ejpam-5915	149	20	and	and	CCONJ
ejpam-5915	149	21	fuzzy	fuzzy	ADJ
ejpam-5915	149	22	hom	hom	NOUN
ejpam-5915	149	23	-	-	PUNCT
ejpam-5915	149	24	normal	normal	ADJ
ejpam-5915	149	25	subgroups	subgroup	NOUN
ejpam-5915	149	26	in	in	ADP
ejpam-5915	149	27	this	this	DET
ejpam-5915	149	28	section	section	NOUN
ejpam-5915	149	29	,	,	PUNCT
ejpam-5915	149	30	we	we	PRON
ejpam-5915	149	31	introduce	introduce	VERB
ejpam-5915	149	32	the	the	DET
ejpam-5915	149	33	concept	concept	NOUN
ejpam-5915	149	34	of	of	ADP
ejpam-5915	149	35	fuzzy	fuzzy	ADJ
ejpam-5915	149	36	hom	hom	NOUN
ejpam-5915	149	37	-	-	PUNCT
ejpam-5915	149	38	subgroups	subgroup	NOUN
ejpam-5915	149	39	(	(	PUNCT
ejpam-5915	149	40	and	and	CCONJ
ejpam-5915	149	41	hom	hom	ADV
ejpam-5915	149	42	-	-	PUNCT
ejpam-5915	149	43	normal	normal	ADJ
ejpam-5915	149	44	subgroups	subgroup	NOUN
ejpam-5915	149	45	)	)	PUNCT
ejpam-5915	149	46	,	,	PUNCT
ejpam-5915	149	47	extending	extend	VERB
ejpam-5915	149	48	the	the	DET
ejpam-5915	149	49	classical	classical	ADJ
ejpam-5915	149	50	notion	notion	NOUN
ejpam-5915	149	51	of	of	ADP
ejpam-5915	149	52	fuzzy	fuzzy	ADJ
ejpam-5915	149	53	subgroups	subgroup	NOUN
ejpam-5915	149	54	(	(	PUNCT
ejpam-5915	149	55	and	and	CCONJ
ejpam-5915	149	56	fuzzy	fuzzy	ADJ
ejpam-5915	149	57	normal	normal	ADJ
ejpam-5915	149	58	subgroups	subgroup	NOUN
ejpam-5915	149	59	)	)	PUNCT
ejpam-5915	149	60	to	to	ADP
ejpam-5915	149	61	the	the	DET
ejpam-5915	149	62	hom	hom	NOUN
ejpam-5915	149	63	-	-	PUNCT
ejpam-5915	149	64	group	group	NOUN
ejpam-5915	149	65	framework	framework	NOUN
ejpam-5915	149	66	.	.	PUNCT
ejpam-5915	150	1	this	this	DET
ejpam-5915	150	2	construction	construction	NOUN
ejpam-5915	150	3	builds	build	VERB
ejpam-5915	150	4	on	on	ADP
ejpam-5915	150	5	the	the	DET
ejpam-5915	150	6	foundations	foundation	NOUN
ejpam-5915	150	7	of	of	ADP
ejpam-5915	150	8	fuzzy	fuzzy	ADJ
ejpam-5915	150	9	set	set	NOUN
ejpam-5915	150	10	theory	theory	NOUN
ejpam-5915	150	11	and	and	CCONJ
ejpam-5915	150	12	hom	hom	NOUN
ejpam-5915	150	13	-	-	PUNCT
ejpam-5915	150	14	groups	group	NOUN
ejpam-5915	150	15	.	.	PUNCT
ejpam-5915	151	1	while	while	SCONJ
ejpam-5915	151	2	fuzzy	fuzzy	ADJ
ejpam-5915	151	3	subsets	subset	NOUN
ejpam-5915	151	4	have	have	AUX
ejpam-5915	151	5	been	be	AUX
ejpam-5915	151	6	widely	widely	ADV
ejpam-5915	151	7	studied	study	VERB
ejpam-5915	151	8	in	in	ADP
ejpam-5915	151	9	algebraic	algebraic	ADJ
ejpam-5915	151	10	structures	structure	NOUN
ejpam-5915	151	11	,	,	PUNCT
ejpam-5915	151	12	their	their	PRON
ejpam-5915	151	13	integration	integration	NOUN
ejpam-5915	151	14	with	with	ADP
ejpam-5915	151	15	hom	hom	NOUN
ejpam-5915	151	16	-	-	PUNCT
ejpam-5915	151	17	group	group	NOUN
ejpam-5915	151	18	structures	structure	NOUN
ejpam-5915	151	19	has	have	AUX
ejpam-5915	151	20	not	not	PART
ejpam-5915	151	21	been	be	AUX
ejpam-5915	151	22	explicitly	explicitly	ADV
ejpam-5915	151	23	explored	explore	VERB
ejpam-5915	151	24	.	.	PUNCT
ejpam-5915	152	1	before	before	ADP
ejpam-5915	152	2	proceeding	proceeding	NOUN
ejpam-5915	152	3	,	,	PUNCT
ejpam-5915	152	4	we	we	PRON
ejpam-5915	152	5	recall	recall	VERB
ejpam-5915	152	6	the	the	DET
ejpam-5915	152	7	definition	definition	NOUN
ejpam-5915	152	8	of	of	ADP
ejpam-5915	152	9	a	a	DET
ejpam-5915	152	10	fuzzy	fuzzy	ADJ
ejpam-5915	152	11	set	set	NOUN
ejpam-5915	152	12	.	.	PUNCT
ejpam-5915	153	1	a	a	DET
ejpam-5915	153	2	fuzzy	fuzzy	ADJ
ejpam-5915	153	3	set	set	VERB
ejpam-5915	153	4	µ	µ	NOUN
ejpam-5915	153	5	on	on	ADP
ejpam-5915	153	6	a	a	DET
ejpam-5915	153	7	set	set	NOUN
ejpam-5915	153	8	g	g	NOUN
ejpam-5915	153	9	is	be	AUX
ejpam-5915	153	10	a	a	DET
ejpam-5915	153	11	function	function	NOUN
ejpam-5915	153	12	µ	µ	NOUN
ejpam-5915	153	13	:	:	PUNCT
ejpam-5915	153	14	g	g	NOUN
ejpam-5915	153	15	→	→	SYM
ejpam-5915	153	16	[	[	X
ejpam-5915	153	17	0	0	NUM
ejpam-5915	153	18	,	,	PUNCT
ejpam-5915	153	19	1	1	NUM
ejpam-5915	153	20	]	]	PUNCT
ejpam-5915	153	21	that	that	PRON
ejpam-5915	153	22	assigns	assign	VERB
ejpam-5915	153	23	to	to	ADP
ejpam-5915	153	24	each	each	DET
ejpam-5915	153	25	element	element	NOUN
ejpam-5915	153	26	g	g	PROPN
ejpam-5915	153	27	∈	∈	PROPN
ejpam-5915	153	28	g	g	ADP
ejpam-5915	153	29	a	a	DET
ejpam-5915	153	30	membership	membership	NOUN
ejpam-5915	153	31	degree	degree	NOUN
ejpam-5915	153	32	µ(g	µ(g	PROPN
ejpam-5915	153	33	)	)	PUNCT
ejpam-5915	153	34	,	,	PUNCT
ejpam-5915	153	35	representing	represent	VERB
ejpam-5915	153	36	the	the	DET
ejpam-5915	153	37	extent	extent	NOUN
ejpam-5915	153	38	to	to	PART
ejpam-5915	153	39	which	which	PRON
ejpam-5915	153	40	g	g	NOUN
ejpam-5915	153	41	belongs	belong	VERB
ejpam-5915	153	42	to	to	ADP
ejpam-5915	153	43	the	the	DET
ejpam-5915	153	44	set	set	NOUN
ejpam-5915	153	45	.	.	PUNCT
ejpam-5915	154	1	using	use	VERB
ejpam-5915	154	2	this	this	DET
ejpam-5915	154	3	foundation	foundation	NOUN
ejpam-5915	154	4	,	,	PUNCT
ejpam-5915	154	5	we	we	PRON
ejpam-5915	154	6	develop	develop	VERB
ejpam-5915	154	7	the	the	DET
ejpam-5915	154	8	notion	notion	NOUN
ejpam-5915	154	9	of	of	ADP
ejpam-5915	154	10	fuzzy	fuzzy	ADJ
ejpam-5915	154	11	hom	hom	NOUN
ejpam-5915	154	12	-	-	PUNCT
ejpam-5915	154	13	subgroups	subgroup	NOUN
ejpam-5915	154	14	and	and	CCONJ
ejpam-5915	154	15	establish	establish	VERB
ejpam-5915	154	16	their	their	PRON
ejpam-5915	154	17	key	key	ADJ
ejpam-5915	154	18	properties	property	NOUN
ejpam-5915	154	19	.	.	PUNCT
ejpam-5915	155	1	definition	definition	NOUN
ejpam-5915	155	2	4	4	NUM
ejpam-5915	155	3	.	.	PUNCT
ejpam-5915	156	1	let	let	VERB
ejpam-5915	156	2	(	(	PUNCT
ejpam-5915	156	3	g	g	NOUN
ejpam-5915	156	4	,	,	PUNCT
ejpam-5915	156	5	·	·	PUNCT
ejpam-5915	156	6	,	,	PUNCT
ejpam-5915	156	7	α	α	X
ejpam-5915	156	8	)	)	PUNCT
ejpam-5915	156	9	be	be	VERB
ejpam-5915	156	10	a	a	DET
ejpam-5915	156	11	hom	hom	NOUN
ejpam-5915	156	12	-	-	PUNCT
ejpam-5915	156	13	group	group	NOUN
ejpam-5915	156	14	and	and	CCONJ
ejpam-5915	156	15	µ	µ	NOUN
ejpam-5915	156	16	:	:	PUNCT
ejpam-5915	156	17	g	g	NOUN
ejpam-5915	156	18	→	→	SYM
ejpam-5915	156	19	[	[	X
ejpam-5915	156	20	0	0	NUM
ejpam-5915	156	21	,	,	PUNCT
ejpam-5915	156	22	1	1	NUM
ejpam-5915	156	23	]	]	PUNCT
ejpam-5915	156	24	be	be	AUX
ejpam-5915	156	25	a	a	DET
ejpam-5915	156	26	fuzzy	fuzzy	ADJ
ejpam-5915	156	27	set	set	NOUN
ejpam-5915	156	28	on	on	ADP
ejpam-5915	156	29	g.	g.	PROPN
ejpam-5915	156	30	the	the	DET
ejpam-5915	156	31	fuzzy	fuzzy	ADJ
ejpam-5915	156	32	set	set	VERB
ejpam-5915	156	33	µ	µ	NOUN
ejpam-5915	156	34	is	be	AUX
ejpam-5915	156	35	called	call	VERB
ejpam-5915	156	36	a	a	DET
ejpam-5915	156	37	fuzzy	fuzzy	ADJ
ejpam-5915	156	38	hom	hom	NOUN
ejpam-5915	156	39	-	-	PUNCT
ejpam-5915	156	40	subgroup	subgroup	NOUN
ejpam-5915	156	41	if	if	SCONJ
ejpam-5915	156	42	for	for	ADP
ejpam-5915	156	43	all	all	DET
ejpam-5915	156	44	g	g	NOUN
ejpam-5915	156	45	,	,	PUNCT
ejpam-5915	156	46	h	h	NOUN
ejpam-5915	156	47	∈	∈	PROPN
ejpam-5915	156	48	g	g	NOUN
ejpam-5915	156	49	:	:	PUNCT
ejpam-5915	156	50	(	(	PUNCT
ejpam-5915	156	51	i	i	NOUN
ejpam-5915	156	52	)	)	PUNCT
ejpam-5915	156	53	µ(g	µ(g	ADP
ejpam-5915	156	54	·	·	SYM
ejpam-5915	156	55	h	h	X
ejpam-5915	156	56	)	)	PUNCT
ejpam-5915	156	57	≥	≥	NOUN
ejpam-5915	156	58	min{µ(g	min{µ(g	PROPN
ejpam-5915	156	59	)	)	PUNCT
ejpam-5915	156	60	,	,	PUNCT
ejpam-5915	156	61	µ(h	µ(h	PROPN
ejpam-5915	156	62	)	)	PUNCT
ejpam-5915	156	63	}	}	PUNCT
ejpam-5915	156	64	,	,	PUNCT
ejpam-5915	156	65	(	(	PUNCT
ejpam-5915	156	66	ii	ii	NOUN
ejpam-5915	156	67	)	)	PUNCT
ejpam-5915	156	68	µ(α(g	µ(α(g	PROPN
ejpam-5915	156	69	)	)	PUNCT
ejpam-5915	156	70	)	)	PUNCT
ejpam-5915	156	71	≥	≥	NOUN
ejpam-5915	156	72	µ(g	µ(g	PROPN
ejpam-5915	156	73	)	)	PUNCT
ejpam-5915	156	74	,	,	PUNCT
ejpam-5915	156	75	(	(	PUNCT
ejpam-5915	156	76	iii	iii	NOUN
ejpam-5915	156	77	)	)	PUNCT
ejpam-5915	156	78	µ(g−1	µ(g−1	PROPN
ejpam-5915	156	79	)	)	PUNCT
ejpam-5915	156	80	≥	≥	NOUN
ejpam-5915	156	81	µ(g	µ(g	PROPN
ejpam-5915	156	82	)	)	PUNCT
ejpam-5915	156	83	.	.	PUNCT
ejpam-5915	157	1	example	example	NOUN
ejpam-5915	158	1	5	5	NUM
ejpam-5915	158	2	.	.	X
ejpam-5915	158	3	consider	consider	VERB
ejpam-5915	158	4	the	the	DET
ejpam-5915	158	5	real	real	ADJ
ejpam-5915	158	6	numbers	number	NOUN
ejpam-5915	158	7	g	g	NOUN
ejpam-5915	158	8	=	=	SYM
ejpam-5915	158	9	r	r	NOUN
ejpam-5915	158	10	equipped	equip	VERB
ejpam-5915	158	11	with	with	ADP
ejpam-5915	158	12	the	the	DET
ejpam-5915	158	13	binary	binary	PROPN
ejpam-5915	158	14	operation	operation	PROPN
ejpam-5915	158	15	⊕	⊕	PROPN
ejpam-5915	158	16	:	:	PUNCT
ejpam-5915	158	17	r×	r×	NOUN
ejpam-5915	158	18	r	r	NOUN
ejpam-5915	158	19	→	→	SYM
ejpam-5915	158	20	r	r	NOUN
ejpam-5915	158	21	defined	define	VERB
ejpam-5915	158	22	by	by	ADP
ejpam-5915	158	23	:	:	PUNCT
ejpam-5915	158	24	a⊕	a⊕	PROPN
ejpam-5915	158	25	b	b	PROPN
ejpam-5915	158	26	=	=	PRON
ejpam-5915	158	27	a+	a+	PUNCT
ejpam-5915	158	28	b	b	PROPN
ejpam-5915	158	29	2	2	NUM
ejpam-5915	158	30	,	,	PUNCT
ejpam-5915	158	31	and	and	CCONJ
ejpam-5915	158	32	the	the	DET
ejpam-5915	158	33	twisting	twisting	NOUN
ejpam-5915	158	34	map	map	NOUN
ejpam-5915	158	35	α	α	NOUN
ejpam-5915	158	36	:	:	PUNCT
ejpam-5915	158	37	r	r	NOUN
ejpam-5915	158	38	→	→	SYM
ejpam-5915	158	39	r	r	NOUN
ejpam-5915	158	40	defined	define	VERB
ejpam-5915	158	41	by	by	ADP
ejpam-5915	158	42	:	:	PUNCT
ejpam-5915	158	43	α(a	α(a	NOUN
ejpam-5915	158	44	)	)	PUNCT
ejpam-5915	158	45	=	=	PUNCT
ejpam-5915	159	1	a	a	DET
ejpam-5915	159	2	2	2	NUM
ejpam-5915	159	3	.	.	PUNCT
ejpam-5915	160	1	s.	s.	PROPN
ejpam-5915	160	2	shaqaqha	shaqaqha	PROPN
ejpam-5915	160	3	/	/	SYM
ejpam-5915	160	4	eur	eur	PROPN
ejpam-5915	160	5	.	.	PUNCT
ejpam-5915	161	1	j.	j.	PROPN
ejpam-5915	161	2	pure	pure	PROPN
ejpam-5915	161	3	appl	appl	PROPN
ejpam-5915	161	4	.	.	PROPN
ejpam-5915	161	5	math	math	PROPN
ejpam-5915	161	6	,	,	PUNCT
ejpam-5915	161	7	18	18	NUM
ejpam-5915	161	8	(	(	PUNCT
ejpam-5915	161	9	2	2	NUM
ejpam-5915	161	10	)	)	PUNCT
ejpam-5915	161	11	(	(	PUNCT
ejpam-5915	161	12	2025	2025	NUM
ejpam-5915	161	13	)	)	PUNCT
ejpam-5915	161	14	,	,	PUNCT
ejpam-5915	161	15	5915	5915	NUM
ejpam-5915	161	16	8	8	NUM
ejpam-5915	161	17	of	of	ADP
ejpam-5915	161	18	16	16	NUM
ejpam-5915	161	19	we	we	PRON
ejpam-5915	161	20	have	have	AUX
ejpam-5915	161	21	shown	show	VERB
ejpam-5915	161	22	in	in	ADP
ejpam-5915	161	23	example	example	NOUN
ejpam-5915	161	24	2	2	NUM
ejpam-5915	161	25	that	that	SCONJ
ejpam-5915	161	26	the	the	DET
ejpam-5915	161	27	pair	pair	NOUN
ejpam-5915	161	28	(	(	PUNCT
ejpam-5915	161	29	g	g	NOUN
ejpam-5915	161	30	=	=	PROPN
ejpam-5915	161	31	r,⊕	r,⊕	PROPN
ejpam-5915	161	32	,	,	PUNCT
ejpam-5915	161	33	α	α	X
ejpam-5915	161	34	)	)	PUNCT
ejpam-5915	161	35	forms	form	VERB
ejpam-5915	161	36	a	a	DET
ejpam-5915	161	37	hom	hom	NOUN
ejpam-5915	161	38	-	-	PUNCT
ejpam-5915	161	39	group	group	NOUN
ejpam-5915	161	40	,	,	PUNCT
ejpam-5915	161	41	satisfying	satisfy	VERB
ejpam-5915	161	42	the	the	DET
ejpam-5915	161	43	properties	property	NOUN
ejpam-5915	161	44	of	of	ADP
ejpam-5915	161	45	hom	hom	NOUN
ejpam-5915	161	46	-	-	PUNCT
ejpam-5915	161	47	associativity	associativity	NOUN
ejpam-5915	161	48	,	,	PUNCT
ejpam-5915	161	49	hom	hom	NOUN
ejpam-5915	161	50	-	-	PUNCT
ejpam-5915	161	51	unitarity	unitarity	NOUN
ejpam-5915	161	52	,	,	PUNCT
ejpam-5915	161	53	and	and	CCONJ
ejpam-5915	161	54	hom	hom	NOUN
ejpam-5915	161	55	-	-	PUNCT
ejpam-5915	161	56	inverses	inverse	NOUN
ejpam-5915	161	57	.	.	PUNCT
ejpam-5915	162	1	now	now	ADV
ejpam-5915	162	2	,	,	PUNCT
ejpam-5915	162	3	let	let	VERB
ejpam-5915	162	4	us	we	PRON
ejpam-5915	162	5	define	define	VERB
ejpam-5915	162	6	a	a	DET
ejpam-5915	162	7	fuzzy	fuzzy	ADJ
ejpam-5915	162	8	set	set	VERB
ejpam-5915	162	9	µ	µ	NOUN
ejpam-5915	162	10	:	:	PUNCT
ejpam-5915	162	11	r	r	NOUN
ejpam-5915	162	12	→	→	SYM
ejpam-5915	163	1	[	[	X
ejpam-5915	163	2	0	0	NUM
ejpam-5915	163	3	,	,	PUNCT
ejpam-5915	163	4	1	1	NUM
ejpam-5915	163	5	]	]	PUNCT
ejpam-5915	163	6	over	over	ADP
ejpam-5915	163	7	this	this	DET
ejpam-5915	163	8	hom	hom	NOUN
ejpam-5915	163	9	-	-	PUNCT
ejpam-5915	163	10	group	group	NOUN
ejpam-5915	163	11	,	,	PUNCT
ejpam-5915	163	12	where	where	SCONJ
ejpam-5915	163	13	µ(a	µ(a	PROPN
ejpam-5915	163	14	)	)	PUNCT
ejpam-5915	163	15	represents	represent	VERB
ejpam-5915	163	16	the	the	DET
ejpam-5915	163	17	degree	degree	NOUN
ejpam-5915	163	18	of	of	ADP
ejpam-5915	163	19	membership	membership	NOUN
ejpam-5915	163	20	of	of	ADP
ejpam-5915	163	21	the	the	DET
ejpam-5915	163	22	element	element	NOUN
ejpam-5915	163	23	a	a	PRON
ejpam-5915	163	24	in	in	ADP
ejpam-5915	163	25	the	the	DET
ejpam-5915	163	26	fuzzy	fuzzy	ADJ
ejpam-5915	163	27	subset	subset	NOUN
ejpam-5915	163	28	.	.	PUNCT
ejpam-5915	164	1	we	we	PRON
ejpam-5915	164	2	define	define	VERB
ejpam-5915	164	3	the	the	DET
ejpam-5915	164	4	fuzzy	fuzzy	ADJ
ejpam-5915	164	5	membership	membership	NOUN
ejpam-5915	164	6	function	function	NOUN
ejpam-5915	164	7	µ	µ	ADV
ejpam-5915	164	8	as	as	ADP
ejpam-5915	164	9	:	:	PUNCT
ejpam-5915	164	10	µ(a	µ(a	PROPN
ejpam-5915	164	11	)	)	PUNCT
ejpam-5915	165	1	=	=	PRON
ejpam-5915	165	2	{	{	PUNCT
ejpam-5915	165	3	1	1	NUM
ejpam-5915	165	4	if	if	SCONJ
ejpam-5915	165	5	a	a	DET
ejpam-5915	165	6	=	=	NOUN
ejpam-5915	165	7	0	0	NUM
ejpam-5915	165	8	,	,	PUNCT
ejpam-5915	165	9	1	1	NUM
ejpam-5915	165	10	1+|a|	1+|a|	NUM
ejpam-5915	165	11	if	if	SCONJ
ejpam-5915	165	12	a	a	DET
ejpam-5915	165	13	̸=	̸=	PROPN
ejpam-5915	165	14	0	0	NUM
ejpam-5915	165	15	.	.	PUNCT
ejpam-5915	166	1	this	this	DET
ejpam-5915	166	2	function	function	NOUN
ejpam-5915	166	3	assigns	assign	VERB
ejpam-5915	166	4	full	full	ADJ
ejpam-5915	166	5	membership	membership	NOUN
ejpam-5915	166	6	to	to	ADP
ejpam-5915	166	7	the	the	DET
ejpam-5915	166	8	identity	identity	NOUN
ejpam-5915	166	9	element	element	NOUN
ejpam-5915	166	10	0	0	NUM
ejpam-5915	166	11	,	,	PUNCT
ejpam-5915	166	12	and	and	CCONJ
ejpam-5915	166	13	the	the	DET
ejpam-5915	166	14	membership	membership	NOUN
ejpam-5915	166	15	decreases	decrease	VERB
ejpam-5915	166	16	as	as	ADP
ejpam-5915	166	17	|a|	|a|	NOUN
ejpam-5915	166	18	increases	increase	NOUN
ejpam-5915	166	19	for	for	ADP
ejpam-5915	166	20	other	other	ADJ
ejpam-5915	166	21	elements	element	NOUN
ejpam-5915	166	22	.	.	PUNCT
ejpam-5915	167	1	verification	verification	NOUN
ejpam-5915	167	2	of	of	ADP
ejpam-5915	167	3	the	the	DET
ejpam-5915	167	4	fuzzy	fuzzy	ADJ
ejpam-5915	167	5	hom	hom	NOUN
ejpam-5915	167	6	-	-	PUNCT
ejpam-5915	167	7	subgroup	subgroup	NOUN
ejpam-5915	167	8	properties	property	NOUN
ejpam-5915	167	9	(	(	PUNCT
ejpam-5915	167	10	i	i	NOUN
ejpam-5915	167	11	)	)	PUNCT
ejpam-5915	167	12	closure	closure	NOUN
ejpam-5915	167	13	under	under	ADP
ejpam-5915	167	14	the	the	DET
ejpam-5915	167	15	operation	operation	NOUN
ejpam-5915	167	16	:	:	PUNCT
ejpam-5915	167	17	we	we	PRON
ejpam-5915	167	18	need	need	VERB
ejpam-5915	167	19	to	to	PART
ejpam-5915	167	20	verify	verify	VERB
ejpam-5915	167	21	that	that	SCONJ
ejpam-5915	167	22	:	:	PUNCT
ejpam-5915	167	23	µ(g	µ(g	ADP
ejpam-5915	167	24	⊕	⊕	PROPN
ejpam-5915	167	25	h	h	PROPN
ejpam-5915	167	26	)	)	PUNCT
ejpam-5915	167	27	≥	≥	NOUN
ejpam-5915	167	28	min{µ(g	min{µ(g	PROPN
ejpam-5915	167	29	)	)	PUNCT
ejpam-5915	167	30	,	,	PUNCT
ejpam-5915	167	31	µ(h	µ(h	PROPN
ejpam-5915	167	32	)	)	PUNCT
ejpam-5915	167	33	}	}	PUNCT
ejpam-5915	167	34	.	.	PUNCT
ejpam-5915	168	1	using	use	VERB
ejpam-5915	168	2	the	the	DET
ejpam-5915	168	3	operation	operation	NOUN
ejpam-5915	168	4	g	g	PROPN
ejpam-5915	168	5	⊕	⊕	PROPN
ejpam-5915	168	6	h	h	PROPN
ejpam-5915	169	1	=	=	SYM
ejpam-5915	169	2	g+h	g+h	PROPN
ejpam-5915	169	3	2	2	NUM
ejpam-5915	169	4	,	,	PUNCT
ejpam-5915	169	5	the	the	DET
ejpam-5915	169	6	membership	membership	NOUN
ejpam-5915	169	7	function	function	NOUN
ejpam-5915	169	8	evaluates	evaluate	VERB
ejpam-5915	169	9	to	to	ADP
ejpam-5915	169	10	:	:	PUNCT
ejpam-5915	169	11	µ	µ	X
ejpam-5915	169	12	(	(	PUNCT
ejpam-5915	169	13	g	g	PROPN
ejpam-5915	169	14	⊕	⊕	PROPN
ejpam-5915	169	15	h	h	PROPN
ejpam-5915	169	16	)	)	PUNCT
ejpam-5915	169	17	=	=	SYM
ejpam-5915	169	18	µ	µ	X
ejpam-5915	169	19	(	(	PUNCT
ejpam-5915	169	20	g	g	PROPN
ejpam-5915	169	21	+	+	NOUN
ejpam-5915	169	22	h	h	NOUN
ejpam-5915	169	23	2	2	X
ejpam-5915	169	24	)	)	PUNCT
ejpam-5915	169	25	=	=	SYM
ejpam-5915	169	26	2	2	NUM
ejpam-5915	169	27	2	2	NUM
ejpam-5915	169	28	+	+	NUM
ejpam-5915	169	29	|g	|g	NOUN
ejpam-5915	169	30	+	+	X
ejpam-5915	169	31	h|	h|	PROPN
ejpam-5915	169	32	.	.	PUNCT
ejpam-5915	170	1	since	since	SCONJ
ejpam-5915	170	2	|g	|g	NOUN
ejpam-5915	170	3	+	+	CCONJ
ejpam-5915	170	4	h|	h|	ADJ
ejpam-5915	170	5	≤	≤	NOUN
ejpam-5915	170	6	2max(|g|	2max(|g|	NUM
ejpam-5915	170	7	,	,	PUNCT
ejpam-5915	170	8	|h|	|h|	PROPN
ejpam-5915	170	9	)	)	PUNCT
ejpam-5915	170	10	,	,	PUNCT
ejpam-5915	170	11	we	we	PRON
ejpam-5915	170	12	have	have	VERB
ejpam-5915	170	13	:	:	PUNCT
ejpam-5915	170	14	2	2	NUM
ejpam-5915	170	15	2	2	NUM
ejpam-5915	170	16	+	+	NUM
ejpam-5915	170	17	|g	|g	NOUN
ejpam-5915	170	18	+	+	X
ejpam-5915	170	19	h|	h|	NOUN
ejpam-5915	170	20	≥	≥	NOUN
ejpam-5915	170	21	2	2	NUM
ejpam-5915	170	22	2	2	NUM
ejpam-5915	170	23	+	+	CCONJ
ejpam-5915	170	24	2max(|g|	2max(|g|	NUM
ejpam-5915	170	25	,	,	PUNCT
ejpam-5915	170	26	|h|	|h|	NOUN
ejpam-5915	170	27	)	)	PUNCT
ejpam-5915	170	28	=	=	SYM
ejpam-5915	171	1	1	1	NUM
ejpam-5915	171	2	1	1	NUM
ejpam-5915	171	3	+	+	NOUN
ejpam-5915	171	4	max(|g|	max(|g|	ADJ
ejpam-5915	171	5	,	,	PUNCT
ejpam-5915	171	6	|h|	|h|	NOUN
ejpam-5915	171	7	)	)	PUNCT
ejpam-5915	172	1	=	=	SYM
ejpam-5915	172	2	min	min	NOUN
ejpam-5915	172	3	(	(	PUNCT
ejpam-5915	172	4	1	1	NUM
ejpam-5915	172	5	1	1	NUM
ejpam-5915	172	6	+	+	CCONJ
ejpam-5915	172	7	|g|	|g|	ADJ
ejpam-5915	172	8	,	,	PUNCT
ejpam-5915	172	9	1	1	NUM
ejpam-5915	172	10	1	1	NUM
ejpam-5915	172	11	+	+	NUM
ejpam-5915	172	12	|h|	|h|	PROPN
ejpam-5915	172	13	)	)	PUNCT
ejpam-5915	172	14	.	.	PUNCT
ejpam-5915	173	1	thus	thus	ADV
ejpam-5915	173	2	,	,	PUNCT
ejpam-5915	173	3	we	we	PRON
ejpam-5915	173	4	have	have	AUX
ejpam-5915	173	5	shown	show	VERB
ejpam-5915	173	6	that	that	SCONJ
ejpam-5915	173	7	:	:	PUNCT
ejpam-5915	173	8	µ(g	µ(g	ADP
ejpam-5915	173	9	⊕	⊕	PROPN
ejpam-5915	173	10	h	h	PROPN
ejpam-5915	173	11	)	)	PUNCT
ejpam-5915	173	12	≥	≥	NOUN
ejpam-5915	173	13	min{µ(g	min{µ(g	PROPN
ejpam-5915	173	14	)	)	PUNCT
ejpam-5915	173	15	,	,	PUNCT
ejpam-5915	173	16	µ(h	µ(h	PROPN
ejpam-5915	173	17	)	)	PUNCT
ejpam-5915	173	18	}	}	PUNCT
ejpam-5915	173	19	,	,	PUNCT
ejpam-5915	173	20	which	which	PRON
ejpam-5915	173	21	confirms	confirm	VERB
ejpam-5915	173	22	that	that	SCONJ
ejpam-5915	173	23	the	the	DET
ejpam-5915	173	24	closure	closure	NOUN
ejpam-5915	173	25	property	property	NOUN
ejpam-5915	173	26	is	be	AUX
ejpam-5915	173	27	satisfied	satisfied	ADJ
ejpam-5915	173	28	.	.	PUNCT
ejpam-5915	174	1	(	(	PUNCT
ejpam-5915	174	2	ii	ii	NOUN
ejpam-5915	174	3	)	)	PUNCT
ejpam-5915	174	4	compatibility	compatibility	NOUN
ejpam-5915	174	5	with	with	ADP
ejpam-5915	174	6	the	the	DET
ejpam-5915	174	7	twisting	twisting	NOUN
ejpam-5915	174	8	map	map	NOUN
ejpam-5915	174	9	:	:	PUNCT
ejpam-5915	174	10	we	we	PRON
ejpam-5915	174	11	need	need	VERB
ejpam-5915	174	12	to	to	PART
ejpam-5915	174	13	verify	verify	VERB
ejpam-5915	174	14	that	that	SCONJ
ejpam-5915	174	15	:	:	PUNCT
ejpam-5915	174	16	µ(α(g	µ(α(g	PROPN
ejpam-5915	174	17	)	)	PUNCT
ejpam-5915	174	18	)	)	PUNCT
ejpam-5915	174	19	≥	≥	NOUN
ejpam-5915	174	20	µ(g	µ(g	PROPN
ejpam-5915	174	21	)	)	PUNCT
ejpam-5915	174	22	.	.	PUNCT
ejpam-5915	175	1	since	since	SCONJ
ejpam-5915	175	2	α(g	α(g	NUM
ejpam-5915	175	3	)	)	PUNCT
ejpam-5915	175	4	=	=	SYM
ejpam-5915	175	5	g	g	PROPN
ejpam-5915	175	6	2	2	NUM
ejpam-5915	175	7	,	,	PUNCT
ejpam-5915	175	8	simple	simple	ADJ
ejpam-5915	175	9	computations	computation	NOUN
ejpam-5915	175	10	show	show	VERB
ejpam-5915	175	11	that	that	SCONJ
ejpam-5915	175	12	:	:	PUNCT
ejpam-5915	175	13	s.	s.	PROPN
ejpam-5915	175	14	shaqaqha	shaqaqha	PROPN
ejpam-5915	175	15	/	/	SYM
ejpam-5915	175	16	eur	eur	PROPN
ejpam-5915	175	17	.	.	PUNCT
ejpam-5915	176	1	j.	j.	PROPN
ejpam-5915	176	2	pure	pure	PROPN
ejpam-5915	176	3	appl	appl	PROPN
ejpam-5915	176	4	.	.	PROPN
ejpam-5915	176	5	math	math	PROPN
ejpam-5915	176	6	,	,	PUNCT
ejpam-5915	176	7	18	18	NUM
ejpam-5915	176	8	(	(	PUNCT
ejpam-5915	176	9	2	2	NUM
ejpam-5915	176	10	)	)	PUNCT
ejpam-5915	176	11	(	(	PUNCT
ejpam-5915	176	12	2025	2025	NUM
ejpam-5915	176	13	)	)	PUNCT
ejpam-5915	176	14	,	,	PUNCT
ejpam-5915	176	15	5915	5915	NUM
ejpam-5915	176	16	9	9	NUM
ejpam-5915	176	17	of	of	ADP
ejpam-5915	176	18	16	16	NUM
ejpam-5915	176	19	µ(α(g	µ(α(g	PROPN
ejpam-5915	176	20	)	)	PUNCT
ejpam-5915	176	21	)	)	PUNCT
ejpam-5915	177	1	=	=	PUNCT
ejpam-5915	177	2	4	4	NUM
ejpam-5915	177	3	4	4	NUM
ejpam-5915	177	4	+	+	CCONJ
ejpam-5915	177	5	|g|	|g|	PROPN
ejpam-5915	177	6	.	.	PUNCT
ejpam-5915	178	1	now	now	ADV
ejpam-5915	178	2	,	,	PUNCT
ejpam-5915	178	3	comparing	compare	VERB
ejpam-5915	178	4	this	this	PRON
ejpam-5915	178	5	with	with	ADP
ejpam-5915	178	6	:	:	PUNCT
ejpam-5915	178	7	µ(g	µ(g	NUM
ejpam-5915	178	8	)	)	PUNCT
ejpam-5915	178	9	=	=	SYM
ejpam-5915	178	10	1	1	NUM
ejpam-5915	178	11	1	1	NUM
ejpam-5915	178	12	+	+	CCONJ
ejpam-5915	178	13	|g|	|g|	PROPN
ejpam-5915	178	14	,	,	PUNCT
ejpam-5915	178	15	we	we	PRON
ejpam-5915	178	16	need	need	VERB
ejpam-5915	178	17	to	to	PART
ejpam-5915	178	18	verify	verify	VERB
ejpam-5915	178	19	the	the	DET
ejpam-5915	178	20	inequality	inequality	NOUN
ejpam-5915	178	21	:	:	PUNCT
ejpam-5915	178	22	4	4	NUM
ejpam-5915	178	23	4	4	NUM
ejpam-5915	178	24	+	+	CCONJ
ejpam-5915	178	25	|g|	|g|	ADJ
ejpam-5915	178	26	≥	≥	NOUN
ejpam-5915	178	27	1	1	NUM
ejpam-5915	178	28	1	1	NUM
ejpam-5915	178	29	+	+	CCONJ
ejpam-5915	178	30	|g|	|g|	PROPN
ejpam-5915	178	31	.	.	PUNCT
ejpam-5915	179	1	to	to	PART
ejpam-5915	179	2	prove	prove	VERB
ejpam-5915	179	3	this	this	DET
ejpam-5915	179	4	inequality	inequality	NOUN
ejpam-5915	179	5	,	,	PUNCT
ejpam-5915	179	6	we	we	PRON
ejpam-5915	179	7	cross	cross	VERB
ejpam-5915	179	8	-	-	VERB
ejpam-5915	179	9	multiply	multiply	ADJ
ejpam-5915	179	10	:	:	PUNCT
ejpam-5915	179	11	4	4	NUM
ejpam-5915	179	12	·	·	PUNCT
ejpam-5915	179	13	(	(	PUNCT
ejpam-5915	179	14	1	1	NUM
ejpam-5915	179	15	+	+	CCONJ
ejpam-5915	179	16	|g|	|g|	ADJ
ejpam-5915	179	17	)	)	PUNCT
ejpam-5915	179	18	≥	≥	NOUN
ejpam-5915	179	19	1	1	NUM
ejpam-5915	179	20	·	·	PUNCT
ejpam-5915	179	21	(	(	PUNCT
ejpam-5915	179	22	4	4	NUM
ejpam-5915	179	23	+	+	CCONJ
ejpam-5915	179	24	|g|	|g|	ADJ
ejpam-5915	179	25	)	)	PUNCT
ejpam-5915	179	26	.	.	PUNCT
ejpam-5915	180	1	expanding	expand	VERB
ejpam-5915	180	2	both	both	DET
ejpam-5915	180	3	sides	side	NOUN
ejpam-5915	180	4	:	:	PUNCT
ejpam-5915	180	5	4	4	NUM
ejpam-5915	180	6	+	+	NUM
ejpam-5915	180	7	4|g|	4|g|	NUM
ejpam-5915	180	8	≥	≥	NUM
ejpam-5915	180	9	4	4	NUM
ejpam-5915	180	10	+	+	CCONJ
ejpam-5915	180	11	|g|	|g|	ADJ
ejpam-5915	180	12	.	.	PUNCT
ejpam-5915	181	1	subtracting	subtract	VERB
ejpam-5915	181	2	4	4	NUM
ejpam-5915	181	3	from	from	ADP
ejpam-5915	181	4	both	both	DET
ejpam-5915	181	5	sides	side	NOUN
ejpam-5915	181	6	:	:	PUNCT
ejpam-5915	181	7	4|g|	4|g|	NUM
ejpam-5915	181	8	≥	≥	NUM
ejpam-5915	181	9	|g|	|g|	ADJ
ejpam-5915	181	10	.	.	PUNCT
ejpam-5915	182	1	this	this	DET
ejpam-5915	182	2	inequality	inequality	NOUN
ejpam-5915	182	3	holds	hold	VERB
ejpam-5915	182	4	for	for	ADP
ejpam-5915	182	5	all	all	DET
ejpam-5915	182	6	g	g	NOUN
ejpam-5915	182	7	∈	∈	NOUN
ejpam-5915	182	8	r	r	NOUN
ejpam-5915	182	9	because	because	SCONJ
ejpam-5915	182	10	4|g|	4|g|	PROPN
ejpam-5915	182	11	is	be	AUX
ejpam-5915	182	12	always	always	ADV
ejpam-5915	182	13	greater	great	ADJ
ejpam-5915	182	14	than	than	ADP
ejpam-5915	182	15	or	or	CCONJ
ejpam-5915	182	16	equal	equal	ADJ
ejpam-5915	182	17	to	to	ADP
ejpam-5915	182	18	|g|	|g|	PROPN
ejpam-5915	182	19	.	.	PUNCT
ejpam-5915	183	1	thus	thus	ADV
ejpam-5915	183	2	,	,	PUNCT
ejpam-5915	183	3	the	the	DET
ejpam-5915	183	4	inequality	inequality	NOUN
ejpam-5915	183	5	:	:	PUNCT
ejpam-5915	183	6	4	4	NUM
ejpam-5915	183	7	4	4	NUM
ejpam-5915	183	8	+	+	CCONJ
ejpam-5915	183	9	|g|	|g|	ADJ
ejpam-5915	183	10	≥	≥	NOUN
ejpam-5915	183	11	1	1	NUM
ejpam-5915	183	12	1	1	NUM
ejpam-5915	183	13	+	+	CCONJ
ejpam-5915	183	14	|g|	|g|	PROPN
ejpam-5915	183	15	is	be	AUX
ejpam-5915	183	16	satisfied	satisfied	ADJ
ejpam-5915	183	17	for	for	ADP
ejpam-5915	183	18	all	all	DET
ejpam-5915	183	19	g	g	PROPN
ejpam-5915	183	20	∈	∈	PROPN
ejpam-5915	183	21	r.	r.	PROPN
ejpam-5915	183	22	hence	hence	ADV
ejpam-5915	183	23	,	,	PUNCT
ejpam-5915	183	24	we	we	PRON
ejpam-5915	183	25	conclude	conclude	VERB
ejpam-5915	183	26	that	that	PRON
ejpam-5915	183	27	:	:	PUNCT
ejpam-5915	183	28	µ(α(g	µ(α(g	PROPN
ejpam-5915	183	29	)	)	PUNCT
ejpam-5915	183	30	)	)	PUNCT
ejpam-5915	183	31	≥	≥	NOUN
ejpam-5915	183	32	µ(g	µ(g	PROPN
ejpam-5915	183	33	)	)	PUNCT
ejpam-5915	183	34	.	.	PUNCT
ejpam-5915	184	1	(	(	PUNCT
ejpam-5915	184	2	iii	iii	X
ejpam-5915	184	3	)	)	PUNCT
ejpam-5915	184	4	closure	closure	NOUN
ejpam-5915	184	5	under	under	ADP
ejpam-5915	184	6	inverses	inverse	NOUN
ejpam-5915	184	7	:	:	PUNCT
ejpam-5915	184	8	we	we	PRON
ejpam-5915	184	9	need	need	VERB
ejpam-5915	184	10	to	to	PART
ejpam-5915	184	11	confirm	confirm	VERB
ejpam-5915	184	12	that	that	PRON
ejpam-5915	184	13	:	:	PUNCT
ejpam-5915	184	14	µ(g∗	µ(g∗	X
ejpam-5915	184	15	)	)	PUNCT
ejpam-5915	184	16	≥	≥	NOUN
ejpam-5915	184	17	µ(g	µ(g	PROPN
ejpam-5915	184	18	)	)	PUNCT
ejpam-5915	184	19	.	.	PUNCT
ejpam-5915	185	1	since	since	SCONJ
ejpam-5915	185	2	g−1	g−1	PROPN
ejpam-5915	185	3	=	=	SYM
ejpam-5915	185	4	−g	−g	NOUN
ejpam-5915	185	5	and	and	CCONJ
ejpam-5915	185	6	the	the	DET
ejpam-5915	185	7	membership	membership	NOUN
ejpam-5915	185	8	function	function	NOUN
ejpam-5915	185	9	depends	depend	VERB
ejpam-5915	185	10	only	only	ADV
ejpam-5915	185	11	on	on	ADP
ejpam-5915	185	12	the	the	DET
ejpam-5915	185	13	absolute	absolute	ADJ
ejpam-5915	185	14	value	value	NOUN
ejpam-5915	185	15	of	of	ADP
ejpam-5915	185	16	the	the	DET
ejpam-5915	185	17	element	element	NOUN
ejpam-5915	185	18	,	,	PUNCT
ejpam-5915	185	19	we	we	PRON
ejpam-5915	185	20	have	have	VERB
ejpam-5915	185	21	:	:	PUNCT
ejpam-5915	185	22	µ(g∗	µ(g∗	NOUN
ejpam-5915	185	23	)	)	PUNCT
ejpam-5915	185	24	=	=	PUNCT
ejpam-5915	185	25	µ(−g	µ(−g	NOUN
ejpam-5915	185	26	)	)	PUNCT
ejpam-5915	185	27	=	=	SYM
ejpam-5915	185	28	µ(g	µ(g	PROPN
ejpam-5915	185	29	)	)	PUNCT
ejpam-5915	185	30	.	.	PUNCT
ejpam-5915	186	1	therefore	therefore	ADV
ejpam-5915	186	2	,	,	PUNCT
ejpam-5915	186	3	the	the	DET
ejpam-5915	186	4	closure	closure	NOUN
ejpam-5915	186	5	under	under	ADP
ejpam-5915	186	6	inverses	inverse	NOUN
ejpam-5915	186	7	is	be	AUX
ejpam-5915	186	8	satisfied	satisfied	ADJ
ejpam-5915	186	9	.	.	PUNCT
ejpam-5915	187	1	the	the	DET
ejpam-5915	187	2	concept	concept	NOUN
ejpam-5915	187	3	of	of	ADP
ejpam-5915	187	4	fuzzy	fuzzy	ADJ
ejpam-5915	187	5	normal	normal	ADJ
ejpam-5915	187	6	subgroups	subgroup	NOUN
ejpam-5915	187	7	has	have	AUX
ejpam-5915	187	8	been	be	AUX
ejpam-5915	187	9	extensively	extensively	ADV
ejpam-5915	187	10	studied	study	VERB
ejpam-5915	187	11	in	in	ADP
ejpam-5915	187	12	classical	classical	ADJ
ejpam-5915	187	13	group	group	NOUN
ejpam-5915	187	14	theory	theory	NOUN
ejpam-5915	187	15	,	,	PUNCT
ejpam-5915	187	16	with	with	ADP
ejpam-5915	187	17	foundational	foundational	ADJ
ejpam-5915	187	18	work	work	NOUN
ejpam-5915	187	19	by	by	ADP
ejpam-5915	187	20	rosenfeld	rosenfeld	PROPN
ejpam-5915	188	1	[	[	X
ejpam-5915	188	2	2	2	NUM
ejpam-5915	188	3	]	]	PUNCT
ejpam-5915	188	4	and	and	CCONJ
ejpam-5915	188	5	later	late	ADJ
ejpam-5915	188	6	refinements	refinement	NOUN
ejpam-5915	188	7	by	by	ADP
ejpam-5915	188	8	ajmal	ajmal	PROPN
ejpam-5915	188	9	and	and	CCONJ
ejpam-5915	188	10	prajapati	prajapati	PROPN
ejpam-5915	189	1	[	[	X
ejpam-5915	189	2	16	16	NUM
ejpam-5915	189	3	]	]	PUNCT
ejpam-5915	189	4	.	.	PUNCT
ejpam-5915	190	1	these	these	DET
ejpam-5915	190	2	studies	study	NOUN
ejpam-5915	190	3	explore	explore	VERB
ejpam-5915	190	4	how	how	SCONJ
ejpam-5915	190	5	fuzzy	fuzzy	ADJ
ejpam-5915	190	6	subsets	subset	NOUN
ejpam-5915	190	7	of	of	ADP
ejpam-5915	190	8	groups	group	NOUN
ejpam-5915	190	9	maintain	maintain	VERB
ejpam-5915	190	10	their	their	PRON
ejpam-5915	190	11	structure	structure	NOUN
ejpam-5915	190	12	under	under	ADP
ejpam-5915	190	13	conjugation	conjugation	NOUN
ejpam-5915	190	14	,	,	PUNCT
ejpam-5915	190	15	forming	form	VERB
ejpam-5915	190	16	the	the	DET
ejpam-5915	190	17	basis	basis	NOUN
ejpam-5915	190	18	for	for	ADP
ejpam-5915	190	19	fuzzy	fuzzy	ADJ
ejpam-5915	190	20	normal	normal	ADJ
ejpam-5915	190	21	subgroups	subgroup	NOUN
ejpam-5915	190	22	.	.	PUNCT
ejpam-5915	191	1	in	in	ADP
ejpam-5915	191	2	this	this	DET
ejpam-5915	191	3	paper	paper	NOUN
ejpam-5915	191	4	,	,	PUNCT
ejpam-5915	191	5	we	we	PRON
ejpam-5915	191	6	extend	extend	VERB
ejpam-5915	191	7	this	this	DET
ejpam-5915	191	8	notion	notion	NOUN
ejpam-5915	191	9	to	to	ADP
ejpam-5915	191	10	hom	hom	NOUN
ejpam-5915	191	11	-	-	PUNCT
ejpam-5915	191	12	groups	group	NOUN
ejpam-5915	191	13	,	,	PUNCT
ejpam-5915	191	14	which	which	PRON
ejpam-5915	191	15	introduce	introduce	VERB
ejpam-5915	191	16	an	an	DET
ejpam-5915	191	17	additional	additional	ADJ
ejpam-5915	191	18	twisting	twisting	NOUN
ejpam-5915	191	19	map	map	NOUN
ejpam-5915	191	20	α	α	NOUN
ejpam-5915	191	21	that	that	PRON
ejpam-5915	191	22	affects	affect	VERB
ejpam-5915	191	23	the	the	DET
ejpam-5915	191	24	algebraic	algebraic	ADJ
ejpam-5915	191	25	structure	structure	NOUN
ejpam-5915	191	26	.	.	PUNCT
ejpam-5915	192	1	as	as	ADV
ejpam-5915	192	2	far	far	ADV
ejpam-5915	192	3	as	as	SCONJ
ejpam-5915	192	4	we	we	PRON
ejpam-5915	192	5	know	know	VERB
ejpam-5915	192	6	,	,	PUNCT
ejpam-5915	192	7	the	the	DET
ejpam-5915	192	8	definition	definition	NOUN
ejpam-5915	192	9	of	of	ADP
ejpam-5915	192	10	fuzzy	fuzzy	ADJ
ejpam-5915	192	11	hom	hom	NOUN
ejpam-5915	192	12	-	-	PUNCT
ejpam-5915	192	13	normal	normal	ADJ
ejpam-5915	192	14	subgroups	subgroup	NOUN
ejpam-5915	192	15	presented	present	VERB
ejpam-5915	192	16	here	here	ADV
ejpam-5915	192	17	is	be	AUX
ejpam-5915	192	18	novel	novel	ADJ
ejpam-5915	192	19	and	and	CCONJ
ejpam-5915	192	20	offers	offer	VERB
ejpam-5915	192	21	a	a	DET
ejpam-5915	192	22	new	new	ADJ
ejpam-5915	192	23	framework	framework	NOUN
ejpam-5915	192	24	for	for	ADP
ejpam-5915	192	25	studying	study	VERB
ejpam-5915	192	26	fuzzy	fuzzy	ADJ
ejpam-5915	192	27	subsets	subset	NOUN
ejpam-5915	192	28	in	in	ADP
ejpam-5915	192	29	hom	hom	NOUN
ejpam-5915	192	30	-	-	PUNCT
ejpam-5915	192	31	group	group	NOUN
ejpam-5915	192	32	theory	theory	NOUN
ejpam-5915	192	33	.	.	PUNCT
ejpam-5915	193	1	below	below	ADV
ejpam-5915	193	2	,	,	PUNCT
ejpam-5915	193	3	we	we	PRON
ejpam-5915	193	4	formalize	formalize	VERB
ejpam-5915	193	5	this	this	DET
ejpam-5915	193	6	concept	concept	NOUN
ejpam-5915	193	7	.	.	PUNCT
ejpam-5915	194	1	s.	s.	PROPN
ejpam-5915	194	2	shaqaqha	shaqaqha	PROPN
ejpam-5915	194	3	/	/	SYM
ejpam-5915	194	4	eur	eur	PROPN
ejpam-5915	194	5	.	.	PUNCT
ejpam-5915	195	1	j.	j.	PROPN
ejpam-5915	195	2	pure	pure	PROPN
ejpam-5915	195	3	appl	appl	PROPN
ejpam-5915	195	4	.	.	PROPN
ejpam-5915	195	5	math	math	PROPN
ejpam-5915	195	6	,	,	PUNCT
ejpam-5915	195	7	18	18	NUM
ejpam-5915	195	8	(	(	PUNCT
ejpam-5915	195	9	2	2	NUM
ejpam-5915	195	10	)	)	PUNCT
ejpam-5915	195	11	(	(	PUNCT
ejpam-5915	195	12	2025	2025	NUM
ejpam-5915	195	13	)	)	PUNCT
ejpam-5915	195	14	,	,	PUNCT
ejpam-5915	195	15	5915	5915	NUM
ejpam-5915	195	16	10	10	NUM
ejpam-5915	195	17	of	of	ADP
ejpam-5915	195	18	16	16	NUM
ejpam-5915	195	19	definition	definition	NOUN
ejpam-5915	195	20	5	5	NUM
ejpam-5915	195	21	.	.	PUNCT
ejpam-5915	196	1	let	let	VERB
ejpam-5915	196	2	(	(	PUNCT
ejpam-5915	196	3	g	g	NOUN
ejpam-5915	196	4	,	,	PUNCT
ejpam-5915	196	5	·	·	PUNCT
ejpam-5915	196	6	,	,	PUNCT
ejpam-5915	196	7	α	α	X
ejpam-5915	196	8	)	)	PUNCT
ejpam-5915	196	9	be	be	VERB
ejpam-5915	196	10	a	a	DET
ejpam-5915	196	11	hom	hom	NOUN
ejpam-5915	196	12	-	-	PUNCT
ejpam-5915	196	13	group	group	NOUN
ejpam-5915	196	14	,	,	PUNCT
ejpam-5915	196	15	and	and	CCONJ
ejpam-5915	196	16	let	let	VERB
ejpam-5915	196	17	µ	µ	X
ejpam-5915	196	18	:	:	PUNCT
ejpam-5915	196	19	g	g	X
ejpam-5915	196	20	→	→	SYM
ejpam-5915	196	21	[	[	X
ejpam-5915	196	22	0	0	NUM
ejpam-5915	196	23	,	,	PUNCT
ejpam-5915	196	24	1	1	NUM
ejpam-5915	196	25	]	]	PUNCT
ejpam-5915	196	26	be	be	AUX
ejpam-5915	196	27	a	a	DET
ejpam-5915	196	28	fuzzy	fuzzy	ADJ
ejpam-5915	196	29	set	set	NOUN
ejpam-5915	196	30	on	on	ADP
ejpam-5915	196	31	g.	g.	PROPN
ejpam-5915	196	32	the	the	DET
ejpam-5915	196	33	fuzzy	fuzzy	ADJ
ejpam-5915	196	34	set	set	VERB
ejpam-5915	196	35	µ	µ	NOUN
ejpam-5915	196	36	is	be	AUX
ejpam-5915	196	37	called	call	VERB
ejpam-5915	196	38	a	a	DET
ejpam-5915	196	39	fuzzy	fuzzy	ADJ
ejpam-5915	196	40	hom	hom	NOUN
ejpam-5915	196	41	-	-	PUNCT
ejpam-5915	196	42	normal	normal	ADJ
ejpam-5915	196	43	subgroup	subgroup	NOUN
ejpam-5915	196	44	if	if	SCONJ
ejpam-5915	196	45	it	it	PRON
ejpam-5915	196	46	is	be	AUX
ejpam-5915	196	47	a	a	DET
ejpam-5915	196	48	fuzzy	fuzzy	ADJ
ejpam-5915	196	49	hom	hom	NOUN
ejpam-5915	196	50	-	-	PUNCT
ejpam-5915	196	51	subgroup	subgroup	NOUN
ejpam-5915	196	52	(	(	PUNCT
ejpam-5915	196	53	definition	definition	NOUN
ejpam-5915	196	54	4	4	NUM
ejpam-5915	196	55	)	)	PUNCT
ejpam-5915	196	56	and	and	CCONJ
ejpam-5915	196	57	satisfies	satisfy	VERB
ejpam-5915	196	58	the	the	DET
ejpam-5915	196	59	additional	additional	ADJ
ejpam-5915	196	60	condition	condition	NOUN
ejpam-5915	196	61	:	:	PUNCT
ejpam-5915	196	62	(	(	PUNCT
ejpam-5915	196	63	fhn3	fhn3	PROPN
ejpam-5915	196	64	)	)	PUNCT
ejpam-5915	196	65	for	for	ADP
ejpam-5915	196	66	all	all	DET
ejpam-5915	196	67	g	g	NOUN
ejpam-5915	196	68	,	,	PUNCT
ejpam-5915	196	69	h	h	NOUN
ejpam-5915	196	70	∈	∈	PROPN
ejpam-5915	196	71	g	g	PROPN
ejpam-5915	196	72	,	,	PUNCT
ejpam-5915	196	73	µ(g	µ(g	ADP
ejpam-5915	196	74	·	·	PUNCT
ejpam-5915	196	75	h	h	PROPN
ejpam-5915	196	76	·	·	PUNCT
ejpam-5915	196	77	g−1	g−1	PROPN
ejpam-5915	196	78	)	)	PUNCT
ejpam-5915	196	79	≥	≥	NOUN
ejpam-5915	196	80	µ(h	µ(h	PROPN
ejpam-5915	196	81	)	)	PUNCT
ejpam-5915	196	82	,	,	PUNCT
ejpam-5915	196	83	ensuring	ensure	VERB
ejpam-5915	196	84	invariance	invariance	NOUN
ejpam-5915	196	85	under	under	ADP
ejpam-5915	196	86	conjugation	conjugation	NOUN
ejpam-5915	196	87	.	.	PUNCT
ejpam-5915	197	1	example	example	NOUN
ejpam-5915	198	1	6	6	NUM
ejpam-5915	198	2	.	.	PUNCT
ejpam-5915	198	3	consider	consider	VERB
ejpam-5915	198	4	the	the	DET
ejpam-5915	198	5	hom	hom	NOUN
ejpam-5915	198	6	-	-	PUNCT
ejpam-5915	198	7	group	group	NOUN
ejpam-5915	198	8	(	(	PUNCT
ejpam-5915	198	9	g	g	NOUN
ejpam-5915	198	10	=	=	PROPN
ejpam-5915	198	11	r,⊕	r,⊕	PROPN
ejpam-5915	198	12	,	,	PUNCT
ejpam-5915	198	13	α	α	NOUN
ejpam-5915	198	14	)	)	PUNCT
ejpam-5915	198	15	,	,	PUNCT
ejpam-5915	198	16	where	where	SCONJ
ejpam-5915	198	17	:	:	PUNCT
ejpam-5915	198	18	a⊕	a⊕	PROPN
ejpam-5915	198	19	b	b	X
ejpam-5915	198	20	=	=	PRON
ejpam-5915	198	21	a+	a+	PUNCT
ejpam-5915	198	22	b	b	PROPN
ejpam-5915	198	23	2	2	NUM
ejpam-5915	198	24	,	,	PUNCT
ejpam-5915	198	25	α(a	α(a	NOUN
ejpam-5915	198	26	)	)	PUNCT
ejpam-5915	198	27	=	=	PUNCT
ejpam-5915	199	1	a	a	DET
ejpam-5915	199	2	2	2	NUM
ejpam-5915	199	3	.	.	PUNCT
ejpam-5915	200	1	define	define	VERB
ejpam-5915	200	2	the	the	DET
ejpam-5915	200	3	fuzzy	fuzzy	ADJ
ejpam-5915	200	4	membership	membership	NOUN
ejpam-5915	200	5	function	function	NOUN
ejpam-5915	200	6	µ	µ	NOUN
ejpam-5915	200	7	:	:	PUNCT
ejpam-5915	200	8	r	r	NOUN
ejpam-5915	200	9	→	→	SYM
ejpam-5915	200	10	[	[	X
ejpam-5915	200	11	0	0	NUM
ejpam-5915	200	12	,	,	PUNCT
ejpam-5915	200	13	1	1	NUM
ejpam-5915	200	14	]	]	PUNCT
ejpam-5915	200	15	as	as	ADP
ejpam-5915	200	16	:	:	PUNCT
ejpam-5915	200	17	µ(a	µ(a	PROPN
ejpam-5915	200	18	)	)	PUNCT
ejpam-5915	200	19	=	=	PRON
ejpam-5915	200	20	{	{	PUNCT
ejpam-5915	200	21	1	1	NUM
ejpam-5915	200	22	if	if	SCONJ
ejpam-5915	200	23	a	a	DET
ejpam-5915	200	24	=	=	NOUN
ejpam-5915	200	25	0	0	NUM
ejpam-5915	200	26	,	,	PUNCT
ejpam-5915	200	27	1	1	NUM
ejpam-5915	200	28	1+|a|	1+|a|	NUM
ejpam-5915	200	29	if	if	SCONJ
ejpam-5915	200	30	a	a	DET
ejpam-5915	200	31	̸=	̸=	PROPN
ejpam-5915	200	32	0	0	NUM
ejpam-5915	200	33	.	.	PUNCT
ejpam-5915	201	1	in	in	ADP
ejpam-5915	201	2	example	example	NOUN
ejpam-5915	201	3	5	5	NUM
ejpam-5915	201	4	,	,	PUNCT
ejpam-5915	201	5	we	we	PRON
ejpam-5915	201	6	verified	verify	VERB
ejpam-5915	201	7	that	that	SCONJ
ejpam-5915	201	8	µ	µ	PRON
ejpam-5915	201	9	satisfies	satisfie	NOUN
ejpam-5915	201	10	the	the	DET
ejpam-5915	201	11	conditions	condition	NOUN
ejpam-5915	201	12	required	require	VERB
ejpam-5915	201	13	for	for	ADP
ejpam-5915	201	14	a	a	DET
ejpam-5915	201	15	fuzzy	fuzzy	ADJ
ejpam-5915	201	16	homsubgroup	homsubgroup	NOUN
ejpam-5915	201	17	of	of	ADP
ejpam-5915	201	18	g	g	NOUN
ejpam-5915	201	19	,	,	PUNCT
ejpam-5915	201	20	specifically	specifically	ADV
ejpam-5915	201	21	:	:	PUNCT
ejpam-5915	201	22	•	•	ADP
ejpam-5915	201	23	closure	closure	NOUN
ejpam-5915	201	24	under	under	ADP
ejpam-5915	201	25	the	the	DET
ejpam-5915	201	26	operation	operation	NOUN
ejpam-5915	201	27	⊕	⊕	PROPN
ejpam-5915	201	28	,	,	PUNCT
ejpam-5915	201	29	•	•	NUM
ejpam-5915	201	30	compatibility	compatibility	NOUN
ejpam-5915	201	31	with	with	ADP
ejpam-5915	201	32	the	the	DET
ejpam-5915	201	33	twisting	twisting	NOUN
ejpam-5915	201	34	map	map	NOUN
ejpam-5915	201	35	α	α	NOUN
ejpam-5915	201	36	,	,	PUNCT
ejpam-5915	201	37	•	•	ADP
ejpam-5915	201	38	closure	closure	NOUN
ejpam-5915	201	39	under	under	ADP
ejpam-5915	201	40	inverses	inverse	NOUN
ejpam-5915	201	41	.	.	PUNCT
ejpam-5915	202	1	to	to	PART
ejpam-5915	202	2	confirm	confirm	VERB
ejpam-5915	202	3	that	that	SCONJ
ejpam-5915	202	4	µ	µ	NOUN
ejpam-5915	202	5	is	be	AUX
ejpam-5915	202	6	a	a	DET
ejpam-5915	202	7	fuzzy	fuzzy	ADJ
ejpam-5915	202	8	hom	hom	NOUN
ejpam-5915	202	9	-	-	PUNCT
ejpam-5915	202	10	normal	normal	ADJ
ejpam-5915	202	11	subgroup	subgroup	NOUN
ejpam-5915	202	12	,	,	PUNCT
ejpam-5915	202	13	we	we	PRON
ejpam-5915	202	14	now	now	ADV
ejpam-5915	202	15	verify	verify	VERB
ejpam-5915	202	16	the	the	DET
ejpam-5915	202	17	additional	additional	ADJ
ejpam-5915	202	18	condition	condition	NOUN
ejpam-5915	202	19	:	:	PUNCT
ejpam-5915	202	20	•	•	NUM
ejpam-5915	202	21	invariance	invariance	NOUN
ejpam-5915	202	22	under	under	ADP
ejpam-5915	202	23	conjugation	conjugation	NOUN
ejpam-5915	202	24	(	(	PUNCT
ejpam-5915	202	25	fhn3	fhn3	PROPN
ejpam-5915	202	26	):	):	PUNCT
ejpam-5915	202	27	for	for	ADP
ejpam-5915	202	28	any	any	DET
ejpam-5915	202	29	g	g	NOUN
ejpam-5915	202	30	,	,	PUNCT
ejpam-5915	202	31	h	h	NOUN
ejpam-5915	202	32	∈	∈	PROPN
ejpam-5915	202	33	g	g	PROPN
ejpam-5915	202	34	,	,	PUNCT
ejpam-5915	202	35	we	we	PRON
ejpam-5915	202	36	need	need	VERB
ejpam-5915	202	37	to	to	PART
ejpam-5915	202	38	show	show	VERB
ejpam-5915	202	39	that	that	SCONJ
ejpam-5915	202	40	:	:	PUNCT
ejpam-5915	202	41	µ(g	µ(g	PROPN
ejpam-5915	202	42	⊕	⊕	PROPN
ejpam-5915	202	43	h⊕	h⊕	PROPN
ejpam-5915	202	44	(	(	PUNCT
ejpam-5915	202	45	−g	−g	NOUN
ejpam-5915	202	46	)	)	PUNCT
ejpam-5915	202	47	)	)	PUNCT
ejpam-5915	202	48	≥	≥	NOUN
ejpam-5915	202	49	µ(h	µ(h	PROPN
ejpam-5915	202	50	)	)	PUNCT
ejpam-5915	202	51	.	.	PUNCT
ejpam-5915	203	1	since	since	SCONJ
ejpam-5915	203	2	g	g	PROPN
ejpam-5915	203	3	⊕	⊕	PROPN
ejpam-5915	203	4	h⊕	h⊕	PROPN
ejpam-5915	203	5	(	(	PUNCT
ejpam-5915	203	6	−g	−g	NOUN
ejpam-5915	203	7	)	)	PUNCT
ejpam-5915	203	8	=	=	SYM
ejpam-5915	203	9	h	h	NOUN
ejpam-5915	203	10	2	2	NUM
ejpam-5915	203	11	,	,	PUNCT
ejpam-5915	203	12	we	we	PRON
ejpam-5915	203	13	calculate	calculate	VERB
ejpam-5915	203	14	:	:	PUNCT
ejpam-5915	203	15	µ	µ	X
ejpam-5915	203	16	(	(	PUNCT
ejpam-5915	203	17	h	h	NOUN
ejpam-5915	203	18	2	2	X
ejpam-5915	203	19	)	)	PUNCT
ejpam-5915	203	20	=	=	SYM
ejpam-5915	203	21	2	2	NUM
ejpam-5915	203	22	2	2	NUM
ejpam-5915	203	23	+	+	SYM
ejpam-5915	203	24	|h|	|h|	NOUN
ejpam-5915	203	25	.	.	PUNCT
ejpam-5915	204	1	given	give	VERB
ejpam-5915	204	2	µ(h	µ(h	NOUN
ejpam-5915	204	3	)	)	PUNCT
ejpam-5915	204	4	=	=	SYM
ejpam-5915	204	5	1	1	NUM
ejpam-5915	204	6	1+|h|	1+|h|	NUM
ejpam-5915	204	7	,	,	PUNCT
ejpam-5915	204	8	we	we	PRON
ejpam-5915	204	9	need	need	VERB
ejpam-5915	204	10	to	to	PART
ejpam-5915	204	11	verify	verify	VERB
ejpam-5915	204	12	that	that	SCONJ
ejpam-5915	204	13	:	:	PUNCT
ejpam-5915	204	14	2	2	NUM
ejpam-5915	204	15	2	2	NUM
ejpam-5915	204	16	+	+	SYM
ejpam-5915	204	17	|h|	|h|	PROPN
ejpam-5915	204	18	≥	≥	NUM
ejpam-5915	204	19	1	1	NUM
ejpam-5915	204	20	1	1	NUM
ejpam-5915	204	21	+	+	NUM
ejpam-5915	204	22	|h|	|h|	PROPN
ejpam-5915	204	23	.	.	PUNCT
ejpam-5915	205	1	to	to	PART
ejpam-5915	205	2	confirm	confirm	VERB
ejpam-5915	205	3	this	this	DET
ejpam-5915	205	4	inequality	inequality	NOUN
ejpam-5915	205	5	,	,	PUNCT
ejpam-5915	205	6	we	we	PRON
ejpam-5915	205	7	cross	cross	VERB
ejpam-5915	205	8	-	-	VERB
ejpam-5915	205	9	multiply	multiply	ADJ
ejpam-5915	205	10	:	:	PUNCT
ejpam-5915	205	11	2(1	2(1	NUM
ejpam-5915	206	1	+	+	CCONJ
ejpam-5915	206	2	|h|	|h|	PROPN
ejpam-5915	206	3	)	)	PUNCT
ejpam-5915	206	4	≥	≥	NOUN
ejpam-5915	206	5	1(2	1(2	NUM
ejpam-5915	206	6	+	+	CCONJ
ejpam-5915	206	7	|h|	|h|	NOUN
ejpam-5915	206	8	)	)	PUNCT
ejpam-5915	206	9	.	.	PUNCT
ejpam-5915	207	1	s.	s.	PROPN
ejpam-5915	207	2	shaqaqha	shaqaqha	PROPN
ejpam-5915	207	3	/	/	SYM
ejpam-5915	207	4	eur	eur	PROPN
ejpam-5915	207	5	.	.	PUNCT
ejpam-5915	208	1	j.	j.	PROPN
ejpam-5915	208	2	pure	pure	PROPN
ejpam-5915	208	3	appl	appl	PROPN
ejpam-5915	208	4	.	.	PROPN
ejpam-5915	208	5	math	math	PROPN
ejpam-5915	208	6	,	,	PUNCT
ejpam-5915	208	7	18	18	NUM
ejpam-5915	208	8	(	(	PUNCT
ejpam-5915	208	9	2	2	NUM
ejpam-5915	208	10	)	)	PUNCT
ejpam-5915	208	11	(	(	PUNCT
ejpam-5915	208	12	2025	2025	NUM
ejpam-5915	208	13	)	)	PUNCT
ejpam-5915	208	14	,	,	PUNCT
ejpam-5915	208	15	5915	5915	NUM
ejpam-5915	208	16	11	11	NUM
ejpam-5915	208	17	of	of	ADP
ejpam-5915	208	18	16	16	NUM
ejpam-5915	208	19	expanding	expand	VERB
ejpam-5915	208	20	both	both	DET
ejpam-5915	208	21	sides	side	NOUN
ejpam-5915	208	22	:	:	PUNCT
ejpam-5915	208	23	2	2	NUM
ejpam-5915	208	24	+	+	SYM
ejpam-5915	208	25	2|h|	2|h|	NUM
ejpam-5915	208	26	≥	≥	NOUN
ejpam-5915	208	27	2	2	NUM
ejpam-5915	208	28	+	+	CCONJ
ejpam-5915	208	29	|h|	|h|	PROPN
ejpam-5915	208	30	,	,	PUNCT
ejpam-5915	208	31	and	and	CCONJ
ejpam-5915	208	32	simplifying	simplifying	NOUN
ejpam-5915	208	33	,	,	PUNCT
ejpam-5915	208	34	we	we	PRON
ejpam-5915	208	35	obtain	obtain	VERB
ejpam-5915	208	36	:	:	PUNCT
ejpam-5915	208	37	2|h|	2|h|	NUM
ejpam-5915	208	38	≥	≥	NOUN
ejpam-5915	208	39	|h|	|h|	NOUN
ejpam-5915	208	40	,	,	PUNCT
ejpam-5915	208	41	which	which	PRON
ejpam-5915	208	42	holds	hold	VERB
ejpam-5915	208	43	for	for	ADP
ejpam-5915	208	44	all	all	DET
ejpam-5915	208	45	h	h	NOUN
ejpam-5915	208	46	∈	∈	PROPN
ejpam-5915	208	47	r.	r.	PROPN
ejpam-5915	208	48	thus	thus	ADV
ejpam-5915	208	49	,	,	PUNCT
ejpam-5915	208	50	the	the	DET
ejpam-5915	208	51	invariance	invariance	NOUN
ejpam-5915	208	52	under	under	ADP
ejpam-5915	208	53	conjugation	conjugation	NOUN
ejpam-5915	208	54	condition	condition	NOUN
ejpam-5915	208	55	(	(	PUNCT
ejpam-5915	208	56	fhn3	fhn3	PROPN
ejpam-5915	208	57	)	)	PUNCT
ejpam-5915	208	58	is	be	AUX
ejpam-5915	208	59	satisfied	satisfied	ADJ
ejpam-5915	208	60	.	.	PUNCT
ejpam-5915	209	1	therefore	therefore	ADV
ejpam-5915	209	2	,	,	PUNCT
ejpam-5915	209	3	µ	µ	PRON
ejpam-5915	209	4	defines	define	VERB
ejpam-5915	209	5	a	a	DET
ejpam-5915	209	6	fuzzy	fuzzy	ADJ
ejpam-5915	209	7	hom	hom	NOUN
ejpam-5915	209	8	-	-	PUNCT
ejpam-5915	209	9	normal	normal	ADJ
ejpam-5915	209	10	subgroup	subgroup	NOUN
ejpam-5915	209	11	for	for	ADP
ejpam-5915	209	12	the	the	DET
ejpam-5915	209	13	hom	hom	NOUN
ejpam-5915	209	14	-	-	PUNCT
ejpam-5915	209	15	group	group	NOUN
ejpam-5915	209	16	(	(	PUNCT
ejpam-5915	209	17	r,⊕	r,⊕	PROPN
ejpam-5915	209	18	,	,	PUNCT
ejpam-5915	209	19	α	α	NOUN
ejpam-5915	209	20	)	)	PUNCT
ejpam-5915	209	21	.	.	PUNCT
ejpam-5915	210	1	4	4	X
ejpam-5915	210	2	.	.	X
ejpam-5915	210	3	relationships	relationship	NOUN
ejpam-5915	210	4	between	between	ADP
ejpam-5915	210	5	fuzzy	fuzzy	ADJ
ejpam-5915	210	6	hom	hom	NOUN
ejpam-5915	210	7	-	-	PUNCT
ejpam-5915	210	8	substructures	substructure	NOUN
ejpam-5915	210	9	and	and	CCONJ
ejpam-5915	210	10	hom	hom	NOUN
ejpam-5915	210	11	-	-	PUNCT
ejpam-5915	210	12	substructures	substructure	NOUN
ejpam-5915	210	13	let	let	VERB
ejpam-5915	210	14	g	g	PRON
ejpam-5915	210	15	be	be	AUX
ejpam-5915	210	16	a	a	DET
ejpam-5915	210	17	hom	hom	NOUN
ejpam-5915	210	18	-	-	PUNCT
ejpam-5915	210	19	group	group	NOUN
ejpam-5915	210	20	equipped	equip	VERB
ejpam-5915	210	21	with	with	ADP
ejpam-5915	210	22	a	a	DET
ejpam-5915	210	23	fuzzy	fuzzy	ADJ
ejpam-5915	210	24	set	set	VERB
ejpam-5915	210	25	µ.	µ.	NOUN
ejpam-5915	210	26	we	we	PRON
ejpam-5915	210	27	define	define	VERB
ejpam-5915	210	28	the	the	DET
ejpam-5915	210	29	set	set	ADJ
ejpam-5915	210	30	u(µ	u(µ	PROPN
ejpam-5915	210	31	,	,	PUNCT
ejpam-5915	210	32	t	t	PROPN
ejpam-5915	210	33	)	)	PUNCT
ejpam-5915	210	34	as	as	ADP
ejpam-5915	210	35	the	the	DET
ejpam-5915	210	36	upper	upper	ADJ
ejpam-5915	210	37	level	level	NOUN
ejpam-5915	210	38	set	set	NOUN
ejpam-5915	210	39	of	of	ADP
ejpam-5915	210	40	µ	µ	NOUN
ejpam-5915	210	41	for	for	ADP
ejpam-5915	210	42	t	t	PROPN
ejpam-5915	210	43	∈	∈	PROPN
ejpam-5915	211	1	[	[	X
ejpam-5915	211	2	0	0	NUM
ejpam-5915	211	3	,	,	PUNCT
ejpam-5915	211	4	1	1	NUM
ejpam-5915	211	5	]	]	PUNCT
ejpam-5915	211	6	,	,	PUNCT
ejpam-5915	211	7	denoted	denote	VERB
ejpam-5915	211	8	by	by	ADP
ejpam-5915	211	9	:	:	PUNCT
ejpam-5915	211	10	u(µ	u(µ	PROPN
ejpam-5915	211	11	,	,	PUNCT
ejpam-5915	211	12	t	t	PROPN
ejpam-5915	211	13	)	)	PUNCT
ejpam-5915	211	14	=	=	PRON
ejpam-5915	211	15	{	{	PUNCT
ejpam-5915	211	16	g	g	PROPN
ejpam-5915	211	17	∈	∈	PROPN
ejpam-5915	211	18	g	g	PROPN
ejpam-5915	211	19	|	|	NOUN
ejpam-5915	211	20	µ(g	µ(g	PROPN
ejpam-5915	211	21	)	)	PUNCT
ejpam-5915	211	22	≥	≥	NOUN
ejpam-5915	211	23	t	t	PROPN
ejpam-5915	211	24	}	}	PUNCT
ejpam-5915	211	25	.	.	PUNCT
ejpam-5915	212	1	in	in	ADP
ejpam-5915	212	2	the	the	DET
ejpam-5915	212	3	setting	setting	NOUN
ejpam-5915	212	4	of	of	ADP
ejpam-5915	212	5	hom	hom	NOUN
ejpam-5915	212	6	-	-	PUNCT
ejpam-5915	212	7	groups	group	NOUN
ejpam-5915	212	8	,	,	PUNCT
ejpam-5915	212	9	we	we	PRON
ejpam-5915	212	10	seek	seek	VERB
ejpam-5915	212	11	to	to	PART
ejpam-5915	212	12	extend	extend	VERB
ejpam-5915	212	13	the	the	DET
ejpam-5915	212	14	result	result	NOUN
ejpam-5915	212	15	of	of	ADP
ejpam-5915	212	16	theorem	theorem	NOUN
ejpam-5915	212	17	1	1	NUM
ejpam-5915	212	18	from	from	ADP
ejpam-5915	212	19	[	[	X
ejpam-5915	212	20	20	20	NUM
ejpam-5915	212	21	]	]	PUNCT
ejpam-5915	212	22	,	,	PUNCT
ejpam-5915	212	23	which	which	PRON
ejpam-5915	212	24	explored	explore	VERB
ejpam-5915	212	25	the	the	DET
ejpam-5915	212	26	relationship	relationship	NOUN
ejpam-5915	212	27	between	between	ADP
ejpam-5915	212	28	fuzzy	fuzzy	ADJ
ejpam-5915	212	29	subsets	subset	NOUN
ejpam-5915	212	30	and	and	CCONJ
ejpam-5915	212	31	structural	structural	ADJ
ejpam-5915	212	32	subgroups	subgroup	NOUN
ejpam-5915	212	33	.	.	PUNCT
ejpam-5915	213	1	by	by	ADP
ejpam-5915	213	2	establishing	establish	VERB
ejpam-5915	213	3	the	the	DET
ejpam-5915	213	4	connection	connection	NOUN
ejpam-5915	213	5	between	between	ADP
ejpam-5915	213	6	fuzzy	fuzzy	ADJ
ejpam-5915	213	7	hom	hom	NOUN
ejpam-5915	213	8	-	-	PUNCT
ejpam-5915	213	9	subgroups	subgroup	NOUN
ejpam-5915	213	10	of	of	ADP
ejpam-5915	213	11	g	g	NOUN
ejpam-5915	213	12	and	and	CCONJ
ejpam-5915	213	13	classical	classical	ADJ
ejpam-5915	213	14	hom	hom	NOUN
ejpam-5915	213	15	-	-	PUNCT
ejpam-5915	213	16	subgroups	subgroup	NOUN
ejpam-5915	213	17	of	of	ADP
ejpam-5915	213	18	g	g	NOUN
ejpam-5915	213	19	,	,	PUNCT
ejpam-5915	213	20	we	we	PRON
ejpam-5915	213	21	can	can	AUX
ejpam-5915	213	22	better	well	ADV
ejpam-5915	213	23	understand	understand	VERB
ejpam-5915	213	24	how	how	SCONJ
ejpam-5915	213	25	fuzzy	fuzzy	ADJ
ejpam-5915	213	26	membership	membership	NOUN
ejpam-5915	213	27	functions	function	NOUN
ejpam-5915	213	28	relate	relate	VERB
ejpam-5915	213	29	to	to	ADP
ejpam-5915	213	30	subgroup	subgroup	NOUN
ejpam-5915	213	31	properties	property	NOUN
ejpam-5915	213	32	within	within	ADP
ejpam-5915	213	33	the	the	DET
ejpam-5915	213	34	framework	framework	NOUN
ejpam-5915	213	35	of	of	ADP
ejpam-5915	213	36	hom	hom	NOUN
ejpam-5915	213	37	-	-	PUNCT
ejpam-5915	213	38	groups	group	NOUN
ejpam-5915	213	39	,	,	PUNCT
ejpam-5915	213	40	supporting	support	VERB
ejpam-5915	213	41	further	further	ADJ
ejpam-5915	213	42	analysis	analysis	NOUN
ejpam-5915	213	43	on	on	ADP
ejpam-5915	213	44	fuzzy	fuzzy	ADJ
ejpam-5915	213	45	structures	structure	NOUN
ejpam-5915	213	46	in	in	ADP
ejpam-5915	213	47	hom	hom	ADV
ejpam-5915	213	48	-	-	PUNCT
ejpam-5915	213	49	algebraic	algebraic	ADJ
ejpam-5915	213	50	systems	system	NOUN
ejpam-5915	213	51	.	.	PUNCT
ejpam-5915	214	1	theorem	theorem	NOUN
ejpam-5915	214	2	2	2	NUM
ejpam-5915	214	3	.	.	PUNCT
ejpam-5915	214	4	let	let	VERB
ejpam-5915	214	5	µ	µ	X
ejpam-5915	214	6	be	be	AUX
ejpam-5915	214	7	a	a	DET
ejpam-5915	214	8	fuzzy	fuzzy	ADJ
ejpam-5915	214	9	subset	subset	NOUN
ejpam-5915	214	10	on	on	ADP
ejpam-5915	214	11	a	a	DET
ejpam-5915	214	12	hom	hom	NOUN
ejpam-5915	214	13	-	-	PUNCT
ejpam-5915	214	14	group	group	NOUN
ejpam-5915	214	15	(	(	PUNCT
ejpam-5915	214	16	g	g	PROPN
ejpam-5915	214	17	,	,	PUNCT
ejpam-5915	214	18	·	·	PUNCT
ejpam-5915	214	19	,	,	PUNCT
ejpam-5915	214	20	α	α	NOUN
ejpam-5915	214	21	)	)	PUNCT
ejpam-5915	214	22	.	.	PUNCT
ejpam-5915	215	1	the	the	DET
ejpam-5915	215	2	following	follow	VERB
ejpam-5915	215	3	statements	statement	NOUN
ejpam-5915	215	4	are	be	AUX
ejpam-5915	215	5	equivalent	equivalent	ADJ
ejpam-5915	215	6	:	:	PUNCT
ejpam-5915	215	7	(	(	PUNCT
ejpam-5915	215	8	i	i	NOUN
ejpam-5915	215	9	)	)	PUNCT
ejpam-5915	215	10	µ	µ	PROPN
ejpam-5915	215	11	is	be	AUX
ejpam-5915	215	12	a	a	DET
ejpam-5915	215	13	fuzzy	fuzzy	ADJ
ejpam-5915	215	14	hom	hom	NOUN
ejpam-5915	215	15	-	-	PUNCT
ejpam-5915	215	16	subgroup	subgroup	NOUN
ejpam-5915	215	17	of	of	ADP
ejpam-5915	215	18	g	g	PROPN
ejpam-5915	215	19	,	,	PUNCT
ejpam-5915	215	20	(	(	PUNCT
ejpam-5915	215	21	ii	ii	NOUN
ejpam-5915	215	22	)	)	PUNCT
ejpam-5915	215	23	for	for	ADP
ejpam-5915	215	24	every	every	DET
ejpam-5915	215	25	t	t	NOUN
ejpam-5915	215	26	∈	∈	PROPN
ejpam-5915	215	27	im(µ	im(µ	PROPN
ejpam-5915	215	28	)	)	PUNCT
ejpam-5915	215	29	,	,	PUNCT
ejpam-5915	215	30	the	the	DET
ejpam-5915	215	31	set	set	NOUN
ejpam-5915	215	32	u(µ	u(µ	PROPN
ejpam-5915	215	33	,	,	PUNCT
ejpam-5915	215	34	t	t	PROPN
ejpam-5915	215	35	)	)	PUNCT
ejpam-5915	215	36	=	=	PRON
ejpam-5915	215	37	{	{	PUNCT
ejpam-5915	215	38	g	g	PROPN
ejpam-5915	215	39	∈	∈	PROPN
ejpam-5915	215	40	g	g	PROPN
ejpam-5915	215	41	|	|	NOUN
ejpam-5915	215	42	µ(g	µ(g	PROPN
ejpam-5915	215	43	)	)	PUNCT
ejpam-5915	215	44	≥	≥	NOUN
ejpam-5915	215	45	t	t	PROPN
ejpam-5915	215	46	}	}	PUNCT
ejpam-5915	215	47	is	be	AUX
ejpam-5915	215	48	a	a	DET
ejpam-5915	215	49	hom	hom	NOUN
ejpam-5915	215	50	-	-	PUNCT
ejpam-5915	215	51	subgroup	subgroup	NOUN
ejpam-5915	215	52	of	of	ADP
ejpam-5915	215	53	g.	g.	PROPN
ejpam-5915	215	54	proof	proof	NOUN
ejpam-5915	215	55	.	.	PUNCT
ejpam-5915	216	1	to	to	PART
ejpam-5915	216	2	show	show	VERB
ejpam-5915	216	3	the	the	DET
ejpam-5915	216	4	equivalence	equivalence	NOUN
ejpam-5915	216	5	,	,	PUNCT
ejpam-5915	216	6	we	we	PRON
ejpam-5915	216	7	proceed	proceed	VERB
ejpam-5915	216	8	as	as	SCONJ
ejpam-5915	216	9	follows	follow	VERB
ejpam-5915	216	10	:	:	PUNCT
ejpam-5915	216	11	(	(	PUNCT
ejpam-5915	216	12	i	i	NOUN
ejpam-5915	216	13	)	)	PUNCT
ejpam-5915	216	14	⇒	⇒	PROPN
ejpam-5915	216	15	(	(	PUNCT
ejpam-5915	216	16	ii	ii	PROPN
ejpam-5915	216	17	):	):	PUNCT
ejpam-5915	216	18	suppose	suppose	VERB
ejpam-5915	216	19	µ	µ	PRON
ejpam-5915	216	20	is	be	AUX
ejpam-5915	216	21	a	a	DET
ejpam-5915	216	22	fuzzy	fuzzy	ADJ
ejpam-5915	216	23	hom	hom	NOUN
ejpam-5915	216	24	-	-	PUNCT
ejpam-5915	216	25	subgroup	subgroup	NOUN
ejpam-5915	216	26	of	of	ADP
ejpam-5915	216	27	g.	g.	PROPN
ejpam-5915	216	28	for	for	ADP
ejpam-5915	216	29	each	each	DET
ejpam-5915	216	30	t	t	PROPN
ejpam-5915	216	31	∈	∈	PROPN
ejpam-5915	216	32	im(µ	im(µ	ADV
ejpam-5915	216	33	)	)	PUNCT
ejpam-5915	216	34	,	,	PUNCT
ejpam-5915	216	35	let	let	VERB
ejpam-5915	216	36	h1	h1	PROPN
ejpam-5915	216	37	,	,	PUNCT
ejpam-5915	216	38	h2	h2	PROPN
ejpam-5915	216	39	∈	∈	PROPN
ejpam-5915	216	40	u(µ	u(µ	PROPN
ejpam-5915	216	41	,	,	PUNCT
ejpam-5915	216	42	t	t	PROPN
ejpam-5915	216	43	)	)	PUNCT
ejpam-5915	216	44	and	and	CCONJ
ejpam-5915	216	45	h	h	NOUN
ejpam-5915	216	46	∈	∈	PROPN
ejpam-5915	216	47	u(µ	u(µ	PROPN
ejpam-5915	216	48	,	,	PUNCT
ejpam-5915	216	49	t	t	PROPN
ejpam-5915	216	50	)	)	PUNCT
ejpam-5915	216	51	.	.	PUNCT
ejpam-5915	217	1	by	by	ADP
ejpam-5915	217	2	definition	definition	NOUN
ejpam-5915	217	3	of	of	ADP
ejpam-5915	217	4	a	a	DET
ejpam-5915	217	5	fuzzy	fuzzy	ADJ
ejpam-5915	217	6	hom	hom	NOUN
ejpam-5915	217	7	-	-	PUNCT
ejpam-5915	217	8	subgroup	subgroup	NOUN
ejpam-5915	217	9	:	:	PUNCT
ejpam-5915	217	10	•	•	NOUN
ejpam-5915	217	11	closure	closure	NOUN
ejpam-5915	217	12	under	under	ADP
ejpam-5915	217	13	·	·	PUNCT
ejpam-5915	217	14	:	:	PUNCT
ejpam-5915	217	15	since	since	SCONJ
ejpam-5915	217	16	µ(h1	µ(h1	NOUN
ejpam-5915	217	17	·	·	SYM
ejpam-5915	217	18	h2	h2	PROPN
ejpam-5915	217	19	)	)	PUNCT
ejpam-5915	217	20	≥	≥	NOUN
ejpam-5915	217	21	min{µ(h1	min{µ(h1	NOUN
ejpam-5915	217	22	)	)	PUNCT
ejpam-5915	217	23	,	,	PUNCT
ejpam-5915	217	24	µ(h2	µ(h2	NOUN
ejpam-5915	217	25	)	)	PUNCT
ejpam-5915	217	26	}	}	PUNCT
ejpam-5915	217	27	≥	≥	PROPN
ejpam-5915	217	28	t	t	PROPN
ejpam-5915	217	29	,	,	PUNCT
ejpam-5915	217	30	we	we	PRON
ejpam-5915	217	31	conclude	conclude	VERB
ejpam-5915	217	32	h1	h1	ADJ
ejpam-5915	217	33	·	·	PUNCT
ejpam-5915	217	34	h2	h2	PROPN
ejpam-5915	217	35	∈	∈	PROPN
ejpam-5915	217	36	u(µ	u(µ	PROPN
ejpam-5915	217	37	,	,	PUNCT
ejpam-5915	217	38	t	t	PROPN
ejpam-5915	217	39	)	)	PUNCT
ejpam-5915	217	40	.	.	PUNCT
ejpam-5915	218	1	•	•	NOUN
ejpam-5915	218	2	closure	closure	NOUN
ejpam-5915	218	3	under	under	ADP
ejpam-5915	218	4	α	α	NOUN
ejpam-5915	218	5	:	:	PUNCT
ejpam-5915	218	6	similarly	similarly	ADV
ejpam-5915	218	7	,	,	PUNCT
ejpam-5915	218	8	α(h	α(h	NOUN
ejpam-5915	218	9	)	)	PUNCT
ejpam-5915	218	10	∈	∈	PROPN
ejpam-5915	218	11	u(µ	u(µ	PROPN
ejpam-5915	218	12	,	,	PUNCT
ejpam-5915	218	13	t	t	PROPN
ejpam-5915	218	14	)	)	PUNCT
ejpam-5915	218	15	as	as	ADP
ejpam-5915	218	16	µ(α(h	µ(α(h	PROPN
ejpam-5915	218	17	)	)	PUNCT
ejpam-5915	218	18	)	)	PUNCT
ejpam-5915	218	19	≥	≥	NOUN
ejpam-5915	218	20	µ(h	µ(h	PROPN
ejpam-5915	218	21	)	)	PUNCT
ejpam-5915	218	22	≥	≥	NOUN
ejpam-5915	218	23	t.	t.	NOUN
ejpam-5915	218	24	•	•	NUM
ejpam-5915	218	25	existence	existence	NOUN
ejpam-5915	218	26	of	of	ADP
ejpam-5915	218	27	inverses	inverse	NOUN
ejpam-5915	218	28	:	:	PUNCT
ejpam-5915	218	29	for	for	ADP
ejpam-5915	218	30	h	h	PROPN
ejpam-5915	218	31	∈	∈	PROPN
ejpam-5915	218	32	u(µ	u(µ	PROPN
ejpam-5915	218	33	,	,	PUNCT
ejpam-5915	218	34	t	t	PROPN
ejpam-5915	218	35	)	)	PUNCT
ejpam-5915	218	36	,	,	PUNCT
ejpam-5915	218	37	h−1	h−1	PROPN
ejpam-5915	218	38	∈	∈	PROPN
ejpam-5915	218	39	u(µ	u(µ	PROPN
ejpam-5915	218	40	,	,	PUNCT
ejpam-5915	218	41	t	t	PROPN
ejpam-5915	218	42	)	)	PUNCT
ejpam-5915	218	43	since	since	SCONJ
ejpam-5915	218	44	µ(h−1	µ(h−1	ADP
ejpam-5915	218	45	)	)	PUNCT
ejpam-5915	218	46	≥	≥	NOUN
ejpam-5915	218	47	t.	t.	PROPN
ejpam-5915	218	48	thus	thus	ADV
ejpam-5915	218	49	,	,	PUNCT
ejpam-5915	218	50	u(µ	u(µ	PROPN
ejpam-5915	218	51	,	,	PUNCT
ejpam-5915	218	52	t	t	PROPN
ejpam-5915	218	53	)	)	PUNCT
ejpam-5915	218	54	is	be	AUX
ejpam-5915	218	55	a	a	DET
ejpam-5915	218	56	hom	hom	NOUN
ejpam-5915	218	57	-	-	PUNCT
ejpam-5915	218	58	subgroup	subgroup	NOUN
ejpam-5915	218	59	of	of	ADP
ejpam-5915	218	60	g.	g.	PROPN
ejpam-5915	218	61	(	(	PUNCT
ejpam-5915	218	62	ii	ii	PROPN
ejpam-5915	218	63	)	)	PUNCT
ejpam-5915	218	64	⇒	⇒	NOUN
ejpam-5915	218	65	(	(	PUNCT
ejpam-5915	218	66	i	i	NOUN
ejpam-5915	218	67	):	):	PUNCT
ejpam-5915	218	68	conversely	conversely	ADV
ejpam-5915	218	69	,	,	PUNCT
ejpam-5915	218	70	assume	assume	VERB
ejpam-5915	218	71	each	each	DET
ejpam-5915	218	72	u(µ	u(µ	PROPN
ejpam-5915	218	73	,	,	PUNCT
ejpam-5915	218	74	t	t	PROPN
ejpam-5915	218	75	)	)	PUNCT
ejpam-5915	218	76	is	be	AUX
ejpam-5915	218	77	a	a	DET
ejpam-5915	218	78	hom	hom	NOUN
ejpam-5915	218	79	-	-	PUNCT
ejpam-5915	218	80	subgroup	subgroup	NOUN
ejpam-5915	218	81	of	of	ADP
ejpam-5915	218	82	g	g	PROPN
ejpam-5915	218	83	for	for	ADP
ejpam-5915	218	84	every	every	DET
ejpam-5915	218	85	t	t	NOUN
ejpam-5915	218	86	∈	∈	PROPN
ejpam-5915	218	87	im(µ	im(µ	ADV
ejpam-5915	218	88	)	)	PUNCT
ejpam-5915	218	89	.	.	PUNCT
ejpam-5915	219	1	we	we	PRON
ejpam-5915	219	2	want	want	VERB
ejpam-5915	219	3	to	to	PART
ejpam-5915	219	4	show	show	VERB
ejpam-5915	219	5	that	that	SCONJ
ejpam-5915	219	6	µ	µ	ADJ
ejpam-5915	219	7	satisfies	satisfie	NOUN
ejpam-5915	219	8	the	the	DET
ejpam-5915	219	9	properties	property	NOUN
ejpam-5915	219	10	required	require	VERB
ejpam-5915	219	11	for	for	ADP
ejpam-5915	219	12	a	a	DET
ejpam-5915	219	13	fuzzy	fuzzy	ADJ
ejpam-5915	219	14	hom	hom	NOUN
ejpam-5915	219	15	-	-	PUNCT
ejpam-5915	219	16	subgroup	subgroup	NOUN
ejpam-5915	219	17	.	.	PUNCT
ejpam-5915	220	1	take	take	VERB
ejpam-5915	220	2	any	any	DET
ejpam-5915	220	3	g	g	NOUN
ejpam-5915	220	4	,	,	PUNCT
ejpam-5915	220	5	h	h	NOUN
ejpam-5915	220	6	∈	∈	PROPN
ejpam-5915	220	7	g	g	PROPN
ejpam-5915	220	8	and	and	CCONJ
ejpam-5915	220	9	let	let	VERB
ejpam-5915	220	10	t1	t1	NOUN
ejpam-5915	220	11	=	=	PROPN
ejpam-5915	220	12	min{µ(g	min{µ(g	PROPN
ejpam-5915	220	13	)	)	PUNCT
ejpam-5915	220	14	,	,	PUNCT
ejpam-5915	220	15	µ(h	µ(h	PROPN
ejpam-5915	220	16	)	)	PUNCT
ejpam-5915	220	17	}	}	PUNCT
ejpam-5915	220	18	.	.	PUNCT
ejpam-5915	221	1	this	this	PRON
ejpam-5915	221	2	implies	imply	VERB
ejpam-5915	221	3	that	that	SCONJ
ejpam-5915	221	4	g	g	PROPN
ejpam-5915	221	5	,	,	PUNCT
ejpam-5915	221	6	h	h	NOUN
ejpam-5915	221	7	∈	∈	PROPN
ejpam-5915	221	8	u(µ	u(µ	PROPN
ejpam-5915	221	9	,	,	PUNCT
ejpam-5915	221	10	t1	t1	NOUN
ejpam-5915	221	11	)	)	PUNCT
ejpam-5915	221	12	since	since	SCONJ
ejpam-5915	221	13	µ(g	µ(g	NUM
ejpam-5915	221	14	)	)	PUNCT
ejpam-5915	221	15	≥	≥	NOUN
ejpam-5915	221	16	t1	t1	NOUN
ejpam-5915	221	17	and	and	CCONJ
ejpam-5915	221	18	µ(h	µ(h	PROPN
ejpam-5915	221	19	)	)	PUNCT
ejpam-5915	221	20	≥	≥	NUM
ejpam-5915	221	21	t1	t1	NOUN
ejpam-5915	221	22	.	.	PUNCT
ejpam-5915	222	1	since	since	SCONJ
ejpam-5915	222	2	u(µ	u(µ	NOUN
ejpam-5915	222	3	,	,	PUNCT
ejpam-5915	222	4	t1	t1	NOUN
ejpam-5915	222	5	)	)	PUNCT
ejpam-5915	222	6	is	be	AUX
ejpam-5915	222	7	a	a	DET
ejpam-5915	222	8	hom	hom	NOUN
ejpam-5915	222	9	-	-	PUNCT
ejpam-5915	222	10	subgroup	subgroup	NOUN
ejpam-5915	222	11	of	of	ADP
ejpam-5915	222	12	g	g	PROPN
ejpam-5915	222	13	,	,	PUNCT
ejpam-5915	222	14	it	it	PRON
ejpam-5915	222	15	has	have	VERB
ejpam-5915	222	16	the	the	DET
ejpam-5915	222	17	following	follow	VERB
ejpam-5915	222	18	properties	property	NOUN
ejpam-5915	222	19	:	:	PUNCT
ejpam-5915	222	20	s.	s.	PROPN
ejpam-5915	222	21	shaqaqha	shaqaqha	PROPN
ejpam-5915	222	22	/	/	SYM
ejpam-5915	222	23	eur	eur	PROPN
ejpam-5915	222	24	.	.	PUNCT
ejpam-5915	223	1	j.	j.	PROPN
ejpam-5915	223	2	pure	pure	PROPN
ejpam-5915	223	3	appl	appl	PROPN
ejpam-5915	223	4	.	.	PROPN
ejpam-5915	223	5	math	math	PROPN
ejpam-5915	223	6	,	,	PUNCT
ejpam-5915	223	7	18	18	NUM
ejpam-5915	223	8	(	(	PUNCT
ejpam-5915	223	9	2	2	NUM
ejpam-5915	223	10	)	)	PUNCT
ejpam-5915	223	11	(	(	PUNCT
ejpam-5915	223	12	2025	2025	NUM
ejpam-5915	223	13	)	)	PUNCT
ejpam-5915	223	14	,	,	PUNCT
ejpam-5915	223	15	5915	5915	NUM
ejpam-5915	223	16	12	12	NUM
ejpam-5915	223	17	of	of	ADP
ejpam-5915	223	18	16	16	NUM
ejpam-5915	223	19	•	•	NOUN
ejpam-5915	223	20	closure	closure	NOUN
ejpam-5915	223	21	under	under	ADP
ejpam-5915	223	22	the	the	DET
ejpam-5915	223	23	operation	operation	NOUN
ejpam-5915	223	24	:	:	PUNCT
ejpam-5915	223	25	since	since	SCONJ
ejpam-5915	223	26	g	g	PROPN
ejpam-5915	223	27	,	,	PUNCT
ejpam-5915	223	28	h	h	NOUN
ejpam-5915	223	29	∈	∈	PROPN
ejpam-5915	223	30	u(µ	u(µ	PROPN
ejpam-5915	223	31	,	,	PUNCT
ejpam-5915	223	32	t1	t1	NOUN
ejpam-5915	223	33	)	)	PUNCT
ejpam-5915	223	34	and	and	CCONJ
ejpam-5915	223	35	u(µ	u(µ	PROPN
ejpam-5915	223	36	,	,	PUNCT
ejpam-5915	223	37	t1	t1	NOUN
ejpam-5915	223	38	)	)	PUNCT
ejpam-5915	223	39	is	be	AUX
ejpam-5915	223	40	closed	close	VERB
ejpam-5915	223	41	under	under	ADP
ejpam-5915	223	42	the	the	DET
ejpam-5915	223	43	operation	operation	NOUN
ejpam-5915	223	44	·	·	PUNCT
ejpam-5915	223	45	,	,	PUNCT
ejpam-5915	223	46	we	we	PRON
ejpam-5915	223	47	have	have	VERB
ejpam-5915	223	48	g	g	NOUN
ejpam-5915	223	49	·	·	PUNCT
ejpam-5915	223	50	h	h	NOUN
ejpam-5915	223	51	∈	∈	PROPN
ejpam-5915	223	52	u(µ	u(µ	PROPN
ejpam-5915	223	53	,	,	PUNCT
ejpam-5915	223	54	t1	t1	NOUN
ejpam-5915	223	55	)	)	PUNCT
ejpam-5915	223	56	.	.	PUNCT
ejpam-5915	224	1	therefore	therefore	ADV
ejpam-5915	224	2	,	,	PUNCT
ejpam-5915	224	3	µ(g	µ(g	PROPN
ejpam-5915	224	4	·	·	SYM
ejpam-5915	224	5	h	h	X
ejpam-5915	224	6	)	)	PUNCT
ejpam-5915	224	7	≥	≥	NOUN
ejpam-5915	224	8	t1	t1	NOUN
ejpam-5915	224	9	=	=	SYM
ejpam-5915	224	10	min{µ(g	min{µ(g	PROPN
ejpam-5915	224	11	)	)	PUNCT
ejpam-5915	224	12	,	,	PUNCT
ejpam-5915	224	13	µ(h	µ(h	PROPN
ejpam-5915	224	14	)	)	PUNCT
ejpam-5915	224	15	}	}	PUNCT
ejpam-5915	224	16	,	,	PUNCT
ejpam-5915	224	17	satisfying	satisfy	VERB
ejpam-5915	224	18	the	the	DET
ejpam-5915	224	19	closure	closure	NOUN
ejpam-5915	224	20	condition	condition	NOUN
ejpam-5915	224	21	for	for	ADP
ejpam-5915	224	22	fuzzy	fuzzy	ADJ
ejpam-5915	224	23	hom	hom	NOUN
ejpam-5915	224	24	-	-	PUNCT
ejpam-5915	224	25	subgroups	subgroup	NOUN
ejpam-5915	224	26	.	.	PUNCT
ejpam-5915	225	1	•	•	NOUN
ejpam-5915	225	2	closure	closure	NOUN
ejpam-5915	225	3	under	under	ADP
ejpam-5915	225	4	the	the	DET
ejpam-5915	225	5	twisting	twisting	NOUN
ejpam-5915	225	6	map	map	NOUN
ejpam-5915	225	7	α	α	NOUN
ejpam-5915	225	8	:	:	PUNCT
ejpam-5915	225	9	since	since	SCONJ
ejpam-5915	225	10	g	g	PROPN
ejpam-5915	225	11	∈	∈	PROPN
ejpam-5915	225	12	u(µ	u(µ	PROPN
ejpam-5915	225	13	,	,	PUNCT
ejpam-5915	225	14	t1	t1	NOUN
ejpam-5915	225	15	)	)	PUNCT
ejpam-5915	225	16	and	and	CCONJ
ejpam-5915	225	17	u(µ	u(µ	PROPN
ejpam-5915	225	18	,	,	PUNCT
ejpam-5915	225	19	t1	t1	NOUN
ejpam-5915	225	20	)	)	PUNCT
ejpam-5915	225	21	is	be	AUX
ejpam-5915	225	22	closed	close	VERB
ejpam-5915	225	23	under	under	ADP
ejpam-5915	225	24	α	α	NOUN
ejpam-5915	225	25	,	,	PUNCT
ejpam-5915	225	26	we	we	PRON
ejpam-5915	225	27	also	also	ADV
ejpam-5915	225	28	have	have	VERB
ejpam-5915	225	29	α(g	α(g	NUM
ejpam-5915	225	30	)	)	PUNCT
ejpam-5915	225	31	∈	∈	PROPN
ejpam-5915	225	32	u(µ	u(µ	PROPN
ejpam-5915	225	33	,	,	PUNCT
ejpam-5915	225	34	t1	t1	NOUN
ejpam-5915	225	35	)	)	PUNCT
ejpam-5915	225	36	,	,	PUNCT
ejpam-5915	225	37	which	which	PRON
ejpam-5915	225	38	implies	imply	VERB
ejpam-5915	225	39	µ(α(g	µ(α(g	PROPN
ejpam-5915	225	40	)	)	PUNCT
ejpam-5915	225	41	)	)	PUNCT
ejpam-5915	225	42	≥	≥	NOUN
ejpam-5915	225	43	µ(g	µ(g	PROPN
ejpam-5915	225	44	)	)	PUNCT
ejpam-5915	225	45	,	,	PUNCT
ejpam-5915	225	46	satisfying	satisfy	VERB
ejpam-5915	225	47	the	the	DET
ejpam-5915	225	48	compatibility	compatibility	NOUN
ejpam-5915	225	49	with	with	ADP
ejpam-5915	225	50	α	α	NOUN
ejpam-5915	225	51	for	for	ADP
ejpam-5915	225	52	fuzzy	fuzzy	ADJ
ejpam-5915	225	53	hom	hom	NOUN
ejpam-5915	225	54	-	-	PUNCT
ejpam-5915	225	55	subgroups	subgroup	NOUN
ejpam-5915	225	56	.	.	PUNCT
ejpam-5915	226	1	•	•	NOUN
ejpam-5915	226	2	closure	closure	NOUN
ejpam-5915	226	3	under	under	ADP
ejpam-5915	226	4	inverses	inverse	NOUN
ejpam-5915	226	5	:	:	PUNCT
ejpam-5915	226	6	for	for	ADP
ejpam-5915	226	7	any	any	DET
ejpam-5915	226	8	g	g	PROPN
ejpam-5915	226	9	∈	∈	PROPN
ejpam-5915	226	10	u(µ	u(µ	PROPN
ejpam-5915	226	11	,	,	PUNCT
ejpam-5915	226	12	t1	t1	NOUN
ejpam-5915	226	13	)	)	PUNCT
ejpam-5915	226	14	,	,	PUNCT
ejpam-5915	226	15	since	since	SCONJ
ejpam-5915	226	16	u(µ	u(µ	NOUN
ejpam-5915	226	17	,	,	PUNCT
ejpam-5915	226	18	t1	t1	NOUN
ejpam-5915	226	19	)	)	PUNCT
ejpam-5915	226	20	is	be	AUX
ejpam-5915	226	21	closed	close	VERB
ejpam-5915	226	22	under	under	ADP
ejpam-5915	226	23	inverses	inverse	NOUN
ejpam-5915	226	24	,	,	PUNCT
ejpam-5915	226	25	we	we	PRON
ejpam-5915	226	26	have	have	VERB
ejpam-5915	226	27	g−1	g−1	PROPN
ejpam-5915	226	28	∈	∈	PROPN
ejpam-5915	226	29	u(µ	u(µ	NOUN
ejpam-5915	226	30	,	,	PUNCT
ejpam-5915	226	31	t1	t1	NOUN
ejpam-5915	226	32	)	)	PUNCT
ejpam-5915	226	33	,	,	PUNCT
ejpam-5915	226	34	which	which	PRON
ejpam-5915	226	35	ensures	ensure	VERB
ejpam-5915	226	36	that	that	SCONJ
ejpam-5915	226	37	µ(g−1	µ(g−1	X
ejpam-5915	226	38	)	)	PUNCT
ejpam-5915	226	39	≥	≥	NOUN
ejpam-5915	226	40	µ(g	µ(g	PROPN
ejpam-5915	226	41	)	)	PUNCT
ejpam-5915	226	42	,	,	PUNCT
ejpam-5915	226	43	fulfilling	fulfil	VERB
ejpam-5915	226	44	the	the	DET
ejpam-5915	226	45	inverse	inverse	NOUN
ejpam-5915	226	46	condition	condition	NOUN
ejpam-5915	226	47	for	for	ADP
ejpam-5915	226	48	fuzzy	fuzzy	ADJ
ejpam-5915	226	49	hom	hom	NOUN
ejpam-5915	226	50	-	-	PUNCT
ejpam-5915	226	51	subgroups	subgroup	NOUN
ejpam-5915	226	52	.	.	PUNCT
ejpam-5915	227	1	thus	thus	ADV
ejpam-5915	227	2	,	,	PUNCT
ejpam-5915	227	3	by	by	ADP
ejpam-5915	227	4	the	the	DET
ejpam-5915	227	5	properties	property	NOUN
ejpam-5915	227	6	of	of	ADP
ejpam-5915	227	7	each	each	DET
ejpam-5915	227	8	upper	upper	ADJ
ejpam-5915	227	9	level	level	NOUN
ejpam-5915	227	10	set	set	VERB
ejpam-5915	227	11	u(µ	u(µ	PROPN
ejpam-5915	227	12	,	,	PUNCT
ejpam-5915	227	13	t	t	PROPN
ejpam-5915	227	14	)	)	PUNCT
ejpam-5915	227	15	as	as	ADP
ejpam-5915	227	16	hom	hom	NOUN
ejpam-5915	227	17	-	-	PUNCT
ejpam-5915	227	18	subgroups	subgroup	NOUN
ejpam-5915	227	19	,	,	PUNCT
ejpam-5915	227	20	µ	µ	PRON
ejpam-5915	227	21	satisfies	satisfie	NOUN
ejpam-5915	227	22	all	all	DET
ejpam-5915	227	23	the	the	DET
ejpam-5915	227	24	conditions	condition	NOUN
ejpam-5915	227	25	to	to	PART
ejpam-5915	227	26	be	be	AUX
ejpam-5915	227	27	a	a	DET
ejpam-5915	227	28	fuzzy	fuzzy	ADJ
ejpam-5915	227	29	hom	hom	NOUN
ejpam-5915	227	30	-	-	PUNCT
ejpam-5915	227	31	subgroup	subgroup	NOUN
ejpam-5915	227	32	.	.	PUNCT
ejpam-5915	228	1	in	in	ADP
ejpam-5915	228	2	analogy	analogy	NOUN
ejpam-5915	228	3	with	with	ADP
ejpam-5915	228	4	the	the	DET
ejpam-5915	228	5	results	result	NOUN
ejpam-5915	228	6	of	of	ADP
ejpam-5915	228	7	theorem	theorem	NOUN
ejpam-5915	228	8	3	3	NUM
ejpam-5915	228	9	in	in	ADP
ejpam-5915	228	10	[	[	X
ejpam-5915	228	11	20	20	NUM
ejpam-5915	228	12	]	]	PUNCT
ejpam-5915	228	13	,	,	PUNCT
ejpam-5915	228	14	which	which	PRON
ejpam-5915	228	15	addressed	address	VERB
ejpam-5915	228	16	strong	strong	ADJ
ejpam-5915	228	17	upper	upper	ADJ
ejpam-5915	228	18	level	level	NOUN
ejpam-5915	228	19	sets	set	NOUN
ejpam-5915	228	20	in	in	ADP
ejpam-5915	228	21	the	the	DET
ejpam-5915	228	22	context	context	NOUN
ejpam-5915	228	23	of	of	ADP
ejpam-5915	228	24	fuzzy	fuzzy	ADJ
ejpam-5915	228	25	subsets	subset	NOUN
ejpam-5915	228	26	,	,	PUNCT
ejpam-5915	228	27	we	we	PRON
ejpam-5915	228	28	investigate	investigate	VERB
ejpam-5915	228	29	similar	similar	ADJ
ejpam-5915	228	30	conditions	condition	NOUN
ejpam-5915	228	31	within	within	ADP
ejpam-5915	228	32	the	the	DET
ejpam-5915	228	33	framework	framework	NOUN
ejpam-5915	228	34	of	of	ADP
ejpam-5915	228	35	hom	hom	NOUN
ejpam-5915	228	36	-	-	PUNCT
ejpam-5915	228	37	groups	group	NOUN
ejpam-5915	228	38	.	.	PUNCT
ejpam-5915	229	1	by	by	ADP
ejpam-5915	229	2	establishing	establish	VERB
ejpam-5915	229	3	a	a	DET
ejpam-5915	229	4	connection	connection	NOUN
ejpam-5915	229	5	between	between	ADP
ejpam-5915	229	6	strong	strong	ADJ
ejpam-5915	229	7	fuzzy	fuzzy	ADJ
ejpam-5915	229	8	hom	hom	NOUN
ejpam-5915	229	9	-	-	PUNCT
ejpam-5915	229	10	subgroups	subgroup	NOUN
ejpam-5915	229	11	of	of	ADP
ejpam-5915	229	12	g	g	NOUN
ejpam-5915	229	13	and	and	CCONJ
ejpam-5915	229	14	strong	strong	ADJ
ejpam-5915	229	15	hom	hom	NOUN
ejpam-5915	229	16	-	-	PUNCT
ejpam-5915	229	17	subgroups	subgroup	NOUN
ejpam-5915	229	18	of	of	ADP
ejpam-5915	229	19	g	g	NOUN
ejpam-5915	229	20	,	,	PUNCT
ejpam-5915	229	21	we	we	PRON
ejpam-5915	229	22	aim	aim	VERB
ejpam-5915	229	23	to	to	PART
ejpam-5915	229	24	understand	understand	VERB
ejpam-5915	229	25	the	the	DET
ejpam-5915	229	26	interplay	interplay	NOUN
ejpam-5915	229	27	between	between	ADP
ejpam-5915	229	28	fuzzy	fuzzy	ADJ
ejpam-5915	229	29	membership	membership	NOUN
ejpam-5915	229	30	functions	function	NOUN
ejpam-5915	229	31	and	and	CCONJ
ejpam-5915	229	32	subgroup	subgroup	NOUN
ejpam-5915	229	33	properties	property	NOUN
ejpam-5915	229	34	in	in	ADP
ejpam-5915	229	35	hom	hom	ADV
ejpam-5915	229	36	-	-	PUNCT
ejpam-5915	229	37	algebraic	algebraic	ADJ
ejpam-5915	229	38	systems	system	NOUN
ejpam-5915	229	39	.	.	PUNCT
ejpam-5915	230	1	theorem	theorem	NOUN
ejpam-5915	230	2	3	3	X
ejpam-5915	230	3	.	.	PUNCT
ejpam-5915	231	1	let	let	VERB
ejpam-5915	231	2	µ	µ	X
ejpam-5915	231	3	be	be	AUX
ejpam-5915	231	4	a	a	DET
ejpam-5915	231	5	fuzzy	fuzzy	ADJ
ejpam-5915	231	6	subset	subset	NOUN
ejpam-5915	231	7	on	on	ADP
ejpam-5915	231	8	a	a	DET
ejpam-5915	231	9	hom	hom	NOUN
ejpam-5915	231	10	-	-	PUNCT
ejpam-5915	231	11	group	group	NOUN
ejpam-5915	231	12	(	(	PUNCT
ejpam-5915	231	13	g	g	PROPN
ejpam-5915	231	14	,	,	PUNCT
ejpam-5915	231	15	·	·	PUNCT
ejpam-5915	231	16	,	,	PUNCT
ejpam-5915	231	17	α	α	NOUN
ejpam-5915	231	18	)	)	PUNCT
ejpam-5915	231	19	.	.	PUNCT
ejpam-5915	232	1	the	the	DET
ejpam-5915	232	2	following	follow	VERB
ejpam-5915	232	3	statements	statement	NOUN
ejpam-5915	232	4	are	be	AUX
ejpam-5915	232	5	equivalent	equivalent	ADJ
ejpam-5915	232	6	:	:	PUNCT
ejpam-5915	232	7	(	(	PUNCT
ejpam-5915	232	8	i	i	NOUN
ejpam-5915	232	9	)	)	PUNCT
ejpam-5915	232	10	µ	µ	PROPN
ejpam-5915	232	11	is	be	AUX
ejpam-5915	232	12	a	a	DET
ejpam-5915	232	13	strong	strong	ADJ
ejpam-5915	232	14	fuzzy	fuzzy	ADJ
ejpam-5915	232	15	hom	hom	NOUN
ejpam-5915	232	16	-	-	PUNCT
ejpam-5915	232	17	subgroup	subgroup	NOUN
ejpam-5915	232	18	of	of	ADP
ejpam-5915	232	19	g	g	PROPN
ejpam-5915	232	20	,	,	PUNCT
ejpam-5915	232	21	(	(	PUNCT
ejpam-5915	232	22	ii	ii	NOUN
ejpam-5915	232	23	)	)	PUNCT
ejpam-5915	232	24	for	for	ADP
ejpam-5915	232	25	every	every	DET
ejpam-5915	232	26	t	t	NOUN
ejpam-5915	232	27	∈	∈	PROPN
ejpam-5915	232	28	(	(	PUNCT
ejpam-5915	232	29	0	0	NUM
ejpam-5915	232	30	,	,	PUNCT
ejpam-5915	232	31	1	1	NUM
ejpam-5915	232	32	]	]	PUNCT
ejpam-5915	232	33	,	,	PUNCT
ejpam-5915	232	34	the	the	DET
ejpam-5915	232	35	set	set	ADJ
ejpam-5915	232	36	u∗(µ	u∗(µ	PROPN
ejpam-5915	232	37	,	,	PUNCT
ejpam-5915	232	38	t	t	PROPN
ejpam-5915	232	39	)	)	PUNCT
ejpam-5915	232	40	=	=	PRON
ejpam-5915	232	41	{	{	PUNCT
ejpam-5915	232	42	g	g	PROPN
ejpam-5915	232	43	∈	∈	PROPN
ejpam-5915	232	44	g	g	PROPN
ejpam-5915	232	45	|	|	NOUN
ejpam-5915	232	46	µ(g	µ(g	PROPN
ejpam-5915	232	47	)	)	PUNCT
ejpam-5915	232	48	>	>	X
ejpam-5915	232	49	t	t	PROPN
ejpam-5915	232	50	}	}	PUNCT
ejpam-5915	232	51	is	be	AUX
ejpam-5915	232	52	a	a	DET
ejpam-5915	232	53	hom	hom	NOUN
ejpam-5915	232	54	-	-	PUNCT
ejpam-5915	232	55	subgroup	subgroup	NOUN
ejpam-5915	232	56	of	of	ADP
ejpam-5915	232	57	g.	g.	PROPN
ejpam-5915	232	58	proof	proof	PROPN
ejpam-5915	232	59	.	.	PUNCT
ejpam-5915	233	1	we	we	PRON
ejpam-5915	233	2	will	will	AUX
ejpam-5915	233	3	show	show	VERB
ejpam-5915	233	4	the	the	DET
ejpam-5915	233	5	equivalence	equivalence	NOUN
ejpam-5915	233	6	between	between	ADP
ejpam-5915	233	7	(	(	PUNCT
ejpam-5915	233	8	i	i	NOUN
ejpam-5915	233	9	)	)	PUNCT
ejpam-5915	233	10	and	and	CCONJ
ejpam-5915	233	11	(	(	PUNCT
ejpam-5915	233	12	ii	ii	NOUN
ejpam-5915	233	13	)	)	PUNCT
ejpam-5915	233	14	as	as	SCONJ
ejpam-5915	233	15	stated	state	VERB
ejpam-5915	233	16	in	in	ADP
ejpam-5915	233	17	the	the	DET
ejpam-5915	233	18	theorem	theorem	NOUN
ejpam-5915	233	19	.	.	PUNCT
ejpam-5915	234	1	(	(	PUNCT
ejpam-5915	234	2	i	i	NOUN
ejpam-5915	234	3	)	)	PUNCT
ejpam-5915	234	4	⇒	⇒	PROPN
ejpam-5915	234	5	(	(	PUNCT
ejpam-5915	234	6	ii	ii	PROPN
ejpam-5915	234	7	):	):	PUNCT
ejpam-5915	234	8	suppose	suppose	VERB
ejpam-5915	234	9	µ	µ	PRON
ejpam-5915	234	10	is	be	AUX
ejpam-5915	234	11	a	a	DET
ejpam-5915	234	12	fuzzy	fuzzy	ADJ
ejpam-5915	234	13	hom	hom	NOUN
ejpam-5915	234	14	-	-	PUNCT
ejpam-5915	234	15	subgroup	subgroup	NOUN
ejpam-5915	234	16	of	of	ADP
ejpam-5915	234	17	g.	g.	PROPN
ejpam-5915	234	18	for	for	ADP
ejpam-5915	234	19	each	each	DET
ejpam-5915	234	20	t	t	PROPN
ejpam-5915	234	21	∈	∈	PROPN
ejpam-5915	234	22	im(µ	im(µ	ADV
ejpam-5915	234	23	)	)	PUNCT
ejpam-5915	234	24	,	,	PUNCT
ejpam-5915	234	25	consider	consider	VERB
ejpam-5915	234	26	the	the	DET
ejpam-5915	234	27	set	set	ADJ
ejpam-5915	234	28	u(µ	u(µ	PROPN
ejpam-5915	234	29	,	,	PUNCT
ejpam-5915	234	30	t	t	PROPN
ejpam-5915	234	31	)	)	PUNCT
ejpam-5915	234	32	=	=	PRON
ejpam-5915	234	33	{	{	PUNCT
ejpam-5915	234	34	g	g	PROPN
ejpam-5915	234	35	∈	∈	PROPN
ejpam-5915	234	36	g	g	PROPN
ejpam-5915	234	37	|	|	NOUN
ejpam-5915	234	38	µ(g	µ(g	PROPN
ejpam-5915	234	39	)	)	PUNCT
ejpam-5915	234	40	≥	≥	NOUN
ejpam-5915	234	41	t	t	PROPN
ejpam-5915	234	42	}	}	PUNCT
ejpam-5915	234	43	.	.	PUNCT
ejpam-5915	235	1	to	to	PART
ejpam-5915	235	2	prove	prove	VERB
ejpam-5915	235	3	that	that	SCONJ
ejpam-5915	235	4	u(µ	u(µ	NOUN
ejpam-5915	235	5	,	,	PUNCT
ejpam-5915	235	6	t	t	PROPN
ejpam-5915	235	7	)	)	PUNCT
ejpam-5915	235	8	is	be	AUX
ejpam-5915	235	9	a	a	DET
ejpam-5915	235	10	hom	hom	NOUN
ejpam-5915	235	11	-	-	PUNCT
ejpam-5915	235	12	subgroup	subgroup	NOUN
ejpam-5915	235	13	,	,	PUNCT
ejpam-5915	235	14	we	we	PRON
ejpam-5915	235	15	verify	verify	VERB
ejpam-5915	235	16	that	that	SCONJ
ejpam-5915	235	17	it	it	PRON
ejpam-5915	235	18	satisfies	satisfy	VERB
ejpam-5915	235	19	the	the	DET
ejpam-5915	235	20	hom	hom	NOUN
ejpam-5915	235	21	-	-	PUNCT
ejpam-5915	235	22	group	group	NOUN
ejpam-5915	235	23	properties	property	NOUN
ejpam-5915	235	24	:	:	PUNCT
ejpam-5915	235	25	•	•	NOUN
ejpam-5915	235	26	closure	closure	NOUN
ejpam-5915	235	27	under	under	ADP
ejpam-5915	235	28	the	the	DET
ejpam-5915	235	29	operation	operation	NOUN
ejpam-5915	235	30	·	·	PUNCT
ejpam-5915	235	31	:	:	PUNCT
ejpam-5915	235	32	for	for	ADP
ejpam-5915	235	33	any	any	DET
ejpam-5915	235	34	g	g	NOUN
ejpam-5915	235	35	,	,	PUNCT
ejpam-5915	235	36	h	h	NOUN
ejpam-5915	235	37	∈	∈	PROPN
ejpam-5915	235	38	u(µ	u(µ	PROPN
ejpam-5915	235	39	,	,	PUNCT
ejpam-5915	235	40	t	t	PROPN
ejpam-5915	235	41	)	)	PUNCT
ejpam-5915	235	42	,	,	PUNCT
ejpam-5915	235	43	since	since	SCONJ
ejpam-5915	235	44	µ	µ	NOUN
ejpam-5915	235	45	is	be	AUX
ejpam-5915	235	46	a	a	DET
ejpam-5915	235	47	fuzzy	fuzzy	ADJ
ejpam-5915	235	48	homsubgroup	homsubgroup	NOUN
ejpam-5915	235	49	,	,	PUNCT
ejpam-5915	235	50	we	we	PRON
ejpam-5915	235	51	have	have	VERB
ejpam-5915	235	52	:	:	PUNCT
ejpam-5915	235	53	µ(g	µ(g	X
ejpam-5915	235	54	·	·	SYM
ejpam-5915	235	55	h	h	X
ejpam-5915	235	56	)	)	PUNCT
ejpam-5915	235	57	≥	≥	NOUN
ejpam-5915	235	58	min{µ(g	min{µ(g	PROPN
ejpam-5915	235	59	)	)	PUNCT
ejpam-5915	235	60	,	,	PUNCT
ejpam-5915	235	61	µ(h	µ(h	PROPN
ejpam-5915	235	62	)	)	PUNCT
ejpam-5915	235	63	}	}	PUNCT
ejpam-5915	235	64	>	>	PUNCT
ejpam-5915	236	1	t.	t.	NOUN
ejpam-5915	236	2	this	this	PRON
ejpam-5915	236	3	implies	imply	VERB
ejpam-5915	236	4	g	g	PROPN
ejpam-5915	236	5	·	·	PUNCT
ejpam-5915	236	6	h	h	NOUN
ejpam-5915	236	7	∈	∈	PROPN
ejpam-5915	236	8	u(µ	u(µ	PROPN
ejpam-5915	236	9	,	,	PUNCT
ejpam-5915	236	10	t	t	PROPN
ejpam-5915	236	11	)	)	PUNCT
ejpam-5915	236	12	,	,	PUNCT
ejpam-5915	236	13	so	so	SCONJ
ejpam-5915	236	14	u(µ	u(µ	PROPN
ejpam-5915	236	15	,	,	PUNCT
ejpam-5915	236	16	t	t	PROPN
ejpam-5915	236	17	)	)	PUNCT
ejpam-5915	236	18	is	be	AUX
ejpam-5915	236	19	closed	close	VERB
ejpam-5915	236	20	under	under	ADP
ejpam-5915	236	21	·	·	PUNCT
ejpam-5915	236	22	.	.	PUNCT
ejpam-5915	237	1	•	•	NOUN
ejpam-5915	237	2	closure	closure	NOUN
ejpam-5915	237	3	under	under	ADP
ejpam-5915	237	4	the	the	DET
ejpam-5915	237	5	twisting	twisting	NOUN
ejpam-5915	237	6	map	map	NOUN
ejpam-5915	237	7	α	α	NOUN
ejpam-5915	237	8	:	:	PUNCT
ejpam-5915	237	9	for	for	ADP
ejpam-5915	237	10	any	any	DET
ejpam-5915	237	11	g	g	PROPN
ejpam-5915	237	12	∈	∈	PROPN
ejpam-5915	237	13	u(µ	u(µ	PROPN
ejpam-5915	237	14	,	,	PUNCT
ejpam-5915	237	15	t	t	PROPN
ejpam-5915	237	16	)	)	PUNCT
ejpam-5915	237	17	,	,	PUNCT
ejpam-5915	237	18	we	we	PRON
ejpam-5915	237	19	have	have	VERB
ejpam-5915	237	20	µ(α(g	µ(α(g	PROPN
ejpam-5915	237	21	)	)	PUNCT
ejpam-5915	237	22	)	)	PUNCT
ejpam-5915	237	23	≥	≥	NOUN
ejpam-5915	237	24	µ(g	µ(g	PROPN
ejpam-5915	237	25	)	)	PUNCT
ejpam-5915	237	26	>	>	X
ejpam-5915	238	1	t.	t.	PROPN
ejpam-5915	238	2	hence	hence	ADV
ejpam-5915	238	3	,	,	PUNCT
ejpam-5915	238	4	α(g	α(g	NUM
ejpam-5915	238	5	)	)	PUNCT
ejpam-5915	238	6	∈	∈	PROPN
ejpam-5915	238	7	u(µ	u(µ	PROPN
ejpam-5915	238	8	,	,	PUNCT
ejpam-5915	238	9	t	t	PROPN
ejpam-5915	238	10	)	)	PUNCT
ejpam-5915	238	11	,	,	PUNCT
ejpam-5915	238	12	confirming	confirm	VERB
ejpam-5915	238	13	closure	closure	NOUN
ejpam-5915	238	14	under	under	ADP
ejpam-5915	238	15	α	α	NOUN
ejpam-5915	238	16	.	.	NOUN
ejpam-5915	238	17	•	•	NUM
ejpam-5915	238	18	closure	closure	NOUN
ejpam-5915	238	19	under	under	ADP
ejpam-5915	238	20	inverses	inverse	NOUN
ejpam-5915	238	21	:	:	PUNCT
ejpam-5915	238	22	for	for	ADP
ejpam-5915	238	23	any	any	DET
ejpam-5915	238	24	g	g	PROPN
ejpam-5915	238	25	∈	∈	PROPN
ejpam-5915	238	26	u(µ	u(µ	PROPN
ejpam-5915	238	27	,	,	PUNCT
ejpam-5915	238	28	t	t	PROPN
ejpam-5915	238	29	)	)	PUNCT
ejpam-5915	238	30	,	,	PUNCT
ejpam-5915	238	31	since	since	SCONJ
ejpam-5915	238	32	µ(g−1	µ(g−1	ADJ
ejpam-5915	238	33	)	)	PUNCT
ejpam-5915	238	34	≥	≥	NOUN
ejpam-5915	238	35	µ(g	µ(g	PROPN
ejpam-5915	238	36	)	)	PUNCT
ejpam-5915	238	37	>	>	X
ejpam-5915	238	38	t	t	PROPN
ejpam-5915	238	39	,	,	PUNCT
ejpam-5915	238	40	we	we	PRON
ejpam-5915	238	41	have	have	VERB
ejpam-5915	238	42	g−1	g−1	PROPN
ejpam-5915	238	43	∈	∈	PROPN
ejpam-5915	238	44	u(µ	u(µ	PROPN
ejpam-5915	238	45	,	,	PUNCT
ejpam-5915	238	46	t	t	PROPN
ejpam-5915	238	47	)	)	PUNCT
ejpam-5915	238	48	.	.	PUNCT
ejpam-5915	239	1	thus	thus	ADV
ejpam-5915	239	2	,	,	PUNCT
ejpam-5915	239	3	u(µ	u(µ	PROPN
ejpam-5915	239	4	,	,	PUNCT
ejpam-5915	239	5	t	t	PROPN
ejpam-5915	239	6	)	)	PUNCT
ejpam-5915	239	7	is	be	AUX
ejpam-5915	239	8	closed	close	VERB
ejpam-5915	239	9	under	under	ADP
ejpam-5915	239	10	inverses	inverse	NOUN
ejpam-5915	239	11	.	.	PUNCT
ejpam-5915	240	1	s.	s.	PROPN
ejpam-5915	240	2	shaqaqha	shaqaqha	PROPN
ejpam-5915	240	3	/	/	SYM
ejpam-5915	240	4	eur	eur	PROPN
ejpam-5915	240	5	.	.	PUNCT
ejpam-5915	241	1	j.	j.	PROPN
ejpam-5915	241	2	pure	pure	PROPN
ejpam-5915	241	3	appl	appl	PROPN
ejpam-5915	241	4	.	.	PROPN
ejpam-5915	241	5	math	math	PROPN
ejpam-5915	241	6	,	,	PUNCT
ejpam-5915	241	7	18	18	NUM
ejpam-5915	241	8	(	(	PUNCT
ejpam-5915	241	9	2	2	NUM
ejpam-5915	241	10	)	)	PUNCT
ejpam-5915	241	11	(	(	PUNCT
ejpam-5915	241	12	2025	2025	NUM
ejpam-5915	241	13	)	)	PUNCT
ejpam-5915	241	14	,	,	PUNCT
ejpam-5915	241	15	5915	5915	NUM
ejpam-5915	241	16	13	13	NUM
ejpam-5915	241	17	of	of	ADP
ejpam-5915	241	18	16	16	NUM
ejpam-5915	241	19	therefore	therefore	ADV
ejpam-5915	241	20	,	,	PUNCT
ejpam-5915	241	21	u(µ	u(µ	PROPN
ejpam-5915	241	22	,	,	PUNCT
ejpam-5915	241	23	t	t	PROPN
ejpam-5915	241	24	)	)	PUNCT
ejpam-5915	241	25	is	be	AUX
ejpam-5915	241	26	a	a	DET
ejpam-5915	241	27	hom	hom	NOUN
ejpam-5915	241	28	-	-	PUNCT
ejpam-5915	241	29	subgroup	subgroup	NOUN
ejpam-5915	241	30	of	of	ADP
ejpam-5915	241	31	g	g	PROPN
ejpam-5915	241	32	for	for	ADP
ejpam-5915	241	33	each	each	DET
ejpam-5915	241	34	t	t	NOUN
ejpam-5915	241	35	∈	∈	PROPN
ejpam-5915	241	36	im(µ	im(µ	ADV
ejpam-5915	241	37	)	)	PUNCT
ejpam-5915	241	38	.	.	PUNCT
ejpam-5915	242	1	(	(	PUNCT
ejpam-5915	242	2	ii	ii	NOUN
ejpam-5915	242	3	)	)	PUNCT
ejpam-5915	242	4	⇒	⇒	NOUN
ejpam-5915	242	5	(	(	PUNCT
ejpam-5915	242	6	i	i	NOUN
ejpam-5915	242	7	):	):	PUNCT
ejpam-5915	242	8	now	now	ADV
ejpam-5915	242	9	,	,	PUNCT
ejpam-5915	242	10	assume	assume	VERB
ejpam-5915	242	11	that	that	SCONJ
ejpam-5915	242	12	for	for	ADP
ejpam-5915	242	13	each	each	DET
ejpam-5915	242	14	t	t	NOUN
ejpam-5915	242	15	∈	∈	PROPN
ejpam-5915	242	16	im(µ	im(µ	ADV
ejpam-5915	242	17	)	)	PUNCT
ejpam-5915	242	18	,	,	PUNCT
ejpam-5915	242	19	the	the	DET
ejpam-5915	242	20	strict	strict	ADJ
ejpam-5915	242	21	upper	upper	ADJ
ejpam-5915	242	22	-	-	PUNCT
ejpam-5915	242	23	level	level	NOUN
ejpam-5915	242	24	set	set	VERB
ejpam-5915	242	25	u∗(µ	u∗(µ	PROPN
ejpam-5915	242	26	,	,	PUNCT
ejpam-5915	242	27	t	t	PROPN
ejpam-5915	242	28	)	)	PUNCT
ejpam-5915	242	29	=	=	PRON
ejpam-5915	242	30	{	{	PUNCT
ejpam-5915	242	31	g	g	PROPN
ejpam-5915	242	32	∈	∈	PROPN
ejpam-5915	242	33	g	g	PROPN
ejpam-5915	242	34	|	|	NOUN
ejpam-5915	242	35	µ(g	µ(g	PROPN
ejpam-5915	242	36	)	)	PUNCT
ejpam-5915	242	37	>	>	X
ejpam-5915	243	1	t	t	PROPN
ejpam-5915	243	2	}	}	PUNCT
ejpam-5915	243	3	is	be	AUX
ejpam-5915	243	4	a	a	DET
ejpam-5915	243	5	hom	hom	NOUN
ejpam-5915	243	6	-	-	PUNCT
ejpam-5915	243	7	subgroup	subgroup	NOUN
ejpam-5915	243	8	of	of	ADP
ejpam-5915	243	9	g.	g.	PROPN
ejpam-5915	243	10	we	we	PRON
ejpam-5915	243	11	want	want	VERB
ejpam-5915	243	12	to	to	PART
ejpam-5915	243	13	show	show	VERB
ejpam-5915	243	14	that	that	SCONJ
ejpam-5915	243	15	µ	µ	ADJ
ejpam-5915	243	16	satisfies	satisfie	NOUN
ejpam-5915	243	17	the	the	DET
ejpam-5915	243	18	conditions	condition	NOUN
ejpam-5915	243	19	required	require	VERB
ejpam-5915	243	20	for	for	ADP
ejpam-5915	243	21	a	a	DET
ejpam-5915	243	22	fuzzy	fuzzy	ADJ
ejpam-5915	243	23	hom	hom	NOUN
ejpam-5915	243	24	-	-	PUNCT
ejpam-5915	243	25	subgroup	subgroup	NOUN
ejpam-5915	243	26	.	.	PUNCT
ejpam-5915	244	1	•	•	NUM
ejpam-5915	244	2	case	case	NOUN
ejpam-5915	244	3	of	of	ADP
ejpam-5915	244	4	zero	zero	NUM
ejpam-5915	244	5	membership	membership	NOUN
ejpam-5915	244	6	:	:	PUNCT
ejpam-5915	244	7	if	if	SCONJ
ejpam-5915	244	8	µ(g	µ(g	ADP
ejpam-5915	244	9	)	)	PUNCT
ejpam-5915	244	10	=	=	SYM
ejpam-5915	244	11	0	0	NUM
ejpam-5915	244	12	or	or	CCONJ
ejpam-5915	244	13	µ(h	µ(h	NOUN
ejpam-5915	244	14	)	)	PUNCT
ejpam-5915	245	1	=	=	SYM
ejpam-5915	245	2	0	0	NUM
ejpam-5915	245	3	,	,	PUNCT
ejpam-5915	245	4	then	then	ADV
ejpam-5915	245	5	by	by	ADP
ejpam-5915	245	6	definition	definition	NOUN
ejpam-5915	245	7	of	of	ADP
ejpam-5915	245	8	a	a	DET
ejpam-5915	245	9	fuzzy	fuzzy	ADJ
ejpam-5915	245	10	hom	hom	NOUN
ejpam-5915	245	11	-	-	PUNCT
ejpam-5915	245	12	subgroup	subgroup	NOUN
ejpam-5915	245	13	,	,	PUNCT
ejpam-5915	245	14	we	we	PRON
ejpam-5915	245	15	have	have	VERB
ejpam-5915	245	16	µ(g	µ(g	ADP
ejpam-5915	245	17	·	·	SYM
ejpam-5915	245	18	h	h	X
ejpam-5915	245	19	)	)	PUNCT
ejpam-5915	245	20	≥	≥	NOUN
ejpam-5915	245	21	0	0	NUM
ejpam-5915	245	22	=	=	SYM
ejpam-5915	245	23	min{µ(g	min{µ(g	PROPN
ejpam-5915	245	24	)	)	PUNCT
ejpam-5915	245	25	,	,	PUNCT
ejpam-5915	245	26	µ(h	µ(h	PROPN
ejpam-5915	245	27	)	)	PUNCT
ejpam-5915	245	28	}	}	PUNCT
ejpam-5915	245	29	,	,	PUNCT
ejpam-5915	245	30	which	which	PRON
ejpam-5915	245	31	fulfills	fulfill	VERB
ejpam-5915	245	32	the	the	DET
ejpam-5915	245	33	closure	closure	NOUN
ejpam-5915	245	34	condition	condition	NOUN
ejpam-5915	245	35	.	.	PUNCT
ejpam-5915	246	1	•	•	NUM
ejpam-5915	246	2	case	case	NOUN
ejpam-5915	246	3	of	of	ADP
ejpam-5915	246	4	positive	positive	ADJ
ejpam-5915	246	5	membership	membership	NOUN
ejpam-5915	246	6	:	:	PUNCT
ejpam-5915	246	7	assume	assume	VERB
ejpam-5915	246	8	µ(g	µ(g	NOUN
ejpam-5915	246	9	)	)	PUNCT
ejpam-5915	246	10	̸=	̸=	PROPN
ejpam-5915	246	11	0	0	NUM
ejpam-5915	246	12	and	and	CCONJ
ejpam-5915	246	13	µ(h	µ(h	NOUN
ejpam-5915	246	14	)	)	PUNCT
ejpam-5915	246	15	̸=	̸=	NOUN
ejpam-5915	246	16	0	0	NUM
ejpam-5915	246	17	.	.	PUNCT
ejpam-5915	247	1	let	let	VERB
ejpam-5915	247	2	t1	t1	NOUN
ejpam-5915	247	3	=	=	PROPN
ejpam-5915	247	4	min{µ(g	min{µ(g	PROPN
ejpam-5915	247	5	)	)	PUNCT
ejpam-5915	247	6	,	,	PUNCT
ejpam-5915	247	7	µ(h	µ(h	PROPN
ejpam-5915	247	8	)	)	PUNCT
ejpam-5915	247	9	}	}	PUNCT
ejpam-5915	247	10	.	.	PUNCT
ejpam-5915	248	1	then	then	ADV
ejpam-5915	248	2	g	g	PROPN
ejpam-5915	248	3	,	,	PUNCT
ejpam-5915	248	4	h	h	NOUN
ejpam-5915	248	5	∈	∈	PROPN
ejpam-5915	248	6	u∗(µ	u∗(µ	PROPN
ejpam-5915	248	7	,	,	PUNCT
ejpam-5915	248	8	t0	t0	PROPN
ejpam-5915	248	9	)	)	PUNCT
ejpam-5915	248	10	for	for	ADP
ejpam-5915	248	11	some	some	DET
ejpam-5915	248	12	t0	t0	PROPN
ejpam-5915	248	13	<	<	X
ejpam-5915	248	14	t1	t1	PROPN
ejpam-5915	248	15	,	,	PUNCT
ejpam-5915	248	16	since	since	SCONJ
ejpam-5915	248	17	µ(g	µ(g	PROPN
ejpam-5915	248	18	)	)	PUNCT
ejpam-5915	248	19	>	>	X
ejpam-5915	248	20	t0	t0	PROPN
ejpam-5915	248	21	and	and	CCONJ
ejpam-5915	248	22	µ(h	µ(h	PROPN
ejpam-5915	248	23	)	)	PUNCT
ejpam-5915	248	24	>	>	X
ejpam-5915	249	1	t0	t0	PROPN
ejpam-5915	249	2	.	.	PUNCT
ejpam-5915	250	1	since	since	SCONJ
ejpam-5915	250	2	u∗(µ	u∗(µ	PROPN
ejpam-5915	250	3	,	,	PUNCT
ejpam-5915	250	4	t0	t0	PROPN
ejpam-5915	250	5	)	)	PUNCT
ejpam-5915	250	6	is	be	AUX
ejpam-5915	250	7	a	a	DET
ejpam-5915	250	8	hom	hom	NOUN
ejpam-5915	250	9	-	-	PUNCT
ejpam-5915	250	10	subgroup	subgroup	NOUN
ejpam-5915	250	11	of	of	ADP
ejpam-5915	250	12	g	g	PROPN
ejpam-5915	250	13	,	,	PUNCT
ejpam-5915	250	14	it	it	PRON
ejpam-5915	250	15	is	be	AUX
ejpam-5915	250	16	closed	close	VERB
ejpam-5915	250	17	under	under	ADP
ejpam-5915	250	18	the	the	DET
ejpam-5915	250	19	group	group	NOUN
ejpam-5915	250	20	operation	operation	NOUN
ejpam-5915	250	21	.	.	PUNCT
ejpam-5915	251	1	assume	assume	VERB
ejpam-5915	251	2	,	,	PUNCT
ejpam-5915	251	3	for	for	ADP
ejpam-5915	251	4	contradiction	contradiction	NOUN
ejpam-5915	251	5	,	,	PUNCT
ejpam-5915	251	6	that	that	SCONJ
ejpam-5915	251	7	µ(g	µ(g	ADP
ejpam-5915	251	8	·	·	SYM
ejpam-5915	251	9	h	h	NOUN
ejpam-5915	251	10	)	)	PUNCT
ejpam-5915	251	11	<	<	X
ejpam-5915	251	12	t1	t1	PROPN
ejpam-5915	251	13	.	.	PUNCT
ejpam-5915	252	1	let	let	VERB
ejpam-5915	252	2	t0	t0	PRON
ejpam-5915	252	3	be	be	AUX
ejpam-5915	252	4	the	the	DET
ejpam-5915	252	5	greatest	greatest	ADV
ejpam-5915	252	6	lower	low	ADJ
ejpam-5915	252	7	bound	bind	VERB
ejpam-5915	252	8	of	of	ADP
ejpam-5915	252	9	values	value	NOUN
ejpam-5915	252	10	strictly	strictly	ADV
ejpam-5915	252	11	below	below	ADP
ejpam-5915	252	12	t1	t1	PROPN
ejpam-5915	252	13	.	.	PUNCT
ejpam-5915	253	1	since	since	SCONJ
ejpam-5915	253	2	g	g	PROPN
ejpam-5915	253	3	,	,	PUNCT
ejpam-5915	253	4	h	h	PROPN
ejpam-5915	253	5	∈	∈	PROPN
ejpam-5915	253	6	u∗(µ	u∗(µ	PROPN
ejpam-5915	253	7	,	,	PUNCT
ejpam-5915	253	8	t0	t0	PROPN
ejpam-5915	253	9	)	)	PUNCT
ejpam-5915	253	10	,	,	PUNCT
ejpam-5915	253	11	it	it	PRON
ejpam-5915	253	12	follows	follow	VERB
ejpam-5915	253	13	that	that	SCONJ
ejpam-5915	253	14	g	g	PROPN
ejpam-5915	253	15	·	·	PUNCT
ejpam-5915	253	16	h	h	PROPN
ejpam-5915	253	17	∈	∈	PROPN
ejpam-5915	253	18	u∗(µ	u∗(µ	PROPN
ejpam-5915	253	19	,	,	PUNCT
ejpam-5915	253	20	t0	t0	PROPN
ejpam-5915	253	21	)	)	PUNCT
ejpam-5915	253	22	as	as	ADV
ejpam-5915	253	23	well	well	ADV
ejpam-5915	253	24	(	(	PUNCT
ejpam-5915	253	25	by	by	ADP
ejpam-5915	253	26	closure	closure	NOUN
ejpam-5915	253	27	under	under	ADP
ejpam-5915	253	28	the	the	DET
ejpam-5915	253	29	operation	operation	NOUN
ejpam-5915	253	30	)	)	PUNCT
ejpam-5915	253	31	.	.	PUNCT
ejpam-5915	254	1	this	this	PRON
ejpam-5915	254	2	implies	imply	VERB
ejpam-5915	254	3	µ(g	µ(g	PROPN
ejpam-5915	254	4	·	·	SYM
ejpam-5915	254	5	h	h	NOUN
ejpam-5915	254	6	)	)	PUNCT
ejpam-5915	254	7	>	>	X
ejpam-5915	254	8	t0	t0	PROPN
ejpam-5915	254	9	,	,	PUNCT
ejpam-5915	254	10	which	which	PRON
ejpam-5915	254	11	contradicts	contradict	VERB
ejpam-5915	254	12	our	our	PRON
ejpam-5915	254	13	assumption	assumption	NOUN
ejpam-5915	254	14	that	that	SCONJ
ejpam-5915	254	15	µ(g	µ(g	ADP
ejpam-5915	254	16	·	·	SYM
ejpam-5915	254	17	h	h	NOUN
ejpam-5915	254	18	)	)	PUNCT
ejpam-5915	254	19	<	<	X
ejpam-5915	254	20	t1	t1	NOUN
ejpam-5915	254	21	because	because	SCONJ
ejpam-5915	254	22	there	there	PRON
ejpam-5915	254	23	would	would	AUX
ejpam-5915	254	24	be	be	AUX
ejpam-5915	254	25	no	no	DET
ejpam-5915	254	26	value	value	NOUN
ejpam-5915	254	27	between	between	ADP
ejpam-5915	254	28	t0	t0	PROPN
ejpam-5915	254	29	and	and	CCONJ
ejpam-5915	254	30	t1	t1	VERB
ejpam-5915	254	31	that	that	SCONJ
ejpam-5915	254	32	µ(g	µ(g	ADP
ejpam-5915	254	33	·	·	SYM
ejpam-5915	254	34	h	h	X
ejpam-5915	254	35	)	)	PUNCT
ejpam-5915	254	36	could	could	AUX
ejpam-5915	254	37	take	take	VERB
ejpam-5915	254	38	.	.	PUNCT
ejpam-5915	255	1	therefore	therefore	ADV
ejpam-5915	255	2	,	,	PUNCT
ejpam-5915	255	3	µ(g	µ(g	PROPN
ejpam-5915	255	4	·	·	SYM
ejpam-5915	255	5	h	h	X
ejpam-5915	255	6	)	)	PUNCT
ejpam-5915	255	7	≥	≥	NOUN
ejpam-5915	255	8	t1	t1	NOUN
ejpam-5915	255	9	,	,	PUNCT
ejpam-5915	255	10	satisfying	satisfy	VERB
ejpam-5915	255	11	the	the	DET
ejpam-5915	255	12	closure	closure	NOUN
ejpam-5915	255	13	condition	condition	NOUN
ejpam-5915	255	14	.	.	PUNCT
ejpam-5915	256	1	•	•	NUM
ejpam-5915	256	2	compatibility	compatibility	NOUN
ejpam-5915	256	3	with	with	ADP
ejpam-5915	256	4	twisting	twisting	NOUN
ejpam-5915	256	5	map	map	NOUN
ejpam-5915	256	6	α	α	NOUN
ejpam-5915	256	7	:	:	PUNCT
ejpam-5915	256	8	similarly	similarly	ADV
ejpam-5915	256	9	,	,	PUNCT
ejpam-5915	256	10	to	to	PART
ejpam-5915	256	11	show	show	VERB
ejpam-5915	256	12	compatibility	compatibility	NOUN
ejpam-5915	256	13	with	with	ADP
ejpam-5915	256	14	α	α	PROPN
ejpam-5915	256	15	,	,	PUNCT
ejpam-5915	256	16	assume	assume	VERB
ejpam-5915	256	17	for	for	ADP
ejpam-5915	256	18	contradiction	contradiction	NOUN
ejpam-5915	256	19	that	that	PRON
ejpam-5915	256	20	µ(α(g	µ(α(g	PROPN
ejpam-5915	256	21	)	)	PUNCT
ejpam-5915	256	22	)	)	PUNCT
ejpam-5915	257	1	<	<	X
ejpam-5915	257	2	µ(g	µ(g	PROPN
ejpam-5915	257	3	)	)	PUNCT
ejpam-5915	257	4	.	.	PUNCT
ejpam-5915	258	1	then	then	ADV
ejpam-5915	258	2	g	g	PROPN
ejpam-5915	258	3	∈	∈	PROPN
ejpam-5915	258	4	u∗(µ	u∗(µ	PROPN
ejpam-5915	258	5	,	,	PUNCT
ejpam-5915	258	6	t0	t0	PROPN
ejpam-5915	258	7	)	)	PUNCT
ejpam-5915	258	8	would	would	AUX
ejpam-5915	258	9	imply	imply	VERB
ejpam-5915	258	10	α(g	α(g	NUM
ejpam-5915	258	11	)	)	PUNCT
ejpam-5915	258	12	∈	∈	PROPN
ejpam-5915	258	13	u∗(µ	u∗(µ	PROPN
ejpam-5915	258	14	,	,	PUNCT
ejpam-5915	258	15	t0	t0	PROPN
ejpam-5915	258	16	)	)	PUNCT
ejpam-5915	258	17	,	,	PUNCT
ejpam-5915	258	18	so	so	ADV
ejpam-5915	258	19	µ(α(g	µ(α(g	PROPN
ejpam-5915	258	20	)	)	PUNCT
ejpam-5915	258	21	)	)	PUNCT
ejpam-5915	259	1	>	>	X
ejpam-5915	259	2	t0	t0	PROPN
ejpam-5915	259	3	,	,	PUNCT
ejpam-5915	259	4	again	again	ADV
ejpam-5915	259	5	contradicting	contradict	VERB
ejpam-5915	259	6	the	the	DET
ejpam-5915	259	7	assumption	assumption	NOUN
ejpam-5915	259	8	µ(α(g	µ(α(g	PROPN
ejpam-5915	259	9	)	)	PUNCT
ejpam-5915	259	10	)	)	PUNCT
ejpam-5915	260	1	<	<	X
ejpam-5915	260	2	µ(g	µ(g	PROPN
ejpam-5915	260	3	)	)	PUNCT
ejpam-5915	260	4	.	.	PUNCT
ejpam-5915	261	1	thus	thus	ADV
ejpam-5915	261	2	,	,	PUNCT
ejpam-5915	261	3	by	by	ADP
ejpam-5915	261	4	showing	show	VERB
ejpam-5915	261	5	that	that	SCONJ
ejpam-5915	261	6	each	each	DET
ejpam-5915	261	7	strict	strict	ADJ
ejpam-5915	261	8	upper	upper	ADJ
ejpam-5915	261	9	level	level	NOUN
ejpam-5915	261	10	set	set	VERB
ejpam-5915	261	11	u∗(µ	u∗(µ	PROPN
ejpam-5915	261	12	,	,	PUNCT
ejpam-5915	261	13	t	t	PROPN
ejpam-5915	261	14	)	)	PUNCT
ejpam-5915	261	15	is	be	AUX
ejpam-5915	261	16	a	a	DET
ejpam-5915	261	17	hom	hom	NOUN
ejpam-5915	261	18	-	-	PUNCT
ejpam-5915	261	19	subgroup	subgroup	NOUN
ejpam-5915	261	20	,	,	PUNCT
ejpam-5915	261	21	we	we	PRON
ejpam-5915	261	22	conclude	conclude	VERB
ejpam-5915	261	23	that	that	SCONJ
ejpam-5915	261	24	µ	µ	PRON
ejpam-5915	261	25	satisfies	satisfie	NOUN
ejpam-5915	261	26	all	all	DET
ejpam-5915	261	27	conditions	condition	NOUN
ejpam-5915	261	28	to	to	PART
ejpam-5915	261	29	be	be	AUX
ejpam-5915	261	30	a	a	DET
ejpam-5915	261	31	fuzzy	fuzzy	ADJ
ejpam-5915	261	32	hom	hom	NOUN
ejpam-5915	261	33	-	-	PUNCT
ejpam-5915	261	34	subgroup	subgroup	NOUN
ejpam-5915	261	35	.	.	PUNCT
ejpam-5915	262	1	we	we	PRON
ejpam-5915	262	2	aim	aim	VERB
ejpam-5915	262	3	to	to	PART
ejpam-5915	262	4	extend	extend	VERB
ejpam-5915	262	5	the	the	DET
ejpam-5915	262	6	relationship	relationship	NOUN
ejpam-5915	262	7	between	between	ADP
ejpam-5915	262	8	fuzzy	fuzzy	ADJ
ejpam-5915	262	9	subsets	subset	NOUN
ejpam-5915	262	10	and	and	CCONJ
ejpam-5915	262	11	structural	structural	ADJ
ejpam-5915	262	12	subgroups	subgroup	NOUN
ejpam-5915	262	13	for	for	ADP
ejpam-5915	262	14	hom	hom	ADV
ejpam-5915	262	15	-	-	PUNCT
ejpam-5915	262	16	normal	normal	ADJ
ejpam-5915	262	17	subgroups	subgroup	NOUN
ejpam-5915	262	18	,	,	PUNCT
ejpam-5915	262	19	building	build	VERB
ejpam-5915	262	20	on	on	ADP
ejpam-5915	262	21	theorem	theorem	NOUN
ejpam-5915	262	22	2	2	NUM
ejpam-5915	262	23	in	in	ADP
ejpam-5915	262	24	[	[	X
ejpam-5915	262	25	20	20	NUM
ejpam-5915	262	26	]	]	PUNCT
ejpam-5915	262	27	.	.	PUNCT
ejpam-5915	263	1	by	by	ADP
ejpam-5915	263	2	establishing	establish	VERB
ejpam-5915	263	3	the	the	DET
ejpam-5915	263	4	connection	connection	NOUN
ejpam-5915	263	5	between	between	ADP
ejpam-5915	263	6	fuzzy	fuzzy	ADJ
ejpam-5915	263	7	hom	hom	NOUN
ejpam-5915	263	8	-	-	PUNCT
ejpam-5915	263	9	normal	normal	ADJ
ejpam-5915	263	10	subgroups	subgroup	NOUN
ejpam-5915	263	11	of	of	ADP
ejpam-5915	263	12	g	g	NOUN
ejpam-5915	263	13	and	and	CCONJ
ejpam-5915	263	14	classical	classical	ADJ
ejpam-5915	263	15	hom	hom	PUNCT
ejpam-5915	263	16	-	-	PUNCT
ejpam-5915	263	17	normal	normal	ADJ
ejpam-5915	263	18	subgroups	subgroup	NOUN
ejpam-5915	263	19	of	of	ADP
ejpam-5915	263	20	g	g	NOUN
ejpam-5915	263	21	,	,	PUNCT
ejpam-5915	263	22	we	we	PRON
ejpam-5915	263	23	can	can	AUX
ejpam-5915	263	24	deepen	deepen	VERB
ejpam-5915	263	25	our	our	PRON
ejpam-5915	263	26	understanding	understanding	NOUN
ejpam-5915	263	27	of	of	ADP
ejpam-5915	263	28	fuzzy	fuzzy	ADJ
ejpam-5915	263	29	membership	membership	NOUN
ejpam-5915	263	30	functions	function	NOUN
ejpam-5915	263	31	within	within	ADP
ejpam-5915	263	32	the	the	DET
ejpam-5915	263	33	hom	hom	NOUN
ejpam-5915	263	34	-	-	PUNCT
ejpam-5915	263	35	group	group	NOUN
ejpam-5915	263	36	framework	framework	NOUN
ejpam-5915	263	37	.	.	PUNCT
ejpam-5915	264	1	theorem	theorem	VERB
ejpam-5915	264	2	4	4	NUM
ejpam-5915	264	3	.	.	PUNCT
ejpam-5915	265	1	let	let	VERB
ejpam-5915	265	2	µ	µ	X
ejpam-5915	265	3	be	be	AUX
ejpam-5915	265	4	a	a	DET
ejpam-5915	265	5	fuzzy	fuzzy	ADJ
ejpam-5915	265	6	subset	subset	NOUN
ejpam-5915	265	7	on	on	ADP
ejpam-5915	265	8	a	a	DET
ejpam-5915	265	9	hom	hom	NOUN
ejpam-5915	265	10	-	-	PUNCT
ejpam-5915	265	11	group	group	NOUN
ejpam-5915	265	12	(	(	PUNCT
ejpam-5915	265	13	g	g	PROPN
ejpam-5915	265	14	,	,	PUNCT
ejpam-5915	265	15	·	·	PUNCT
ejpam-5915	265	16	,	,	PUNCT
ejpam-5915	265	17	α	α	NOUN
ejpam-5915	265	18	)	)	PUNCT
ejpam-5915	265	19	.	.	PUNCT
ejpam-5915	266	1	the	the	DET
ejpam-5915	266	2	following	follow	VERB
ejpam-5915	266	3	statements	statement	NOUN
ejpam-5915	266	4	are	be	AUX
ejpam-5915	266	5	equivalent	equivalent	ADJ
ejpam-5915	266	6	:	:	PUNCT
ejpam-5915	266	7	(	(	PUNCT
ejpam-5915	266	8	i	i	NOUN
ejpam-5915	266	9	)	)	PUNCT
ejpam-5915	266	10	µ	µ	PROPN
ejpam-5915	266	11	is	be	AUX
ejpam-5915	266	12	a	a	DET
ejpam-5915	266	13	fuzzy	fuzzy	ADJ
ejpam-5915	266	14	hom	hom	NOUN
ejpam-5915	266	15	-	-	PUNCT
ejpam-5915	266	16	normal	normal	ADJ
ejpam-5915	266	17	subgroup	subgroup	NOUN
ejpam-5915	266	18	of	of	ADP
ejpam-5915	266	19	g	g	PROPN
ejpam-5915	266	20	,	,	PUNCT
ejpam-5915	266	21	(	(	PUNCT
ejpam-5915	266	22	ii	ii	NOUN
ejpam-5915	266	23	)	)	PUNCT
ejpam-5915	266	24	for	for	ADP
ejpam-5915	266	25	every	every	DET
ejpam-5915	266	26	t	t	NOUN
ejpam-5915	266	27	∈	∈	PROPN
ejpam-5915	266	28	im(µ	im(µ	PROPN
ejpam-5915	266	29	)	)	PUNCT
ejpam-5915	266	30	,	,	PUNCT
ejpam-5915	266	31	the	the	DET
ejpam-5915	266	32	set	set	NOUN
ejpam-5915	266	33	u(µ	u(µ	PROPN
ejpam-5915	266	34	,	,	PUNCT
ejpam-5915	266	35	t	t	PROPN
ejpam-5915	266	36	)	)	PUNCT
ejpam-5915	266	37	=	=	PRON
ejpam-5915	266	38	{	{	PUNCT
ejpam-5915	266	39	g	g	PROPN
ejpam-5915	266	40	∈	∈	PROPN
ejpam-5915	266	41	g	g	PROPN
ejpam-5915	266	42	|	|	NOUN
ejpam-5915	266	43	µ(g	µ(g	PROPN
ejpam-5915	266	44	)	)	PUNCT
ejpam-5915	266	45	≥	≥	NOUN
ejpam-5915	266	46	t	t	PROPN
ejpam-5915	266	47	}	}	PUNCT
ejpam-5915	266	48	is	be	AUX
ejpam-5915	266	49	a	a	DET
ejpam-5915	266	50	hom	hom	ADV
ejpam-5915	266	51	-	-	PUNCT
ejpam-5915	266	52	normal	normal	ADJ
ejpam-5915	266	53	subgroup	subgroup	NOUN
ejpam-5915	266	54	of	of	ADP
ejpam-5915	266	55	g.	g.	PROPN
ejpam-5915	266	56	proof	proof	PROPN
ejpam-5915	266	57	.	.	PUNCT
ejpam-5915	267	1	(	(	PUNCT
ejpam-5915	267	2	i	i	NOUN
ejpam-5915	267	3	)	)	PUNCT
ejpam-5915	267	4	⇒	⇒	PROPN
ejpam-5915	267	5	(	(	PUNCT
ejpam-5915	267	6	ii	ii	PROPN
ejpam-5915	267	7	):	):	PUNCT
ejpam-5915	267	8	suppose	suppose	VERB
ejpam-5915	267	9	µ	µ	PRON
ejpam-5915	267	10	is	be	AUX
ejpam-5915	267	11	a	a	DET
ejpam-5915	267	12	fuzzy	fuzzy	ADJ
ejpam-5915	267	13	hom	hom	NOUN
ejpam-5915	267	14	-	-	PUNCT
ejpam-5915	267	15	normal	normal	ADJ
ejpam-5915	267	16	subgroup	subgroup	NOUN
ejpam-5915	267	17	of	of	ADP
ejpam-5915	267	18	g.	g.	PROPN
ejpam-5915	267	19	for	for	ADP
ejpam-5915	267	20	each	each	DET
ejpam-5915	267	21	t	t	PROPN
ejpam-5915	267	22	∈	∈	PROPN
ejpam-5915	267	23	im(µ	im(µ	ADV
ejpam-5915	267	24	)	)	PUNCT
ejpam-5915	267	25	,	,	PUNCT
ejpam-5915	267	26	we	we	PRON
ejpam-5915	267	27	need	need	VERB
ejpam-5915	267	28	to	to	PART
ejpam-5915	267	29	show	show	VERB
ejpam-5915	267	30	that	that	SCONJ
ejpam-5915	267	31	u(µ	u(µ	NOUN
ejpam-5915	267	32	,	,	PUNCT
ejpam-5915	267	33	t	t	PROPN
ejpam-5915	267	34	)	)	PUNCT
ejpam-5915	267	35	is	be	AUX
ejpam-5915	267	36	a	a	DET
ejpam-5915	267	37	hom	hom	ADV
ejpam-5915	267	38	-	-	PUNCT
ejpam-5915	267	39	normal	normal	ADJ
ejpam-5915	267	40	subgroup	subgroup	NOUN
ejpam-5915	267	41	of	of	ADP
ejpam-5915	267	42	g.	g.	PROPN
ejpam-5915	267	43	since	since	SCONJ
ejpam-5915	267	44	µ	µ	PROPN
ejpam-5915	267	45	is	be	AUX
ejpam-5915	267	46	a	a	DET
ejpam-5915	267	47	fuzzy	fuzzy	ADJ
ejpam-5915	267	48	hom	hom	NOUN
ejpam-5915	267	49	-	-	PUNCT
ejpam-5915	267	50	normal	normal	ADJ
ejpam-5915	267	51	subgroup	subgroup	NOUN
ejpam-5915	267	52	,	,	PUNCT
ejpam-5915	267	53	we	we	PRON
ejpam-5915	267	54	know	know	VERB
ejpam-5915	267	55	from	from	ADP
ejpam-5915	267	56	the	the	DET
ejpam-5915	267	57	fuzzy	fuzzy	ADJ
ejpam-5915	267	58	hom	hom	NOUN
ejpam-5915	267	59	-	-	PUNCT
ejpam-5915	267	60	subgroup	subgroup	NOUN
ejpam-5915	267	61	conditions	condition	NOUN
ejpam-5915	267	62	that	that	SCONJ
ejpam-5915	267	63	u(µ	u(µ	NOUN
ejpam-5915	267	64	,	,	PUNCT
ejpam-5915	267	65	t	t	PROPN
ejpam-5915	267	66	)	)	PUNCT
ejpam-5915	267	67	satisfies	satisfie	NOUN
ejpam-5915	267	68	closure	closure	NOUN
ejpam-5915	267	69	under	under	ADP
ejpam-5915	267	70	the	the	DET
ejpam-5915	267	71	operation	operation	NOUN
ejpam-5915	267	72	·	·	PUNCT
ejpam-5915	267	73	,	,	PUNCT
ejpam-5915	267	74	closure	closure	NOUN
ejpam-5915	267	75	under	under	ADP
ejpam-5915	267	76	α	α	NOUN
ejpam-5915	267	77	,	,	PUNCT
ejpam-5915	267	78	and	and	CCONJ
ejpam-5915	267	79	closure	closure	NOUN
ejpam-5915	267	80	under	under	ADP
ejpam-5915	267	81	inverses	inverse	NOUN
ejpam-5915	267	82	.	.	PUNCT
ejpam-5915	268	1	to	to	PART
ejpam-5915	268	2	establish	establish	VERB
ejpam-5915	268	3	that	that	SCONJ
ejpam-5915	268	4	u(µ	u(µ	NOUN
ejpam-5915	268	5	,	,	PUNCT
ejpam-5915	268	6	t	t	PROPN
ejpam-5915	268	7	)	)	PUNCT
ejpam-5915	268	8	is	be	AUX
ejpam-5915	268	9	also	also	ADV
ejpam-5915	268	10	normal	normal	ADJ
ejpam-5915	268	11	,	,	PUNCT
ejpam-5915	268	12	we	we	PRON
ejpam-5915	268	13	verify	verify	VERB
ejpam-5915	268	14	:	:	PUNCT
ejpam-5915	268	15	s.	s.	PROPN
ejpam-5915	268	16	shaqaqha	shaqaqha	PROPN
ejpam-5915	268	17	/	/	SYM
ejpam-5915	268	18	eur	eur	PROPN
ejpam-5915	268	19	.	.	PUNCT
ejpam-5915	269	1	j.	j.	PROPN
ejpam-5915	269	2	pure	pure	PROPN
ejpam-5915	269	3	appl	appl	PROPN
ejpam-5915	269	4	.	.	PROPN
ejpam-5915	269	5	math	math	PROPN
ejpam-5915	269	6	,	,	PUNCT
ejpam-5915	269	7	18	18	NUM
ejpam-5915	269	8	(	(	PUNCT
ejpam-5915	269	9	2	2	NUM
ejpam-5915	269	10	)	)	PUNCT
ejpam-5915	269	11	(	(	PUNCT
ejpam-5915	269	12	2025	2025	NUM
ejpam-5915	269	13	)	)	PUNCT
ejpam-5915	269	14	,	,	PUNCT
ejpam-5915	269	15	5915	5915	NUM
ejpam-5915	269	16	14	14	NUM
ejpam-5915	269	17	of	of	ADP
ejpam-5915	269	18	16	16	NUM
ejpam-5915	269	19	invariance	invariance	NOUN
ejpam-5915	269	20	under	under	ADP
ejpam-5915	269	21	conjugation	conjugation	NOUN
ejpam-5915	269	22	:	:	PUNCT
ejpam-5915	269	23	for	for	ADP
ejpam-5915	269	24	any	any	DET
ejpam-5915	269	25	g	g	NOUN
ejpam-5915	269	26	,	,	PUNCT
ejpam-5915	269	27	h	h	NOUN
ejpam-5915	269	28	∈	∈	PROPN
ejpam-5915	269	29	g	g	PROPN
ejpam-5915	269	30	,	,	PUNCT
ejpam-5915	269	31	we	we	PRON
ejpam-5915	269	32	have	have	AUX
ejpam-5915	269	33	µ(g	µ(g	ADP
ejpam-5915	269	34	·	·	PUNCT
ejpam-5915	269	35	h	h	PROPN
ejpam-5915	269	36	·	·	PUNCT
ejpam-5915	269	37	g−1	g−1	PROPN
ejpam-5915	269	38	)	)	PUNCT
ejpam-5915	269	39	≥	≥	NOUN
ejpam-5915	269	40	µ(h	µ(h	PROPN
ejpam-5915	269	41	)	)	PUNCT
ejpam-5915	269	42	.	.	PUNCT
ejpam-5915	270	1	thus	thus	ADV
ejpam-5915	270	2	,	,	PUNCT
ejpam-5915	270	3	if	if	SCONJ
ejpam-5915	270	4	h	h	PROPN
ejpam-5915	270	5	∈	∈	PROPN
ejpam-5915	270	6	u(µ	u(µ	PROPN
ejpam-5915	270	7	,	,	PUNCT
ejpam-5915	270	8	t	t	PROPN
ejpam-5915	270	9	)	)	PUNCT
ejpam-5915	270	10	(	(	PUNCT
ejpam-5915	270	11	i.e.	i.e.	X
ejpam-5915	270	12	,	,	PUNCT
ejpam-5915	270	13	µ(h	µ(h	PROPN
ejpam-5915	270	14	)	)	PUNCT
ejpam-5915	270	15	≥	≥	PROPN
ejpam-5915	270	16	t	t	PROPN
ejpam-5915	270	17	)	)	PUNCT
ejpam-5915	270	18	,	,	PUNCT
ejpam-5915	270	19	it	it	PRON
ejpam-5915	270	20	follows	follow	VERB
ejpam-5915	270	21	that	that	SCONJ
ejpam-5915	270	22	g	g	PROPN
ejpam-5915	270	23	·	·	PUNCT
ejpam-5915	270	24	h	h	NOUN
ejpam-5915	270	25	·	·	PUNCT
ejpam-5915	270	26	g−1	g−1	PROPN
ejpam-5915	270	27	∈	∈	PROPN
ejpam-5915	270	28	u(µ	u(µ	PROPN
ejpam-5915	270	29	,	,	PUNCT
ejpam-5915	270	30	t	t	PROPN
ejpam-5915	270	31	)	)	PUNCT
ejpam-5915	270	32	,	,	PUNCT
ejpam-5915	270	33	ensuring	ensure	VERB
ejpam-5915	270	34	u(µ	u(µ	NOUN
ejpam-5915	270	35	,	,	PUNCT
ejpam-5915	270	36	t	t	PROPN
ejpam-5915	270	37	)	)	PUNCT
ejpam-5915	270	38	is	be	AUX
ejpam-5915	270	39	closed	close	VERB
ejpam-5915	270	40	under	under	ADP
ejpam-5915	270	41	conjugation	conjugation	NOUN
ejpam-5915	270	42	.	.	PUNCT
ejpam-5915	271	1	therefore	therefore	ADV
ejpam-5915	271	2	,	,	PUNCT
ejpam-5915	271	3	u(µ	u(µ	PROPN
ejpam-5915	271	4	,	,	PUNCT
ejpam-5915	271	5	t	t	PROPN
ejpam-5915	271	6	)	)	PUNCT
ejpam-5915	271	7	is	be	AUX
ejpam-5915	271	8	a	a	DET
ejpam-5915	271	9	hom	hom	ADV
ejpam-5915	271	10	-	-	PUNCT
ejpam-5915	271	11	normal	normal	ADJ
ejpam-5915	271	12	subgroup	subgroup	NOUN
ejpam-5915	271	13	of	of	ADP
ejpam-5915	271	14	g	g	PROPN
ejpam-5915	271	15	for	for	ADP
ejpam-5915	271	16	each	each	DET
ejpam-5915	271	17	t	t	NOUN
ejpam-5915	271	18	∈	∈	PROPN
ejpam-5915	271	19	im(µ	im(µ	ADV
ejpam-5915	271	20	)	)	PUNCT
ejpam-5915	271	21	.	.	PUNCT
ejpam-5915	272	1	(	(	PUNCT
ejpam-5915	272	2	ii	ii	NOUN
ejpam-5915	272	3	)	)	PUNCT
ejpam-5915	272	4	⇒	⇒	NOUN
ejpam-5915	272	5	(	(	PUNCT
ejpam-5915	272	6	i	i	NOUN
ejpam-5915	272	7	):	):	PUNCT
ejpam-5915	272	8	conversely	conversely	ADV
ejpam-5915	272	9	,	,	PUNCT
ejpam-5915	272	10	assume	assume	VERB
ejpam-5915	272	11	that	that	SCONJ
ejpam-5915	272	12	each	each	DET
ejpam-5915	272	13	u(µ	u(µ	PROPN
ejpam-5915	272	14	,	,	PUNCT
ejpam-5915	272	15	t	t	PROPN
ejpam-5915	272	16	)	)	PUNCT
ejpam-5915	272	17	is	be	AUX
ejpam-5915	272	18	a	a	DET
ejpam-5915	272	19	hom	hom	ADV
ejpam-5915	272	20	-	-	PUNCT
ejpam-5915	272	21	normal	normal	ADJ
ejpam-5915	272	22	subgroup	subgroup	NOUN
ejpam-5915	272	23	of	of	ADP
ejpam-5915	272	24	g	g	PROPN
ejpam-5915	272	25	for	for	ADP
ejpam-5915	272	26	every	every	DET
ejpam-5915	272	27	t	t	NOUN
ejpam-5915	272	28	∈	∈	PROPN
ejpam-5915	272	29	im(µ	im(µ	ADV
ejpam-5915	272	30	)	)	PUNCT
ejpam-5915	272	31	.	.	PUNCT
ejpam-5915	273	1	we	we	PRON
ejpam-5915	273	2	want	want	VERB
ejpam-5915	273	3	to	to	PART
ejpam-5915	273	4	show	show	VERB
ejpam-5915	273	5	that	that	SCONJ
ejpam-5915	273	6	µ	µ	ADJ
ejpam-5915	273	7	satisfies	satisfie	NOUN
ejpam-5915	273	8	the	the	DET
ejpam-5915	273	9	properties	property	NOUN
ejpam-5915	273	10	of	of	ADP
ejpam-5915	273	11	a	a	DET
ejpam-5915	273	12	fuzzy	fuzzy	ADJ
ejpam-5915	273	13	hom	hom	NOUN
ejpam-5915	273	14	-	-	PUNCT
ejpam-5915	273	15	normal	normal	ADJ
ejpam-5915	273	16	subgroup	subgroup	NOUN
ejpam-5915	273	17	.	.	PUNCT
ejpam-5915	274	1	since	since	SCONJ
ejpam-5915	274	2	each	each	DET
ejpam-5915	274	3	u(µ	u(µ	PROPN
ejpam-5915	274	4	,	,	PUNCT
ejpam-5915	274	5	t	t	PROPN
ejpam-5915	274	6	)	)	PUNCT
ejpam-5915	274	7	is	be	AUX
ejpam-5915	274	8	a	a	DET
ejpam-5915	274	9	hom	hom	NOUN
ejpam-5915	274	10	-	-	PUNCT
ejpam-5915	274	11	subgroup	subgroup	NOUN
ejpam-5915	274	12	,	,	PUNCT
ejpam-5915	274	13	we	we	PRON
ejpam-5915	274	14	know	know	VERB
ejpam-5915	274	15	from	from	ADP
ejpam-5915	274	16	the	the	DET
ejpam-5915	274	17	previous	previous	ADJ
ejpam-5915	274	18	case	case	NOUN
ejpam-5915	274	19	that	that	SCONJ
ejpam-5915	274	20	µ	µ	PRON
ejpam-5915	274	21	satisfies	satisfy	VERB
ejpam-5915	274	22	the	the	DET
ejpam-5915	274	23	closure	closure	NOUN
ejpam-5915	274	24	conditions	condition	NOUN
ejpam-5915	274	25	for	for	ADP
ejpam-5915	274	26	a	a	DET
ejpam-5915	274	27	fuzzy	fuzzy	ADJ
ejpam-5915	274	28	hom	hom	NOUN
ejpam-5915	274	29	-	-	PUNCT
ejpam-5915	274	30	subgroup	subgroup	NOUN
ejpam-5915	274	31	.	.	PUNCT
ejpam-5915	275	1	to	to	PART
ejpam-5915	275	2	verify	verify	VERB
ejpam-5915	275	3	the	the	DET
ejpam-5915	275	4	additional	additional	ADJ
ejpam-5915	275	5	normality	normality	NOUN
ejpam-5915	275	6	condition	condition	NOUN
ejpam-5915	275	7	:	:	PUNCT
ejpam-5915	275	8	invariance	invariance	NOUN
ejpam-5915	275	9	under	under	ADP
ejpam-5915	275	10	conjugation	conjugation	NOUN
ejpam-5915	275	11	:	:	PUNCT
ejpam-5915	275	12	take	take	VERB
ejpam-5915	275	13	any	any	DET
ejpam-5915	275	14	g	g	NOUN
ejpam-5915	275	15	,	,	PUNCT
ejpam-5915	275	16	h	h	NOUN
ejpam-5915	275	17	∈	∈	PROPN
ejpam-5915	275	18	g	g	PROPN
ejpam-5915	275	19	and	and	CCONJ
ejpam-5915	275	20	let	let	VERB
ejpam-5915	275	21	t1	t1	NOUN
ejpam-5915	275	22	=	=	PROPN
ejpam-5915	275	23	min{µ(g	min{µ(g	PROPN
ejpam-5915	275	24	)	)	PUNCT
ejpam-5915	275	25	,	,	PUNCT
ejpam-5915	275	26	µ(h	µ(h	PROPN
ejpam-5915	275	27	)	)	PUNCT
ejpam-5915	275	28	}	}	PUNCT
ejpam-5915	275	29	.	.	PUNCT
ejpam-5915	276	1	then	then	ADV
ejpam-5915	276	2	g	g	PROPN
ejpam-5915	276	3	,	,	PUNCT
ejpam-5915	276	4	h	h	NOUN
ejpam-5915	276	5	∈	∈	PROPN
ejpam-5915	276	6	u(µ	u(µ	PROPN
ejpam-5915	276	7	,	,	PUNCT
ejpam-5915	276	8	t1	t1	NOUN
ejpam-5915	276	9	)	)	PUNCT
ejpam-5915	276	10	,	,	PUNCT
ejpam-5915	276	11	and	and	CCONJ
ejpam-5915	276	12	because	because	SCONJ
ejpam-5915	276	13	u(µ	u(µ	NOUN
ejpam-5915	276	14	,	,	PUNCT
ejpam-5915	276	15	t1	t1	NOUN
ejpam-5915	276	16	)	)	PUNCT
ejpam-5915	276	17	is	be	AUX
ejpam-5915	276	18	a	a	DET
ejpam-5915	276	19	hom	hom	ADV
ejpam-5915	276	20	-	-	PUNCT
ejpam-5915	276	21	normal	normal	ADJ
ejpam-5915	276	22	subgroup	subgroup	NOUN
ejpam-5915	276	23	,	,	PUNCT
ejpam-5915	276	24	it	it	PRON
ejpam-5915	276	25	follows	follow	VERB
ejpam-5915	276	26	that	that	SCONJ
ejpam-5915	276	27	g	g	PROPN
ejpam-5915	276	28	·	·	PUNCT
ejpam-5915	276	29	h	h	NOUN
ejpam-5915	276	30	·	·	PUNCT
ejpam-5915	276	31	g−1	g−1	PROPN
ejpam-5915	276	32	∈	∈	PROPN
ejpam-5915	276	33	u(µ	u(µ	PROPN
ejpam-5915	276	34	,	,	PUNCT
ejpam-5915	276	35	t1	t1	NOUN
ejpam-5915	276	36	)	)	PUNCT
ejpam-5915	276	37	,	,	PUNCT
ejpam-5915	276	38	which	which	PRON
ejpam-5915	276	39	implies	imply	VERB
ejpam-5915	276	40	µ(g	µ(g	PROPN
ejpam-5915	276	41	·	·	PUNCT
ejpam-5915	276	42	h	h	PROPN
ejpam-5915	276	43	·	·	PUNCT
ejpam-5915	276	44	g−1	g−1	PROPN
ejpam-5915	276	45	)	)	PUNCT
ejpam-5915	276	46	≥	≥	NOUN
ejpam-5915	276	47	µ(h	µ(h	PROPN
ejpam-5915	276	48	)	)	PUNCT
ejpam-5915	276	49	.	.	PUNCT
ejpam-5915	277	1	this	this	PRON
ejpam-5915	277	2	shows	show	VERB
ejpam-5915	277	3	that	that	SCONJ
ejpam-5915	277	4	µ	µ	DET
ejpam-5915	277	5	satisfies	satisfie	NOUN
ejpam-5915	277	6	the	the	DET
ejpam-5915	277	7	conjugation	conjugation	NOUN
ejpam-5915	277	8	invariance	invariance	NOUN
ejpam-5915	277	9	required	require	VERB
ejpam-5915	277	10	for	for	ADP
ejpam-5915	277	11	a	a	DET
ejpam-5915	277	12	fuzzy	fuzzy	ADJ
ejpam-5915	277	13	hom	hom	NOUN
ejpam-5915	277	14	-	-	PUNCT
ejpam-5915	277	15	normal	normal	ADJ
ejpam-5915	277	16	subgroup	subgroup	NOUN
ejpam-5915	277	17	.	.	PUNCT
ejpam-5915	278	1	in	in	ADP
ejpam-5915	278	2	analogy	analogy	NOUN
ejpam-5915	278	3	with	with	ADP
ejpam-5915	278	4	the	the	DET
ejpam-5915	278	5	results	result	NOUN
ejpam-5915	278	6	established	establish	VERB
ejpam-5915	278	7	for	for	ADP
ejpam-5915	278	8	strong	strong	ADJ
ejpam-5915	278	9	fuzzy	fuzzy	ADJ
ejpam-5915	278	10	hom	hom	NOUN
ejpam-5915	278	11	-	-	PUNCT
ejpam-5915	278	12	subgroups	subgroup	NOUN
ejpam-5915	278	13	in	in	ADP
ejpam-5915	278	14	theorem	theorem	NOUN
ejpam-5915	278	15	3	3	NUM
ejpam-5915	278	16	,	,	PUNCT
ejpam-5915	278	17	we	we	PRON
ejpam-5915	278	18	extend	extend	VERB
ejpam-5915	278	19	our	our	PRON
ejpam-5915	278	20	investigation	investigation	NOUN
ejpam-5915	278	21	to	to	ADP
ejpam-5915	278	22	strong	strong	ADJ
ejpam-5915	278	23	fuzzy	fuzzy	ADJ
ejpam-5915	278	24	hom	hom	NOUN
ejpam-5915	278	25	-	-	PUNCT
ejpam-5915	278	26	normal	normal	ADJ
ejpam-5915	278	27	subgroups	subgroup	NOUN
ejpam-5915	278	28	.	.	PUNCT
ejpam-5915	279	1	the	the	DET
ejpam-5915	279	2	proof	proof	NOUN
ejpam-5915	279	3	of	of	ADP
ejpam-5915	279	4	the	the	DET
ejpam-5915	279	5	following	follow	VERB
ejpam-5915	279	6	theorem	theorem	NOUN
ejpam-5915	279	7	follows	follow	VERB
ejpam-5915	279	8	the	the	DET
ejpam-5915	279	9	same	same	ADJ
ejpam-5915	279	10	reasoning	reasoning	NOUN
ejpam-5915	279	11	and	and	CCONJ
ejpam-5915	279	12	structure	structure	NOUN
ejpam-5915	279	13	as	as	ADP
ejpam-5915	279	14	that	that	PRON
ejpam-5915	279	15	for	for	ADP
ejpam-5915	279	16	strong	strong	ADJ
ejpam-5915	279	17	fuzzy	fuzzy	ADJ
ejpam-5915	279	18	hom	hom	NOUN
ejpam-5915	279	19	-	-	PUNCT
ejpam-5915	279	20	subgroups	subgroup	NOUN
ejpam-5915	279	21	,	,	PUNCT
ejpam-5915	279	22	with	with	ADP
ejpam-5915	279	23	the	the	DET
ejpam-5915	279	24	only	only	ADJ
ejpam-5915	279	25	additional	additional	ADJ
ejpam-5915	279	26	consideration	consideration	NOUN
ejpam-5915	279	27	being	be	AUX
ejpam-5915	279	28	the	the	DET
ejpam-5915	279	29	normality	normality	NOUN
ejpam-5915	279	30	condition	condition	NOUN
ejpam-5915	279	31	.	.	PUNCT
ejpam-5915	280	1	therefore	therefore	ADV
ejpam-5915	280	2	,	,	PUNCT
ejpam-5915	280	3	we	we	PRON
ejpam-5915	280	4	omit	omit	VERB
ejpam-5915	280	5	repetitive	repetitive	ADJ
ejpam-5915	280	6	details	detail	NOUN
ejpam-5915	280	7	and	and	CCONJ
ejpam-5915	280	8	focus	focus	VERB
ejpam-5915	280	9	only	only	ADV
ejpam-5915	280	10	on	on	ADP
ejpam-5915	280	11	the	the	DET
ejpam-5915	280	12	modifications	modification	NOUN
ejpam-5915	280	13	necessary	necessary	ADJ
ejpam-5915	280	14	for	for	ADP
ejpam-5915	280	15	the	the	DET
ejpam-5915	280	16	hom	hom	ADV
ejpam-5915	280	17	-	-	PUNCT
ejpam-5915	280	18	normal	normal	ADJ
ejpam-5915	280	19	subgroup	subgroup	NOUN
ejpam-5915	280	20	case	case	NOUN
ejpam-5915	280	21	.	.	PUNCT
ejpam-5915	281	1	theorem	theorem	NOUN
ejpam-5915	281	2	5	5	NUM
ejpam-5915	281	3	.	.	PUNCT
ejpam-5915	282	1	let	let	VERB
ejpam-5915	282	2	µ	µ	X
ejpam-5915	282	3	be	be	AUX
ejpam-5915	282	4	a	a	DET
ejpam-5915	282	5	fuzzy	fuzzy	ADJ
ejpam-5915	282	6	subset	subset	NOUN
ejpam-5915	282	7	on	on	ADP
ejpam-5915	282	8	a	a	DET
ejpam-5915	282	9	hom	hom	NOUN
ejpam-5915	282	10	-	-	PUNCT
ejpam-5915	282	11	group	group	NOUN
ejpam-5915	282	12	(	(	PUNCT
ejpam-5915	282	13	g	g	PROPN
ejpam-5915	282	14	,	,	PUNCT
ejpam-5915	282	15	·	·	PUNCT
ejpam-5915	282	16	,	,	PUNCT
ejpam-5915	282	17	α	α	NOUN
ejpam-5915	282	18	)	)	PUNCT
ejpam-5915	282	19	.	.	PUNCT
ejpam-5915	283	1	the	the	DET
ejpam-5915	283	2	following	follow	VERB
ejpam-5915	283	3	statements	statement	NOUN
ejpam-5915	283	4	are	be	AUX
ejpam-5915	283	5	equivalent	equivalent	ADJ
ejpam-5915	283	6	:	:	PUNCT
ejpam-5915	283	7	(	(	PUNCT
ejpam-5915	283	8	i	i	NOUN
ejpam-5915	283	9	)	)	PUNCT
ejpam-5915	283	10	µ	µ	PROPN
ejpam-5915	283	11	is	be	AUX
ejpam-5915	283	12	a	a	DET
ejpam-5915	283	13	strong	strong	ADJ
ejpam-5915	283	14	fuzzy	fuzzy	ADJ
ejpam-5915	283	15	hom	hom	NOUN
ejpam-5915	283	16	-	-	ADJ
ejpam-5915	283	17	normal	normal	ADJ
ejpam-5915	283	18	subgroup	subgroup	NOUN
ejpam-5915	283	19	of	of	ADP
ejpam-5915	283	20	g	g	PROPN
ejpam-5915	283	21	,	,	PUNCT
ejpam-5915	283	22	(	(	PUNCT
ejpam-5915	283	23	ii	ii	NOUN
ejpam-5915	283	24	)	)	PUNCT
ejpam-5915	283	25	for	for	ADP
ejpam-5915	283	26	every	every	DET
ejpam-5915	283	27	t	t	NOUN
ejpam-5915	283	28	∈	∈	PROPN
ejpam-5915	283	29	(	(	PUNCT
ejpam-5915	283	30	0	0	NUM
ejpam-5915	283	31	,	,	PUNCT
ejpam-5915	283	32	1	1	NUM
ejpam-5915	283	33	]	]	PUNCT
ejpam-5915	283	34	,	,	PUNCT
ejpam-5915	283	35	the	the	DET
ejpam-5915	283	36	set	set	ADJ
ejpam-5915	283	37	u∗(µ	u∗(µ	PROPN
ejpam-5915	283	38	,	,	PUNCT
ejpam-5915	283	39	t	t	PROPN
ejpam-5915	283	40	)	)	PUNCT
ejpam-5915	283	41	=	=	PRON
ejpam-5915	283	42	{	{	PUNCT
ejpam-5915	283	43	g	g	PROPN
ejpam-5915	283	44	∈	∈	PROPN
ejpam-5915	283	45	g	g	PROPN
ejpam-5915	283	46	|	|	NOUN
ejpam-5915	283	47	µ(g	µ(g	PROPN
ejpam-5915	283	48	)	)	PUNCT
ejpam-5915	283	49	>	>	X
ejpam-5915	283	50	t	t	PROPN
ejpam-5915	283	51	}	}	PUNCT
ejpam-5915	283	52	is	be	AUX
ejpam-5915	283	53	a	a	DET
ejpam-5915	283	54	hom	hom	ADV
ejpam-5915	283	55	-	-	PUNCT
ejpam-5915	283	56	normal	normal	ADJ
ejpam-5915	283	57	subgroup	subgroup	NOUN
ejpam-5915	283	58	of	of	ADP
ejpam-5915	283	59	g	g	PROPN
ejpam-5915	283	60	,	,	PUNCT
ejpam-5915	283	61	with	with	ADP
ejpam-5915	283	62	equality	equality	NOUN
ejpam-5915	283	63	in	in	ADP
ejpam-5915	283	64	the	the	DET
ejpam-5915	283	65	conjugation	conjugation	NOUN
ejpam-5915	283	66	condition	condition	NOUN
ejpam-5915	283	67	.	.	PUNCT
ejpam-5915	284	1	5	5	X
ejpam-5915	284	2	.	.	X
ejpam-5915	284	3	conclusions	conclusion	NOUN
ejpam-5915	284	4	this	this	DET
ejpam-5915	284	5	paper	paper	NOUN
ejpam-5915	284	6	introduced	introduce	VERB
ejpam-5915	284	7	the	the	DET
ejpam-5915	284	8	concept	concept	NOUN
ejpam-5915	284	9	of	of	ADP
ejpam-5915	284	10	fuzzy	fuzzy	ADJ
ejpam-5915	284	11	hom	hom	NOUN
ejpam-5915	284	12	-	-	PUNCT
ejpam-5915	284	13	groups	group	NOUN
ejpam-5915	284	14	,	,	PUNCT
ejpam-5915	284	15	combining	combine	VERB
ejpam-5915	284	16	fuzzy	fuzzy	ADJ
ejpam-5915	284	17	set	set	NOUN
ejpam-5915	284	18	theory	theory	NOUN
ejpam-5915	284	19	with	with	ADP
ejpam-5915	284	20	the	the	DET
ejpam-5915	284	21	structural	structural	ADJ
ejpam-5915	284	22	flexibility	flexibility	NOUN
ejpam-5915	284	23	of	of	ADP
ejpam-5915	284	24	hom	hom	NOUN
ejpam-5915	284	25	-	-	PUNCT
ejpam-5915	284	26	groups	group	NOUN
ejpam-5915	284	27	.	.	PUNCT
ejpam-5915	285	1	by	by	ADP
ejpam-5915	285	2	defining	define	VERB
ejpam-5915	285	3	fuzzy	fuzzy	ADJ
ejpam-5915	285	4	hom	hom	NOUN
ejpam-5915	285	5	-	-	PUNCT
ejpam-5915	285	6	subgroups	subgroup	NOUN
ejpam-5915	285	7	and	and	CCONJ
ejpam-5915	285	8	fuzzy	fuzzy	ADJ
ejpam-5915	285	9	hom	hom	NOUN
ejpam-5915	285	10	-	-	PUNCT
ejpam-5915	285	11	normal	normal	ADJ
ejpam-5915	285	12	subgroups	subgroup	NOUN
ejpam-5915	285	13	,	,	PUNCT
ejpam-5915	285	14	we	we	PRON
ejpam-5915	285	15	extended	extend	VERB
ejpam-5915	285	16	classical	classical	ADJ
ejpam-5915	285	17	fuzzy	fuzzy	ADJ
ejpam-5915	285	18	group	group	NOUN
ejpam-5915	285	19	theory	theory	NOUN
ejpam-5915	285	20	to	to	ADP
ejpam-5915	285	21	the	the	DET
ejpam-5915	285	22	hom	hom	ADV
ejpam-5915	285	23	-	-	PUNCT
ejpam-5915	285	24	algebraic	algebraic	ADJ
ejpam-5915	285	25	setting	setting	NOUN
ejpam-5915	285	26	,	,	PUNCT
ejpam-5915	285	27	where	where	SCONJ
ejpam-5915	285	28	the	the	DET
ejpam-5915	285	29	twisting	twisting	NOUN
ejpam-5915	285	30	map	map	NOUN
ejpam-5915	285	31	α	α	NOUN
ejpam-5915	285	32	modifies	modify	VERB
ejpam-5915	285	33	associativity	associativity	NOUN
ejpam-5915	285	34	and	and	CCONJ
ejpam-5915	285	35	identity	identity	NOUN
ejpam-5915	285	36	conditions	condition	NOUN
ejpam-5915	285	37	.	.	PUNCT
ejpam-5915	286	1	we	we	PRON
ejpam-5915	286	2	established	establish	VERB
ejpam-5915	286	3	key	key	ADJ
ejpam-5915	286	4	properties	property	NOUN
ejpam-5915	286	5	,	,	PUNCT
ejpam-5915	286	6	including	include	VERB
ejpam-5915	286	7	structural	structural	ADJ
ejpam-5915	286	8	behaviors	behavior	NOUN
ejpam-5915	286	9	and	and	CCONJ
ejpam-5915	286	10	the	the	DET
ejpam-5915	286	11	connection	connection	NOUN
ejpam-5915	286	12	between	between	ADP
ejpam-5915	286	13	fuzzy	fuzzy	ADJ
ejpam-5915	286	14	hom	hom	NOUN
ejpam-5915	286	15	-	-	PUNCT
ejpam-5915	286	16	subgroups	subgroup	NOUN
ejpam-5915	286	17	and	and	CCONJ
ejpam-5915	286	18	upper	upper	ADJ
ejpam-5915	286	19	-	-	PUNCT
ejpam-5915	286	20	level	level	NOUN
ejpam-5915	286	21	sets	set	NOUN
ejpam-5915	286	22	of	of	ADP
ejpam-5915	286	23	classical	classical	ADJ
ejpam-5915	286	24	hom	hom	NOUN
ejpam-5915	286	25	-	-	PUNCT
ejpam-5915	286	26	subgroups	subgroup	NOUN
ejpam-5915	286	27	.	.	PUNCT
ejpam-5915	287	1	our	our	PRON
ejpam-5915	287	2	results	result	NOUN
ejpam-5915	287	3	contribute	contribute	VERB
ejpam-5915	287	4	to	to	ADP
ejpam-5915	287	5	the	the	DET
ejpam-5915	287	6	growing	grow	VERB
ejpam-5915	287	7	research	research	NOUN
ejpam-5915	287	8	on	on	ADP
ejpam-5915	287	9	fuzzy	fuzzy	ADJ
ejpam-5915	287	10	algebraic	algebraic	ADJ
ejpam-5915	287	11	systems	system	NOUN
ejpam-5915	287	12	and	and	CCONJ
ejpam-5915	287	13	homalgebras	homalgebra	NOUN
ejpam-5915	287	14	,	,	PUNCT
ejpam-5915	287	15	building	build	VERB
ejpam-5915	287	16	on	on	ADP
ejpam-5915	287	17	rosenfeld	rosenfeld	PROPN
ejpam-5915	287	18	’s	’s	PART
ejpam-5915	287	19	fuzzy	fuzzy	ADJ
ejpam-5915	287	20	groups	group	NOUN
ejpam-5915	287	21	[	[	X
ejpam-5915	287	22	2	2	X
ejpam-5915	287	23	]	]	PUNCT
ejpam-5915	287	24	and	and	CCONJ
ejpam-5915	287	25	chen	chen	PROPN
ejpam-5915	287	26	et	et	PROPN
ejpam-5915	287	27	al	al	PROPN
ejpam-5915	287	28	.	.	PROPN
ejpam-5915	287	29	’s	’s	PART
ejpam-5915	287	30	hom	hom	NOUN
ejpam-5915	287	31	-	-	PUNCT
ejpam-5915	287	32	groups	group	NOUN
ejpam-5915	287	33	[	[	X
ejpam-5915	287	34	14	14	NUM
ejpam-5915	287	35	]	]	PUNCT
ejpam-5915	287	36	.	.	PUNCT
ejpam-5915	288	1	the	the	DET
ejpam-5915	288	2	examples	example	NOUN
ejpam-5915	288	3	provided	provide	VERB
ejpam-5915	288	4	illustrate	illustrate	VERB
ejpam-5915	288	5	how	how	SCONJ
ejpam-5915	288	6	fuzzy	fuzzy	ADJ
ejpam-5915	288	7	hom	hom	NOUN
ejpam-5915	288	8	-	-	PUNCT
ejpam-5915	288	9	subgroups	subgroup	NOUN
ejpam-5915	288	10	behave	behave	VERB
ejpam-5915	288	11	under	under	ADP
ejpam-5915	288	12	different	different	ADJ
ejpam-5915	288	13	algebraic	algebraic	ADJ
ejpam-5915	288	14	conditions	condition	NOUN
ejpam-5915	288	15	.	.	PUNCT
ejpam-5915	289	1	in	in	ADP
ejpam-5915	289	2	particular	particular	ADJ
ejpam-5915	289	3	,	,	PUNCT
ejpam-5915	289	4	the	the	DET
ejpam-5915	289	5	generalization	generalization	NOUN
ejpam-5915	289	6	using	use	VERB
ejpam-5915	289	7	a	a	DET
ejpam-5915	289	8	parameter	parameter	NOUN
ejpam-5915	289	9	k	k	PROPN
ejpam-5915	289	10	̸=	̸=	PROPN
ejpam-5915	289	11	0	0	NUM
ejpam-5915	289	12	demonstrates	demonstrate	VERB
ejpam-5915	289	13	the	the	DET
ejpam-5915	289	14	adaptability	adaptability	NOUN
ejpam-5915	289	15	of	of	ADP
ejpam-5915	289	16	fuzzy	fuzzy	ADJ
ejpam-5915	289	17	hom	hom	NOUN
ejpam-5915	289	18	-	-	PUNCT
ejpam-5915	289	19	group	group	NOUN
ejpam-5915	289	20	structures	structure	NOUN
ejpam-5915	289	21	beyond	beyond	ADP
ejpam-5915	289	22	previously	previously	ADV
ejpam-5915	289	23	known	know	VERB
ejpam-5915	289	24	cases	case	NOUN
ejpam-5915	289	25	.	.	PUNCT
ejpam-5915	290	1	future	future	ADJ
ejpam-5915	290	2	research	research	NOUN
ejpam-5915	290	3	directions	direction	NOUN
ejpam-5915	290	4	include	include	VERB
ejpam-5915	290	5	:	:	PUNCT
ejpam-5915	290	6	s.	s.	PROPN
ejpam-5915	290	7	shaqaqha	shaqaqha	PROPN
ejpam-5915	290	8	/	/	SYM
ejpam-5915	290	9	eur	eur	PROPN
ejpam-5915	290	10	.	.	PUNCT
ejpam-5915	291	1	j.	j.	PROPN
ejpam-5915	291	2	pure	pure	PROPN
ejpam-5915	291	3	appl	appl	PROPN
ejpam-5915	291	4	.	.	PROPN
ejpam-5915	291	5	math	math	PROPN
ejpam-5915	291	6	,	,	PUNCT
ejpam-5915	291	7	18	18	NUM
ejpam-5915	291	8	(	(	PUNCT
ejpam-5915	291	9	2	2	NUM
ejpam-5915	291	10	)	)	PUNCT
ejpam-5915	291	11	(	(	PUNCT
ejpam-5915	291	12	2025	2025	NUM
ejpam-5915	291	13	)	)	PUNCT
ejpam-5915	291	14	,	,	PUNCT
ejpam-5915	291	15	5915	5915	NUM
ejpam-5915	291	16	15	15	NUM
ejpam-5915	291	17	of	of	ADP
ejpam-5915	291	18	16	16	NUM
ejpam-5915	291	19	•	•	NOUN
ejpam-5915	291	20	extending	extend	VERB
ejpam-5915	291	21	fuzzy	fuzzy	ADJ
ejpam-5915	291	22	hom	hom	NOUN
ejpam-5915	291	23	-	-	PUNCT
ejpam-5915	291	24	groups	group	NOUN
ejpam-5915	291	25	to	to	ADP
ejpam-5915	291	26	other	other	ADJ
ejpam-5915	291	27	algebraic	algebraic	ADJ
ejpam-5915	291	28	structures	structure	NOUN
ejpam-5915	291	29	,	,	PUNCT
ejpam-5915	291	30	such	such	ADJ
ejpam-5915	291	31	as	as	ADP
ejpam-5915	291	32	fuzzy	fuzzy	ADJ
ejpam-5915	291	33	hom	hom	NOUN
ejpam-5915	291	34	-	-	PUNCT
ejpam-5915	291	35	lie	lie	NOUN
ejpam-5915	291	36	algebras	algebra	NOUN
ejpam-5915	291	37	and	and	CCONJ
ejpam-5915	291	38	fuzzy	fuzzy	ADJ
ejpam-5915	291	39	hom	hom	NOUN
ejpam-5915	291	40	-	-	PUNCT
ejpam-5915	291	41	rings	ring	NOUN
ejpam-5915	291	42	.	.	PUNCT
ejpam-5915	292	1	•	•	NUM
ejpam-5915	292	2	investigating	investigate	VERB
ejpam-5915	292	3	fuzzy	fuzzy	ADJ
ejpam-5915	292	4	nilpotent	nilpotent	NOUN
ejpam-5915	292	5	and	and	CCONJ
ejpam-5915	292	6	fuzzy	fuzzy	ADJ
ejpam-5915	292	7	solvable	solvable	ADJ
ejpam-5915	292	8	hom	hom	NOUN
ejpam-5915	292	9	-	-	PUNCT
ejpam-5915	292	10	groups	group	NOUN
ejpam-5915	292	11	,	,	PUNCT
ejpam-5915	292	12	inspired	inspire	VERB
ejpam-5915	292	13	by	by	ADP
ejpam-5915	292	14	classical	classical	ADJ
ejpam-5915	292	15	fuzzy	fuzzy	ADJ
ejpam-5915	292	16	group	group	NOUN
ejpam-5915	292	17	results	result	NOUN
ejpam-5915	292	18	[	[	X
ejpam-5915	292	19	4	4	NUM
ejpam-5915	292	20	,	,	PUNCT
ejpam-5915	292	21	6	6	NUM
ejpam-5915	292	22	]	]	PUNCT
ejpam-5915	292	23	.	.	PUNCT
ejpam-5915	293	1	•	•	PUNCT
ejpam-5915	293	2	exploring	explore	VERB
ejpam-5915	293	3	advanced	advanced	ADJ
ejpam-5915	293	4	fuzzy	fuzzy	ADJ
ejpam-5915	293	5	extensions	extension	NOUN
ejpam-5915	293	6	,	,	PUNCT
ejpam-5915	293	7	including	include	VERB
ejpam-5915	293	8	:	:	PUNCT
ejpam-5915	293	9	–	–	PUNCT
ejpam-5915	293	10	complex	complex	ADJ
ejpam-5915	293	11	fuzzy	fuzzy	ADJ
ejpam-5915	293	12	hom	hom	NOUN
ejpam-5915	293	13	-	-	PUNCT
ejpam-5915	293	14	groups	group	NOUN
ejpam-5915	293	15	,	,	PUNCT
ejpam-5915	293	16	with	with	ADP
ejpam-5915	293	17	membership	membership	NOUN
ejpam-5915	293	18	values	value	NOUN
ejpam-5915	293	19	in	in	ADP
ejpam-5915	293	20	the	the	DET
ejpam-5915	293	21	complex	complex	ADJ
ejpam-5915	293	22	unit	unit	NOUN
ejpam-5915	293	23	disk	disk	NOUN
ejpam-5915	293	24	.	.	PUNCT
ejpam-5915	294	1	–	–	PUNCT
ejpam-5915	294	2	intuitionistic	intuitionistic	ADJ
ejpam-5915	294	3	fuzzy	fuzzy	ADJ
ejpam-5915	294	4	hom	hom	NOUN
ejpam-5915	294	5	-	-	PUNCT
ejpam-5915	294	6	groups	group	NOUN
ejpam-5915	294	7	,	,	PUNCT
ejpam-5915	294	8	incorporating	incorporate	VERB
ejpam-5915	294	9	membership	membership	NOUN
ejpam-5915	294	10	and	and	CCONJ
ejpam-5915	294	11	non	non	ADJ
ejpam-5915	294	12	-	-	ADJ
ejpam-5915	294	13	membership	membership	ADJ
ejpam-5915	294	14	degrees	degree	NOUN
ejpam-5915	294	15	.	.	PUNCT
ejpam-5915	295	1	–	–	PUNCT
ejpam-5915	295	2	bipolar	bipolar	ADJ
ejpam-5915	295	3	fuzzy	fuzzy	ADJ
ejpam-5915	295	4	hom	hom	NOUN
ejpam-5915	295	5	-	-	PUNCT
ejpam-5915	295	6	groups	group	NOUN
ejpam-5915	295	7	,	,	PUNCT
ejpam-5915	295	8	handling	handle	VERB
ejpam-5915	295	9	positive	positive	ADJ
ejpam-5915	295	10	and	and	CCONJ
ejpam-5915	295	11	negative	negative	ADJ
ejpam-5915	295	12	membership	membership	NOUN
ejpam-5915	295	13	values	value	NOUN
ejpam-5915	295	14	.	.	PUNCT
ejpam-5915	296	1	–	–	PUNCT
ejpam-5915	296	2	pythagorean	pythagorean	VERB
ejpam-5915	296	3	fuzzy	fuzzy	ADJ
ejpam-5915	296	4	hom	hom	NOUN
ejpam-5915	296	5	-	-	PUNCT
ejpam-5915	296	6	groups	group	NOUN
ejpam-5915	296	7	,	,	PUNCT
ejpam-5915	296	8	allowing	allow	VERB
ejpam-5915	296	9	squared	square	VERB
ejpam-5915	296	10	membership	membership	NOUN
ejpam-5915	296	11	and	and	CCONJ
ejpam-5915	296	12	non	non	ADJ
ejpam-5915	296	13	-	-	ADJ
ejpam-5915	296	14	membership	membership	ADJ
ejpam-5915	296	15	values	value	NOUN
ejpam-5915	296	16	.	.	PUNCT
ejpam-5915	297	1	–	–	PUNCT
ejpam-5915	297	2	interval	interval	NOUN
ejpam-5915	297	3	-	-	PUNCT
ejpam-5915	297	4	valued	value	VERB
ejpam-5915	297	5	fuzzy	fuzzy	ADJ
ejpam-5915	297	6	hom	hom	NOUN
ejpam-5915	297	7	-	-	PUNCT
ejpam-5915	297	8	groups	group	NOUN
ejpam-5915	297	9	,	,	PUNCT
ejpam-5915	297	10	considering	consider	VERB
ejpam-5915	297	11	ranges	range	NOUN
ejpam-5915	297	12	of	of	ADP
ejpam-5915	297	13	membership	membership	NOUN
ejpam-5915	297	14	values	value	NOUN
ejpam-5915	297	15	.	.	PUNCT
ejpam-5915	298	1	–	–	PUNCT
ejpam-5915	298	2	picture	picture	NOUN
ejpam-5915	298	3	fuzzy	fuzzy	ADJ
ejpam-5915	298	4	hom	hom	NOUN
ejpam-5915	298	5	-	-	PUNCT
ejpam-5915	298	6	groups	group	NOUN
ejpam-5915	298	7	,	,	PUNCT
ejpam-5915	298	8	introducing	introduce	VERB
ejpam-5915	298	9	a	a	DET
ejpam-5915	298	10	neutral	neutral	ADJ
ejpam-5915	298	11	membership	membership	NOUN
ejpam-5915	298	12	degree	degree	NOUN
ejpam-5915	298	13	alongside	alongside	ADP
ejpam-5915	298	14	standard	standard	ADJ
ejpam-5915	298	15	values	value	NOUN
ejpam-5915	298	16	.	.	PUNCT
ejpam-5915	299	1	•	•	PUNCT
ejpam-5915	299	2	investigating	investigate	VERB
ejpam-5915	299	3	fuzzy	fuzzy	ADJ
ejpam-5915	299	4	hom	hom	NOUN
ejpam-5915	299	5	-	-	PUNCT
ejpam-5915	299	6	groups	group	NOUN
ejpam-5915	299	7	in	in	ADP
ejpam-5915	299	8	automorphism	automorphism	NOUN
ejpam-5915	299	9	theory	theory	NOUN
ejpam-5915	299	10	to	to	PART
ejpam-5915	299	11	understand	understand	VERB
ejpam-5915	299	12	symmetry	symmetry	NOUN
ejpam-5915	299	13	and	and	CCONJ
ejpam-5915	299	14	invariance	invariance	NOUN
ejpam-5915	299	15	under	under	ADP
ejpam-5915	299	16	fuzzy	fuzzy	ADJ
ejpam-5915	299	17	hom	hom	NOUN
ejpam-5915	299	18	-	-	PUNCT
ejpam-5915	299	19	structures	structure	NOUN
ejpam-5915	299	20	.	.	PUNCT
ejpam-5915	300	1	•	•	NUM
ejpam-5915	300	2	developing	develop	VERB
ejpam-5915	300	3	computational	computational	ADJ
ejpam-5915	300	4	approaches	approach	NOUN
ejpam-5915	300	5	for	for	ADP
ejpam-5915	300	6	fuzzy	fuzzy	ADJ
ejpam-5915	300	7	hom	hom	NOUN
ejpam-5915	300	8	-	-	PUNCT
ejpam-5915	300	9	groups	group	NOUN
ejpam-5915	300	10	,	,	PUNCT
ejpam-5915	300	11	with	with	ADP
ejpam-5915	300	12	applications	application	NOUN
ejpam-5915	300	13	in	in	ADP
ejpam-5915	300	14	decision	decision	NOUN
ejpam-5915	300	15	-	-	PUNCT
ejpam-5915	300	16	making	making	NOUN
ejpam-5915	300	17	,	,	PUNCT
ejpam-5915	300	18	fuzzy	fuzzy	ADJ
ejpam-5915	300	19	control	control	NOUN
ejpam-5915	300	20	systems	system	NOUN
ejpam-5915	300	21	,	,	PUNCT
ejpam-5915	300	22	and	and	CCONJ
ejpam-5915	300	23	uncertainty	uncertainty	NOUN
ejpam-5915	300	24	modeling	modeling	NOUN
ejpam-5915	300	25	.	.	PUNCT
ejpam-5915	301	1	this	this	DET
ejpam-5915	301	2	work	work	NOUN
ejpam-5915	301	3	lays	lay	VERB
ejpam-5915	301	4	a	a	DET
ejpam-5915	301	5	foundation	foundation	NOUN
ejpam-5915	301	6	for	for	ADP
ejpam-5915	301	7	further	further	ADJ
ejpam-5915	301	8	studies	study	NOUN
ejpam-5915	301	9	in	in	ADP
ejpam-5915	301	10	the	the	DET
ejpam-5915	301	11	intersection	intersection	NOUN
ejpam-5915	301	12	of	of	ADP
ejpam-5915	301	13	fuzzy	fuzzy	ADJ
ejpam-5915	301	14	set	set	NOUN
ejpam-5915	301	15	theory	theory	NOUN
ejpam-5915	301	16	and	and	CCONJ
ejpam-5915	301	17	hom	hom	ADV
ejpam-5915	301	18	-	-	PUNCT
ejpam-5915	301	19	algebraic	algebraic	ADJ
ejpam-5915	301	20	structures	structure	NOUN
ejpam-5915	301	21	.	.	PUNCT
ejpam-5915	302	1	we	we	PRON
ejpam-5915	302	2	anticipate	anticipate	VERB
ejpam-5915	302	3	that	that	SCONJ
ejpam-5915	302	4	fuzzy	fuzzy	ADJ
ejpam-5915	302	5	hom	hom	NOUN
ejpam-5915	302	6	-	-	PUNCT
ejpam-5915	302	7	groups	group	NOUN
ejpam-5915	302	8	will	will	AUX
ejpam-5915	302	9	continue	continue	VERB
ejpam-5915	302	10	to	to	PART
ejpam-5915	302	11	inspire	inspire	VERB
ejpam-5915	302	12	research	research	NOUN
ejpam-5915	302	13	in	in	ADP
ejpam-5915	302	14	both	both	CCONJ
ejpam-5915	302	15	theoretical	theoretical	ADJ
ejpam-5915	302	16	and	and	CCONJ
ejpam-5915	302	17	applied	applied	ADJ
ejpam-5915	302	18	mathematics	mathematic	NOUN
ejpam-5915	302	19	.	.	PUNCT
ejpam-5915	303	1	acknowledgements	acknowledgement	VERB
ejpam-5915	303	2	the	the	DET
ejpam-5915	303	3	author	author	NOUN
ejpam-5915	303	4	sincerely	sincerely	ADV
ejpam-5915	303	5	thanks	thank	NOUN
ejpam-5915	303	6	the	the	DET
ejpam-5915	303	7	anonymous	anonymous	ADJ
ejpam-5915	303	8	reviewers	reviewer	NOUN
ejpam-5915	303	9	for	for	ADP
ejpam-5915	303	10	their	their	PRON
ejpam-5915	303	11	insightful	insightful	ADJ
ejpam-5915	303	12	comments	comment	NOUN
ejpam-5915	303	13	and	and	CCONJ
ejpam-5915	303	14	constructive	constructive	ADJ
ejpam-5915	303	15	suggestions	suggestion	NOUN
ejpam-5915	303	16	,	,	PUNCT
ejpam-5915	303	17	which	which	PRON
ejpam-5915	303	18	have	have	AUX
ejpam-5915	303	19	significantly	significantly	ADV
ejpam-5915	303	20	enhanced	enhance	VERB
ejpam-5915	303	21	the	the	DET
ejpam-5915	303	22	clarity	clarity	NOUN
ejpam-5915	303	23	and	and	CCONJ
ejpam-5915	303	24	overall	overall	ADJ
ejpam-5915	303	25	quality	quality	NOUN
ejpam-5915	303	26	of	of	ADP
ejpam-5915	303	27	this	this	DET
ejpam-5915	303	28	manuscript	manuscript	NOUN
ejpam-5915	303	29	.	.	PUNCT
ejpam-5915	304	1	references	reference	NOUN
ejpam-5915	304	2	[	[	X
ejpam-5915	304	3	1	1	NUM
ejpam-5915	304	4	]	]	PUNCT
ejpam-5915	304	5	l.	l.	PROPN
ejpam-5915	304	6	zadeh	zadeh	PROPN
ejpam-5915	304	7	.	.	PUNCT
ejpam-5915	305	1	fuzzy	fuzzy	ADJ
ejpam-5915	305	2	sets	set	NOUN
ejpam-5915	305	3	.	.	PUNCT
ejpam-5915	306	1	inform	inform	NOUN
ejpam-5915	306	2	.	.	PUNCT
ejpam-5915	307	1	control	control	NOUN
ejpam-5915	307	2	,	,	PUNCT
ejpam-5915	307	3	8:338–358	8:338–358	NOUN
ejpam-5915	307	4	,	,	PUNCT
ejpam-5915	307	5	1965	1965	NUM
ejpam-5915	307	6	.	.	PUNCT
ejpam-5915	308	1	[	[	X
ejpam-5915	308	2	2	2	NUM
ejpam-5915	308	3	]	]	PUNCT
ejpam-5915	308	4	a.	a.	NOUN
ejpam-5915	308	5	rosenfeld	rosenfeld	PROPN
ejpam-5915	308	6	.	.	PUNCT
ejpam-5915	309	1	fuzzy	fuzzy	ADJ
ejpam-5915	309	2	groups	group	NOUN
ejpam-5915	309	3	.	.	PUNCT
ejpam-5915	310	1	journal	journal	PROPN
ejpam-5915	310	2	of	of	ADP
ejpam-5915	310	3	mathematical	mathematical	ADJ
ejpam-5915	310	4	analysis	analysis	NOUN
ejpam-5915	310	5	and	and	CCONJ
ejpam-5915	310	6	applications	application	NOUN
ejpam-5915	310	7	,	,	PUNCT
ejpam-5915	310	8	35(3):512–517	35(3):512–517	PROPN
ejpam-5915	310	9	,	,	PUNCT
ejpam-5915	310	10	1971	1971	NUM
ejpam-5915	310	11	.	.	PUNCT
ejpam-5915	311	1	[	[	X
ejpam-5915	311	2	3	3	X
ejpam-5915	311	3	]	]	PUNCT
ejpam-5915	312	1	p.	p.	NOUN
ejpam-5915	312	2	s.	s.	PROPN
ejpam-5915	312	3	das	das	PROPN
ejpam-5915	312	4	.	.	PROPN
ejpam-5915	313	1	fuzzy	fuzzy	ADJ
ejpam-5915	313	2	groups	group	NOUN
ejpam-5915	313	3	and	and	CCONJ
ejpam-5915	313	4	level	level	NOUN
ejpam-5915	313	5	subgroups	subgroup	NOUN
ejpam-5915	313	6	.	.	PUNCT
ejpam-5915	314	1	journal	journal	PROPN
ejpam-5915	314	2	of	of	ADP
ejpam-5915	314	3	mathematical	mathematical	ADJ
ejpam-5915	314	4	analysis	analysis	NOUN
ejpam-5915	314	5	and	and	CCONJ
ejpam-5915	314	6	applications	application	NOUN
ejpam-5915	314	7	,	,	PUNCT
ejpam-5915	314	8	84(2):264–269	84(2):264–269	NOUN
ejpam-5915	314	9	,	,	PUNCT
ejpam-5915	314	10	1981	1981	NUM
ejpam-5915	314	11	.	.	PUNCT
ejpam-5915	315	1	[	[	X
ejpam-5915	315	2	4	4	X
ejpam-5915	315	3	]	]	PUNCT
ejpam-5915	315	4	k.	k.	PROPN
ejpam-5915	315	5	c.	c.	PROPN
ejpam-5915	315	6	gupta	gupta	PROPN
ejpam-5915	315	7	and	and	CCONJ
ejpam-5915	315	8	b.	b.	PROPN
ejpam-5915	315	9	k.	k.	PROPN
ejpam-5915	315	10	sarma	sarma	PROPN
ejpam-5915	315	11	.	.	PUNCT
ejpam-5915	316	1	nilpotent	nilpotent	ADJ
ejpam-5915	316	2	fuzzy	fuzzy	ADJ
ejpam-5915	316	3	groups	group	NOUN
ejpam-5915	316	4	.	.	PUNCT
ejpam-5915	317	1	fuzzy	fuzzy	ADJ
ejpam-5915	317	2	sets	set	NOUN
ejpam-5915	317	3	and	and	CCONJ
ejpam-5915	317	4	systems	system	NOUN
ejpam-5915	317	5	,	,	PUNCT
ejpam-5915	317	6	101(2):167–176	101(2):167–176	NUM
ejpam-5915	317	7	,	,	PUNCT
ejpam-5915	317	8	1999	1999	NUM
ejpam-5915	317	9	.	.	PUNCT
ejpam-5915	318	1	s.	s.	PROPN
ejpam-5915	318	2	shaqaqha	shaqaqha	PROPN
ejpam-5915	318	3	/	/	SYM
ejpam-5915	318	4	eur	eur	PROPN
ejpam-5915	318	5	.	.	PUNCT
ejpam-5915	319	1	j.	j.	PROPN
ejpam-5915	319	2	pure	pure	PROPN
ejpam-5915	319	3	appl	appl	PROPN
ejpam-5915	319	4	.	.	PROPN
ejpam-5915	319	5	math	math	PROPN
ejpam-5915	319	6	,	,	PUNCT
ejpam-5915	319	7	18	18	NUM
ejpam-5915	319	8	(	(	PUNCT
ejpam-5915	319	9	2	2	NUM
ejpam-5915	319	10	)	)	PUNCT
ejpam-5915	319	11	(	(	PUNCT
ejpam-5915	319	12	2025	2025	NUM
ejpam-5915	319	13	)	)	PUNCT
ejpam-5915	319	14	,	,	PUNCT
ejpam-5915	319	15	5915	5915	NUM
ejpam-5915	319	16	16	16	NUM
ejpam-5915	319	17	of	of	ADP
ejpam-5915	319	18	16	16	NUM
ejpam-5915	319	19	[	[	X
ejpam-5915	319	20	5	5	NUM
ejpam-5915	319	21	]	]	PUNCT
ejpam-5915	319	22	sarka	sarka	PROPN
ejpam-5915	319	23	hoskova	hoskova	PROPN
ejpam-5915	319	24	-	-	PUNCT
ejpam-5915	319	25	mayerova	mayerova	PROPN
ejpam-5915	319	26	and	and	CCONJ
ejpam-5915	319	27	madeline	madeline	PROPN
ejpam-5915	319	28	al	al	PROPN
ejpam-5915	319	29	tahan	tahan	PROPN
ejpam-5915	319	30	.	.	PUNCT
ejpam-5915	320	1	anti	anti	ADJ
ejpam-5915	320	2	-	-	ADJ
ejpam-5915	320	3	fuzzy	fuzzy	ADJ
ejpam-5915	320	4	multi	multi	NOUN
ejpam-5915	320	5	-	-	NOUN
ejpam-5915	320	6	ideals	ideal	NOUN
ejpam-5915	320	7	of	of	ADP
ejpam-5915	320	8	near	near	ADJ
ejpam-5915	320	9	ring	ring	NOUN
ejpam-5915	320	10	.	.	PUNCT
ejpam-5915	321	1	mathematics	mathematic	NOUN
ejpam-5915	321	2	,	,	PUNCT
ejpam-5915	321	3	9(5):494	9(5):494	NUM
ejpam-5915	321	4	,	,	PUNCT
ejpam-5915	321	5	2021	2021	NUM
ejpam-5915	321	6	.	.	PUNCT
ejpam-5915	322	1	[	[	X
ejpam-5915	322	2	6	6	NUM
ejpam-5915	322	3	]	]	PUNCT
ejpam-5915	322	4	b.	b.	PROPN
ejpam-5915	322	5	k.	k.	PROPN
ejpam-5915	322	6	sarma	sarma	PROPN
ejpam-5915	322	7	.	.	PUNCT
ejpam-5915	323	1	solvable	solvable	ADJ
ejpam-5915	323	2	fuzzy	fuzzy	ADJ
ejpam-5915	323	3	groups	group	NOUN
ejpam-5915	323	4	.	.	PUNCT
ejpam-5915	324	1	fuzzy	fuzzy	ADJ
ejpam-5915	324	2	sets	set	NOUN
ejpam-5915	324	3	and	and	CCONJ
ejpam-5915	324	4	systems	system	NOUN
ejpam-5915	324	5	,	,	PUNCT
ejpam-5915	324	6	106(4):463–467	106(4):463–467	NUM
ejpam-5915	324	7	,	,	PUNCT
ejpam-5915	324	8	1999	1999	NUM
ejpam-5915	324	9	.	.	PUNCT
ejpam-5915	325	1	[	[	X
ejpam-5915	325	2	7	7	NUM
ejpam-5915	325	3	]	]	PUNCT
ejpam-5915	325	4	a.	a.	NOUN
ejpam-5915	325	5	jain	jain	PROPN
ejpam-5915	325	6	.	.	PUNCT
ejpam-5915	326	1	fuzzy	fuzzy	ADJ
ejpam-5915	326	2	subgroups	subgroup	NOUN
ejpam-5915	326	3	and	and	CCONJ
ejpam-5915	326	4	certain	certain	ADJ
ejpam-5915	326	5	equivalence	equivalence	NOUN
ejpam-5915	326	6	relations	relation	NOUN
ejpam-5915	326	7	.	.	PUNCT
ejpam-5915	327	1	iranian	iranian	ADJ
ejpam-5915	327	2	journal	journal	PROPN
ejpam-5915	327	3	of	of	ADP
ejpam-5915	327	4	fuzzy	fuzzy	ADJ
ejpam-5915	327	5	systems	system	NOUN
ejpam-5915	327	6	,	,	PUNCT
ejpam-5915	327	7	3(2):75–91	3(2):75–91	NUM
ejpam-5915	327	8	,	,	PUNCT
ejpam-5915	327	9	2006	2006	NUM
ejpam-5915	327	10	.	.	PUNCT
ejpam-5915	328	1	[	[	X
ejpam-5915	328	2	8	8	NUM
ejpam-5915	328	3	]	]	PUNCT
ejpam-5915	328	4	m.	m.	NOUN
ejpam-5915	328	5	tarnauceanu	tarnauceanu	NOUN
ejpam-5915	328	6	.	.	PUNCT
ejpam-5915	329	1	classifying	classify	VERB
ejpam-5915	329	2	fuzzy	fuzzy	ADJ
ejpam-5915	329	3	normal	normal	ADJ
ejpam-5915	329	4	subgroups	subgroup	NOUN
ejpam-5915	329	5	of	of	ADP
ejpam-5915	329	6	finite	finite	ADJ
ejpam-5915	329	7	groups	group	NOUN
ejpam-5915	329	8	.	.	PUNCT
ejpam-5915	330	1	iranian	iranian	ADJ
ejpam-5915	330	2	journal	journal	PROPN
ejpam-5915	330	3	of	of	ADP
ejpam-5915	330	4	fuzzy	fuzzy	ADJ
ejpam-5915	330	5	systems	system	NOUN
ejpam-5915	330	6	,	,	PUNCT
ejpam-5915	330	7	12(2):107–115	12(2):107–115	NUM
ejpam-5915	330	8	,	,	PUNCT
ejpam-5915	330	9	2015	2015	NUM
ejpam-5915	330	10	.	.	PUNCT
ejpam-5915	331	1	[	[	X
ejpam-5915	331	2	9	9	X
ejpam-5915	331	3	]	]	X
ejpam-5915	331	4	y.	y.	NOUN
ejpam-5915	331	5	frégier	frégier	NOUN
ejpam-5915	331	6	and	and	CCONJ
ejpam-5915	331	7	a.	a.	NOUN
ejpam-5915	331	8	gohr	gohr	PROPN
ejpam-5915	331	9	.	.	PUNCT
ejpam-5915	332	1	on	on	ADP
ejpam-5915	332	2	hom	hom	NOUN
ejpam-5915	332	3	-	-	PUNCT
ejpam-5915	332	4	type	type	NOUN
ejpam-5915	332	5	algebras	algebra	NOUN
ejpam-5915	332	6	.	.	PUNCT
ejpam-5915	332	7	journal	journal	PROPN
ejpam-5915	332	8	of	of	ADP
ejpam-5915	332	9	generalized	generalized	ADJ
ejpam-5915	332	10	lie	lie	NOUN
ejpam-5915	332	11	theory	theory	NOUN
ejpam-5915	332	12	and	and	CCONJ
ejpam-5915	332	13	applications	application	NOUN
ejpam-5915	332	14	,	,	PUNCT
ejpam-5915	332	15	4:1–16	4:1–16	NUM
ejpam-5915	332	16	,	,	PUNCT
ejpam-5915	332	17	2010	2010	NUM
ejpam-5915	332	18	.	.	PUNCT
ejpam-5915	333	1	[	[	X
ejpam-5915	333	2	10	10	NUM
ejpam-5915	333	3	]	]	PUNCT
ejpam-5915	333	4	a.	a.	NOUN
ejpam-5915	333	5	makhlouf	makhlouf	PROPN
ejpam-5915	333	6	and	and	CCONJ
ejpam-5915	333	7	s.	s.	PROPN
ejpam-5915	333	8	silvestrov	silvestrov	PROPN
ejpam-5915	333	9	.	.	PUNCT
ejpam-5915	334	1	hom	hom	X
ejpam-5915	334	2	-	-	PUNCT
ejpam-5915	334	3	algebra	algebra	NOUN
ejpam-5915	334	4	structures	structure	NOUN
ejpam-5915	334	5	.	.	PUNCT
ejpam-5915	335	1	journal	journal	NOUN
ejpam-5915	335	2	of	of	ADP
ejpam-5915	335	3	generalized	generalized	ADJ
ejpam-5915	335	4	lie	lie	NOUN
ejpam-5915	335	5	theory	theory	NOUN
ejpam-5915	335	6	and	and	CCONJ
ejpam-5915	335	7	applications	application	NOUN
ejpam-5915	335	8	,	,	PUNCT
ejpam-5915	335	9	2(2):51–64	2(2):51–64	NUM
ejpam-5915	335	10	,	,	PUNCT
ejpam-5915	335	11	2008	2008	NUM
ejpam-5915	335	12	.	.	PUNCT
ejpam-5915	336	1	[	[	X
ejpam-5915	336	2	11	11	NUM
ejpam-5915	336	3	]	]	PUNCT
ejpam-5915	336	4	s.	s.	PROPN
ejpam-5915	336	5	shaqaqha	shaqaqha	PROPN
ejpam-5915	336	6	.	.	PUNCT
ejpam-5915	337	1	restricted	restrict	VERB
ejpam-5915	337	2	hom	hom	NOUN
ejpam-5915	337	3	-	-	PUNCT
ejpam-5915	337	4	lie	lie	NOUN
ejpam-5915	337	5	superalgebras	superalgebras	PROPN
ejpam-5915	337	6	.	.	PUNCT
ejpam-5915	338	1	jordan	jordan	PROPN
ejpam-5915	338	2	journal	journal	PROPN
ejpam-5915	338	3	of	of	ADP
ejpam-5915	338	4	mathematics	mathematics	PROPN
ejpam-5915	338	5	and	and	CCONJ
ejpam-5915	338	6	statistics	statistic	NOUN
ejpam-5915	338	7	(	(	PUNCT
ejpam-5915	338	8	jjms	jjms	NOUN
ejpam-5915	338	9	)	)	PUNCT
ejpam-5915	338	10	,	,	PUNCT
ejpam-5915	338	11	12(2):233–252	12(2):233–252	NUM
ejpam-5915	338	12	,	,	PUNCT
ejpam-5915	338	13	2019	2019	NUM
ejpam-5915	338	14	.	.	PUNCT
ejpam-5915	339	1	[	[	X
ejpam-5915	339	2	12	12	NUM
ejpam-5915	339	3	]	]	X
ejpam-5915	339	4	s.	s.	PROPN
ejpam-5915	339	5	shaqaqha	shaqaqha	PROPN
ejpam-5915	339	6	and	and	CCONJ
ejpam-5915	339	7	n.	n.	PROPN
ejpam-5915	339	8	kdaisat	kdaisat	PROPN
ejpam-5915	339	9	.	.	PUNCT
ejpam-5915	340	1	extraction	extraction	NOUN
ejpam-5915	340	2	algorithm	algorithm	NOUN
ejpam-5915	340	3	of	of	ADP
ejpam-5915	340	4	hom	hom	NOUN
ejpam-5915	340	5	–	–	PUNCT
ejpam-5915	340	6	lie	lie	NOUN
ejpam-5915	340	7	algebras	algebras	PROPN
ejpam-5915	340	8	based	base	VERB
ejpam-5915	340	9	on	on	ADP
ejpam-5915	340	10	solvable	solvable	ADJ
ejpam-5915	340	11	and	and	CCONJ
ejpam-5915	340	12	nilpotent	nilpotent	ADJ
ejpam-5915	340	13	groups	group	NOUN
ejpam-5915	340	14	.	.	PUNCT
ejpam-5915	341	1	international	international	ADJ
ejpam-5915	341	2	journal	journal	PROPN
ejpam-5915	341	3	of	of	ADP
ejpam-5915	341	4	mathematics	mathematics	PROPN
ejpam-5915	341	5	and	and	CCONJ
ejpam-5915	341	6	mathematical	mathematical	ADJ
ejpam-5915	341	7	sciences	science	NOUN
ejpam-5915	341	8	,	,	PUNCT
ejpam-5915	341	9	2023(1):6633715	2023(1):6633715	NOUN
ejpam-5915	341	10	,	,	PUNCT
ejpam-5915	341	11	2023	2023	NUM
ejpam-5915	341	12	.	.	PUNCT
ejpam-5915	342	1	[	[	X
ejpam-5915	342	2	13	13	NUM
ejpam-5915	342	3	]	]	PUNCT
ejpam-5915	342	4	s.	s.	PROPN
ejpam-5915	342	5	shaqaqha	shaqaqha	PROPN
ejpam-5915	342	6	and	and	CCONJ
ejpam-5915	342	7	n.	n.	PROPN
ejpam-5915	342	8	kdaisat	kdaisat	PROPN
ejpam-5915	342	9	.	.	PUNCT
ejpam-5915	343	1	more	more	ADJ
ejpam-5915	343	2	properties	property	NOUN
ejpam-5915	343	3	of	of	ADP
ejpam-5915	343	4	(	(	PUNCT
ejpam-5915	343	5	multiplicative)-hom	multiplicative)-hom	ADJ
ejpam-5915	343	6	-	-	PUNCT
ejpam-5915	343	7	lie	lie	NOUN
ejpam-5915	343	8	algebras	algebras	PROPN
ejpam-5915	343	9	.	.	PUNCT
ejpam-5915	344	1	palestine	palestine	PROPN
ejpam-5915	344	2	journal	journal	PROPN
ejpam-5915	344	3	of	of	ADP
ejpam-5915	344	4	mathematics	mathematic	NOUN
ejpam-5915	344	5	,	,	PUNCT
ejpam-5915	344	6	12(2):565–577	12(2):565–577	PROPN
ejpam-5915	344	7	,	,	PUNCT
ejpam-5915	344	8	2023	2023	NUM
ejpam-5915	344	9	.	.	PUNCT
ejpam-5915	345	1	[	[	X
ejpam-5915	345	2	14	14	NUM
ejpam-5915	345	3	]	]	PUNCT
ejpam-5915	345	4	l.	l.	PROPN
ejpam-5915	345	5	y.	y.	PROPN
ejpam-5915	345	6	chen	chen	PROPN
ejpam-5915	345	7	,	,	PUNCT
ejpam-5915	345	8	t.	t.	PROPN
ejpam-5915	345	9	q.	q.	PROPN
ejpam-5915	345	10	feng	feng	PROPN
ejpam-5915	345	11	,	,	PUNCT
ejpam-5915	345	12	y.	y.	PROPN
ejpam-5915	345	13	ma	ma	PROPN
ejpam-5915	345	14	,	,	PUNCT
ejpam-5915	345	15	r.	r.	PROPN
ejpam-5915	345	16	saha	saha	PROPN
ejpam-5915	345	17	,	,	PUNCT
ejpam-5915	345	18	and	and	CCONJ
ejpam-5915	345	19	h.	h.	PROPN
ejpam-5915	345	20	y.	y.	PROPN
ejpam-5915	345	21	zhang	zhang	PROPN
ejpam-5915	345	22	.	.	PUNCT
ejpam-5915	346	1	on	on	ADP
ejpam-5915	346	2	hom	hom	NOUN
ejpam-5915	346	3	-	-	PUNCT
ejpam-5915	346	4	groups	group	NOUN
ejpam-5915	346	5	and	and	CCONJ
ejpam-5915	346	6	hom	hom	NOUN
ejpam-5915	346	7	-	-	PUNCT
ejpam-5915	346	8	group	group	NOUN
ejpam-5915	346	9	actions	action	NOUN
ejpam-5915	346	10	.	.	PUNCT
ejpam-5915	347	1	acta	acta	PROPN
ejpam-5915	347	2	mathematica	mathematica	PROPN
ejpam-5915	347	3	sinica	sinica	PROPN
ejpam-5915	347	4	,	,	PUNCT
ejpam-5915	347	5	english	english	ADJ
ejpam-5915	347	6	series	series	NOUN
ejpam-5915	347	7	,	,	PUNCT
ejpam-5915	347	8	39:1887–1906	39:1887–1906	NUM
ejpam-5915	347	9	,	,	PUNCT
ejpam-5915	347	10	2023	2023	NUM
ejpam-5915	347	11	.	.	PUNCT
ejpam-5915	348	1	[	[	X
ejpam-5915	348	2	15	15	NUM
ejpam-5915	348	3	]	]	X
ejpam-5915	348	4	h.	h.	PROPN
ejpam-5915	348	5	bouremel	bouremel	PROPN
ejpam-5915	348	6	,	,	PUNCT
ejpam-5915	348	7	z.	z.	PROPN
ejpam-5915	348	8	chebel	chebel	PROPN
ejpam-5915	348	9	,	,	PUNCT
ejpam-5915	348	10	and	and	CCONJ
ejpam-5915	348	11	h.	h.	PROPN
ejpam-5915	348	12	adimi	adimi	PROPN
ejpam-5915	348	13	.	.	PROPN
ejpam-5915	348	14	ordered	order	VERB
ejpam-5915	348	15	hom	hom	NOUN
ejpam-5915	348	16	-	-	PUNCT
ejpam-5915	348	17	groups	group	NOUN
ejpam-5915	348	18	based	base	VERB
ejpam-5915	348	19	on	on	ADP
ejpam-5915	348	20	homcompatibility	homcompatibility	NOUN
ejpam-5915	348	21	.	.	PUNCT
ejpam-5915	349	1	rendiconti	rendiconti	PROPN
ejpam-5915	349	2	del	del	PROPN
ejpam-5915	349	3	circolo	circolo	PROPN
ejpam-5915	349	4	matematico	matematico	NOUN
ejpam-5915	349	5	di	di	PROPN
ejpam-5915	349	6	palermo	palermo	PROPN
ejpam-5915	349	7	series	series	PROPN
ejpam-5915	349	8	2	2	NUM
ejpam-5915	349	9	,	,	PUNCT
ejpam-5915	349	10	72:3377–3398	72:3377–3398	NUM
ejpam-5915	349	11	,	,	PUNCT
ejpam-5915	349	12	2023	2023	NUM
ejpam-5915	349	13	.	.	PUNCT
ejpam-5915	350	1	[	[	X
ejpam-5915	350	2	16	16	NUM
ejpam-5915	350	3	]	]	X
ejpam-5915	350	4	n.	n.	PROPN
ejpam-5915	350	5	ajmal	ajmal	PROPN
ejpam-5915	350	6	and	and	CCONJ
ejpam-5915	350	7	a.	a.	PROPN
ejpam-5915	350	8	s.	s.	PROPN
ejpam-5915	350	9	prajapati	prajapati	PROPN
ejpam-5915	350	10	.	.	PUNCT
ejpam-5915	351	1	fuzzy	fuzzy	ADJ
ejpam-5915	351	2	cosets	coset	NOUN
ejpam-5915	351	3	and	and	CCONJ
ejpam-5915	351	4	fuzzy	fuzzy	ADJ
ejpam-5915	351	5	normal	normal	ADJ
ejpam-5915	351	6	subgroups	subgroup	NOUN
ejpam-5915	351	7	.	.	PUNCT
ejpam-5915	352	1	information	information	NOUN
ejpam-5915	352	2	sciences	sciences	PROPN
ejpam-5915	352	3	,	,	PUNCT
ejpam-5915	352	4	64:17–25	64:17–25	NUM
ejpam-5915	352	5	,	,	PUNCT
ejpam-5915	352	6	1992	1992	NUM
ejpam-5915	352	7	.	.	PUNCT
ejpam-5915	353	1	[	[	X
ejpam-5915	353	2	17	17	NUM
ejpam-5915	353	3	]	]	X
ejpam-5915	353	4	e.	e.	PROPN
ejpam-5915	353	5	eslami	eslami	PROPN
ejpam-5915	353	6	.	.	PUNCT
ejpam-5915	354	1	an	an	DET
ejpam-5915	354	2	algebraic	algebraic	ADJ
ejpam-5915	354	3	structure	structure	NOUN
ejpam-5915	354	4	for	for	ADP
ejpam-5915	354	5	intuitionistic	intuitionistic	ADJ
ejpam-5915	354	6	fuzzy	fuzzy	ADJ
ejpam-5915	354	7	logic	logic	NOUN
ejpam-5915	354	8	.	.	PUNCT
ejpam-5915	355	1	iranian	iranian	ADJ
ejpam-5915	355	2	journal	journal	PROPN
ejpam-5915	355	3	of	of	ADP
ejpam-5915	355	4	fuzzy	fuzzy	ADJ
ejpam-5915	355	5	systems	system	NOUN
ejpam-5915	355	6	,	,	PUNCT
ejpam-5915	355	7	9(6):31–41	9(6):31–41	NUM
ejpam-5915	355	8	,	,	PUNCT
ejpam-5915	355	9	2012	2012	NUM
ejpam-5915	355	10	.	.	PUNCT
ejpam-5915	356	1	[	[	X
ejpam-5915	356	2	18	18	NUM
ejpam-5915	356	3	]	]	X
ejpam-5915	356	4	s.	s.	PROPN
ejpam-5915	356	5	shaqaqha	shaqaqha	PROPN
ejpam-5915	356	6	.	.	PUNCT
ejpam-5915	357	1	complex	complex	ADJ
ejpam-5915	357	2	fuzzy	fuzzy	ADJ
ejpam-5915	357	3	lie	lie	NOUN
ejpam-5915	357	4	algebras	algebras	PROPN
ejpam-5915	357	5	.	.	PUNCT
ejpam-5915	358	1	jordan	jordan	PROPN
ejpam-5915	358	2	journal	journal	PROPN
ejpam-5915	358	3	of	of	ADP
ejpam-5915	358	4	mathematics	mathematics	PROPN
ejpam-5915	358	5	and	and	CCONJ
ejpam-5915	358	6	statistics	statistic	NOUN
ejpam-5915	358	7	,	,	PUNCT
ejpam-5915	358	8	13(2):231–247	13(2):231–247	PROPN
ejpam-5915	358	9	,	,	PUNCT
ejpam-5915	358	10	2020	2020	NUM
ejpam-5915	358	11	.	.	PUNCT
ejpam-5915	359	1	[	[	X
ejpam-5915	359	2	19	19	NUM
ejpam-5915	359	3	]	]	PUNCT
ejpam-5915	359	4	s.	s.	PROPN
ejpam-5915	359	5	shaqaqha	shaqaqha	PROPN
ejpam-5915	359	6	.	.	PUNCT
ejpam-5915	360	1	on	on	ADP
ejpam-5915	360	2	fuzzification	fuzzification	NOUN
ejpam-5915	360	3	of	of	ADP
ejpam-5915	360	4	n	n	CCONJ
ejpam-5915	360	5	-	-	PUNCT
ejpam-5915	360	6	lie	lie	NOUN
ejpam-5915	360	7	algebras	algebra	NOUN
ejpam-5915	360	8	.	.	PUNCT
ejpam-5915	361	1	jordan	jordan	PROPN
ejpam-5915	361	2	journal	journal	PROPN
ejpam-5915	361	3	of	of	ADP
ejpam-5915	361	4	mathematics	mathematics	PROPN
ejpam-5915	361	5	and	and	CCONJ
ejpam-5915	361	6	statistics	statistic	NOUN
ejpam-5915	361	7	,	,	PUNCT
ejpam-5915	361	8	15(3a):523–540	15(3a):523–540	NUM
ejpam-5915	361	9	,	,	PUNCT
ejpam-5915	361	10	2022	2022	NUM
ejpam-5915	361	11	.	.	PUNCT
ejpam-5915	362	1	[	[	X
ejpam-5915	362	2	20	20	NUM
ejpam-5915	362	3	]	]	PUNCT
ejpam-5915	362	4	s.	s.	PROPN
ejpam-5915	362	5	shaqaqha	shaqaqha	PROPN
ejpam-5915	362	6	.	.	PUNCT
ejpam-5915	363	1	fuzzy	fuzzy	PROPN
ejpam-5915	363	2	hom	hom	NOUN
ejpam-5915	363	3	–	–	PUNCT
ejpam-5915	363	4	lie	lie	NOUN
ejpam-5915	363	5	ideals	ideal	NOUN
ejpam-5915	363	6	of	of	ADP
ejpam-5915	363	7	hom	hom	NOUN
ejpam-5915	363	8	–	–	PUNCT
ejpam-5915	363	9	lie	lie	NOUN
ejpam-5915	363	10	algebras	algebra	NOUN
ejpam-5915	363	11	.	.	PUNCT
ejpam-5915	364	1	axioms	axiom	NOUN
ejpam-5915	364	2	,	,	PUNCT
ejpam-5915	364	3	12(7):630	12(7):630	NUM
ejpam-5915	364	4	,	,	PUNCT
ejpam-5915	364	5	2023	2023	NUM
ejpam-5915	364	6	.	.	PUNCT
ejpam-5915	365	1	[	[	X
ejpam-5915	365	2	21	21	NUM
ejpam-5915	365	3	]	]	X
ejpam-5915	365	4	s.	s.	PROPN
ejpam-5915	365	5	shaqaqha	shaqaqha	PROPN
ejpam-5915	365	6	.	.	PUNCT
ejpam-5915	366	1	characterizations	characterization	NOUN
ejpam-5915	366	2	of	of	ADP
ejpam-5915	366	3	artinian	artinian	ADJ
ejpam-5915	366	4	and	and	CCONJ
ejpam-5915	366	5	noetherian	noetherian	ADJ
ejpam-5915	366	6	gamma	gamma	NOUN
ejpam-5915	366	7	rings	ring	NOUN
ejpam-5915	366	8	in	in	ADP
ejpam-5915	366	9	terms	term	NOUN
ejpam-5915	366	10	of	of	ADP
ejpam-5915	366	11	homogeneous	homogeneous	ADJ
ejpam-5915	366	12	complex	complex	ADJ
ejpam-5915	366	13	fuzzy	fuzzy	ADJ
ejpam-5915	366	14	ideals	ideal	NOUN
ejpam-5915	366	15	.	.	PUNCT
ejpam-5915	367	1	palestine	palestine	PROPN
ejpam-5915	367	2	journal	journal	PROPN
ejpam-5915	367	3	of	of	ADP
ejpam-5915	367	4	mathematics	mathematics	PROPN
ejpam-5915	367	5	,	,	PUNCT
ejpam-5915	367	6	11(4):167–171	11(4):167–171	PROPN
ejpam-5915	367	7	,	,	PUNCT
ejpam-5915	367	8	2022	2022	NUM
ejpam-5915	367	9	.	.	PUNCT
ejpam-5915	368	1	[	[	X
ejpam-5915	368	2	22	22	NUM
ejpam-5915	368	3	]	]	PUNCT
ejpam-5915	368	4	s.	s.	PROPN
ejpam-5915	368	5	shaqaqha	shaqaqha	PROPN
ejpam-5915	368	6	.	.	PUNCT
ejpam-5915	369	1	isomorphism	isomorphism	NOUN
ejpam-5915	369	2	theorems	theorem	NOUN
ejpam-5915	369	3	of	of	ADP
ejpam-5915	369	4	complex	complex	ADJ
ejpam-5915	369	5	fuzzy	fuzzy	ADJ
ejpam-5915	369	6	γ	γ	NOUN
ejpam-5915	369	7	-	-	PUNCT
ejpam-5915	369	8	rings	ring	NOUN
ejpam-5915	369	9	.	.	PUNCT
ejpam-5915	370	1	missouri	missouri	PROPN
ejpam-5915	370	2	journal	journal	PROPN
ejpam-5915	370	3	of	of	ADP
ejpam-5915	370	4	mathematical	mathematical	ADJ
ejpam-5915	370	5	sciences	sciences	PROPN
ejpam-5915	370	6	,	,	PUNCT
ejpam-5915	370	7	34(2):196–207	34(2):196–207	NOUN
ejpam-5915	370	8	,	,	PUNCT
ejpam-5915	370	9	2022	2022	NUM
ejpam-5915	370	10	.	.	PUNCT