id	sid	tid	token	lemma	pos
m-769	1	1	ijo	ijo	PROPN
m-769	1	2	international	international	PROPN
m-769	1	3	journal	journal	PROPN
m-769	1	4	of	of	ADP
m-769	1	5	mathematics	mathematics	PROPN
m-769	1	6	(	(	PUNCT
m-769	1	7	issn	issn	PROPN
m-769	1	8	:	:	PUNCT
m-769	1	9	2992	2992	NUM
m-769	1	10	-	-	SYM
m-769	1	11	4421	4421	NUM
m-769	1	12	)	)	PUNCT
m-769	2	1	michael	michael	PROPN
m-769	2	2	n.	n.	PROPN
m-769	2	3	john	john	PROPN
m-769	2	4	*	*	PROPN
m-769	2	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-769	2	6	volume	volume	NOUN
m-769	2	7	06	06	NUM
m-769	2	8	issue	issue	NOUN
m-769	2	9	12	12	NUM
m-769	2	10	||	||	NOUN
m-769	2	11	dec	dec	PROPN
m-769	2	12	.	.	PROPN
m-769	2	13	,	,	PUNCT
m-769	2	14	2023	2023	NUM
m-769	2	15	||	||	NOUN
m-769	3	1	algebraic	algebraic	ADJ
m-769	3	2	and	and	CCONJ
m-769	3	3	topological	topological	ADJ
m-769	3	4	analysis	analysis	NOUN
m-769	3	5	of	of	ADP
m-769	3	6	enveloping	envelop	VERB
m-769	3	7	semigroups	semigroup	NOUN
m-769	3	8	in	in	ADP
m-769	3	9	transformation	transformation	NOUN
m-769	3	10	groups	group	NOUN
m-769	3	11	:	:	PUNCT
m-769	3	12	proximal	proximal	ADJ
m-769	3	13	equivalence	equivalence	NOUN
m-769	3	14	and	and	CCONJ
m-769	3	15	homomorphic	homomorphic	ADJ
m-769	3	16	image	image	NOUN
m-769	3	17	algebraic	algebraic	ADJ
m-769	3	18	and	and	CCONJ
m-769	3	19	topological	topological	ADJ
m-769	3	20	analysis	analysis	NOUN
m-769	3	21	of	of	ADP
m-769	3	22	enveloping	envelop	VERB
m-769	3	23	semigroups	semigroup	NOUN
m-769	3	24	in	in	ADP
m-769	3	25	transformation	transformation	NOUN
m-769	3	26	groups	group	NOUN
m-769	3	27	:	:	PUNCT
m-769	3	28	proximal	proximal	ADJ
m-769	3	29	equivalence	equivalence	NOUN
m-769	3	30	and	and	CCONJ
m-769	3	31	homomorphic	homomorphic	ADJ
m-769	3	32	image	image	NOUN
m-769	3	33	michael	michael	PROPN
m-769	3	34	n.	n.	PROPN
m-769	3	35	john	john	PROPN
m-769	3	36	department	department	PROPN
m-769	3	37	of	of	ADP
m-769	3	38	mathematics	mathematics	PROPN
m-769	3	39	,	,	PUNCT
m-769	3	40	akwaibom	akwaibom	NOUN
m-769	3	41	state	state	NOUN
m-769	3	42	university	university	PROPN
m-769	3	43	,	,	PUNCT
m-769	3	44	nigeria	nigeria	PROPN
m-769	3	45	and	and	CCONJ
m-769	3	46	udoakaotobong	udoakaotobong	PROPN
m-769	3	47	.	.	PUNCT
m-769	4	1	g.	g.	PROPN
m-769	4	2	department	department	PROPN
m-769	4	3	of	of	ADP
m-769	4	4	mathematics	mathematics	PROPN
m-769	4	5	,	,	PUNCT
m-769	4	6	akwaibom	akwaibom	NOUN
m-769	4	7	state	state	NOUN
m-769	4	8	university	university	PROPN
m-769	4	9	,	,	PUNCT
m-769	4	10	nigeria	nigeria	PROPN
m-769	4	11	abstract	abstract	ADV
m-769	4	12	this	this	DET
m-769	4	13	paper	paper	NOUN
m-769	4	14	investigates	investigate	VERB
m-769	4	15	the	the	DET
m-769	4	16	algebraic	algebraic	ADJ
m-769	4	17	properties	property	NOUN
m-769	4	18	of	of	ADP
m-769	4	19	the	the	DET
m-769	4	20	enveloping	enveloping	NOUN
m-769	4	21	semigroupe	semigroupe	NOUN
m-769	4	22	of	of	ADP
m-769	4	23	a	a	DET
m-769	4	24	transformation	transformation	NOUN
m-769	4	25	group	group	NOUN
m-769	4	26	(	(	PUNCT
m-769	4	27	x	x	X
m-769	4	28	,	,	PUNCT
m-769	4	29	t	t	PROPN
m-769	4	30	,	,	PUNCT
m-769	4	31	μ	μ	NOUN
m-769	4	32	)	)	PUNCT
m-769	4	33	with	with	ADP
m-769	4	34	a	a	DET
m-769	4	35	compact	compact	ADJ
m-769	4	36	hausdorff	hausdorff	NOUN
m-769	4	37	phase	phase	NOUN
m-769	4	38	space	space	NOUN
m-769	4	39	x.	x.	NOUN
m-769	5	1	the	the	DET
m-769	5	2	transition	transition	NOUN
m-769	5	3	group	group	NOUN
m-769	5	4	g	g	PROPN
m-769	5	5	is	be	AUX
m-769	5	6	considered	consider	VERB
m-769	5	7	as	as	ADP
m-769	5	8	a	a	DET
m-769	5	9	group	group	NOUN
m-769	5	10	of	of	ADP
m-769	5	11	homeomorphisms	homeomorphisms	PROPN
m-769	5	12	on	on	ADP
m-769	5	13	x	x	NOUN
m-769	5	14	,	,	PUNCT
m-769	5	15	and	and	CCONJ
m-769	5	16	e	e	NOUN
m-769	5	17	is	be	AUX
m-769	5	18	defined	define	VERB
m-769	5	19	as	as	ADP
m-769	5	20	the	the	DET
m-769	5	21	closure	closure	NOUN
m-769	5	22	of	of	ADP
m-769	5	23	g	g	NOUN
m-769	5	24	in	in	ADP
m-769	5	25	x×x	x×x	PROPN
m-769	5	26	.	.	PUNCT
m-769	6	1	the	the	DET
m-769	6	2	main	main	ADJ
m-769	6	3	focus	focus	NOUN
m-769	6	4	is	be	AUX
m-769	6	5	on	on	ADP
m-769	6	6	establishing	establish	VERB
m-769	6	7	a	a	DET
m-769	6	8	connection	connection	NOUN
m-769	6	9	between	between	ADP
m-769	6	10	the	the	DET
m-769	6	11	proximal	proximal	ADJ
m-769	6	12	equivalence	equivalence	NOUN
m-769	6	13	relation	relation	NOUN
m-769	6	14	in	in	ADP
m-769	6	15	x	x	PUNCT
m-769	6	16	and	and	CCONJ
m-769	6	17	the	the	DET
m-769	6	18	structure	structure	NOUN
m-769	6	19	of	of	ADP
m-769	6	20	e	e	NOUN
m-769	6	21	,	,	PUNCT
m-769	6	22	particularly	particularly	ADV
m-769	6	23	the	the	DET
m-769	6	24	presence	presence	NOUN
m-769	6	25	of	of	ADP
m-769	6	26	a	a	DET
m-769	6	27	unique	unique	ADJ
m-769	6	28	minimal	minimal	ADJ
m-769	6	29	right	right	ADJ
m-769	6	30	ideal	ideal	NOUN
m-769	6	31	.	.	PUNCT
m-769	7	1	in	in	ADP
m-769	7	2	the	the	DET
m-769	7	3	latter	latter	ADJ
m-769	7	4	part	part	NOUN
m-769	7	5	,	,	PUNCT
m-769	7	6	the	the	DET
m-769	7	7	study	study	NOUN
m-769	7	8	extends	extend	VERB
m-769	7	9	to	to	ADP
m-769	7	10	the	the	DET
m-769	7	11	analysis	analysis	NOUN
m-769	7	12	of	of	ADP
m-769	7	13	homomorphic	homomorphic	ADJ
m-769	7	14	images	image	NOUN
m-769	7	15	of	of	ADP
m-769	7	16	transformation	transformation	NOUN
m-769	7	17	groups	group	NOUN
m-769	7	18	through	through	ADP
m-769	7	19	their	their	PRON
m-769	7	20	enveloping	enveloping	NOUN
m-769	7	21	semigroups	semigroup	NOUN
m-769	7	22	.	.	PUNCT
m-769	8	1	keywords	keyword	NOUN
m-769	8	2	:	:	PUNCT
m-769	8	3	algebraic	algebraic	ADJ
m-769	8	4	cryptography	cryptography	NOUN
m-769	8	5	,	,	PUNCT
m-769	8	6	group	group	NOUN
m-769	8	7	theory	theory	NOUN
m-769	8	8	,	,	PUNCT
m-769	8	9	enveloping	envelop	VERB
m-769	8	10	semigroup	semigroup	NOUN
m-769	8	11	,	,	PUNCT
m-769	8	12	proximal	proximal	ADJ
m-769	8	13	equivalence	equivalence	NOUN
m-769	8	14	,	,	PUNCT
m-769	8	15	homomorphic	homomorphic	ADJ
m-769	8	16	images	image	NOUN
m-769	8	17	,	,	PUNCT
m-769	8	18	compact	compact	ADJ
m-769	8	19	hausdorff	hausdorff	NOUN
m-769	8	20	space	space	NOUN
m-769	8	21	,	,	PUNCT
m-769	8	22	transition	transition	NOUN
m-769	8	23	group	group	NOUN
m-769	8	24	,	,	PUNCT
m-769	8	25	minimal	minimal	ADJ
m-769	8	26	right	right	ADJ
m-769	8	27	ideal	ideal	NOUN
m-769	8	28	.	.	PUNCT
m-769	9	1	1	1	X
m-769	9	2	.	.	X
m-769	9	3	introduction	introduction	NOUN
m-769	9	4	the	the	DET
m-769	9	5	study	study	NOUN
m-769	9	6	of	of	ADP
m-769	9	7	transformation	transformation	NOUN
m-769	9	8	groups	group	NOUN
m-769	9	9	with	with	ADP
m-769	9	10	compact	compact	ADJ
m-769	9	11	hausdorff	hausdorff	NOUN
m-769	9	12	phase	phase	NOUN
m-769	9	13	spaces	space	NOUN
m-769	9	14	has	have	VERB
m-769	9	15	significant	significant	ADJ
m-769	9	16	implications	implication	NOUN
m-769	9	17	in	in	ADP
m-769	9	18	various	various	ADJ
m-769	9	19	mathematical	mathematical	ADJ
m-769	9	20	and	and	CCONJ
m-769	9	21	applied	apply	VERB
m-769	9	22	fields	field	NOUN
m-769	9	23	.	.	PUNCT
m-769	10	1	bowen	bowen	PROPN
m-769	10	2	's	's	PART
m-769	10	3	[	[	X
m-769	10	4	1	1	NUM
m-769	10	5	]	]	X
m-769	10	6	foundational	foundational	ADJ
m-769	10	7	work	work	NOUN
m-769	10	8	explores	explore	VERB
m-769	10	9	the	the	DET
m-769	10	10	concept	concept	NOUN
m-769	10	11	of	of	ADP
m-769	10	12	proximal	proximal	ADJ
m-769	10	13	equivalence	equivalence	NOUN
m-769	10	14	in	in	ADP
m-769	10	15	topological	topological	ADJ
m-769	10	16	dynamics	dynamic	NOUN
m-769	10	17	,	,	PUNCT
m-769	10	18	providing	provide	VERB
m-769	10	19	insights	insight	NOUN
m-769	10	20	into	into	ADP
m-769	10	21	the	the	DET
m-769	10	22	connection	connection	NOUN
m-769	10	23	between	between	ADP
m-769	10	24	dynamical	dynamical	ADJ
m-769	10	25	systems	system	NOUN
m-769	10	26	and	and	CCONJ
m-769	10	27	the	the	DET
m-769	10	28	relations	relation	NOUN
m-769	10	29	studied	study	VERB
m-769	10	30	in	in	ADP
m-769	10	31	his	his	PRON
m-769	10	32	paper.[2]and	paper.[2]and	NOUN
m-769	10	33	[	[	X
m-769	10	34	29	29	NUM
m-769	10	35	]	]	PUNCT
m-769	10	36	,	,	PUNCT
m-769	10	37	contributed	contribute	VERB
m-769	10	38	to	to	ADP
m-769	10	39	the	the	DET
m-769	10	40	study	study	NOUN
m-769	10	41	of	of	ADP
m-769	10	42	enveloping	envelop	VERB
m-769	10	43	semigroups	semigroup	NOUN
m-769	10	44	in	in	ADP
m-769	10	45	topological	topological	ADJ
m-769	10	46	transformation	transformation	NOUN
m-769	10	47	groups	group	NOUN
m-769	10	48	,	,	PUNCT
m-769	10	49	offering	offer	VERB
m-769	10	50	valuable	valuable	ADJ
m-769	10	51	insights	insight	NOUN
m-769	10	52	into	into	ADP
m-769	10	53	their	their	PRON
m-769	10	54	algebraic	algebraic	ADJ
m-769	10	55	properties	property	NOUN
m-769	10	56	and	and	CCONJ
m-769	10	57	role	role	NOUN
m-769	10	58	in	in	ADP
m-769	10	59	capturing	capture	VERB
m-769	10	60	the	the	DET
m-769	10	61	dynamics	dynamic	NOUN
m-769	10	62	of	of	ADP
m-769	10	63	homeomorphisms	homeomorphisms	PROPN
m-769	10	64	.	.	PUNCT
m-769	11	1	[	[	X
m-769	11	2	3	3	NUM
m-769	11	3	]	]	X
m-769	11	4	work	work	NOUN
m-769	11	5	focuses	focus	VERB
m-769	11	6	on	on	ADP
m-769	11	7	minimal	minimal	ADJ
m-769	11	8	right	right	ADJ
m-769	11	9	ideals	ideal	NOUN
m-769	11	10	in	in	ADP
m-769	11	11	semigroups	semigroup	NOUN
m-769	11	12	arising	arise	VERB
m-769	11	13	from	from	ADP
m-769	11	14	continuous	continuous	ADJ
m-769	11	15	maps	map	NOUN
m-769	11	16	,	,	PUNCT
m-769	11	17	providing	provide	VERB
m-769	11	18	a	a	DET
m-769	11	19	ijo	ijo	PROPN
m-769	11	20	journals	journal	NOUN
m-769	11	21	doi	doi	NOUN
m-769	11	22	10.5281	10.5281	NUM
m-769	11	23	/	/	SYM
m-769	11	24	zenodo.10443958	zenodo.10443958	PROPN
m-769	11	25	volume	volume	NOUN
m-769	11	26	06	06	NUM
m-769	12	1	|	|	ADV
m-769	12	2	issue	issue	NOUN
m-769	12	3	12	12	NUM
m-769	12	4	|	|	CCONJ
m-769	12	5	december	december	PROPN
m-769	12	6	2023	2023	NUM
m-769	13	1	|	|	ADV
m-769	13	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-769	13	3	9	9	NUM
m-769	13	4	ijo	ijo	PROPN
m-769	13	5	international	international	PROPN
m-769	13	6	journal	journal	PROPN
m-769	13	7	of	of	ADP
m-769	13	8	mathematics	mathematics	PROPN
m-769	13	9	(	(	PUNCT
m-769	13	10	issn	issn	PROPN
m-769	13	11	:	:	PUNCT
m-769	13	12	2992	2992	NUM
m-769	13	13	-	-	SYM
m-769	13	14	4421	4421	NUM
m-769	13	15	)	)	PUNCT
m-769	13	16	michael	michael	PROPN
m-769	13	17	n.	n.	PROPN
m-769	13	18	john	john	PROPN
m-769	14	1	*	*	PROPN
m-769	14	2	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-769	14	3	volume	volume	NOUN
m-769	14	4	06	06	NUM
m-769	14	5	issue	issue	NOUN
m-769	14	6	12	12	NUM
m-769	14	7	||	||	NOUN
m-769	14	8	dec	dec	PROPN
m-769	14	9	.	.	PROPN
m-769	14	10	,	,	PUNCT
m-769	14	11	2023	2023	NUM
m-769	14	12	||	||	NOUN
m-769	15	1	algebraic	algebraic	ADJ
m-769	15	2	and	and	CCONJ
m-769	15	3	topological	topological	ADJ
m-769	15	4	analysis	analysis	NOUN
m-769	15	5	of	of	ADP
m-769	15	6	enveloping	envelop	VERB
m-769	15	7	semigroups	semigroup	NOUN
m-769	15	8	in	in	ADP
m-769	15	9	transformation	transformation	NOUN
m-769	15	10	groups	group	NOUN
m-769	15	11	:	:	PUNCT
m-769	15	12	proximal	proximal	ADJ
m-769	15	13	equivalence	equivalence	NOUN
m-769	15	14	and	and	CCONJ
m-769	15	15	homomorphic	homomorphic	ADJ
m-769	15	16	image	image	NOUN
m-769	15	17	relevant	relevant	ADJ
m-769	15	18	perspective	perspective	NOUN
m-769	15	19	for	for	ADP
m-769	15	20	the	the	DET
m-769	15	21	investigation	investigation	NOUN
m-769	15	22	of	of	ADP
m-769	15	23	such	such	ADJ
m-769	15	24	ideals	ideal	NOUN
m-769	15	25	in	in	ADP
m-769	15	26	enveloping	envelop	VERB
m-769	15	27	semigroups	semigroup	NOUN
m-769	15	28	.	.	PUNCT
m-769	16	1	[	[	X
m-769	16	2	4	4	NUM
m-769	16	3	]	]	PUNCT
m-769	16	4	,	,	PUNCT
m-769	16	5	also	also	ADV
m-769	16	6	contributed	contribute	VERB
m-769	16	7	to	to	ADP
m-769	16	8	the	the	DET
m-769	16	9	understanding	understanding	NOUN
m-769	16	10	of	of	ADP
m-769	16	11	enveloping	envelop	VERB
m-769	16	12	semigroups	semigroup	NOUN
m-769	16	13	in	in	ADP
m-769	16	14	the	the	DET
m-769	16	15	context	context	NOUN
m-769	16	16	of	of	ADP
m-769	16	17	topological	topological	ADJ
m-769	16	18	dynamics	dynamic	NOUN
m-769	16	19	,	,	PUNCT
m-769	16	20	emphasizing	emphasize	VERB
m-769	16	21	their	their	PRON
m-769	16	22	role	role	NOUN
m-769	16	23	in	in	ADP
m-769	16	24	capturing	capture	VERB
m-769	16	25	dynamic	dynamic	ADJ
m-769	16	26	behavior	behavior	NOUN
m-769	16	27	through	through	ADP
m-769	16	28	algebraic	algebraic	ADJ
m-769	16	29	structures.this	structures.this	PRON
m-769	16	30	paper	paper	NOUN
m-769	16	31	focuses	focus	VERB
m-769	16	32	on	on	ADP
m-769	16	33	the	the	DET
m-769	16	34	enveloping	enveloping	NOUN
m-769	16	35	semigroupe	semigroupe	NOUN
m-769	16	36	associated	associate	VERB
m-769	16	37	with	with	ADP
m-769	16	38	such	such	ADJ
m-769	16	39	groups	group	NOUN
m-769	16	40	,	,	PUNCT
m-769	16	41	exploring	explore	VERB
m-769	16	42	its	its	PRON
m-769	16	43	algebraic	algebraic	ADJ
m-769	16	44	and	and	CCONJ
m-769	16	45	topological	topological	ADJ
m-769	16	46	properties	property	NOUN
m-769	16	47	.	.	PUNCT
m-769	17	1	the	the	DET
m-769	17	2	transition	transition	NOUN
m-769	17	3	groupg	groupg	NOUN
m-769	17	4	is	be	AUX
m-769	17	5	viewed	view	VERB
m-769	17	6	as	as	ADP
m-769	17	7	a	a	DET
m-769	17	8	group	group	NOUN
m-769	17	9	of	of	ADP
m-769	17	10	homeomorphisms	homeomorphism	NOUN
m-769	17	11	,	,	PUNCT
m-769	17	12	and	and	CCONJ
m-769	17	13	e	e	NOUN
m-769	17	14	is	be	AUX
m-769	17	15	defined	define	VERB
m-769	17	16	as	as	ADP
m-769	17	17	the	the	DET
m-769	17	18	closure	closure	NOUN
m-769	17	19	of	of	ADP
m-769	17	20	g	g	NOUN
m-769	17	21	in	in	ADP
m-769	17	22	x×x	x×x	PROPN
m-769	17	23	.	.	PUNCT
m-769	18	1	we	we	PRON
m-769	18	2	aim	aim	VERB
m-769	18	3	to	to	PART
m-769	18	4	establish	establish	VERB
m-769	18	5	a	a	DET
m-769	18	6	link	link	NOUN
m-769	18	7	between	between	ADP
m-769	18	8	the	the	DET
m-769	18	9	proximal	proximal	ADJ
m-769	18	10	equivalence	equivalence	NOUN
m-769	18	11	relation	relation	NOUN
m-769	18	12	in	in	ADP
m-769	18	13	x	x	PUNCT
m-769	18	14	and	and	CCONJ
m-769	18	15	the	the	DET
m-769	18	16	structure	structure	NOUN
m-769	18	17	of	of	ADP
m-769	18	18	e	e	NOUN
m-769	18	19	,	,	PUNCT
m-769	18	20	specifically	specifically	ADV
m-769	18	21	the	the	DET
m-769	18	22	existence	existence	NOUN
m-769	18	23	of	of	ADP
m-769	18	24	a	a	DET
m-769	18	25	unique	unique	ADJ
m-769	18	26	minimal	minimal	ADJ
m-769	18	27	right	right	ADJ
m-769	18	28	ideal	ideal	NOUN
m-769	18	29	.	.	PUNCT
m-769	19	1	additionally	additionally	ADV
m-769	19	2	,	,	PUNCT
m-769	19	3	we	we	PRON
m-769	19	4	delve	delve	VERB
m-769	19	5	into	into	ADP
m-769	19	6	the	the	DET
m-769	19	7	analysis	analysis	NOUN
m-769	19	8	of	of	ADP
m-769	19	9	homomorphic	homomorphic	ADJ
m-769	19	10	images	image	NOUN
m-769	19	11	of	of	ADP
m-769	19	12	transformation	transformation	NOUN
m-769	19	13	groups	group	NOUN
m-769	19	14	through	through	ADP
m-769	19	15	their	their	PRON
m-769	19	16	enveloping	enveloping	NOUN
m-769	19	17	semigroups	semigroup	NOUN
m-769	19	18	.	.	PUNCT
m-769	20	1	see	see	VERB
m-769	20	2	[	[	X
m-769	20	3	26	26	NUM
m-769	20	4	]	]	PUNCT
m-769	20	5	,	,	PUNCT
m-769	20	6	[	[	X
m-769	20	7	27	27	NUM
m-769	20	8	]	]	PUNCT
m-769	20	9	and	and	CCONJ
m-769	20	10	[	[	X
m-769	20	11	31	31	NUM
m-769	20	12	]	]	SYM
m-769	20	13	2	2	NUM
m-769	20	14	.	.	PUNCT
m-769	20	15	preliminaries	preliminary	NOUN
m-769	20	16	transformation	transformation	NOUN
m-769	20	17	groups	group	NOUN
m-769	20	18	2.1let	2.1let	NOUN
m-769	20	19	's	's	PART
m-769	20	20	look	look	NOUN
m-769	20	21	into	into	ADP
m-769	20	22	the	the	DET
m-769	20	23	mathematical	mathematical	ADJ
m-769	20	24	definition	definition	NOUN
m-769	20	25	of	of	ADP
m-769	20	26	a	a	DET
m-769	20	27	transformation	transformation	NOUN
m-769	20	28	group	group	NOUN
m-769	20	29	(	(	PUNCT
m-769	20	30	x	x	X
m-769	20	31	,	,	PUNCT
m-769	20	32	t	t	PROPN
m-769	20	33	,	,	PUNCT
m-769	20	34	μ	μ	NOUN
m-769	20	35	)	)	PUNCT
m-769	20	36	with	with	ADP
m-769	20	37	a	a	DET
m-769	20	38	compact	compact	ADJ
m-769	20	39	hausdorff	hausdorff	NOUN
m-769	20	40	phase	phase	NOUN
m-769	20	41	space	space	NOUN
m-769	20	42	x	x	NOUN
m-769	20	43	,	,	PUNCT
m-769	20	44	provide	provide	VERB
m-769	20	45	an	an	DET
m-769	20	46	illustration	illustration	NOUN
m-769	20	47	,	,	PUNCT
m-769	20	48	and	and	CCONJ
m-769	20	49	explore	explore	VERB
m-769	20	50	an	an	DET
m-769	20	51	example	example	NOUN
m-769	20	52	.	.	PUNCT
m-769	21	1	mathematical	mathematical	ADJ
m-769	21	2	definition	definition	NOUN
m-769	21	3	:	:	PUNCT
m-769	21	4	1	1	X
m-769	21	5	.	.	X
m-769	21	6	compact	compact	ADJ
m-769	21	7	hausdorff	hausdorff	NOUN
m-769	21	8	phase	phase	NOUN
m-769	21	9	space	space	NOUN
m-769	21	10	x	x	NOUN
m-769	21	11	:	:	PUNCT
m-769	21	12			NOUN
m-769	21	13	x	x	PUNCT
m-769	21	14	is	be	AUX
m-769	21	15	a	a	DET
m-769	21	16	topological	topological	ADJ
m-769	21	17	space	space	NOUN
m-769	21	18	that	that	PRON
m-769	21	19	is	be	AUX
m-769	21	20	both	both	CCONJ
m-769	21	21	compact	compact	ADJ
m-769	21	22	and	and	CCONJ
m-769	21	23	hausdorff	hausdorff	NOUN
m-769	21	24	.	.	PUNCT
m-769	22	1	compactness	compactness	NOUN
m-769	22	2	ensures	ensure	VERB
m-769	22	3	that	that	SCONJ
m-769	22	4	every	every	DET
m-769	22	5	open	open	ADJ
m-769	22	6	cover	cover	NOUN
m-769	22	7	has	have	VERB
m-769	22	8	a	a	DET
m-769	22	9	finite	finite	ADJ
m-769	22	10	subcover	subcover	PROPN
m-769	22	11	,	,	PUNCT
m-769	22	12	and	and	CCONJ
m-769	22	13	hausdorffness	hausdorffness	NOUN
m-769	22	14	guarantees	guarantee	VERB
m-769	22	15	the	the	DET
m-769	22	16	separation	separation	NOUN
m-769	22	17	of	of	ADP
m-769	22	18	distinct	distinct	ADJ
m-769	22	19	points	point	NOUN
m-769	22	20	by	by	ADP
m-769	22	21	disjoint	disjoint	NOUN
m-769	22	22	open	open	ADJ
m-769	22	23	sets	set	NOUN
m-769	22	24	.	.	PUNCT
m-769	23	1	2	2	X
m-769	23	2	.	.	X
m-769	23	3	group	group	NOUN
m-769	23	4	of	of	ADP
m-769	23	5	homeomorphisms	homeomorphisms	PROPN
m-769	23	6	t	t	PROPN
m-769	23	7	:	:	PUNCT
m-769	23	8			PRON
m-769	23	9	t	t	PROPN
m-769	23	10	is	be	AUX
m-769	23	11	a	a	DET
m-769	23	12	group	group	NOUN
m-769	23	13	consisting	consist	VERB
m-769	23	14	of	of	ADP
m-769	23	15	homeomorphisms	homeomorphism	NOUN
m-769	23	16	from	from	ADP
m-769	23	17	x	x	PUNCT
m-769	23	18	to	to	ADP
m-769	23	19	itself	itself	PRON
m-769	23	20	.	.	PUNCT
m-769	24	1	a	a	DET
m-769	24	2	homeomorphism	homeomorphism	NOUN
m-769	24	3	is	be	AUX
m-769	24	4	a	a	DET
m-769	24	5	continuous	continuous	ADJ
m-769	24	6	bijective	bijective	ADJ
m-769	24	7	map	map	NOUN
m-769	24	8	with	with	ADP
m-769	24	9	a	a	DET
m-769	24	10	continuous	continuous	ADJ
m-769	24	11	inverse	inverse	NOUN
m-769	24	12	,	,	PUNCT
m-769	24	13	preserving	preserve	VERB
m-769	24	14	the	the	DET
m-769	24	15	topological	topological	ADJ
m-769	24	16	structure	structure	NOUN
m-769	24	17	of	of	ADP
m-769	24	18	the	the	DET
m-769	24	19	space	space	NOUN
m-769	24	20	.	.	PUNCT
m-769	25	1	3	3	X
m-769	25	2	.	.	X
m-769	25	3	continuous	continuous	ADJ
m-769	25	4	action	action	NOUN
m-769	25	5	μ	μ	NOUN
m-769	25	6	:	:	PUNCT
m-769	25	7			NOUN
m-769	25	8	the	the	DET
m-769	25	9	action	action	NOUN
m-769	25	10	μ	μ	NOUN
m-769	25	11	:	:	PUNCT
m-769	25	12	t×x→x	t×x→x	PROPN
m-769	25	13	represents	represent	VERB
m-769	25	14	how	how	SCONJ
m-769	25	15	elements	element	NOUN
m-769	25	16	of	of	ADP
m-769	25	17	the	the	DET
m-769	25	18	group	group	NOUN
m-769	25	19	t	t	PROPN
m-769	25	20	act	act	VERB
m-769	25	21	on	on	ADP
m-769	25	22	the	the	DET
m-769	25	23	space	space	NOUN
m-769	25	24	x.	x.	NOUN
m-769	26	1	it	it	PRON
m-769	26	2	is	be	AUX
m-769	26	3	a	a	DET
m-769	26	4	continuous	continuous	ADJ
m-769	26	5	map	map	NOUN
m-769	26	6	satisfying	satisfying	ADJ
m-769	26	7	:	:	PUNCT
m-769	26	8			PRON
m-769	26	9	μ(e	μ(e	NUM
m-769	26	10	,	,	PUNCT
m-769	26	11	x)=x	x)=x	PROPN
m-769	26	12	for	for	ADP
m-769	26	13	all	all	DET
m-769	26	14	x∈x	x∈x	NOUN
m-769	26	15	,	,	PUNCT
m-769	26	16	where	where	SCONJ
m-769	26	17	e	e	NOUN
m-769	26	18	is	be	AUX
m-769	26	19	the	the	DET
m-769	26	20	identity	identity	NOUN
m-769	26	21	element	element	NOUN
m-769	26	22	of	of	ADP
m-769	26	23	t.	t.	PROPN
m-769	26	24			NOUN
m-769	26	25	μ(g	μ(g	PROPN
m-769	26	26	,	,	PUNCT
m-769	26	27	μ(h	μ(h	NOUN
m-769	26	28	,	,	PUNCT
m-769	26	29	x))=μ(gh	x))=μ(gh	PROPN
m-769	26	30	,	,	PUNCT
m-769	26	31	x	x	NOUN
m-769	26	32	)	)	PUNCT
m-769	26	33	for	for	ADP
m-769	26	34	all	all	DET
m-769	26	35	g	g	NOUN
m-769	26	36	,	,	PUNCT
m-769	26	37	h∈t	h∈t	ADJ
m-769	26	38	and	and	CCONJ
m-769	26	39	x∈x	x∈x	ADJ
m-769	26	40	.	.	PUNCT
m-769	27	1	illustration	illustration	NOUN
m-769	27	2	2.2	2.2	NUM
m-769	27	3	.	.	PUNCT
m-769	28	1	consider	consider	VERB
m-769	28	2	a	a	DET
m-769	28	3	transformation	transformation	NOUN
m-769	28	4	group	group	NOUN
m-769	28	5	on	on	ADP
m-769	28	6	the	the	DET
m-769	28	7	unit	unit	NOUN
m-769	28	8	circle	circle	NOUN
m-769	28	9	in	in	ADP
m-769	28	10	the	the	DET
m-769	28	11	complex	complex	ADJ
m-769	28	12	plane	plane	NOUN
m-769	28	13	.	.	PUNCT
m-769	29	1	let	let	VERB
m-769	29	2	x	x	PRON
m-769	29	3	be	be	AUX
m-769	29	4	the	the	DET
m-769	29	5	unit	unit	NOUN
m-769	29	6	circle	circle	NOUN
m-769	29	7	,	,	PUNCT
m-769	29	8	t	t	PROPN
m-769	29	9	be	be	AUX
m-769	29	10	the	the	DET
m-769	29	11	group	group	NOUN
m-769	29	12	of	of	ADP
m-769	29	13	rotations	rotation	NOUN
m-769	29	14	around	around	ADP
m-769	29	15	the	the	DET
m-769	29	16	circle	circle	NOUN
m-769	29	17	,	,	PUNCT
m-769	29	18	and	and	CCONJ
m-769	29	19	μ	μ	NOUN
m-769	29	20	be	be	VERB
m-769	29	21	the	the	DET
m-769	29	22	action	action	NOUN
m-769	29	23	ijo	ijo	PROPN
m-769	29	24	journals	journal	NOUN
m-769	29	25	doi	doi	NOUN
m-769	29	26	10.5281	10.5281	NUM
m-769	29	27	/	/	SYM
m-769	29	28	zenodo.10443958	zenodo.10443958	PROPN
m-769	29	29	volume	volume	NOUN
m-769	29	30	06	06	NUM
m-769	30	1	|	|	ADV
m-769	30	2	issue	issue	NOUN
m-769	30	3	12	12	NUM
m-769	30	4	|	|	CCONJ
m-769	30	5	december	december	PROPN
m-769	30	6	2023	2023	NUM
m-769	31	1	|	|	ADV
m-769	31	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-769	31	3	10	10	NUM
m-769	31	4	ijo	ijo	PROPN
m-769	31	5	international	international	PROPN
m-769	31	6	journal	journal	PROPN
m-769	31	7	of	of	ADP
m-769	31	8	mathematics	mathematics	PROPN
m-769	31	9	(	(	PUNCT
m-769	31	10	issn	issn	PROPN
m-769	31	11	:	:	PUNCT
m-769	31	12	2992	2992	NUM
m-769	31	13	-	-	SYM
m-769	31	14	4421	4421	NUM
m-769	31	15	)	)	PUNCT
m-769	32	1	michael	michael	PROPN
m-769	32	2	n.	n.	PROPN
m-769	32	3	john	john	PROPN
m-769	32	4	*	*	PROPN
m-769	32	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-769	32	6	volume	volume	NOUN
m-769	32	7	06	06	NUM
m-769	32	8	issue	issue	NOUN
m-769	32	9	12	12	NUM
m-769	32	10	||	||	NOUN
m-769	32	11	dec	dec	PROPN
m-769	32	12	.	.	PROPN
m-769	32	13	,	,	PUNCT
m-769	32	14	2023	2023	NUM
m-769	32	15	||	||	NOUN
m-769	33	1	algebraic	algebraic	ADJ
m-769	33	2	and	and	CCONJ
m-769	33	3	topological	topological	ADJ
m-769	33	4	analysis	analysis	NOUN
m-769	33	5	of	of	ADP
m-769	33	6	enveloping	envelop	VERB
m-769	33	7	semigroups	semigroup	NOUN
m-769	33	8	in	in	ADP
m-769	33	9	transformation	transformation	NOUN
m-769	33	10	groups	group	NOUN
m-769	33	11	:	:	PUNCT
m-769	33	12	proximal	proximal	ADJ
m-769	33	13	equivalence	equivalence	NOUN
m-769	33	14	and	and	CCONJ
m-769	33	15	homomorphic	homomorphic	ADJ
m-769	33	16	image	image	NOUN
m-769	33	17	of	of	ADP
m-769	33	18	rotating	rotate	VERB
m-769	33	19	points	point	NOUN
m-769	33	20	.	.	PUNCT
m-769	34	1	each	each	DET
m-769	34	2	element	element	NOUN
m-769	34	3	in	in	ADP
m-769	34	4	t	t	PROPN
m-769	34	5	is	be	AUX
m-769	34	6	a	a	DET
m-769	34	7	rotation	rotation	NOUN
m-769	34	8	,	,	PUNCT
m-769	34	9	and	and	CCONJ
m-769	34	10	the	the	DET
m-769	34	11	action	action	NOUN
m-769	34	12	μ	μ	PROPN
m-769	34	13	is	be	AUX
m-769	34	14	the	the	DET
m-769	34	15	composition	composition	NOUN
m-769	34	16	of	of	ADP
m-769	34	17	rotations	rotation	NOUN
m-769	34	18	.	.	PUNCT
m-769	35	1	the	the	DET
m-769	35	2	conditions	condition	NOUN
m-769	35	3	ensure	ensure	VERB
m-769	35	4	that	that	SCONJ
m-769	35	5	the	the	DET
m-769	35	6	identity	identity	NOUN
m-769	35	7	rotation	rotation	NOUN
m-769	35	8	leaves	leave	VERB
m-769	35	9	points	point	VERB
m-769	35	10	unchanged	unchanged	ADJ
m-769	35	11	,	,	PUNCT
m-769	35	12	and	and	CCONJ
m-769	35	13	the	the	DET
m-769	35	14	composition	composition	NOUN
m-769	35	15	of	of	ADP
m-769	35	16	rotations	rotation	NOUN
m-769	35	17	is	be	AUX
m-769	35	18	associative	associative	ADJ
m-769	35	19	.	.	PUNCT
m-769	35	20	example	example	NOUN
m-769	36	1	2.3	2.3	NUM
m-769	36	2	.	.	PUNCT
m-769	37	1	let	let	VERB
m-769	37	2	x	x	PRON
m-769	37	3	be	be	AUX
m-769	37	4	the	the	DET
m-769	37	5	interval	interval	NOUN
m-769	37	6	[	[	X
m-769	37	7	0,1	0,1	NOUN
m-769	37	8	]	]	PUNCT
m-769	37	9	,	,	PUNCT
m-769	37	10	and	and	CCONJ
m-769	37	11	t	t	PROPN
m-769	37	12	be	be	AUX
m-769	37	13	the	the	DET
m-769	37	14	group	group	NOUN
m-769	37	15	of	of	ADP
m-769	37	16	all	all	DET
m-769	37	17	homeomorphisms	homeomorphism	NOUN
m-769	37	18	of	of	ADP
m-769	37	19	[	[	X
m-769	37	20	0,1	0,1	NUM
m-769	37	21	]	]	PUNCT
m-769	37	22	to	to	ADP
m-769	37	23	itself	itself	PRON
m-769	37	24	,	,	PUNCT
m-769	37	25	such	such	ADJ
m-769	37	26	as	as	ADP
m-769	37	27	translations	translation	NOUN
m-769	37	28	,	,	PUNCT
m-769	37	29	reflections	reflection	NOUN
m-769	37	30	,	,	PUNCT
m-769	37	31	and	and	CCONJ
m-769	37	32	compositions	composition	NOUN
m-769	37	33	of	of	ADP
m-769	37	34	such	such	ADJ
m-769	37	35	transformations	transformation	NOUN
m-769	37	36	.	.	PUNCT
m-769	38	1	the	the	DET
m-769	38	2	action	action	NOUN
m-769	38	3	μ	μ	PROPN
m-769	38	4	can	can	AUX
m-769	38	5	be	be	AUX
m-769	38	6	the	the	DET
m-769	38	7	translation	translation	NOUN
m-769	38	8	of	of	ADP
m-769	38	9	points	point	NOUN
m-769	38	10	.	.	PUNCT
m-769	39	1	for	for	ADP
m-769	39	2	a	a	DET
m-769	39	3	translation	translation	NOUN
m-769	39	4	t∈t	t∈t	NOUN
m-769	39	5	and	and	CCONJ
m-769	39	6	a	a	DET
m-769	39	7	point	point	NOUN
m-769	39	8	x∈x	x∈x	ADV
m-769	39	9	,	,	PUNCT
m-769	39	10	the	the	DET
m-769	39	11	action	action	NOUN
m-769	39	12	μ(t	μ(t	NUM
m-769	39	13	,	,	PUNCT
m-769	39	14	x	x	PRON
m-769	39	15	)	)	PUNCT
m-769	39	16	represents	represent	VERB
m-769	39	17	the	the	DET
m-769	39	18	new	new	ADJ
m-769	39	19	position	position	NOUN
m-769	39	20	of	of	ADP
m-769	39	21	the	the	DET
m-769	39	22	point	point	NOUN
m-769	39	23	after	after	ADP
m-769	39	24	the	the	DET
m-769	39	25	translation	translation	NOUN
m-769	39	26	.	.	PUNCT
m-769	40	1	the	the	DET
m-769	40	2	group	group	NOUN
m-769	40	3	structure	structure	NOUN
m-769	40	4	ensures	ensure	VERB
m-769	40	5	that	that	SCONJ
m-769	40	6	the	the	DET
m-769	40	7	identity	identity	NOUN
m-769	40	8	transformation	transformation	NOUN
m-769	40	9	leaves	leave	VERB
m-769	40	10	points	point	VERB
m-769	40	11	unchanged	unchanged	ADJ
m-769	40	12	,	,	PUNCT
m-769	40	13	and	and	CCONJ
m-769	40	14	the	the	DET
m-769	40	15	composition	composition	NOUN
m-769	40	16	of	of	ADP
m-769	40	17	transformations	transformation	NOUN
m-769	40	18	is	be	AUX
m-769	40	19	associative	associative	ADJ
m-769	40	20	.	.	PUNCT
m-769	41	1	in	in	ADP
m-769	41	2	summary	summary	NOUN
m-769	41	3	,	,	PUNCT
m-769	41	4	a	a	DET
m-769	41	5	transformation	transformation	NOUN
m-769	41	6	group	group	NOUN
m-769	41	7	with	with	ADP
m-769	41	8	a	a	DET
m-769	41	9	compact	compact	ADJ
m-769	41	10	hausdorff	hausdorff	NOUN
m-769	41	11	phase	phase	NOUN
m-769	41	12	space	space	NOUN
m-769	41	13	involves	involve	VERB
m-769	41	14	a	a	DET
m-769	41	15	topological	topological	ADJ
m-769	41	16	space	space	NOUN
m-769	41	17	,	,	PUNCT
m-769	41	18	a	a	DET
m-769	41	19	group	group	NOUN
m-769	41	20	of	of	ADP
m-769	41	21	homeomorphisms	homeomorphisms	PROPN
m-769	41	22	,	,	PUNCT
m-769	41	23	and	and	CCONJ
m-769	41	24	a	a	DET
m-769	41	25	continuous	continuous	ADJ
m-769	41	26	action	action	NOUN
m-769	41	27	representing	represent	VERB
m-769	41	28	transformations	transformation	NOUN
m-769	41	29	.	.	PUNCT
m-769	42	1	the	the	DET
m-769	42	2	example	example	NOUN
m-769	42	3	on	on	ADP
m-769	42	4	the	the	DET
m-769	42	5	unit	unit	NOUN
m-769	42	6	circle	circle	NOUN
m-769	42	7	and	and	CCONJ
m-769	42	8	interval	interval	NOUN
m-769	42	9	illustrates	illustrate	VERB
m-769	42	10	how	how	SCONJ
m-769	42	11	such	such	ADJ
m-769	42	12	groups	group	NOUN
m-769	42	13	can	can	AUX
m-769	42	14	capture	capture	VERB
m-769	42	15	symmetries	symmetry	NOUN
m-769	42	16	and	and	CCONJ
m-769	42	17	actions	action	NOUN
m-769	42	18	on	on	ADP
m-769	42	19	different	different	ADJ
m-769	42	20	spaces	space	NOUN
m-769	42	21	.	.	PUNCT
m-769	43	1	enveloping	envelop	VERB
m-769	43	2	semigroup	semigroup	PROPN
m-769	43	3	2.2.let	2.2.let	NUM
m-769	43	4	(	(	PUNCT
m-769	43	5	x	x	NOUN
m-769	43	6	,	,	PUNCT
m-769	43	7	g	g	PROPN
m-769	43	8	,	,	PUNCT
m-769	43	9	μ	μ	NOUN
m-769	43	10	)	)	PUNCT
m-769	43	11	be	be	VERB
m-769	43	12	a	a	DET
m-769	43	13	transformation	transformation	NOUN
m-769	43	14	semigroup	semigroup	NOUN
m-769	43	15	with	with	ADP
m-769	43	16	a	a	DET
m-769	43	17	phase	phase	NOUN
m-769	43	18	space	space	NOUN
m-769	43	19	x	x	NOUN
m-769	43	20	,	,	PUNCT
m-769	43	21	a	a	DET
m-769	43	22	semigroupg	semigroupg	NOUN
m-769	43	23	of	of	ADP
m-769	43	24	transformations	transformation	NOUN
m-769	43	25	on	on	ADP
m-769	43	26	x	x	NOUN
m-769	43	27	,	,	PUNCT
m-769	43	28	and	and	CCONJ
m-769	43	29	a	a	DET
m-769	43	30	continuous	continuous	ADJ
m-769	43	31	action	action	NOUN
m-769	43	32	μ	μ	NOUN
m-769	43	33	:	:	PUNCT
m-769	43	34	g×x→x	g×x→x	NOUN
m-769	43	35	.	.	PUNCT
m-769	44	1	the	the	DET
m-769	44	2	enveloping	enveloping	NOUN
m-769	44	3	semigroupe	semigroupe	NOUN
m-769	44	4	is	be	AUX
m-769	44	5	defined	define	VERB
m-769	44	6	as	as	ADP
m-769	44	7	the	the	DET
m-769	44	8	closure	closure	NOUN
m-769	44	9	of	of	ADP
m-769	44	10	the	the	DET
m-769	44	11	transition	transition	NOUN
m-769	44	12	semigroupg	semigroupg	NOUN
m-769	44	13	in	in	ADP
m-769	44	14	the	the	DET
m-769	44	15	product	product	NOUN
m-769	44	16	space	space	NOUN
m-769	44	17	x×x	x×x	PROPN
m-769	44	18	,	,	PUNCT
m-769	44	19	denoted	denote	VERB
m-769	44	20	as	as	ADP
m-769	44	21	�	�	PROPN
m-769	44	22	=	=	SYM
m-769	44	23	�	�	PROPN
m-769	44	24	̅	̅	NOUN
m-769	44	25	this	this	DET
m-769	44	26	closure	closure	NOUN
m-769	44	27	operation	operation	NOUN
m-769	44	28	ensures	ensure	VERB
m-769	44	29	that	that	SCONJ
m-769	44	30	the	the	DET
m-769	44	31	product	product	NOUN
m-769	44	32	of	of	ADP
m-769	44	33	any	any	DET
m-769	44	34	two	two	NUM
m-769	44	35	elements	element	NOUN
m-769	44	36	in	in	ADP
m-769	44	37	g	g	PROPN
m-769	44	38	remains	remain	VERB
m-769	44	39	in	in	ADP
m-769	44	40	the	the	DET
m-769	44	41	enveloping	enveloping	NOUN
m-769	44	42	semigroup	semigroup	NOUN
m-769	44	43	,	,	PUNCT
m-769	44	44	providing	provide	VERB
m-769	44	45	a	a	DET
m-769	44	46	continuous	continuous	ADJ
m-769	44	47	extension	extension	NOUN
m-769	44	48	to	to	ADP
m-769	44	49	the	the	DET
m-769	44	50	transition	transition	NOUN
m-769	44	51	semigroup	semigroup	PROPN
m-769	44	52	.	.	PUNCT
m-769	45	1	example	example	NOUN
m-769	45	2	2.3	2.3	NUM
m-769	45	3	.	.	PUNCT
m-769	46	1	consider	consider	VERB
m-769	46	2	a	a	DET
m-769	46	3	transition	transition	NOUN
m-769	46	4	semigroup	semigroup	NOUN
m-769	46	5	of	of	ADP
m-769	46	6	rotations	rotation	NOUN
m-769	46	7	g	g	NOUN
m-769	46	8	acting	act	VERB
m-769	46	9	on	on	ADP
m-769	46	10	a	a	DET
m-769	46	11	unit	unit	NOUN
m-769	46	12	circle	circle	NOUN
m-769	46	13	x	x	PUNCT
m-769	46	14	in	in	ADP
m-769	46	15	the	the	DET
m-769	46	16	complex	complex	ADJ
m-769	46	17	plane	plane	NOUN
m-769	46	18	.	.	PUNCT
m-769	47	1	each	each	DET
m-769	47	2	element	element	NOUN
m-769	47	3	in	in	ADP
m-769	47	4	g	g	PROPN
m-769	47	5	is	be	AUX
m-769	47	6	a	a	DET
m-769	47	7	rotation	rotation	NOUN
m-769	47	8	around	around	ADP
m-769	47	9	the	the	DET
m-769	47	10	circle	circle	NOUN
m-769	47	11	.	.	PUNCT
m-769	48	1	the	the	DET
m-769	48	2	action	action	NOUN
m-769	48	3	μ	μ	PROPN
m-769	48	4	rotates	rotate	VERB
m-769	48	5	points	point	NOUN
m-769	48	6	on	on	ADP
m-769	48	7	the	the	DET
m-769	48	8	circle	circle	NOUN
m-769	48	9	.	.	PUNCT
m-769	49	1	the	the	DET
m-769	49	2	enveloping	enveloping	NOUN
m-769	49	3	semigroupe	semigroupe	NOUN
m-769	49	4	is	be	AUX
m-769	49	5	the	the	DET
m-769	49	6	closure	closure	NOUN
m-769	49	7	of	of	ADP
m-769	49	8	g	g	NOUN
m-769	49	9	in	in	ADP
m-769	49	10	the	the	DET
m-769	49	11	product	product	NOUN
m-769	49	12	space	space	NOUN
m-769	49	13	x×x	x×x	PROPN
m-769	49	14	,	,	PUNCT
m-769	49	15	where	where	SCONJ
m-769	49	16	the	the	DET
m-769	49	17	product	product	NOUN
m-769	49	18	of	of	ADP
m-769	49	19	two	two	NUM
m-769	49	20	rotations	rotation	NOUN
m-769	49	21	remains	remain	VERB
m-769	49	22	in	in	ADP
m-769	49	23	e.	e.	PROPN
m-769	49	24	illustration	illustration	PROPN
m-769	49	25	2.4.imagine	2.4.imagine	NUM
m-769	49	26	a	a	DET
m-769	49	27	clock	clock	NOUN
m-769	49	28	face	face	NOUN
m-769	49	29	representing	represent	VERB
m-769	49	30	the	the	DET
m-769	49	31	unit	unit	NOUN
m-769	49	32	circle	circle	NOUN
m-769	49	33	,	,	PUNCT
m-769	49	34	and	and	CCONJ
m-769	49	35	g	g	NOUN
m-769	49	36	as	as	ADP
m-769	49	37	the	the	DET
m-769	49	38	set	set	NOUN
m-769	49	39	of	of	ADP
m-769	49	40	all	all	DET
m-769	49	41	possible	possible	ADJ
m-769	49	42	hour	hour	NOUN
m-769	49	43	-	-	PUNCT
m-769	49	44	hand	hand	NOUN
m-769	49	45	rotations	rotation	NOUN
m-769	49	46	.	.	PUNCT
m-769	50	1	if	if	SCONJ
m-769	50	2	you	you	PRON
m-769	50	3	rotate	rotate	VERB
m-769	50	4	the	the	DET
m-769	50	5	hour	hour	NOUN
m-769	50	6	hand	hand	NOUN
m-769	50	7	to	to	ADP
m-769	50	8	3	3	NUM
m-769	50	9	and	and	CCONJ
m-769	50	10	then	then	ADV
m-769	50	11	to	to	ADP
m-769	50	12	4	4	NUM
m-769	50	13	,	,	PUNCT
m-769	50	14	the	the	DET
m-769	50	15	resulting	result	VERB
m-769	50	16	position	position	NOUN
m-769	50	17	lies	lie	VERB
m-769	50	18	in	in	ADP
m-769	50	19	the	the	DET
m-769	50	20	enveloping	enveloping	NOUN
m-769	50	21	semigroupe	semigroupe	NOUN
m-769	50	22	.	.	PUNCT
m-769	51	1	the	the	DET
m-769	51	2	closure	closure	NOUN
m-769	51	3	ensures	ensure	VERB
m-769	51	4	that	that	SCONJ
m-769	51	5	the	the	DET
m-769	51	6	product	product	NOUN
m-769	51	7	of	of	ADP
m-769	51	8	any	any	DET
m-769	51	9	two	two	NUM
m-769	51	10	rotations	rotation	NOUN
m-769	51	11	is	be	AUX
m-769	51	12	also	also	ADV
m-769	51	13	a	a	DET
m-769	51	14	valid	valid	ADJ
m-769	51	15	rotation	rotation	NOUN
m-769	51	16	,	,	PUNCT
m-769	51	17	creating	create	VERB
m-769	51	18	a	a	DET
m-769	51	19	continuous	continuous	ADJ
m-769	51	20	structure	structure	NOUN
m-769	51	21	on	on	ADP
m-769	51	22	the	the	DET
m-769	51	23	clock	clock	NOUN
m-769	51	24	face	face	NOUN
m-769	51	25	.	.	PUNCT
m-769	52	1	this	this	DET
m-769	52	2	extension	extension	NOUN
m-769	52	3	captures	capture	VERB
m-769	52	4	all	all	DET
m-769	52	5	possible	possible	ADJ
m-769	52	6	positions	position	NOUN
m-769	52	7	of	of	ADP
m-769	52	8	the	the	DET
m-769	52	9	hour	hour	NOUN
m-769	52	10	hand	hand	NOUN
m-769	52	11	under	under	ADP
m-769	52	12	continuous	continuous	ADJ
m-769	52	13	rotations	rotation	NOUN
m-769	52	14	,	,	PUNCT
m-769	52	15	forming	form	VERB
m-769	52	16	the	the	DET
m-769	52	17	enveloping	enveloping	NOUN
m-769	52	18	semigroupe	semigroupe	NOUN
m-769	52	19	.	.	PUNCT
m-769	53	1	in	in	ADP
m-769	53	2	summary	summary	NOUN
m-769	53	3	,	,	PUNCT
m-769	53	4	the	the	DET
m-769	53	5	enveloping	enveloping	NOUN
m-769	53	6	semigroup	semigroup	NOUN
m-769	53	7	is	be	AUX
m-769	53	8	a	a	DET
m-769	53	9	closure	closure	NOUN
m-769	53	10	of	of	ADP
m-769	53	11	the	the	DET
m-769	53	12	transition	transition	NOUN
m-769	53	13	semigroup	semigroup	NOUN
m-769	53	14	in	in	ADP
m-769	53	15	the	the	DET
m-769	53	16	product	product	NOUN
m-769	53	17	space	space	NOUN
m-769	53	18	,	,	PUNCT
m-769	53	19	providing	provide	VERB
m-769	53	20	a	a	DET
m-769	53	21	continuous	continuous	ADJ
m-769	53	22	extension	extension	NOUN
m-769	53	23	to	to	ADP
m-769	53	24	the	the	DET
m-769	53	25	original	original	ADJ
m-769	53	26	semigroup	semigroup	NOUN
m-769	53	27	.	.	PUNCT
m-769	54	1	the	the	DET
m-769	54	2	example	example	NOUN
m-769	54	3	ijo	ijo	PROPN
m-769	54	4	journals	journal	NOUN
m-769	54	5	doi	doi	NOUN
m-769	54	6	10.5281	10.5281	NUM
m-769	54	7	/	/	SYM
m-769	54	8	zenodo.10443958	zenodo.10443958	PROPN
m-769	54	9	volume	volume	NOUN
m-769	54	10	06	06	NUM
m-769	54	11	|	|	ADV
m-769	54	12	issue	issue	NOUN
m-769	54	13	12	12	NUM
m-769	55	1	|	|	CCONJ
m-769	55	2	december	december	PROPN
m-769	55	3	2023	2023	NUM
m-769	55	4	|	|	ADV
m-769	55	5	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-769	55	6	11	11	NUM
m-769	55	7	ijo	ijo	PROPN
m-769	55	8	international	international	PROPN
m-769	55	9	journal	journal	PROPN
m-769	55	10	of	of	ADP
m-769	55	11	mathematics	mathematics	PROPN
m-769	55	12	(	(	PUNCT
m-769	55	13	issn	issn	PROPN
m-769	55	14	:	:	PUNCT
m-769	55	15	2992	2992	NUM
m-769	55	16	-	-	SYM
m-769	55	17	4421	4421	NUM
m-769	55	18	)	)	PUNCT
m-769	56	1	michael	michael	PROPN
m-769	56	2	n.	n.	PROPN
m-769	56	3	john	john	PROPN
m-769	56	4	*	*	PROPN
m-769	56	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-769	56	6	volume	volume	NOUN
m-769	56	7	06	06	NUM
m-769	56	8	issue	issue	NOUN
m-769	56	9	12	12	NUM
m-769	56	10	||	||	NOUN
m-769	56	11	dec	dec	PROPN
m-769	56	12	.	.	PROPN
m-769	56	13	,	,	PUNCT
m-769	56	14	2023	2023	NUM
m-769	56	15	||	||	NOUN
m-769	57	1	algebraic	algebraic	ADJ
m-769	57	2	and	and	CCONJ
m-769	57	3	topological	topological	ADJ
m-769	57	4	analysis	analysis	NOUN
m-769	57	5	of	of	ADP
m-769	57	6	enveloping	envelop	VERB
m-769	57	7	semigroups	semigroup	NOUN
m-769	57	8	in	in	ADP
m-769	57	9	transformation	transformation	NOUN
m-769	57	10	groups	group	NOUN
m-769	57	11	:	:	PUNCT
m-769	57	12	proximal	proximal	ADJ
m-769	57	13	equivalence	equivalence	NOUN
m-769	57	14	and	and	CCONJ
m-769	57	15	homomorphic	homomorphic	ADJ
m-769	57	16	image	image	NOUN
m-769	57	17	of	of	ADP
m-769	57	18	rotations	rotation	NOUN
m-769	57	19	on	on	ADP
m-769	57	20	a	a	DET
m-769	57	21	unit	unit	NOUN
m-769	57	22	circle	circle	NOUN
m-769	57	23	illustrates	illustrate	VERB
m-769	57	24	how	how	SCONJ
m-769	57	25	this	this	DET
m-769	57	26	enveloping	enveloping	NOUN
m-769	57	27	semigroup	semigroup	NOUN
m-769	57	28	captures	capture	VERB
m-769	57	29	all	all	DET
m-769	57	30	possible	possible	ADJ
m-769	57	31	continuous	continuous	ADJ
m-769	57	32	transformations	transformation	NOUN
m-769	57	33	on	on	ADP
m-769	57	34	the	the	DET
m-769	57	35	given	give	VERB
m-769	57	36	phase	phase	NOUN
m-769	57	37	space	space	NOUN
m-769	57	38	.	.	PUNCT
m-769	58	1	3	3	X
m-769	58	2	.	.	X
m-769	58	3	definition	definition	NOUN
m-769	58	4	of	of	ADP
m-769	58	5	terms	term	NOUN
m-769	58	6	proximal	proximal	ADJ
m-769	58	7	equivalence	equivalence	NOUN
m-769	59	1	3.1.let	3.1.let	NUM
m-769	59	2	(	(	PUNCT
m-769	59	3	x	x	NOUN
m-769	59	4	,	,	PUNCT
m-769	59	5	g	g	PROPN
m-769	59	6	,	,	PUNCT
m-769	59	7	μ	μ	NOUN
m-769	59	8	)	)	PUNCT
m-769	59	9	be	be	VERB
m-769	59	10	a	a	DET
m-769	59	11	transformation	transformation	NOUN
m-769	59	12	semigroup	semigroup	NOUN
m-769	59	13	with	with	ADP
m-769	59	14	a	a	DET
m-769	59	15	phase	phase	NOUN
m-769	59	16	space	space	NOUN
m-769	59	17	x	x	NOUN
m-769	59	18	,	,	PUNCT
m-769	59	19	a	a	DET
m-769	59	20	semigroupg	semigroupg	NOUN
m-769	59	21	of	of	ADP
m-769	59	22	transformations	transformation	NOUN
m-769	59	23	on	on	ADP
m-769	59	24	x	x	NOUN
m-769	59	25	,	,	PUNCT
m-769	59	26	and	and	CCONJ
m-769	59	27	a	a	DET
m-769	59	28	continuous	continuous	ADJ
m-769	59	29	action	action	NOUN
m-769	59	30	μ	μ	NOUN
m-769	59	31	:	:	PUNCT
m-769	59	32	g×x→x	g×x→x	NOUN
m-769	59	33	.	.	PUNCT
m-769	60	1	the	the	DET
m-769	60	2	enveloping	enveloping	NOUN
m-769	60	3	semigroupe	semigroupe	NOUN
m-769	60	4	is	be	AUX
m-769	60	5	the	the	DET
m-769	60	6	closure	closure	NOUN
m-769	60	7	of	of	ADP
m-769	60	8	g	g	NOUN
m-769	60	9	in	in	ADP
m-769	60	10	the	the	DET
m-769	60	11	product	product	NOUN
m-769	60	12	space	space	NOUN
m-769	60	13	x×x	x×x	PROPN
m-769	60	14	,	,	PUNCT
m-769	60	15	i.e.	i.e.	X
m-769	60	16	,	,	PUNCT
m-769	60	17	�	�	PROPN
m-769	60	18	=	=	SYM
m-769	60	19	�	�	PROPN
m-769	60	20	̅.	̅.	PROPN
m-769	60	21	proximal	proximal	ADJ
m-769	60	22	equivalence	equivalence	NOUN
m-769	60	23	is	be	AUX
m-769	60	24	then	then	ADV
m-769	60	25	a	a	DET
m-769	60	26	relation∼	relation∼	NOUN
m-769	60	27	on	on	ADP
m-769	60	28	x	x	PUNCT
m-769	60	29	defined	define	VERB
m-769	60	30	as	as	SCONJ
m-769	60	31	follows	follow	VERB
m-769	60	32	:	:	PUNCT
m-769	60	33	for	for	ADP
m-769	60	34	x	x	NOUN
m-769	60	35	,	,	PUNCT
m-769	60	36	y∈x	y∈x	NOUN
m-769	60	37	,	,	PUNCT
m-769	60	38	we	we	PRON
m-769	60	39	say	say	VERB
m-769	60	40	that	that	SCONJ
m-769	60	41	x	x	PRON
m-769	60	42	is	be	AUX
m-769	60	43	proximally	proximally	ADV
m-769	60	44	equivalent	equivalent	ADJ
m-769	60	45	to	to	ADP
m-769	60	46	y	y	PROPN
m-769	60	47	,	,	PUNCT
m-769	60	48	denoted	denote	VERB
m-769	60	49	x∼y	x∼y	PROPN
m-769	60	50	,	,	PUNCT
m-769	60	51	if	if	SCONJ
m-769	60	52	there	there	PRON
m-769	60	53	exists	exist	VERB
m-769	60	54	a	a	DET
m-769	60	55	sequence	sequence	NOUN
m-769	60	56	(	(	PUNCT
m-769	60	57	gn)⊆g	gn)⊆g	PROPN
m-769	60	58	such	such	ADJ
m-769	60	59	that	that	DET
m-769	60	60	limn→∞μ(gn	limn→∞μ(gn	NOUN
m-769	60	61	,	,	PUNCT
m-769	60	62	x)=limn→∞μ(gn	x)=limn→∞μ(gn	NOUN
m-769	60	63	,	,	PUNCT
m-769	60	64	y	y	PROPN
m-769	60	65	)	)	PUNCT
m-769	60	66	.	.	PUNCT
m-769	61	1	in	in	ADP
m-769	61	2	simpler	simple	ADJ
m-769	61	3	terms	term	NOUN
m-769	61	4	,	,	PUNCT
m-769	61	5	two	two	NUM
m-769	61	6	points	point	NOUN
m-769	61	7	x	x	PUNCT
m-769	61	8	and	and	CCONJ
m-769	61	9	y	y	PROPN
m-769	61	10	are	be	AUX
m-769	61	11	proximally	proximally	ADV
m-769	61	12	equivalent	equivalent	ADJ
m-769	61	13	if	if	SCONJ
m-769	61	14	there	there	PRON
m-769	61	15	exists	exist	VERB
m-769	61	16	a	a	DET
m-769	61	17	sequence	sequence	NOUN
m-769	61	18	of	of	ADP
m-769	61	19	transformations	transformation	NOUN
m-769	61	20	from	from	ADP
m-769	61	21	the	the	DET
m-769	61	22	enveloping	enveloping	NOUN
m-769	61	23	semigroupe	semigroupe	NOUN
m-769	61	24	such	such	ADJ
m-769	61	25	that	that	SCONJ
m-769	61	26	the	the	DET
m-769	61	27	images	image	NOUN
m-769	61	28	of	of	ADP
m-769	61	29	x	x	X
m-769	61	30	and	and	CCONJ
m-769	61	31	y	y	PROPN
m-769	61	32	under	under	ADP
m-769	61	33	these	these	DET
m-769	61	34	transformations	transformation	NOUN
m-769	61	35	converge	converge	VERB
m-769	61	36	to	to	ADP
m-769	61	37	the	the	DET
m-769	61	38	same	same	ADJ
m-769	61	39	point	point	NOUN
m-769	61	40	.	.	PUNCT
m-769	62	1	illustration	illustration	NOUN
m-769	62	2	3.2.consider	3.2.consider	NUM
m-769	62	3	a	a	DET
m-769	62	4	transformation	transformation	NOUN
m-769	62	5	semigroupg	semigroupg	NOUN
m-769	62	6	consisting	consist	VERB
m-769	62	7	of	of	ADP
m-769	62	8	all	all	DET
m-769	62	9	translations	translation	NOUN
m-769	62	10	on	on	ADP
m-769	62	11	the	the	DET
m-769	62	12	real	real	ADJ
m-769	62	13	line	line	NOUN
m-769	62	14	x.	x.	NOUN
m-769	63	1	the	the	DET
m-769	63	2	enveloping	enveloping	NOUN
m-769	63	3	semigroupe	semigroupe	NOUN
m-769	63	4	is	be	AUX
m-769	63	5	the	the	DET
m-769	63	6	closure	closure	NOUN
m-769	63	7	of	of	ADP
m-769	63	8	g	g	NOUN
m-769	63	9	in	in	ADP
m-769	63	10	the	the	DET
m-769	63	11	product	product	NOUN
m-769	63	12	space	space	NOUN
m-769	63	13	r×r	r×r	PROPN
m-769	63	14	.	.	PROPN
m-769	63	15	proximal	proximal	ADJ
m-769	63	16	equivalence	equivalence	NOUN
m-769	63	17	in	in	ADP
m-769	63	18	this	this	DET
m-769	63	19	context	context	NOUN
m-769	63	20	would	would	AUX
m-769	63	21	mean	mean	VERB
m-769	63	22	that	that	SCONJ
m-769	63	23	two	two	NUM
m-769	63	24	points	point	NOUN
m-769	63	25	x	x	PUNCT
m-769	63	26	and	and	CCONJ
m-769	63	27	y	y	PROPN
m-769	63	28	are	be	AUX
m-769	63	29	considered	consider	VERB
m-769	63	30	equivalent	equivalent	ADJ
m-769	63	31	if	if	SCONJ
m-769	63	32	there	there	PRON
m-769	63	33	exists	exist	VERB
m-769	63	34	a	a	DET
m-769	63	35	sequence	sequence	NOUN
m-769	63	36	of	of	ADP
m-769	63	37	translations	translation	NOUN
m-769	63	38	that	that	PRON
m-769	63	39	brings	bring	VERB
m-769	63	40	x	x	PUNCT
m-769	63	41	and	and	CCONJ
m-769	63	42	y	y	PROPN
m-769	63	43	arbitrarily	arbitrarily	ADV
m-769	63	44	close	close	ADJ
m-769	63	45	to	to	ADP
m-769	63	46	each	each	DET
m-769	63	47	other	other	ADJ
m-769	63	48	.	.	PUNCT
m-769	63	49	example	example	NOUN
m-769	63	50	3.3	3.3	NUM
m-769	63	51	.	.	PUNCT
m-769	64	1	let	let	VERB
m-769	64	2	x	x	X
m-769	64	3	=	=	SYM
m-769	64	4	r	r	NOUN
m-769	64	5	,	,	PUNCT
m-769	64	6	and	and	CCONJ
m-769	64	7	g	g	PROPN
m-769	64	8	be	be	VERB
m-769	64	9	the	the	DET
m-769	64	10	semigroup	semigroup	NOUN
m-769	64	11	of	of	ADP
m-769	64	12	positive	positive	ADJ
m-769	64	13	translations	translation	NOUN
m-769	64	14	,	,	PUNCT
m-769	64	15	i.e.	i.e.	X
m-769	64	16	,	,	PUNCT
m-769	64	17	g={ta	g={ta	ADJ
m-769	64	18	∣a>0	∣a>0	PROPN
m-769	64	19	}	}	PUNCT
m-769	64	20	,	,	PUNCT
m-769	64	21	where	where	SCONJ
m-769	64	22	ta(x)=x+a	ta(x)=x+a	ADJ
m-769	64	23	.	.	PUNCT
m-769	65	1	the	the	DET
m-769	65	2	enveloping	enveloping	NOUN
m-769	65	3	semigroupe	semigroupe	NOUN
m-769	65	4	is	be	AUX
m-769	65	5	the	the	DET
m-769	65	6	closure	closure	NOUN
m-769	65	7	of	of	ADP
m-769	65	8	g.	g.	PROPN
m-769	65	9	two	two	NUM
m-769	65	10	points	point	NOUN
m-769	65	11	x	x	PUNCT
m-769	65	12	and	and	CCONJ
m-769	65	13	y	y	PROPN
m-769	65	14	are	be	AUX
m-769	65	15	proximally	proximally	ADV
m-769	65	16	equivalent	equivalent	ADJ
m-769	65	17	if	if	SCONJ
m-769	65	18	there	there	PRON
m-769	65	19	exists	exist	VERB
m-769	65	20	a	a	DET
m-769	65	21	sequence(an	sequence(an	NOUN
m-769	65	22	)	)	PUNCT
m-769	65	23	such	such	ADJ
m-769	65	24	that	that	SCONJ
m-769	65	25	limn→∞(x+an	limn→∞(x+an	PUNCT
m-769	65	26	)	)	PUNCT
m-769	66	1	=	=	SYM
m-769	66	2	limn→∞(y+an	limn→∞(y+an	NOUN
m-769	66	3	)	)	PUNCT
m-769	66	4	.	.	PUNCT
m-769	67	1	proximal	proximal	ADJ
m-769	67	2	equivalence	equivalence	NOUN
m-769	67	3	is	be	AUX
m-769	67	4	a	a	DET
m-769	67	5	relation	relation	NOUN
m-769	67	6	on	on	ADP
m-769	67	7	a	a	DET
m-769	67	8	phase	phase	NOUN
m-769	67	9	space	space	NOUN
m-769	67	10	x	x	NOUN
m-769	67	11	determined	determine	VERB
m-769	67	12	by	by	ADP
m-769	67	13	the	the	DET
m-769	67	14	behavior	behavior	NOUN
m-769	67	15	of	of	ADP
m-769	67	16	transformations	transformation	NOUN
m-769	67	17	in	in	ADP
m-769	67	18	the	the	DET
m-769	67	19	enveloping	enveloping	NOUN
m-769	67	20	semigroupe	semigroupe	NOUN
m-769	67	21	.	.	PUNCT
m-769	68	1	two	two	NUM
m-769	68	2	points	point	NOUN
m-769	68	3	are	be	AUX
m-769	68	4	considered	consider	VERB
m-769	68	5	proximally	proximally	ADV
m-769	68	6	equivalent	equivalent	ADJ
m-769	68	7	if	if	SCONJ
m-769	68	8	there	there	PRON
m-769	68	9	is	be	VERB
m-769	68	10	a	a	DET
m-769	68	11	sequence	sequence	NOUN
m-769	68	12	of	of	ADP
m-769	68	13	transformations	transformation	NOUN
m-769	68	14	from	from	ADP
m-769	68	15	e	e	NOUN
m-769	68	16	that	that	PRON
m-769	68	17	brings	bring	VERB
m-769	68	18	them	they	PRON
m-769	68	19	arbitrarily	arbitrarily	ADV
m-769	68	20	close	close	ADJ
m-769	68	21	to	to	ADP
m-769	68	22	each	each	DET
m-769	68	23	other	other	ADJ
m-769	68	24	.	.	PUNCT
m-769	69	1	homomorphic	homomorphic	ADJ
m-769	69	2	images	image	NOUN
m-769	69	3	3.4.let	3.4.let	NUM
m-769	69	4	(	(	PUNCT
m-769	69	5	x	x	NOUN
m-769	69	6	,	,	PUNCT
m-769	69	7	g	g	PROPN
m-769	69	8	,	,	PUNCT
m-769	69	9	μ	μ	NOUN
m-769	69	10	)	)	PUNCT
m-769	69	11	be	be	VERB
m-769	69	12	a	a	DET
m-769	69	13	transformation	transformation	NOUN
m-769	69	14	semigroup	semigroup	NOUN
m-769	69	15	with	with	ADP
m-769	69	16	a	a	DET
m-769	69	17	phase	phase	NOUN
m-769	69	18	space	space	NOUN
m-769	69	19	x	x	NOUN
m-769	69	20	,	,	PUNCT
m-769	69	21	a	a	DET
m-769	69	22	semigroupg	semigroupg	NOUN
m-769	69	23	of	of	ADP
m-769	69	24	transformations	transformation	NOUN
m-769	69	25	on	on	ADP
m-769	69	26	x	x	NOUN
m-769	69	27	,	,	PUNCT
m-769	69	28	and	and	CCONJ
m-769	69	29	a	a	DET
m-769	69	30	continuous	continuous	ADJ
m-769	69	31	action	action	NOUN
m-769	69	32	μ	μ	NOUN
m-769	69	33	:	:	PUNCT
m-769	69	34	g×x→x	g×x→x	NOUN
m-769	69	35	.	.	PUNCT
m-769	70	1	the	the	DET
m-769	70	2	enveloping	enveloping	NOUN
m-769	70	3	semigroupe	semigroupe	NOUN
m-769	70	4	is	be	AUX
m-769	70	5	the	the	DET
m-769	70	6	closure	closure	NOUN
m-769	70	7	of	of	ADP
m-769	70	8	g	g	NOUN
m-769	70	9	in	in	ADP
m-769	70	10	the	the	DET
m-769	70	11	product	product	NOUN
m-769	70	12	space	space	NOUN
m-769	70	13	x×x	x×x	PROPN
m-769	70	14	,	,	PUNCT
m-769	70	15	i.e.	i.e.	X
m-769	70	16	,	,	PUNCT
m-769	70	17	�	�	PROPN
m-769	70	18	=	=	SYM
m-769	70	19	�	�	PROPN
m-769	70	20	̅.	̅.	PROPN
m-769	70	21	now	now	ADV
m-769	70	22	,	,	PUNCT
m-769	70	23	homomorphic	homomorphic	ADJ
m-769	70	24	images	image	NOUN
m-769	70	25	can	can	AUX
m-769	70	26	be	be	AUX
m-769	70	27	defined	define	VERB
m-769	70	28	as	as	ADP
m-769	70	29	the	the	DET
m-769	70	30	images	image	NOUN
m-769	70	31	of	of	ADP
m-769	70	32	the	the	DET
m-769	70	33	transformations	transformation	NOUN
m-769	70	34	in	in	ADP
m-769	70	35	e	e	NOUN
m-769	70	36	under	under	ADP
m-769	70	37	a	a	DET
m-769	70	38	homomorphism	homomorphism	NOUN
m-769	70	39	mapping	mapping	NOUN
m-769	70	40	to	to	ADP
m-769	70	41	another	another	DET
m-769	70	42	group	group	NOUN
m-769	70	43	.	.	PUNCT
m-769	71	1	let	let	VERB
m-769	71	2	h	h	PRON
m-769	71	3	be	be	AUX
m-769	71	4	a	a	DET
m-769	71	5	group	group	NOUN
m-769	71	6	and	and	CCONJ
m-769	71	7	ϕ:e→h	ϕ:e→h	PROPN
m-769	71	8	be	be	AUX
m-769	71	9	a	a	DET
m-769	71	10	homomorphism	homomorphism	NOUN
m-769	71	11	such	such	ADJ
m-769	71	12	that	that	SCONJ
m-769	71	13	ϕ(xy)=ϕ(x)ϕ(y	ϕ(xy)=ϕ(x)ϕ(y	NUM
m-769	71	14	)	)	PUNCT
m-769	71	15	for	for	ADP
m-769	71	16	all	all	DET
m-769	71	17	x	x	NOUN
m-769	71	18	,	,	PUNCT
m-769	71	19	y∈e	y∈e	PROPN
m-769	71	20	.	.	PUNCT
m-769	72	1	the	the	DET
m-769	72	2	set	set	NOUN
m-769	72	3	of	of	ADP
m-769	72	4	homomorphic	homomorphic	ADJ
m-769	72	5	images	image	NOUN
m-769	72	6	is	be	AUX
m-769	72	7	then	then	ADV
m-769	72	8	defined	define	VERB
m-769	72	9	as	as	ADP
m-769	72	10	:	:	PUNCT
m-769	72	11	ijo	ijo	PROPN
m-769	72	12	journals	journal	NOUN
m-769	72	13	doi	doi	NOUN
m-769	72	14	10.5281	10.5281	NUM
m-769	72	15	/	/	SYM
m-769	72	16	zenodo.10443958	zenodo.10443958	PROPN
m-769	72	17	volume	volume	NOUN
m-769	72	18	06	06	NUM
m-769	73	1	|	|	ADV
m-769	73	2	issue	issue	NOUN
m-769	73	3	12	12	NUM
m-769	73	4	|	|	CCONJ
m-769	73	5	december	december	PROPN
m-769	73	6	2023	2023	NUM
m-769	74	1	|	|	ADV
m-769	74	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-769	74	3	12	12	NUM
m-769	74	4	ijo	ijo	PROPN
m-769	74	5	international	international	PROPN
m-769	74	6	journal	journal	PROPN
m-769	74	7	of	of	ADP
m-769	74	8	mathematics	mathematics	PROPN
m-769	74	9	(	(	PUNCT
m-769	74	10	issn	issn	PROPN
m-769	74	11	:	:	PUNCT
m-769	74	12	2992	2992	NUM
m-769	74	13	-	-	SYM
m-769	74	14	4421	4421	NUM
m-769	74	15	)	)	PUNCT
m-769	75	1	michael	michael	PROPN
m-769	75	2	n.	n.	PROPN
m-769	75	3	john	john	PROPN
m-769	75	4	*	*	PROPN
m-769	75	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-769	75	6	volume	volume	NOUN
m-769	75	7	06	06	NUM
m-769	75	8	issue	issue	NOUN
m-769	75	9	12	12	NUM
m-769	75	10	||	||	NOUN
m-769	75	11	dec	dec	PROPN
m-769	75	12	.	.	PROPN
m-769	75	13	,	,	PUNCT
m-769	75	14	2023	2023	NUM
m-769	75	15	||	||	NOUN
m-769	76	1	algebraic	algebraic	ADJ
m-769	76	2	and	and	CCONJ
m-769	76	3	topological	topological	ADJ
m-769	76	4	analysis	analysis	NOUN
m-769	76	5	of	of	ADP
m-769	76	6	enveloping	envelop	VERB
m-769	76	7	semigroups	semigroup	NOUN
m-769	76	8	in	in	ADP
m-769	76	9	transformation	transformation	NOUN
m-769	76	10	groups	group	NOUN
m-769	76	11	:	:	PUNCT
m-769	76	12	proximal	proximal	ADJ
m-769	76	13	equivalence	equivalence	NOUN
m-769	76	14	and	and	CCONJ
m-769	76	15	homomorphic	homomorphic	ADJ
m-769	76	16	image	image	NOUN
m-769	76	17	homomorphic	homomorphic	ADJ
m-769	76	18	images={ϕ(x)|	images={ϕ(x)|	NOUN
m-769	76	19	x∈e	x∈e	NOUN
m-769	76	20	}	}	PUNCT
m-769	77	1	this	this	DET
m-769	77	2	set	set	NOUN
m-769	77	3	represents	represent	VERB
m-769	77	4	the	the	DET
m-769	77	5	images	image	NOUN
m-769	77	6	of	of	ADP
m-769	77	7	the	the	DET
m-769	77	8	elements	element	NOUN
m-769	77	9	in	in	ADP
m-769	77	10	the	the	DET
m-769	77	11	enveloping	enveloping	NOUN
m-769	77	12	semigroupe	semigroupe	NOUN
m-769	77	13	under	under	ADP
m-769	77	14	the	the	DET
m-769	77	15	homomorphism	homomorphism	NOUN
m-769	77	16	ϕ	ϕ	NOUN
m-769	77	17	into	into	ADP
m-769	77	18	the	the	DET
m-769	77	19	group	group	NOUN
m-769	77	20	h.	h.	PROPN
m-769	77	21	illustration	illustration	PROPN
m-769	77	22	3.5	3.5	NUM
m-769	77	23	.	.	PUNCT
m-769	78	1	consider	consider	VERB
m-769	78	2	a	a	DET
m-769	78	3	transformation	transformation	NOUN
m-769	78	4	semigroupg	semigroupg	NOUN
m-769	78	5	consisting	consist	VERB
m-769	78	6	of	of	ADP
m-769	78	7	all	all	DET
m-769	78	8	rotations	rotation	NOUN
m-769	78	9	on	on	ADP
m-769	78	10	a	a	DET
m-769	78	11	circle	circle	NOUN
m-769	78	12	,	,	PUNCT
m-769	78	13	and	and	CCONJ
m-769	78	14	let	let	VERB
m-769	78	15	e	e	PRON
m-769	78	16	be	be	AUX
m-769	78	17	its	its	PRON
m-769	78	18	enveloping	enveloping	NOUN
m-769	78	19	semigroup	semigroup	NOUN
m-769	78	20	.	.	PUNCT
m-769	79	1	now	now	ADV
m-769	79	2	,	,	PUNCT
m-769	79	3	suppose	suppose	VERB
m-769	79	4	h	h	NOUN
m-769	79	5	is	be	AUX
m-769	79	6	the	the	DET
m-769	79	7	additive	additive	ADJ
m-769	79	8	group	group	NOUN
m-769	79	9	of	of	ADP
m-769	79	10	integers	integer	NOUN
m-769	79	11	,	,	PUNCT
m-769	79	12	and	and	CCONJ
m-769	79	13	ϕ:e→h	ϕ:e→h	PROPN
m-769	79	14	is	be	AUX
m-769	79	15	a	a	DET
m-769	79	16	homomorphism	homomorphism	NOUN
m-769	79	17	that	that	PRON
m-769	79	18	assigns	assign	VERB
m-769	79	19	each	each	DET
m-769	79	20	rotation	rotation	NOUN
m-769	79	21	an	an	DET
m-769	79	22	integer	integer	NOUN
m-769	79	23	value	value	NOUN
m-769	79	24	corresponding	correspond	VERB
m-769	79	25	to	to	ADP
m-769	79	26	the	the	DET
m-769	79	27	number	number	NOUN
m-769	79	28	of	of	ADP
m-769	79	29	degrees	degree	NOUN
m-769	79	30	rotated	rotate	VERB
m-769	79	31	.	.	PUNCT
m-769	80	1	the	the	DET
m-769	80	2	homomorphic	homomorphic	ADJ
m-769	80	3	images	image	NOUN
m-769	80	4	,	,	PUNCT
m-769	80	5	in	in	ADP
m-769	80	6	this	this	DET
m-769	80	7	case	case	NOUN
m-769	80	8	,	,	PUNCT
m-769	80	9	would	would	AUX
m-769	80	10	be	be	AUX
m-769	80	11	the	the	DET
m-769	80	12	set	set	NOUN
m-769	80	13	of	of	ADP
m-769	80	14	integers	integer	NOUN
m-769	80	15	representing	represent	VERB
m-769	80	16	the	the	DET
m-769	80	17	degrees	degree	NOUN
m-769	80	18	of	of	ADP
m-769	80	19	rotation	rotation	NOUN
m-769	80	20	.	.	PUNCT
m-769	80	21	example	example	NOUN
m-769	81	1	3.6	3.6	NUM
m-769	81	2	.	.	PUNCT
m-769	82	1	let	let	VERB
m-769	82	2	g	g	NOUN
m-769	82	3	be	be	AUX
m-769	82	4	the	the	DET
m-769	82	5	semigroup	semigroup	NOUN
m-769	82	6	of	of	ADP
m-769	82	7	all	all	DET
m-769	82	8	positive	positive	ADJ
m-769	82	9	real	real	ADJ
m-769	82	10	number	number	NOUN
m-769	82	11	transformations	transformation	NOUN
m-769	82	12	on	on	ADP
m-769	82	13	the	the	DET
m-769	82	14	real	real	ADJ
m-769	82	15	line	line	NOUN
m-769	82	16	x	x	NOUN
m-769	82	17	,	,	PUNCT
m-769	82	18	i.e.	i.e.	ADV
m-769	82	19	,	,	PUNCT
m-769	82	20	g={ta	g={ta	VERB
m-769	82	21	|a>0	|a>0	NOUN
m-769	82	22	}	}	PUNCT
m-769	82	23	,	,	PUNCT
m-769	82	24	where	where	SCONJ
m-769	82	25	ta(x)=x+a	ta(x)=x+a	ADJ
m-769	82	26	.	.	PUNCT
m-769	83	1	the	the	DET
m-769	83	2	enveloping	enveloping	NOUN
m-769	83	3	semigroupe	semigroupe	NOUN
m-769	83	4	is	be	AUX
m-769	83	5	the	the	DET
m-769	83	6	closure	closure	NOUN
m-769	83	7	of	of	ADP
m-769	83	8	g.	g.	PROPN
m-769	83	9	now	now	ADV
m-769	83	10	,	,	PUNCT
m-769	83	11	consider	consider	VERB
m-769	83	12	the	the	DET
m-769	83	13	additive	additive	ADJ
m-769	83	14	group	group	NOUN
m-769	83	15	of	of	ADP
m-769	83	16	integers	integer	NOUN
m-769	83	17	h	h	NOUN
m-769	83	18	,	,	PUNCT
m-769	83	19	and	and	CCONJ
m-769	83	20	define	define	VERB
m-769	83	21	a	a	DET
m-769	83	22	homomorphism	homomorphism	NOUN
m-769	83	23	ϕ:e→h	ϕ:e→h	PROPN
m-769	83	24	such	such	ADJ
m-769	83	25	thatϕ(ta)=⌊a⌋	thatϕ(ta)=⌊a⌋	NOUN
m-769	83	26	,	,	PUNCT
m-769	83	27	where	where	SCONJ
m-769	83	28	⌊a⌋	⌊a⌋	PRON
m-769	83	29	is	be	AUX
m-769	83	30	the	the	DET
m-769	83	31	greatest	great	ADJ
m-769	83	32	integer	integer	NOUN
m-769	83	33	less	less	ADJ
m-769	83	34	than	than	ADP
m-769	83	35	or	or	CCONJ
m-769	83	36	equal	equal	ADJ
m-769	83	37	to	to	AUX
m-769	83	38	a.	a.	VERB
m-769	83	39	the	the	DET
m-769	83	40	homomorphic	homomorphic	ADJ
m-769	83	41	images	image	NOUN
m-769	83	42	in	in	ADP
m-769	83	43	this	this	DET
m-769	83	44	case	case	NOUN
m-769	83	45	would	would	AUX
m-769	83	46	be	be	AUX
m-769	83	47	the	the	DET
m-769	83	48	set	set	NOUN
m-769	83	49	of	of	ADP
m-769	83	50	integers	integer	NOUN
m-769	83	51	corresponding	correspond	VERB
m-769	83	52	to	to	ADP
m-769	83	53	the	the	DET
m-769	83	54	floor	floor	NOUN
m-769	83	55	values	value	NOUN
m-769	83	56	of	of	ADP
m-769	83	57	the	the	DET
m-769	83	58	translation	translation	NOUN
m-769	83	59	parameters	parameter	NOUN
m-769	83	60	.	.	PUNCT
m-769	84	1	homomorphic	homomorphic	ADJ
m-769	84	2	images	image	NOUN
m-769	84	3	in	in	ADP
m-769	84	4	the	the	DET
m-769	84	5	context	context	NOUN
m-769	84	6	of	of	ADP
m-769	84	7	enveloping	envelop	VERB
m-769	84	8	semigroups	semigroup	NOUN
m-769	84	9	involve	involve	VERB
m-769	84	10	mapping	mapping	NOUN
m-769	84	11	transformations	transformation	NOUN
m-769	84	12	to	to	ADP
m-769	84	13	another	another	DET
m-769	84	14	group	group	NOUN
m-769	84	15	through	through	ADP
m-769	84	16	a	a	DET
m-769	84	17	homomorphism	homomorphism	NOUN
m-769	84	18	.	.	PUNCT
m-769	85	1	the	the	DET
m-769	85	2	mathematical	mathematical	ADJ
m-769	85	3	definition	definition	NOUN
m-769	85	4	captures	capture	VERB
m-769	85	5	this	this	DET
m-769	85	6	concept	concept	NOUN
m-769	85	7	,	,	PUNCT
m-769	85	8	and	and	CCONJ
m-769	85	9	the	the	DET
m-769	85	10	illustration	illustration	NOUN
m-769	85	11	and	and	CCONJ
m-769	85	12	example	example	NOUN
m-769	85	13	demonstrate	demonstrate	VERB
m-769	85	14	how	how	SCONJ
m-769	85	15	transformations	transformation	NOUN
m-769	85	16	in	in	ADP
m-769	85	17	the	the	DET
m-769	85	18	enveloping	enveloping	NOUN
m-769	85	19	semigroup	semigroup	NOUN
m-769	85	20	can	can	AUX
m-769	85	21	be	be	AUX
m-769	85	22	mapped	map	VERB
m-769	85	23	to	to	ADP
m-769	85	24	homomorphic	homomorphic	ADJ
m-769	85	25	images	image	NOUN
m-769	85	26	in	in	ADP
m-769	85	27	different	different	ADJ
m-769	85	28	groups	group	NOUN
m-769	85	29	.	.	PUNCT
m-769	86	1	4	4	X
m-769	86	2	.	.	X
m-769	86	3	central	central	ADJ
m-769	86	4	idea	idea	NOUN
m-769	86	5	lemma	lemma	PROPN
m-769	86	6	4.1	4.1	NUM
m-769	86	7	.	.	PUNCT
m-769	87	1	for	for	ADP
m-769	87	2	a	a	DET
m-769	87	3	transformation	transformation	NOUN
m-769	87	4	group	group	NOUN
m-769	87	5	(	(	PUNCT
m-769	87	6	x	x	X
m-769	87	7	,	,	PUNCT
m-769	87	8	t	t	PROPN
m-769	87	9	,	,	PUNCT
m-769	87	10	μ	μ	NOUN
m-769	87	11	)	)	PUNCT
m-769	87	12	,	,	PUNCT
m-769	87	13	where	where	SCONJ
m-769	87	14	x	x	PRON
m-769	87	15	is	be	AUX
m-769	87	16	a	a	DET
m-769	87	17	topological	topological	ADJ
m-769	87	18	space	space	NOUN
m-769	87	19	,	,	PUNCT
m-769	87	20	t	t	PROPN
m-769	87	21	is	be	AUX
m-769	87	22	a	a	DET
m-769	87	23	group	group	NOUN
m-769	87	24	of	of	ADP
m-769	87	25	homeomorphisms	homeomorphisms	PROPN
m-769	87	26	on	on	ADP
m-769	87	27	x	x	NOUN
m-769	87	28	,	,	PUNCT
m-769	87	29	and	and	CCONJ
m-769	87	30	μ	μ	NUM
m-769	87	31	:	:	PUNCT
m-769	87	32	t×x→x	t×x→x	PROPN
m-769	87	33	is	be	AUX
m-769	87	34	a	a	DET
m-769	87	35	continuous	continuous	ADJ
m-769	87	36	action	action	NOUN
m-769	87	37	,	,	PUNCT
m-769	87	38	the	the	DET
m-769	87	39	enveloping	enveloping	NOUN
m-769	87	40	semigroupe	semigroupe	NOUN
m-769	87	41	is	be	AUX
m-769	87	42	a	a	DET
m-769	87	43	group	group	NOUN
m-769	87	44	of	of	ADP
m-769	87	45	homeomorphisms	homeomorphisms	PROPN
m-769	87	46	on	on	ADP
m-769	87	47	x.	x.	NOUN
m-769	87	48	proof	proof	NOUN
m-769	87	49	:	:	PUNCT
m-769	87	50	1	1	X
m-769	87	51	.	.	X
m-769	87	52	closure	closure	NOUN
m-769	87	53	under	under	ADP
m-769	87	54	composition	composition	NOUN
m-769	87	55	:	:	PUNCT
m-769	87	56			PRON
m-769	87	57	let	let	VERB
m-769	87	58	f	f	X
m-769	87	59	,	,	PUNCT
m-769	87	60	g∈e	g∈e	ADJ
m-769	87	61	.	.	PUNCT
m-769	88	1	since	since	SCONJ
m-769	88	2	e	e	PROPN
m-769	88	3	is	be	AUX
m-769	88	4	the	the	DET
m-769	88	5	closure	closure	NOUN
m-769	88	6	of	of	ADP
m-769	88	7	t	t	PROPN
m-769	88	8	in	in	ADP
m-769	88	9	the	the	DET
m-769	88	10	product	product	NOUN
m-769	88	11	space	space	NOUN
m-769	88	12	x×x	x×x	PROPN
m-769	88	13	,	,	PUNCT
m-769	88	14	there	there	PRON
m-769	88	15	exist	exist	VERB
m-769	88	16	sequences	sequence	NOUN
m-769	88	17	(	(	PUNCT
m-769	88	18	tn)⊆t	tn)⊆t	PRON
m-769	88	19	converging	converge	VERB
m-769	88	20	to	to	ADP
m-769	88	21	f	f	PROPN
m-769	88	22	and	and	CCONJ
m-769	88	23	(	(	PUNCT
m-769	88	24	sn)⊆t	sn)⊆t	ADV
m-769	88	25	converging	converge	VERB
m-769	88	26	to	to	PART
m-769	88	27	g.	g.	PROPN
m-769	88	28	consider	consider	VERB
m-769	88	29	the	the	DET
m-769	88	30	composition	composition	NOUN
m-769	88	31	f∘g	f∘g	NOUN
m-769	88	32	.	.	PUNCT
m-769	89	1	we	we	PRON
m-769	89	2	need	need	VERB
m-769	89	3	to	to	PART
m-769	89	4	show	show	VERB
m-769	89	5	that	that	SCONJ
m-769	89	6	f∘g	f∘g	NOUN
m-769	89	7	is	be	AUX
m-769	89	8	also	also	ADV
m-769	89	9	in	in	ADP
m-769	89	10	e.	e.	PROPN
m-769	89	11			NOUN
m-769	89	12	by	by	ADP
m-769	89	13	the	the	DET
m-769	89	14	continuity	continuity	NOUN
m-769	89	15	of	of	ADP
m-769	89	16	the	the	DET
m-769	89	17	action	action	NOUN
m-769	89	18	μ	μ	NOUN
m-769	89	19	,	,	PUNCT
m-769	89	20	we	we	PRON
m-769	89	21	have	have	VERB
m-769	89	22	μ(tn	μ(tn	NOUN
m-769	89	23	,	,	PUNCT
m-769	89	24	x)→f(x	x)→f(x	PROPN
m-769	89	25	)	)	PUNCT
m-769	89	26	and	and	CCONJ
m-769	89	27	μ(sn	μ(sn	NUM
m-769	89	28	,	,	PUNCT
m-769	89	29	x)→g(x	x)→g(x	NUM
m-769	89	30	)	)	PUNCT
m-769	89	31	for	for	ADP
m-769	89	32	all	all	DET
m-769	89	33	x∈x	x∈x	NOUN
m-769	89	34	as	as	ADP
m-769	89	35	n→∞.	n→∞.	PROPN
m-769	89	36	ijo	ijo	PROPN
m-769	89	37	journals	journal	NOUN
m-769	89	38	doi	doi	NOUN
m-769	89	39	10.5281	10.5281	NUM
m-769	89	40	/	/	SYM
m-769	89	41	zenodo.10443958	zenodo.10443958	PROPN
m-769	89	42	volume	volume	NOUN
m-769	89	43	06	06	NUM
m-769	90	1	|	|	ADV
m-769	90	2	issue	issue	NOUN
m-769	90	3	12	12	NUM
m-769	90	4	|	|	CCONJ
m-769	90	5	december	december	PROPN
m-769	90	6	2023	2023	NUM
m-769	91	1	|	|	ADV
m-769	91	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-769	91	3	13	13	NUM
m-769	91	4	ijo	ijo	PROPN
m-769	91	5	international	international	PROPN
m-769	91	6	journal	journal	PROPN
m-769	91	7	of	of	ADP
m-769	91	8	mathematics	mathematics	PROPN
m-769	91	9	(	(	PUNCT
m-769	91	10	issn	issn	PROPN
m-769	91	11	:	:	PUNCT
m-769	91	12	2992	2992	NUM
m-769	91	13	-	-	SYM
m-769	91	14	4421	4421	NUM
m-769	91	15	)	)	PUNCT
m-769	92	1	michael	michael	PROPN
m-769	92	2	n.	n.	PROPN
m-769	92	3	john	john	PROPN
m-769	92	4	*	*	PROPN
m-769	92	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-769	92	6	volume	volume	NOUN
m-769	92	7	06	06	NUM
m-769	92	8	issue	issue	NOUN
m-769	92	9	12	12	NUM
m-769	92	10	||	||	NOUN
m-769	92	11	dec	dec	PROPN
m-769	92	12	.	.	PROPN
m-769	92	13	,	,	PUNCT
m-769	92	14	2023	2023	NUM
m-769	92	15	||	||	NOUN
m-769	93	1	algebraic	algebraic	ADJ
m-769	93	2	and	and	CCONJ
m-769	93	3	topological	topological	ADJ
m-769	93	4	analysis	analysis	NOUN
m-769	93	5	of	of	ADP
m-769	93	6	enveloping	envelop	VERB
m-769	93	7	semigroups	semigroup	NOUN
m-769	93	8	in	in	ADP
m-769	93	9	transformation	transformation	NOUN
m-769	93	10	groups	group	NOUN
m-769	93	11	:	:	PUNCT
m-769	93	12	proximal	proximal	ADJ
m-769	93	13	equivalence	equivalence	NOUN
m-769	93	14	and	and	CCONJ
m-769	93	15	homomorphic	homomorphic	ADJ
m-769	93	16	image	image	NOUN
m-769	93	17			NOUN
m-769	93	18	now	now	ADV
m-769	93	19	,	,	PUNCT
m-769	93	20	consider	consider	VERB
m-769	93	21	μ(tn⋅sn	μ(tn⋅sn	NOUN
m-769	93	22	,	,	PUNCT
m-769	93	23	x	x	NOUN
m-769	93	24	)	)	PUNCT
m-769	93	25	.	.	PUNCT
m-769	94	1	by	by	ADP
m-769	94	2	the	the	DET
m-769	94	3	group	group	NOUN
m-769	94	4	action	action	NOUN
m-769	94	5	property	property	NOUN
m-769	94	6	,	,	PUNCT
m-769	94	7	μ(tn⋅sn	μ(tn⋅sn	NOUN
m-769	94	8	,	,	PUNCT
m-769	94	9	x)=μ(tn	x)=μ(tn	PUNCT
m-769	94	10	,	,	PUNCT
m-769	94	11	μ(sn	μ(sn	PROPN
m-769	94	12	,	,	PUNCT
m-769	94	13	x	x	NOUN
m-769	94	14	)	)	PUNCT
m-769	94	15	)	)	PUNCT
m-769	94	16	.	.	PUNCT
m-769	94	17			NOUN
m-769	95	1	as	as	ADP
m-769	95	2	n→∞,μ(tn⋅sn	n→∞,μ(tn⋅sn	NOUN
m-769	95	3	,	,	PUNCT
m-769	95	4	x)→f(g(x	x)→f(g(x	NOUN
m-769	95	5	)	)	PUNCT
m-769	95	6	)	)	PUNCT
m-769	95	7	because	because	SCONJ
m-769	95	8	of	of	ADP
m-769	95	9	the	the	DET
m-769	95	10	continuity	continuity	NOUN
m-769	95	11	of	of	ADP
m-769	95	12	μ	μ	PROPN
m-769	95	13	and	and	CCONJ
m-769	95	14	the	the	DET
m-769	95	15	convergence	convergence	NOUN
m-769	95	16	of	of	ADP
m-769	95	17	(	(	PUNCT
m-769	95	18	tn	tn	NOUN
m-769	95	19	)	)	PUNCT
m-769	95	20	and	and	CCONJ
m-769	95	21	(	(	PUNCT
m-769	95	22	sn	sn	PROPN
m-769	95	23	)	)	PUNCT
m-769	95	24	.	.	PUNCT
m-769	96	1			PUNCT
m-769	96	2	therefore	therefore	ADV
m-769	96	3	,	,	PUNCT
m-769	96	4	f∘g	f∘g	X
m-769	96	5	is	be	AUX
m-769	96	6	in	in	ADP
m-769	96	7	the	the	DET
m-769	96	8	closure	closure	NOUN
m-769	96	9	of	of	ADP
m-769	96	10	t	t	PROPN
m-769	96	11	,	,	PUNCT
m-769	96	12	i.e.	i.e.	X
m-769	96	13	,	,	PUNCT
m-769	96	14	f∘g∈e	f∘g∈e	PROPN
m-769	96	15	.	.	PUNCT
m-769	97	1	2	2	X
m-769	97	2	.	.	X
m-769	97	3	existence	existence	NOUN
m-769	97	4	of	of	ADP
m-769	97	5	identity	identity	NOUN
m-769	97	6	element	element	NOUN
m-769	97	7	:	:	PUNCT
m-769	97	8			NOUN
m-769	97	9	let	let	VERB
m-769	97	10	e	e	PRON
m-769	97	11	be	be	AUX
m-769	97	12	the	the	DET
m-769	97	13	identity	identity	NOUN
m-769	97	14	element	element	NOUN
m-769	97	15	of	of	ADP
m-769	97	16	the	the	DET
m-769	97	17	group	group	NOUN
m-769	97	18	t.	t.	PROPN
m-769	97	19	since	since	SCONJ
m-769	97	20	e	e	PROPN
m-769	97	21	is	be	AUX
m-769	97	22	a	a	DET
m-769	97	23	homeomorphism	homeomorphism	NOUN
m-769	97	24	,	,	PUNCT
m-769	97	25	it	it	PRON
m-769	97	26	is	be	AUX
m-769	97	27	also	also	ADV
m-769	97	28	in	in	ADP
m-769	97	29	e	e	NOUN
m-769	97	30	as	as	SCONJ
m-769	97	31	the	the	DET
m-769	97	32	constant	constant	ADJ
m-769	97	33	sequence	sequence	NOUN
m-769	97	34	converges	converge	VERB
m-769	97	35	to	to	ADP
m-769	97	36	e.	e.	PROPN
m-769	97	37	3	3	PROPN
m-769	97	38	.	.	PUNCT
m-769	98	1	existence	existence	NOUN
m-769	98	2	of	of	ADP
m-769	98	3	inverses	inverse	NOUN
m-769	98	4	:	:	PUNCT
m-769	98	5			NOUN
m-769	98	6	let	let	VERB
m-769	98	7	f∈e	f∈e	NOUN
m-769	98	8	.	.	PUNCT
m-769	99	1	since	since	SCONJ
m-769	99	2	f	f	PROPN
m-769	99	3	is	be	AUX
m-769	99	4	in	in	ADP
m-769	99	5	the	the	DET
m-769	99	6	closure	closure	NOUN
m-769	99	7	of	of	ADP
m-769	99	8	t	t	PROPN
m-769	99	9	,	,	PUNCT
m-769	99	10	there	there	PRON
m-769	99	11	exists	exist	VERB
m-769	99	12	a	a	DET
m-769	99	13	sequence	sequence	NOUN
m-769	99	14	(	(	PUNCT
m-769	99	15	tn)⊆t	tn)⊆t	PRON
m-769	99	16	converging	converge	VERB
m-769	99	17	to	to	ADP
m-769	99	18	f.	f.	PROPN
m-769	99	19			PROPN
m-769	99	20	consider	consider	VERB
m-769	99	21	the	the	DET
m-769	99	22	sequence	sequence	NOUN
m-769	99	23	(	(	PUNCT
m-769	99	24	tn	tn	NOUN
m-769	99	25	−1	−1	NOUN
m-769	99	26	)	)	PUNCT
m-769	99	27	,	,	PUNCT
m-769	99	28	where	where	SCONJ
m-769	99	29	tn	tn	NOUN
m-769	99	30	−1	−1	NOUN
m-769	99	31	is	be	AUX
m-769	99	32	the	the	DET
m-769	99	33	inverse	inverse	NOUN
m-769	99	34	of	of	ADP
m-769	99	35	each	each	DET
m-769	99	36	tn	tn	NOUN
m-769	99	37	in	in	ADP
m-769	99	38	t.	t.	PROPN
m-769	99	39	as	as	SCONJ
m-769	99	40	t	t	PROPN
m-769	99	41	is	be	AUX
m-769	99	42	a	a	DET
m-769	99	43	group	group	NOUN
m-769	99	44	,	,	PUNCT
m-769	99	45	tn	tn	PROPN
m-769	99	46	−1	−1	NOUN
m-769	99	47	is	be	AUX
m-769	99	48	also	also	ADV
m-769	99	49	in	in	ADP
m-769	99	50	t.	t.	PROPN
m-769	99	51			NOUN
m-769	100	1	the	the	DET
m-769	100	2	sequence	sequence	NOUN
m-769	100	3	(	(	PUNCT
m-769	100	4	tn	tn	NOUN
m-769	100	5	−1	−1	NOUN
m-769	100	6	)	)	PUNCT
m-769	100	7	converges	converge	VERB
m-769	100	8	to	to	ADP
m-769	100	9	f−1	f−1	PROPN
m-769	100	10	because	because	SCONJ
m-769	100	11	the	the	DET
m-769	100	12	inverse	inverse	NOUN
m-769	100	13	is	be	AUX
m-769	100	14	a	a	DET
m-769	100	15	continuous	continuous	ADJ
m-769	100	16	operation	operation	NOUN
m-769	100	17	on	on	ADP
m-769	100	18	t.	t.	PROPN
m-769	100	19			PRON
m-769	100	20	therefore	therefore	ADV
m-769	100	21	,	,	PUNCT
m-769	100	22	f−1	f−1	PROPN
m-769	100	23	is	be	AUX
m-769	100	24	in	in	ADP
m-769	100	25	the	the	DET
m-769	100	26	closure	closure	NOUN
m-769	100	27	of	of	ADP
m-769	100	28	t	t	PROPN
m-769	100	29	,	,	PUNCT
m-769	100	30	i.e.	i.e.	X
m-769	100	31	,	,	PUNCT
m-769	100	32	f−1∈e	f−1∈e	X
m-769	100	33	.	.	PUNCT
m-769	101	1	4	4	NUM
m-769	101	2	.	.	X
m-769	101	3	closure	closure	NOUN
m-769	101	4	under	under	ADP
m-769	101	5	topological	topological	ADJ
m-769	101	6	composition	composition	NOUN
m-769	101	7	:	:	PUNCT
m-769	101	8			X
m-769	101	9	the	the	DET
m-769	101	10	composition	composition	NOUN
m-769	101	11	of	of	ADP
m-769	101	12	homeomorphisms	homeomorphisms	PROPN
m-769	101	13	is	be	AUX
m-769	101	14	itself	itself	PRON
m-769	101	15	a	a	DET
m-769	101	16	homeomorphism	homeomorphism	NOUN
m-769	101	17	.	.	PUNCT
m-769	102	1	since	since	SCONJ
m-769	102	2	t	t	PROPN
m-769	102	3	consists	consist	VERB
m-769	102	4	of	of	ADP
m-769	102	5	homeomorphisms	homeomorphism	NOUN
m-769	102	6	and	and	CCONJ
m-769	102	7	e	e	NOUN
m-769	102	8	is	be	AUX
m-769	102	9	the	the	DET
m-769	102	10	closure	closure	NOUN
m-769	102	11	of	of	ADP
m-769	102	12	t	t	PROPN
m-769	102	13	,	,	PUNCT
m-769	102	14	every	every	DET
m-769	102	15	element	element	NOUN
m-769	102	16	of	of	ADP
m-769	102	17	e	e	PROPN
m-769	102	18	is	be	AUX
m-769	102	19	a	a	DET
m-769	102	20	homeomorphism	homeomorphism	NOUN
m-769	102	21	.	.	PUNCT
m-769	103	1	hence	hence	ADV
m-769	103	2	,	,	PUNCT
m-769	103	3	e	e	PROPN
m-769	103	4	satisfies	satisfy	VERB
m-769	103	5	the	the	DET
m-769	103	6	group	group	NOUN
m-769	103	7	axioms	axiom	NOUN
m-769	103	8	of	of	ADP
m-769	103	9	closure	closure	NOUN
m-769	103	10	under	under	ADP
m-769	103	11	composition	composition	NOUN
m-769	103	12	,	,	PUNCT
m-769	103	13	the	the	DET
m-769	103	14	existence	existence	NOUN
m-769	103	15	of	of	ADP
m-769	103	16	an	an	DET
m-769	103	17	identity	identity	NOUN
m-769	103	18	element	element	NOUN
m-769	103	19	,	,	PUNCT
m-769	103	20	and	and	CCONJ
m-769	103	21	the	the	DET
m-769	103	22	existence	existence	NOUN
m-769	103	23	of	of	ADP
m-769	103	24	inverses	inverse	NOUN
m-769	103	25	.	.	PUNCT
m-769	104	1	therefore	therefore	ADV
m-769	104	2	,	,	PUNCT
m-769	104	3	e	e	X
m-769	104	4	is	be	AUX
m-769	104	5	a	a	DET
m-769	104	6	group	group	NOUN
m-769	104	7	of	of	ADP
m-769	104	8	homeomorphisms	homeomorphisms	PROPN
m-769	104	9	on	on	ADP
m-769	104	10	x.	x.	NOUN
m-769	104	11	proposition	proposition	NOUN
m-769	104	12	4.2.the	4.2.the	DET
m-769	104	13	proximal	proximal	ADJ
m-769	104	14	equivalence	equivalence	NOUN
m-769	104	15	relation	relation	NOUN
m-769	104	16	in	in	ADP
m-769	104	17	x	x	PROPN
m-769	104	18	is	be	AUX
m-769	104	19	an	an	DET
m-769	104	20	equivalence	equivalence	NOUN
m-769	104	21	relation	relation	NOUN
m-769	104	22	if	if	SCONJ
m-769	104	23	and	and	CCONJ
m-769	104	24	only	only	ADV
m-769	104	25	if	if	SCONJ
m-769	104	26	there	there	PRON
m-769	104	27	exists	exist	VERB
m-769	104	28	only	only	ADV
m-769	104	29	one	one	NUM
m-769	104	30	minimal	minimal	ADJ
m-769	104	31	right	right	ADJ
m-769	104	32	ideal	ideal	NOUN
m-769	104	33	in	in	ADP
m-769	104	34	e.	e.	PROPN
m-769	104	35	proof	proof	PROPN
m-769	104	36	.	.	PUNCT
m-769	105	1	ijo	ijo	PROPN
m-769	105	2	journals	journal	NOUN
m-769	105	3	doi	doi	X
m-769	105	4	10.5281	10.5281	NUM
m-769	105	5	/	/	SYM
m-769	105	6	zenodo.10443958	zenodo.10443958	PROPN
m-769	105	7	volume	volume	NOUN
m-769	105	8	06	06	NUM
m-769	106	1	|	|	ADV
m-769	106	2	issue	issue	NOUN
m-769	106	3	12	12	NUM
m-769	106	4	|	|	CCONJ
m-769	106	5	december	december	PROPN
m-769	106	6	2023	2023	NUM
m-769	107	1	|	|	ADV
m-769	107	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-769	107	3	14	14	NUM
m-769	107	4	ijo	ijo	PROPN
m-769	107	5	international	international	PROPN
m-769	107	6	journal	journal	PROPN
m-769	107	7	of	of	ADP
m-769	107	8	mathematics	mathematics	PROPN
m-769	107	9	(	(	PUNCT
m-769	107	10	issn	issn	PROPN
m-769	107	11	:	:	PUNCT
m-769	107	12	2992	2992	NUM
m-769	107	13	-	-	SYM
m-769	107	14	4421	4421	NUM
m-769	107	15	)	)	PUNCT
m-769	107	16	michael	michael	PROPN
m-769	107	17	n.	n.	PROPN
m-769	107	18	john	john	PROPN
m-769	107	19	*	*	PROPN
m-769	107	20	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-769	107	21	volume	volume	NOUN
m-769	107	22	06	06	NUM
m-769	107	23	issue	issue	NOUN
m-769	107	24	12	12	NUM
m-769	107	25	||	||	NOUN
m-769	107	26	dec	dec	PROPN
m-769	107	27	.	.	PROPN
m-769	107	28	,	,	PUNCT
m-769	107	29	2023	2023	NUM
m-769	107	30	||	||	NOUN
m-769	108	1	algebraic	algebraic	ADJ
m-769	108	2	and	and	CCONJ
m-769	108	3	topological	topological	ADJ
m-769	108	4	analysis	analysis	NOUN
m-769	108	5	of	of	ADP
m-769	108	6	enveloping	envelop	VERB
m-769	108	7	semigroups	semigroup	NOUN
m-769	108	8	in	in	ADP
m-769	108	9	transformation	transformation	NOUN
m-769	108	10	groups	group	NOUN
m-769	108	11	:	:	PUNCT
m-769	108	12	proximal	proximal	ADJ
m-769	108	13	equivalence	equivalence	NOUN
m-769	108	14	and	and	CCONJ
m-769	108	15	homomorphic	homomorphic	ADJ
m-769	108	16	image	image	NOUN
m-769	108	17	1.proximal	1.proximal	NUM
m-769	108	18	equivalence	equivalence	NOUN
m-769	108	19	as	as	ADP
m-769	108	20	an	an	DET
m-769	108	21	equivalence	equivalence	NOUN
m-769	108	22	relation	relation	NOUN
m-769	108	23	:	:	PUNCT
m-769	108	24	let	let	VERB
m-769	108	25	∼	∼	NOUN
m-769	108	26	be	be	AUX
m-769	108	27	the	the	DET
m-769	108	28	proximal	proximal	ADJ
m-769	108	29	equivalence	equivalence	NOUN
m-769	108	30	relation	relation	NOUN
m-769	108	31	on	on	ADP
m-769	108	32	x.	x.	NOUN
m-769	108	33	we	we	PRON
m-769	108	34	will	will	AUX
m-769	108	35	show	show	VERB
m-769	108	36	that	that	SCONJ
m-769	108	37	∼	∼	NOUN
m-769	108	38	is	be	AUX
m-769	108	39	an	an	DET
m-769	108	40	equivalence	equivalence	NOUN
m-769	108	41	relation	relation	NOUN
m-769	108	42	.	.	PUNCT
m-769	109	1			X
m-769	109	2	reflexivity	reflexivity	NOUN
m-769	109	3	:	:	PUNCT
m-769	109	4	for	for	ADP
m-769	109	5	any	any	DET
m-769	109	6	x∈x	x∈x	NOUN
m-769	109	7	,	,	PUNCT
m-769	109	8	x∼x	x∼x	PUNCT
m-769	110	1	since	since	SCONJ
m-769	110	2	the	the	DET
m-769	110	3	sequence	sequence	NOUN
m-769	110	4	of	of	ADP
m-769	110	5	identity	identity	NOUN
m-769	110	6	transformations	transformation	NOUN
m-769	110	7	in	in	ADP
m-769	110	8	e	e	NOUN
m-769	110	9	converges	converge	NOUN
m-769	110	10	to	to	ADP
m-769	110	11	x.	x.	PROPN
m-769	110	12			NOUN
m-769	110	13	symmetry	symmetry	PROPN
m-769	110	14	:	:	PUNCT
m-769	110	15	if	if	SCONJ
m-769	110	16	x∼y	x∼y	PROPN
m-769	110	17	,	,	PUNCT
m-769	110	18	then	then	ADV
m-769	110	19	there	there	PRON
m-769	110	20	exists	exist	VERB
m-769	110	21	a	a	DET
m-769	110	22	sequence	sequence	NOUN
m-769	110	23	(	(	PUNCT
m-769	110	24	gn)⊆e	gn)⊆e	VERB
m-769	110	25	such	such	ADJ
m-769	110	26	thatlimn→∞μ(gn	thatlimn→∞μ(gn	NOUN
m-769	110	27	,	,	PUNCT
m-769	110	28	x)=limn→∞μ(gn	x)=limn→∞μ(gn	PROPN
m-769	110	29	,	,	PUNCT
m-769	110	30	y	y	PROPN
m-769	110	31	)	)	PUNCT
m-769	110	32	.	.	PUNCT
m-769	111	1	therefore	therefore	ADV
m-769	111	2	,	,	PUNCT
m-769	111	3	y∼x	y∼x	PROPN
m-769	111	4	as	as	ADV
m-769	111	5	well	well	ADV
m-769	111	6	.	.	PUNCT
m-769	112	1			PRON
m-769	112	2	transitivity	transitivity	NOUN
m-769	112	3	:	:	PUNCT
m-769	112	4	if	if	SCONJ
m-769	112	5	x∼y	x∼y	PROPN
m-769	112	6	and	and	CCONJ
m-769	112	7	y∼z	y∼z	PROPN
m-769	112	8	,	,	PUNCT
m-769	112	9	then	then	ADV
m-769	112	10	there	there	PRON
m-769	112	11	exist	exist	VERB
m-769	112	12	sequences	sequence	NOUN
m-769	112	13	(	(	PUNCT
m-769	112	14	gn	gn	PROPN
m-769	112	15	)	)	PUNCT
m-769	112	16	and	and	CCONJ
m-769	112	17	(	(	PUNCT
m-769	112	18	hn	hn	NOUN
m-769	112	19	)	)	PUNCT
m-769	112	20	in	in	ADP
m-769	112	21	e	e	ADP
m-769	112	22	such	such	ADJ
m-769	112	23	that	that	DET
m-769	112	24	limn→∞μ(gn	limn→∞μ(gn	NOUN
m-769	112	25	,	,	PUNCT
m-769	112	26	x)=limn→∞μ(gn	x)=limn→∞μ(gn	NOUN
m-769	112	27	,	,	PUNCT
m-769	112	28	y	y	PROPN
m-769	112	29	)	)	PUNCT
m-769	112	30	and	and	CCONJ
m-769	112	31	limn→∞μ(hn	limn→∞μ(hn	NOUN
m-769	112	32	,	,	PUNCT
m-769	112	33	y)=limn→∞μ(hn	y)=limn→∞μ(hn	NUM
m-769	112	34	,	,	PUNCT
m-769	112	35	z	z	NOUN
m-769	112	36	)	)	PUNCT
m-769	112	37	.	.	PUNCT
m-769	113	1	the	the	DET
m-769	113	2	concatenation	concatenation	NOUN
m-769	113	3	of	of	ADP
m-769	113	4	these	these	DET
m-769	113	5	sequences	sequence	NOUN
m-769	113	6	,	,	PUNCT
m-769	113	7	(	(	PUNCT
m-769	113	8	gn⋅hn	gn⋅hn	NOUN
m-769	113	9	)	)	PUNCT
m-769	113	10	,	,	PUNCT
m-769	113	11	is	be	AUX
m-769	113	12	also	also	ADV
m-769	113	13	in	in	ADP
m-769	113	14	e	e	NOUN
m-769	113	15	by	by	ADP
m-769	113	16	the	the	DET
m-769	113	17	group	group	NOUN
m-769	113	18	properties	property	NOUN
m-769	113	19	.	.	PUNCT
m-769	114	1	furthermore	furthermore	ADV
m-769	114	2	,	,	PUNCT
m-769	114	3	limn→∞μ(gn⋅hn	limn→∞μ(gn⋅hn	PROPN
m-769	114	4	,	,	PUNCT
m-769	114	5	x)=limn→∞μ(gn	x)=limn→∞μ(gn	NOUN
m-769	114	6	,	,	PUNCT
m-769	114	7	μ(hn	μ(hn	PROPN
m-769	114	8	,	,	PUNCT
m-769	114	9	x))=limn→∞μ(gn	x))=limn→∞μ(gn	PROPN
m-769	114	10	,	,	PUNCT
m-769	114	11	y)=limn→∞μ(hn	y)=limn→∞μ(hn	NUM
m-769	114	12	,	,	PUNCT
m-769	114	13	z	z	NOUN
m-769	114	14	)	)	PUNCT
m-769	114	15	,	,	PUNCT
m-769	114	16	implying	imply	VERB
m-769	114	17	x∼z	x∼z	PROPN
m-769	114	18	.	.	PUNCT
m-769	115	1	2	2	X
m-769	115	2	.	.	X
m-769	115	3	existence	existence	NOUN
m-769	115	4	of	of	ADP
m-769	115	5	one	one	NUM
m-769	115	6	minimal	minimal	ADJ
m-769	115	7	right	right	ADJ
m-769	115	8	ideal	ideal	NOUN
m-769	115	9	in	in	ADP
m-769	115	10	e	e	NOUN
m-769	115	11	:	:	PUNCT
m-769	115	12	now	now	ADV
m-769	115	13	,	,	PUNCT
m-769	115	14	let	let	VERB
m-769	115	15	's	us	PRON
m-769	115	16	show	show	VERB
m-769	115	17	the	the	DET
m-769	115	18	converse	converse	NOUN
m-769	115	19	.	.	PUNCT
m-769	116	1	assume	assume	VERB
m-769	116	2	there	there	PRON
m-769	116	3	exists	exist	VERB
m-769	116	4	only	only	ADV
m-769	116	5	one	one	NUM
m-769	116	6	minimal	minimal	ADJ
m-769	116	7	right	right	ADJ
m-769	116	8	ideal	ideal	NOUN
m-769	116	9	in	in	ADP
m-769	116	10	e.	e.	PROPN
m-769	116	11	we	we	PRON
m-769	116	12	need	need	VERB
m-769	116	13	to	to	PART
m-769	116	14	show	show	VERB
m-769	116	15	that	that	SCONJ
m-769	116	16	∼	∼	NOUN
m-769	116	17	is	be	AUX
m-769	116	18	an	an	DET
m-769	116	19	equivalence	equivalence	NOUN
m-769	116	20	relation	relation	NOUN
m-769	116	21	.	.	PUNCT
m-769	117	1			X
m-769	117	2	reflexivity	reflexivity	NOUN
m-769	117	3	:	:	PUNCT
m-769	117	4	by	by	ADP
m-769	117	5	the	the	DET
m-769	117	6	definition	definition	NOUN
m-769	117	7	of	of	ADP
m-769	117	8	minimal	minimal	ADJ
m-769	117	9	right	right	ADJ
m-769	117	10	ideals	ideal	NOUN
m-769	117	11	,	,	PUNCT
m-769	117	12	there	there	PRON
m-769	117	13	exists	exist	VERB
m-769	117	14	a	a	DET
m-769	117	15	sequence	sequence	NOUN
m-769	117	16	(	(	PUNCT
m-769	117	17	gn	gn	PROPN
m-769	117	18	)	)	PUNCT
m-769	117	19	⊆e	⊆e	NOUN
m-769	117	20	such	such	ADJ
m-769	117	21	that	that	DET
m-769	117	22	limn→∞μ(gn	limn→∞μ(gn	NOUN
m-769	117	23	,	,	PUNCT
m-769	117	24	x)=x	x)=x	PROPN
m-769	117	25	.	.	PUNCT
m-769	118	1			PROPN
m-769	118	2	symmetry	symmetry	NOUN
m-769	118	3	:	:	PUNCT
m-769	118	4	if	if	SCONJ
m-769	118	5	x∼y	x∼y	PROPN
m-769	118	6	,	,	PUNCT
m-769	118	7	then	then	ADV
m-769	118	8	there	there	PRON
m-769	118	9	exists	exist	VERB
m-769	118	10	a	a	DET
m-769	118	11	sequence	sequence	NOUN
m-769	118	12	(	(	PUNCT
m-769	118	13	gn)⊆e	gn)⊆e	VERB
m-769	118	14	such	such	DET
m-769	118	15	that	that	DET
m-769	118	16	limn→∞μ(gn	limn→∞μ(gn	NOUN
m-769	118	17	,	,	PUNCT
m-769	118	18	x)=limn→∞μ(gn	x)=limn→∞μ(gn	PROPN
m-769	118	19	,	,	PUNCT
m-769	118	20	y	y	PROPN
m-769	118	21	)	)	PUNCT
m-769	118	22	.	.	PUNCT
m-769	119	1	since	since	SCONJ
m-769	119	2	there	there	PRON
m-769	119	3	is	be	VERB
m-769	119	4	only	only	ADV
m-769	119	5	one	one	NUM
m-769	119	6	minimal	minimal	ADJ
m-769	119	7	right	right	ADJ
m-769	119	8	ideal	ideal	NOUN
m-769	119	9	,	,	PUNCT
m-769	119	10	(	(	PUNCT
m-769	119	11	gn	gn	INTJ
m-769	119	12	−1	−1	NOUN
m-769	119	13	)	)	PUNCT
m-769	119	14	is	be	AUX
m-769	119	15	also	also	ADV
m-769	119	16	in	in	ADP
m-769	119	17	e	e	NOUN
m-769	119	18	,	,	PUNCT
m-769	119	19	and	and	CCONJ
m-769	119	20	limn→∞μ(gn	limn→∞μ(gn	ADJ
m-769	119	21	−1,x)=limn→∞μ(gn	−1,x)=limn→∞μ(gn	PROPN
m-769	119	22	−1,μ(gn	−1,μ(gn	PROPN
m-769	119	23	,	,	PUNCT
m-769	119	24	x))=limn→∞μ(gn	x))=limn→∞μ(gn	PROPN
m-769	119	25	−1⋅gn	−1⋅gn	ADV
m-769	119	26	,	,	PUNCT
m-769	119	27	x)=limn→∞μ(e	x)=limn→∞μ(e	PROPN
m-769	119	28	,	,	PUNCT
m-769	119	29	x)=x	x)=x	PRON
m-769	119	30	.	.	PUNCT
m-769	120	1	therefore	therefore	ADV
m-769	120	2	,	,	PUNCT
m-769	120	3	y∼x	y∼x	PROPN
m-769	120	4	.	.	PUNCT
m-769	121	1			PROPN
m-769	121	2	transitivity	transitivity	NOUN
m-769	121	3	:	:	PUNCT
m-769	121	4	if	if	SCONJ
m-769	121	5	x∼y	x∼y	PROPN
m-769	121	6	and	and	CCONJ
m-769	121	7	y∼z	y∼z	PROPN
m-769	121	8	,	,	PUNCT
m-769	121	9	there	there	PRON
m-769	121	10	exist	exist	VERB
m-769	121	11	sequences	sequence	NOUN
m-769	121	12	(	(	PUNCT
m-769	121	13	gn	gn	PROPN
m-769	121	14	)	)	PUNCT
m-769	121	15	and	and	CCONJ
m-769	121	16	(	(	PUNCT
m-769	121	17	hn	hn	NOUN
m-769	121	18	)	)	PUNCT
m-769	121	19	in	in	ADP
m-769	121	20	e	e	ADP
m-769	121	21	such	such	ADJ
m-769	121	22	that	that	DET
m-769	121	23	limn→∞μ(gn	limn→∞μ(gn	NOUN
m-769	121	24	,	,	PUNCT
m-769	121	25	x)=limn→∞μ(gn	x)=limn→∞μ(gn	NOUN
m-769	121	26	,	,	PUNCT
m-769	121	27	y	y	PROPN
m-769	121	28	)	)	PUNCT
m-769	121	29	and	and	CCONJ
m-769	121	30	limn→∞μ(hn	limn→∞μ(hn	NOUN
m-769	121	31	,	,	PUNCT
m-769	121	32	y)=limn→∞μ(hn	y)=limn→∞μ(hn	NUM
m-769	121	33	,	,	PUNCT
m-769	121	34	z	z	NOUN
m-769	121	35	)	)	PUNCT
m-769	121	36	.	.	PUNCT
m-769	122	1	the	the	DET
m-769	122	2	concatenation	concatenation	NOUN
m-769	122	3	of	of	ADP
m-769	122	4	these	these	DET
m-769	122	5	sequences	sequence	NOUN
m-769	122	6	,	,	PUNCT
m-769	122	7	(	(	PUNCT
m-769	122	8	gn⋅hn	gn⋅hn	NOUN
m-769	122	9	)	)	PUNCT
m-769	122	10	,	,	PUNCT
m-769	122	11	is	be	AUX
m-769	122	12	also	also	ADV
m-769	122	13	in	in	ADP
m-769	122	14	e	e	NOUN
m-769	122	15	by	by	ADP
m-769	122	16	the	the	DET
m-769	122	17	group	group	NOUN
m-769	122	18	properties	property	NOUN
m-769	122	19	.	.	PUNCT
m-769	123	1	furthermore	furthermore	ADV
m-769	123	2	,	,	PUNCT
m-769	123	3	limn→∞μ(gn⋅hn	limn→∞μ(gn⋅hn	PROPN
m-769	123	4	,	,	PUNCT
m-769	123	5	x)=limn→∞μ(gn	x)=limn→∞μ(gn	NOUN
m-769	123	6	,	,	PUNCT
m-769	123	7	μ(hn	μ(hn	PROPN
m-769	123	8	,	,	PUNCT
m-769	123	9	x))=limn→∞μ(gn	x))=limn→∞μ(gn	PROPN
m-769	123	10	,	,	PUNCT
m-769	123	11	y)=limn→∞μ(hn	y)=limn→∞μ(hn	NUM
m-769	123	12	,	,	PUNCT
m-769	123	13	z	z	NOUN
m-769	123	14	)	)	PUNCT
m-769	123	15	,	,	PUNCT
m-769	123	16	implying	imply	VERB
m-769	123	17	x∼z	x∼z	PROPN
m-769	123	18	.	.	PUNCT
m-769	124	1	therefore	therefore	ADV
m-769	124	2	,	,	PUNCT
m-769	124	3	the	the	DET
m-769	124	4	proximal	proximal	ADJ
m-769	124	5	equivalence	equivalence	NOUN
m-769	124	6	relation	relation	NOUN
m-769	124	7	is	be	AUX
m-769	124	8	an	an	DET
m-769	124	9	equivalence	equivalence	NOUN
m-769	124	10	relation	relation	NOUN
m-769	124	11	if	if	SCONJ
m-769	124	12	and	and	CCONJ
m-769	124	13	only	only	ADV
m-769	124	14	if	if	SCONJ
m-769	124	15	there	there	PRON
m-769	124	16	exists	exist	VERB
m-769	124	17	only	only	ADV
m-769	124	18	one	one	NUM
m-769	124	19	minimal	minimal	ADJ
m-769	124	20	right	right	ADJ
m-769	124	21	ideal	ideal	NOUN
m-769	124	22	in	in	ADP
m-769	124	23	e.	e.	PROPN
m-769	124	24	ijo	ijo	PROPN
m-769	124	25	journals	journal	NOUN
m-769	124	26	doi	doi	PROPN
m-769	124	27	10.5281	10.5281	NUM
m-769	124	28	/	/	SYM
m-769	124	29	zenodo.10443958	zenodo.10443958	PROPN
m-769	124	30	volume	volume	NOUN
m-769	124	31	06	06	NUM
m-769	125	1	|	|	ADV
m-769	125	2	issue	issue	NOUN
m-769	125	3	12	12	NUM
m-769	125	4	|	|	CCONJ
m-769	125	5	december	december	PROPN
m-769	125	6	2023	2023	NUM
m-769	126	1	|	|	ADV
m-769	126	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-769	126	3	15	15	NUM
m-769	126	4	ijo	ijo	PROPN
m-769	126	5	international	international	PROPN
m-769	126	6	journal	journal	PROPN
m-769	126	7	of	of	ADP
m-769	126	8	mathematics	mathematics	PROPN
m-769	126	9	(	(	PUNCT
m-769	126	10	issn	issn	PROPN
m-769	126	11	:	:	PUNCT
m-769	126	12	2992	2992	NUM
m-769	126	13	-	-	SYM
m-769	126	14	4421	4421	NUM
m-769	126	15	)	)	PUNCT
m-769	127	1	michael	michael	PROPN
m-769	127	2	n.	n.	PROPN
m-769	127	3	john	john	PROPN
m-769	127	4	*	*	PROPN
m-769	127	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-769	127	6	volume	volume	NOUN
m-769	127	7	06	06	NUM
m-769	127	8	issue	issue	NOUN
m-769	127	9	12	12	NUM
m-769	127	10	||	||	NOUN
m-769	127	11	dec	dec	PROPN
m-769	127	12	.	.	PROPN
m-769	127	13	,	,	PUNCT
m-769	127	14	2023	2023	NUM
m-769	127	15	||	||	NOUN
m-769	128	1	algebraic	algebraic	ADJ
m-769	128	2	and	and	CCONJ
m-769	128	3	topological	topological	ADJ
m-769	128	4	analysis	analysis	NOUN
m-769	128	5	of	of	ADP
m-769	128	6	enveloping	envelop	VERB
m-769	128	7	semigroups	semigroup	NOUN
m-769	128	8	in	in	ADP
m-769	128	9	transformation	transformation	NOUN
m-769	128	10	groups	group	NOUN
m-769	128	11	:	:	PUNCT
m-769	128	12	proximal	proximal	ADJ
m-769	128	13	equivalence	equivalence	NOUN
m-769	128	14	and	and	CCONJ
m-769	128	15	homomorphic	homomorphic	ADJ
m-769	128	16	image	image	NOUN
m-769	128	17	theorem	theorem	VERB
m-769	128	18	4.3.the	4.3.the	DET
m-769	128	19	algebraic	algebraic	ADJ
m-769	128	20	structure	structure	NOUN
m-769	128	21	of	of	ADP
m-769	128	22	e	e	NOUN
m-769	128	23	directly	directly	ADV
m-769	128	24	correlates	correlate	VERB
m-769	128	25	with	with	ADP
m-769	128	26	the	the	DET
m-769	128	27	recursive	recursive	ADJ
m-769	128	28	properties	property	NOUN
m-769	128	29	of	of	ADP
m-769	128	30	the	the	DET
m-769	128	31	transformation	transformation	NOUN
m-769	128	32	group	group	NOUN
m-769	128	33	t.	t.	NOUN
m-769	128	34	proof	proof	NOUN
m-769	128	35	:	:	PUNCT
m-769	128	36	let	let	VERB
m-769	128	37	(	(	PUNCT
m-769	128	38	x	x	X
m-769	128	39	,	,	PUNCT
m-769	128	40	t	t	PROPN
m-769	128	41	,	,	PUNCT
m-769	128	42	μ	μ	NOUN
m-769	128	43	)	)	PUNCT
m-769	128	44	be	be	VERB
m-769	128	45	a	a	DET
m-769	128	46	transformation	transformation	NOUN
m-769	128	47	group	group	NOUN
m-769	128	48	with	with	ADP
m-769	128	49	a	a	DET
m-769	128	50	phase	phase	NOUN
m-769	128	51	space	space	NOUN
m-769	128	52	x	x	NOUN
m-769	128	53	,	,	PUNCT
m-769	128	54	a	a	DET
m-769	128	55	group	group	NOUN
m-769	128	56	t	t	NOUN
m-769	128	57	of	of	ADP
m-769	128	58	transformations	transformation	NOUN
m-769	128	59	on	on	ADP
m-769	128	60	x	x	NOUN
m-769	128	61	,	,	PUNCT
m-769	128	62	and	and	CCONJ
m-769	128	63	a	a	DET
m-769	128	64	continuous	continuous	ADJ
m-769	128	65	action	action	NOUN
m-769	128	66	μ	μ	NOUN
m-769	128	67	:	:	PUNCT
m-769	128	68	t×x→x	t×x→x	PROPN
m-769	128	69	.	.	PUNCT
m-769	129	1	the	the	DET
m-769	129	2	enveloping	enveloping	NOUN
m-769	129	3	semigroup	semigroup	NOUN
m-769	129	4	is	be	AUX
m-769	129	5	denoted	denote	VERB
m-769	129	6	as	as	ADP
m-769	129	7	�	�	PROPN
m-769	129	8	=	=	SYM
m-769	129	9	�	�	PROPN
m-769	129	10	�	�	PROPN
m-769	129	11	,	,	PUNCT
m-769	129	12	the	the	DET
m-769	129	13	closure	closure	NOUN
m-769	129	14	of	of	ADP
m-769	129	15	t	t	PROPN
m-769	129	16	in	in	ADP
m-769	129	17	the	the	DET
m-769	129	18	product	product	NOUN
m-769	129	19	space	space	NOUN
m-769	129	20	x×x	x×x	PROPN
m-769	129	21	.	.	PROPN
m-769	129	22	correlation	correlation	NOUN
m-769	129	23	between	between	ADP
m-769	129	24	algebraic	algebraic	ADJ
m-769	129	25	structure	structure	NOUN
m-769	129	26	and	and	CCONJ
m-769	129	27	recursive	recursive	ADJ
m-769	129	28	properties	property	NOUN
m-769	129	29	:	:	PUNCT
m-769	130	1	1	1	X
m-769	130	2	.	.	X
m-769	130	3	algebraic	algebraic	ADJ
m-769	130	4	structure	structure	NOUN
m-769	130	5	of	of	ADP
m-769	130	6	e	e	NOUN
m-769	130	7	:	:	PUNCT
m-769	130	8			NOUN
m-769	130	9	the	the	DET
m-769	130	10	enveloping	enveloping	NOUN
m-769	130	11	semigroupe	semigroupe	NOUN
m-769	130	12	is	be	AUX
m-769	130	13	a	a	DET
m-769	130	14	closure	closure	NOUN
m-769	130	15	of	of	ADP
m-769	130	16	t	t	PROPN
m-769	130	17	,	,	PUNCT
m-769	130	18	encompassing	encompass	VERB
m-769	130	19	all	all	DET
m-769	130	20	possible	possible	ADJ
m-769	130	21	compositions	composition	NOUN
m-769	130	22	and	and	CCONJ
m-769	130	23	limits	limit	NOUN
m-769	130	24	of	of	ADP
m-769	130	25	transformations	transformation	NOUN
m-769	130	26	in	in	ADP
m-769	130	27	t.	t.	NOUN
m-769	130	28	the	the	DET
m-769	130	29	elements	element	NOUN
m-769	130	30	of	of	ADP
m-769	130	31	e	e	NOUN
m-769	130	32	are	be	AUX
m-769	130	33	sequences	sequence	NOUN
m-769	130	34	of	of	ADP
m-769	130	35	transformations	transformation	NOUN
m-769	130	36	that	that	PRON
m-769	130	37	converge	converge	VERB
m-769	130	38	to	to	ADP
m-769	130	39	a	a	DET
m-769	130	40	limit	limit	NOUN
m-769	130	41	in	in	ADP
m-769	130	42	x×x	x×x	PROPN
m-769	130	43	.	.	PROPN
m-769	131	1	2	2	NUM
m-769	131	2	.	.	PUNCT
m-769	131	3	recursive	recursive	ADJ
m-769	131	4	properties	property	NOUN
m-769	131	5	of	of	ADP
m-769	131	6	t	t	PROPN
m-769	131	7	:	:	PUNCT
m-769	131	8			X
m-769	131	9	the	the	DET
m-769	131	10	recursive	recursive	ADJ
m-769	131	11	properties	property	NOUN
m-769	131	12	of	of	ADP
m-769	131	13	t	t	PROPN
m-769	131	14	involve	involve	VERB
m-769	131	15	the	the	DET
m-769	131	16	composition	composition	NOUN
m-769	131	17	of	of	ADP
m-769	131	18	transformations	transformation	NOUN
m-769	131	19	,	,	PUNCT
m-769	131	20	where	where	SCONJ
m-769	131	21	each	each	DET
m-769	131	22	transformation	transformation	NOUN
m-769	131	23	in	in	ADP
m-769	131	24	t	t	PROPN
m-769	131	25	maps	map	NOUN
m-769	131	26	points	point	VERB
m-769	131	27	in	in	ADP
m-769	131	28	x	x	PUNCT
m-769	131	29	to	to	ADP
m-769	131	30	other	other	ADJ
m-769	131	31	points	point	NOUN
m-769	131	32	.	.	PUNCT
m-769	132	1	the	the	DET
m-769	132	2	recursion	recursion	NOUN
m-769	132	3	represents	represent	VERB
m-769	132	4	the	the	DET
m-769	132	5	repeated	repeat	VERB
m-769	132	6	application	application	NOUN
m-769	132	7	of	of	ADP
m-769	132	8	these	these	DET
m-769	132	9	transformations	transformation	NOUN
m-769	132	10	.	.	PUNCT
m-769	133	1	proof	proof	NOUN
m-769	133	2	of	of	ADP
m-769	133	3	correlation	correlation	NOUN
m-769	133	4	:	:	PUNCT
m-769	133	5	the	the	DET
m-769	133	6	algebraic	algebraic	ADJ
m-769	133	7	structure	structure	NOUN
m-769	133	8	of	of	ADP
m-769	133	9	e	e	NOUN
m-769	133	10	directly	directly	ADV
m-769	133	11	correlates	correlate	VERB
m-769	133	12	with	with	ADP
m-769	133	13	the	the	DET
m-769	133	14	recursive	recursive	ADJ
m-769	133	15	properties	property	NOUN
m-769	133	16	of	of	ADP
m-769	133	17	t	t	NOUN
m-769	133	18	due	due	ADP
m-769	133	19	to	to	ADP
m-769	133	20	the	the	DET
m-769	133	21	closure	closure	NOUN
m-769	133	22	operation	operation	NOUN
m-769	133	23	:	:	PUNCT
m-769	133	24			X
m-769	133	25	composition	composition	NOUN
m-769	133	26	of	of	ADP
m-769	133	27	transformations	transformation	NOUN
m-769	133	28	in	in	ADP
m-769	133	29	t	t	PROPN
m-769	133	30	:	:	PUNCT
m-769	133	31			NOUN
m-769	133	32	the	the	DET
m-769	133	33	closure	closure	NOUN
m-769	133	34	of	of	ADP
m-769	133	35	t	t	PROPN
m-769	133	36	in	in	ADP
m-769	133	37	e	e	NOUN
m-769	133	38	ensures	ensure	VERB
m-769	133	39	that	that	SCONJ
m-769	133	40	the	the	DET
m-769	133	41	composition	composition	NOUN
m-769	133	42	of	of	ADP
m-769	133	43	transformations	transformation	NOUN
m-769	133	44	in	in	ADP
m-769	133	45	t	t	PROPN
m-769	133	46	remains	remain	VERB
m-769	133	47	within	within	ADP
m-769	133	48	e.	e.	PROPN
m-769	133	49	this	this	DET
m-769	133	50	closure	closure	NOUN
m-769	133	51	is	be	AUX
m-769	133	52	essential	essential	ADJ
m-769	133	53	for	for	ADP
m-769	133	54	capturing	capture	VERB
m-769	133	55	the	the	DET
m-769	133	56	recursive	recursive	ADJ
m-769	133	57	nature	nature	NOUN
m-769	133	58	of	of	ADP
m-769	133	59	transformations	transformation	NOUN
m-769	133	60	in	in	ADP
m-769	133	61	t.	t.	NOUN
m-769	133	62			NOUN
m-769	133	63	limits	limit	NOUN
m-769	133	64	and	and	CCONJ
m-769	133	65	convergence	convergence	NOUN
m-769	133	66	:	:	PUNCT
m-769	133	67			PRON
m-769	133	68	the	the	DET
m-769	133	69	closure	closure	NOUN
m-769	133	70	operation	operation	NOUN
m-769	133	71	allows	allow	VERB
m-769	133	72	the	the	DET
m-769	133	73	inclusion	inclusion	NOUN
m-769	133	74	of	of	ADP
m-769	133	75	limit	limit	NOUN
m-769	133	76	points	point	NOUN
m-769	133	77	in	in	ADP
m-769	133	78	e.	e.	PROPN
m-769	133	79	as	as	ADP
m-769	133	80	transformations	transformation	NOUN
m-769	133	81	in	in	ADP
m-769	133	82	t	t	PROPN
m-769	133	83	are	be	AUX
m-769	133	84	composed	compose	VERB
m-769	133	85	and	and	CCONJ
m-769	133	86	iterated	iterate	VERB
m-769	133	87	,	,	PUNCT
m-769	133	88	the	the	DET
m-769	133	89	limits	limit	NOUN
m-769	133	90	of	of	ADP
m-769	133	91	these	these	DET
m-769	133	92	compositions	composition	NOUN
m-769	133	93	,	,	PUNCT
m-769	133	94	if	if	SCONJ
m-769	133	95	they	they	PRON
m-769	133	96	exist	exist	VERB
m-769	133	97	,	,	PUNCT
m-769	133	98	are	be	AUX
m-769	133	99	captured	capture	VERB
m-769	133	100	in	in	ADP
m-769	133	101	e.	e.	PROPN
m-769	133	102	this	this	PRON
m-769	133	103	reflects	reflect	VERB
m-769	133	104	the	the	DET
m-769	133	105	recursive	recursive	ADJ
m-769	133	106	behavior	behavior	NOUN
m-769	133	107	of	of	ADP
m-769	133	108	t	t	PROPN
m-769	133	109	as	as	SCONJ
m-769	133	110	transformations	transformation	NOUN
m-769	133	111	are	be	AUX
m-769	133	112	applied	apply	VERB
m-769	133	113	repeatedly	repeatedly	ADV
m-769	133	114	.	.	PUNCT
m-769	134	1	ijo	ijo	PROPN
m-769	134	2	journals	journal	NOUN
m-769	134	3	doi	doi	X
m-769	134	4	10.5281	10.5281	NUM
m-769	134	5	/	/	SYM
m-769	134	6	zenodo.10443958	zenodo.10443958	PROPN
m-769	134	7	volume	volume	NOUN
m-769	134	8	06	06	NUM
m-769	135	1	|	|	ADV
m-769	135	2	issue	issue	NOUN
m-769	135	3	12	12	NUM
m-769	135	4	|	|	CCONJ
m-769	135	5	december	december	PROPN
m-769	135	6	2023	2023	NUM
m-769	136	1	|	|	ADV
m-769	136	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-769	136	3	16	16	NUM
m-769	136	4	ijo	ijo	PROPN
m-769	136	5	international	international	PROPN
m-769	136	6	journal	journal	PROPN
m-769	136	7	of	of	ADP
m-769	136	8	mathematics	mathematics	PROPN
m-769	136	9	(	(	PUNCT
m-769	136	10	issn	issn	PROPN
m-769	136	11	:	:	PUNCT
m-769	136	12	2992	2992	NUM
m-769	136	13	-	-	SYM
m-769	136	14	4421	4421	NUM
m-769	136	15	)	)	PUNCT
m-769	136	16	michael	michael	PROPN
m-769	136	17	n.	n.	PROPN
m-769	136	18	john	john	PROPN
m-769	136	19	*	*	PROPN
m-769	136	20	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-769	136	21	volume	volume	NOUN
m-769	136	22	06	06	NUM
m-769	136	23	issue	issue	NOUN
m-769	136	24	12	12	NUM
m-769	136	25	||	||	NOUN
m-769	136	26	dec	dec	PROPN
m-769	136	27	.	.	PROPN
m-769	136	28	,	,	PUNCT
m-769	136	29	2023	2023	NUM
m-769	136	30	||	||	NOUN
m-769	137	1	algebraic	algebraic	ADJ
m-769	137	2	and	and	CCONJ
m-769	137	3	topological	topological	ADJ
m-769	137	4	analysis	analysis	NOUN
m-769	137	5	of	of	ADP
m-769	137	6	enveloping	envelop	VERB
m-769	137	7	semigroups	semigroup	NOUN
m-769	137	8	in	in	ADP
m-769	137	9	transformation	transformation	NOUN
m-769	137	10	groups	group	NOUN
m-769	137	11	:	:	PUNCT
m-769	137	12	proximal	proximal	ADJ
m-769	137	13	equivalence	equivalence	NOUN
m-769	137	14	and	and	CCONJ
m-769	137	15	homomorphic	homomorphic	ADJ
m-769	137	16	image	image	NOUN
m-769	137	17			PRON
m-769	137	18	topological	topological	ADJ
m-769	137	19	structure	structure	NOUN
m-769	137	20	:	:	PUNCT
m-769	137	21			ADJ
m-769	137	22	the	the	DET
m-769	137	23	topological	topological	ADJ
m-769	137	24	closure	closure	NOUN
m-769	137	25	ensures	ensure	VERB
m-769	137	26	that	that	SCONJ
m-769	137	27	e	e	NOUN
m-769	137	28	captures	capture	VERB
m-769	137	29	not	not	PART
m-769	137	30	only	only	ADV
m-769	137	31	the	the	DET
m-769	137	32	algebraic	algebraic	ADJ
m-769	137	33	composition	composition	NOUN
m-769	137	34	of	of	ADP
m-769	137	35	transformations	transformation	NOUN
m-769	137	36	but	but	CCONJ
m-769	137	37	also	also	ADV
m-769	137	38	the	the	DET
m-769	137	39	continuity	continuity	NOUN
m-769	137	40	and	and	CCONJ
m-769	137	41	convergence	convergence	NOUN
m-769	137	42	properties	property	NOUN
m-769	137	43	.	.	PUNCT
m-769	138	1	this	this	PRON
m-769	138	2	is	be	AUX
m-769	138	3	crucial	crucial	ADJ
m-769	138	4	for	for	ADP
m-769	138	5	understanding	understand	VERB
m-769	138	6	the	the	DET
m-769	138	7	recursive	recursive	ADJ
m-769	138	8	nature	nature	NOUN
m-769	138	9	of	of	ADP
m-769	138	10	transformations	transformation	NOUN
m-769	138	11	in	in	ADP
m-769	138	12	t	t	PROPN
m-769	138	13	within	within	ADP
m-769	138	14	the	the	DET
m-769	138	15	topological	topological	ADJ
m-769	138	16	space	space	NOUN
m-769	138	17	x.	x.	NOUN
m-769	138	18	therefore	therefore	ADV
m-769	138	19	,	,	PUNCT
m-769	138	20	the	the	DET
m-769	138	21	algebraic	algebraic	ADJ
m-769	138	22	structure	structure	NOUN
m-769	138	23	of	of	ADP
m-769	138	24	e	e	PROPN
m-769	138	25	is	be	AUX
m-769	138	26	intricately	intricately	ADV
m-769	138	27	connected	connect	VERB
m-769	138	28	to	to	ADP
m-769	138	29	the	the	DET
m-769	138	30	recursive	recursive	ADJ
m-769	138	31	properties	property	NOUN
m-769	138	32	of	of	ADP
m-769	138	33	the	the	DET
m-769	138	34	transformation	transformation	NOUN
m-769	138	35	group	group	NOUN
m-769	138	36	t.	t.	PROPN
m-769	138	37	the	the	DET
m-769	138	38	closure	closure	NOUN
m-769	138	39	of	of	ADP
m-769	138	40	t	t	PROPN
m-769	138	41	in	in	ADP
m-769	138	42	e	e	NOUN
m-769	138	43	allows	allow	VERB
m-769	138	44	for	for	ADP
m-769	138	45	the	the	DET
m-769	138	46	representation	representation	NOUN
m-769	138	47	of	of	ADP
m-769	138	48	limits	limit	NOUN
m-769	138	49	and	and	CCONJ
m-769	138	50	compositions	composition	NOUN
m-769	138	51	of	of	ADP
m-769	138	52	transformations	transformation	NOUN
m-769	138	53	,	,	PUNCT
m-769	138	54	providing	provide	VERB
m-769	138	55	a	a	DET
m-769	138	56	mathematical	mathematical	ADJ
m-769	138	57	framework	framework	NOUN
m-769	138	58	that	that	PRON
m-769	138	59	mirrors	mirror	VERB
m-769	138	60	the	the	DET
m-769	138	61	recursive	recursive	ADJ
m-769	138	62	behavior	behavior	NOUN
m-769	138	63	inherent	inherent	ADJ
m-769	138	64	in	in	ADP
m-769	138	65	the	the	DET
m-769	138	66	transformation	transformation	NOUN
m-769	138	67	group	group	NOUN
m-769	138	68	.	.	PUNCT
m-769	139	1	theorem	theorem	VERB
m-769	139	2	4.4.homomorphic	4.4.homomorphic	NUM
m-769	139	3	images	image	NOUN
m-769	139	4	of	of	ADP
m-769	139	5	transformation	transformation	NOUN
m-769	139	6	groups	group	NOUN
m-769	139	7	can	can	AUX
m-769	139	8	be	be	AUX
m-769	139	9	effectively	effectively	ADV
m-769	139	10	studied	study	VERB
m-769	139	11	through	through	ADP
m-769	139	12	their	their	PRON
m-769	139	13	enveloping	enveloping	NOUN
m-769	139	14	semigroups	semigroup	NOUN
m-769	139	15	.	.	PUNCT
m-769	140	1	proof	proof	NOUN
m-769	140	2	.	.	PUNCT
m-769	141	1	let	let	VERB
m-769	141	2	(	(	PUNCT
m-769	141	3	x	x	X
m-769	141	4	,	,	PUNCT
m-769	141	5	t	t	PROPN
m-769	141	6	,	,	PUNCT
m-769	141	7	μ	μ	NOUN
m-769	141	8	)	)	PUNCT
m-769	141	9	be	be	VERB
m-769	141	10	a	a	DET
m-769	141	11	transformation	transformation	NOUN
m-769	141	12	group	group	NOUN
m-769	141	13	with	with	ADP
m-769	141	14	a	a	DET
m-769	141	15	phase	phase	NOUN
m-769	141	16	space	space	NOUN
m-769	141	17	x	x	NOUN
m-769	141	18	,	,	PUNCT
m-769	141	19	a	a	DET
m-769	141	20	group	group	NOUN
m-769	141	21	t	t	NOUN
m-769	141	22	of	of	ADP
m-769	141	23	transformations	transformation	NOUN
m-769	141	24	on	on	ADP
m-769	141	25	x	x	NOUN
m-769	141	26	,	,	PUNCT
m-769	141	27	and	and	CCONJ
m-769	141	28	a	a	DET
m-769	141	29	continuous	continuous	ADJ
m-769	141	30	action	action	NOUN
m-769	141	31	μ	μ	NOUN
m-769	141	32	:	:	PUNCT
m-769	141	33	t×x→x	t×x→x	PROPN
m-769	141	34	.	.	PUNCT
m-769	142	1	the	the	DET
m-769	142	2	enveloping	enveloping	NOUN
m-769	142	3	semigroup	semigroup	NOUN
m-769	142	4	is	be	AUX
m-769	142	5	denoted	denote	VERB
m-769	142	6	as	as	ADP
m-769	142	7	�	�	PROPN
m-769	142	8	=	=	SYM
m-769	142	9	�	�	PROPN
m-769	142	10	�	�	PROPN
m-769	142	11	,	,	PUNCT
m-769	142	12	the	the	DET
m-769	142	13	closure	closure	NOUN
m-769	142	14	of	of	ADP
m-769	142	15	t	t	PROPN
m-769	142	16	in	in	ADP
m-769	142	17	the	the	DET
m-769	142	18	product	product	NOUN
m-769	142	19	space	space	NOUN
m-769	142	20	x×x	x×x	PROPN
m-769	142	21	.	.	PUNCT
m-769	143	1	studying	study	VERB
m-769	143	2	homomorphic	homomorphic	ADJ
m-769	143	3	images	image	NOUN
m-769	143	4	1	1	NUM
m-769	143	5	.	.	PUNCT
m-769	143	6	definition	definition	NOUN
m-769	143	7	of	of	ADP
m-769	143	8	homomorphic	homomorphic	ADJ
m-769	143	9	images	image	NOUN
m-769	143	10	:	:	PUNCT
m-769	143	11			X
m-769	143	12	a	a	DET
m-769	143	13	homomorphism	homomorphism	NOUN
m-769	143	14	ϕ:e→h	ϕ:e→h	PROPN
m-769	143	15	maps	map	VERB
m-769	143	16	elements	element	NOUN
m-769	143	17	from	from	ADP
m-769	143	18	the	the	DET
m-769	143	19	enveloping	enveloping	NOUN
m-769	143	20	semigroupe	semigroupe	NOUN
m-769	143	21	to	to	ADP
m-769	143	22	a	a	DET
m-769	143	23	target	target	NOUN
m-769	143	24	group	group	NOUN
m-769	143	25	h	h	NOUN
m-769	143	26	in	in	ADP
m-769	143	27	a	a	DET
m-769	143	28	way	way	NOUN
m-769	143	29	that	that	PRON
m-769	143	30	preserves	preserve	VERB
m-769	143	31	the	the	DET
m-769	143	32	group	group	NOUN
m-769	143	33	structure	structure	NOUN
m-769	143	34	.	.	PUNCT
m-769	144	1	mathematically	mathematically	ADV
m-769	144	2	,	,	PUNCT
m-769	144	3	ϕ(xy)=ϕ(x)ϕ(y	ϕ(xy)=ϕ(x)ϕ(y	NUM
m-769	144	4	)	)	PUNCT
m-769	144	5	for	for	ADP
m-769	144	6	all	all	DET
m-769	144	7	x	x	NOUN
m-769	144	8	,	,	PUNCT
m-769	144	9	y∈e	y∈e	NOUN
m-769	144	10	.	.	PROPN
m-769	145	1	2	2	NUM
m-769	145	2	.	.	X
m-769	145	3	effective	effective	ADJ
m-769	145	4	study	study	NOUN
m-769	145	5	through	through	ADP
m-769	145	6	enveloping	envelop	VERB
m-769	145	7	semigroups	semigroup	NOUN
m-769	145	8	:	:	PUNCT
m-769	145	9			NOUN
m-769	145	10	the	the	DET
m-769	145	11	enveloping	enveloping	NOUN
m-769	145	12	semigroupe	semigroupe	NOUN
m-769	145	13	contains	contain	VERB
m-769	145	14	all	all	DET
m-769	145	15	possible	possible	ADJ
m-769	145	16	compositions	composition	NOUN
m-769	145	17	and	and	CCONJ
m-769	145	18	limits	limit	NOUN
m-769	145	19	of	of	ADP
m-769	145	20	transformations	transformation	NOUN
m-769	145	21	in	in	ADP
m-769	145	22	t.	t.	NOUN
m-769	145	23	since	since	SCONJ
m-769	145	24	homomorphisms	homomorphism	NOUN
m-769	145	25	preserve	preserve	VERB
m-769	145	26	group	group	NOUN
m-769	145	27	operations	operation	NOUN
m-769	145	28	,	,	PUNCT
m-769	145	29	studying	study	VERB
m-769	145	30	homomorphic	homomorphic	ADJ
m-769	145	31	images	image	NOUN
m-769	145	32	through	through	ADP
m-769	145	33	e	e	NOUN
m-769	145	34	allows	allow	VERB
m-769	145	35	us	we	PRON
m-769	145	36	to	to	PART
m-769	145	37	analyze	analyze	VERB
m-769	145	38	how	how	SCONJ
m-769	145	39	these	these	DET
m-769	145	40	compositions	composition	NOUN
m-769	145	41	and	and	CCONJ
m-769	145	42	limits	limit	NOUN
m-769	145	43	are	be	AUX
m-769	145	44	mapped	map	VERB
m-769	145	45	to	to	ADP
m-769	145	46	the	the	DET
m-769	145	47	target	target	NOUN
m-769	145	48	group	group	NOUN
m-769	145	49	h.	h.	NOUN
m-769	145	50	proof	proof	NOUN
m-769	145	51	of	of	ADP
m-769	145	52	effectiveness	effectiveness	NOUN
m-769	145	53			NOUN
m-769	145	54	closure	closure	NOUN
m-769	145	55	under	under	ADP
m-769	145	56	composition	composition	NOUN
m-769	145	57	:	:	PUNCT
m-769	145	58			PRON
m-769	145	59	the	the	DET
m-769	145	60	closure	closure	NOUN
m-769	145	61	of	of	ADP
m-769	145	62	t	t	PROPN
m-769	145	63	in	in	ADP
m-769	145	64	e	e	NOUN
m-769	145	65	ensures	ensure	VERB
m-769	145	66	that	that	SCONJ
m-769	145	67	the	the	DET
m-769	145	68	composition	composition	NOUN
m-769	145	69	of	of	ADP
m-769	145	70	transformations	transformation	NOUN
m-769	145	71	remains	remain	VERB
m-769	145	72	within	within	ADP
m-769	145	73	e.	e.	PROPN
m-769	146	1	this	this	DET
m-769	146	2	closure	closure	NOUN
m-769	146	3	property	property	NOUN
m-769	146	4	is	be	AUX
m-769	146	5	preserved	preserve	VERB
m-769	146	6	under	under	ADP
m-769	146	7	ijo	ijo	PROPN
m-769	146	8	journals	journal	NOUN
m-769	146	9	doi	doi	NOUN
m-769	146	10	10.5281	10.5281	NUM
m-769	146	11	/	/	SYM
m-769	146	12	zenodo.10443958	zenodo.10443958	PROPN
m-769	146	13	volume	volume	NOUN
m-769	146	14	06	06	NUM
m-769	147	1	|	|	ADV
m-769	147	2	issue	issue	NOUN
m-769	147	3	12	12	NUM
m-769	147	4	|	|	CCONJ
m-769	147	5	december	december	PROPN
m-769	147	6	2023	2023	NUM
m-769	148	1	|	|	ADV
m-769	148	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-769	148	3	17	17	NUM
m-769	148	4	ijo	ijo	PROPN
m-769	148	5	international	international	PROPN
m-769	148	6	journal	journal	PROPN
m-769	148	7	of	of	ADP
m-769	148	8	mathematics	mathematics	PROPN
m-769	148	9	(	(	PUNCT
m-769	148	10	issn	issn	PROPN
m-769	148	11	:	:	PUNCT
m-769	148	12	2992	2992	NUM
m-769	148	13	-	-	SYM
m-769	148	14	4421	4421	NUM
m-769	148	15	)	)	PUNCT
m-769	149	1	michael	michael	PROPN
m-769	149	2	n.	n.	PROPN
m-769	149	3	john	john	PROPN
m-769	149	4	*	*	PROPN
m-769	149	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-769	149	6	volume	volume	NOUN
m-769	149	7	06	06	NUM
m-769	149	8	issue	issue	NOUN
m-769	149	9	12	12	NUM
m-769	149	10	||	||	NOUN
m-769	149	11	dec	dec	PROPN
m-769	149	12	.	.	PROPN
m-769	149	13	,	,	PUNCT
m-769	149	14	2023	2023	NUM
m-769	149	15	||	||	NOUN
m-769	150	1	algebraic	algebraic	ADJ
m-769	150	2	and	and	CCONJ
m-769	150	3	topological	topological	ADJ
m-769	150	4	analysis	analysis	NOUN
m-769	150	5	of	of	ADP
m-769	150	6	enveloping	envelop	VERB
m-769	150	7	semigroups	semigroup	NOUN
m-769	150	8	in	in	ADP
m-769	150	9	transformation	transformation	NOUN
m-769	150	10	groups	group	NOUN
m-769	150	11	:	:	PUNCT
m-769	150	12	proximal	proximal	ADJ
m-769	150	13	equivalence	equivalence	NOUN
m-769	150	14	and	and	CCONJ
m-769	150	15	homomorphic	homomorphic	ADJ
m-769	150	16	image	image	NOUN
m-769	150	17	homomorphisms	homomorphism	NOUN
m-769	150	18	,	,	PUNCT
m-769	150	19	allowing	allow	VERB
m-769	150	20	for	for	ADP
m-769	150	21	the	the	DET
m-769	150	22	effective	effective	ADJ
m-769	150	23	study	study	NOUN
m-769	150	24	of	of	ADP
m-769	150	25	compositions	composition	NOUN
m-769	150	26	in	in	ADP
m-769	150	27	the	the	DET
m-769	150	28	target	target	NOUN
m-769	150	29	group	group	NOUN
m-769	150	30	h.	h.	PROPN
m-769	150	31			NOUN
m-769	150	32	limits	limit	NOUN
m-769	150	33	and	and	CCONJ
m-769	150	34	convergence	convergence	NOUN
m-769	150	35	:	:	PUNCT
m-769	150	36			PRON
m-769	150	37	the	the	DET
m-769	150	38	closure	closure	NOUN
m-769	150	39	operation	operation	NOUN
m-769	150	40	in	in	ADP
m-769	150	41	e	e	PROPN
m-769	150	42	captures	capture	NOUN
m-769	150	43	limit	limit	VERB
m-769	150	44	points	point	NOUN
m-769	150	45	and	and	CCONJ
m-769	150	46	convergence	convergence	NOUN
m-769	150	47	of	of	ADP
m-769	150	48	sequences	sequence	NOUN
m-769	150	49	of	of	ADP
m-769	150	50	transformations	transformation	NOUN
m-769	150	51	in	in	ADP
m-769	150	52	t.	t.	PROPN
m-769	150	53	homomorphisms	homomorphism	NOUN
m-769	150	54	then	then	ADV
m-769	150	55	preserve	preserve	VERB
m-769	150	56	these	these	DET
m-769	150	57	limit	limit	NOUN
m-769	150	58	properties	property	NOUN
m-769	150	59	when	when	SCONJ
m-769	150	60	mapping	map	VERB
m-769	150	61	to	to	ADP
m-769	150	62	the	the	DET
m-769	150	63	target	target	NOUN
m-769	150	64	group	group	NOUN
m-769	150	65	h	h	NOUN
m-769	150	66	,	,	PUNCT
m-769	150	67	providing	provide	VERB
m-769	150	68	insight	insight	NOUN
m-769	150	69	into	into	ADP
m-769	150	70	how	how	SCONJ
m-769	150	71	limits	limit	NOUN
m-769	150	72	are	be	AUX
m-769	150	73	transformed	transform	VERB
m-769	150	74	.	.	PUNCT
m-769	151	1			X
m-769	151	2	algebraic	algebraic	ADJ
m-769	151	3	structure	structure	NOUN
m-769	151	4	:	:	PUNCT
m-769	151	5			ADJ
m-769	151	6	the	the	DET
m-769	151	7	algebraic	algebraic	ADJ
m-769	151	8	structure	structure	NOUN
m-769	151	9	of	of	ADP
m-769	151	10	e	e	PROPN
m-769	151	11	reflects	reflect	VERB
m-769	151	12	the	the	DET
m-769	151	13	algebraic	algebraic	ADJ
m-769	151	14	properties	property	NOUN
m-769	151	15	of	of	ADP
m-769	151	16	the	the	DET
m-769	151	17	transformation	transformation	NOUN
m-769	151	18	group	group	NOUN
m-769	151	19	t.	t.	PROPN
m-769	151	20	homomorphisms	homomorphisms	PROPN
m-769	151	21	retain	retain	VERB
m-769	151	22	this	this	DET
m-769	151	23	structure	structure	NOUN
m-769	151	24	in	in	ADP
m-769	151	25	the	the	DET
m-769	151	26	target	target	NOUN
m-769	151	27	group	group	NOUN
m-769	151	28	h	h	NOUN
m-769	151	29	,	,	PUNCT
m-769	151	30	facilitating	facilitate	VERB
m-769	151	31	the	the	DET
m-769	151	32	study	study	NOUN
m-769	151	33	of	of	ADP
m-769	151	34	the	the	DET
m-769	151	35	algebraic	algebraic	ADJ
m-769	151	36	properties	property	NOUN
m-769	151	37	of	of	ADP
m-769	151	38	homomorphic	homomorphic	ADJ
m-769	151	39	images	image	NOUN
m-769	151	40	.	.	PUNCT
m-769	152	1			X
m-769	152	2	topological	topological	ADJ
m-769	152	3	structure	structure	NOUN
m-769	152	4	:	:	PUNCT
m-769	152	5			NOUN
m-769	152	6	as	as	SCONJ
m-769	152	7	e	e	PROPN
m-769	152	8	is	be	AUX
m-769	152	9	equipped	equip	VERB
m-769	152	10	with	with	ADP
m-769	152	11	a	a	DET
m-769	152	12	topological	topological	ADJ
m-769	152	13	structure	structure	NOUN
m-769	152	14	,	,	PUNCT
m-769	152	15	studying	study	VERB
m-769	152	16	homomorphic	homomorphic	ADJ
m-769	152	17	images	image	NOUN
m-769	152	18	through	through	ADP
m-769	152	19	e	e	NOUN
m-769	152	20	allows	allow	VERB
m-769	152	21	for	for	ADP
m-769	152	22	the	the	DET
m-769	152	23	consideration	consideration	NOUN
m-769	152	24	of	of	ADP
m-769	152	25	topological	topological	ADJ
m-769	152	26	properties	property	NOUN
m-769	152	27	and	and	CCONJ
m-769	152	28	continuity	continuity	NOUN
m-769	152	29	in	in	ADP
m-769	152	30	the	the	DET
m-769	152	31	target	target	NOUN
m-769	152	32	group	group	NOUN
m-769	152	33	h.	h.	PROPN
m-769	152	34	therefore	therefore	ADV
m-769	152	35	,	,	PUNCT
m-769	152	36	homomorphic	homomorphic	ADJ
m-769	152	37	images	image	NOUN
m-769	152	38	of	of	ADP
m-769	152	39	transformation	transformation	NOUN
m-769	152	40	groups	group	NOUN
m-769	152	41	can	can	AUX
m-769	152	42	be	be	AUX
m-769	152	43	effectively	effectively	ADV
m-769	152	44	studied	study	VERB
m-769	152	45	through	through	ADP
m-769	152	46	their	their	PRON
m-769	152	47	enveloping	enveloping	NOUN
m-769	152	48	semigroups	semigroup	NOUN
m-769	152	49	.	.	PUNCT
m-769	153	1	the	the	DET
m-769	153	2	closure	closure	NOUN
m-769	153	3	,	,	PUNCT
m-769	153	4	composition	composition	NOUN
m-769	153	5	,	,	PUNCT
m-769	153	6	limit	limit	NOUN
m-769	153	7	properties	property	NOUN
m-769	153	8	,	,	PUNCT
m-769	153	9	and	and	CCONJ
m-769	153	10	algebraic	algebraic	ADJ
m-769	153	11	structure	structure	NOUN
m-769	153	12	present	present	ADJ
m-769	153	13	in	in	ADP
m-769	153	14	the	the	DET
m-769	153	15	enveloping	enveloping	NOUN
m-769	153	16	semigroup	semigroup	NOUN
m-769	153	17	provide	provide	VERB
m-769	153	18	a	a	DET
m-769	153	19	comprehensive	comprehensive	ADJ
m-769	153	20	framework	framework	NOUN
m-769	153	21	for	for	ADP
m-769	153	22	understanding	understand	VERB
m-769	153	23	how	how	SCONJ
m-769	153	24	transformations	transformation	NOUN
m-769	153	25	are	be	AUX
m-769	153	26	mapped	map	VERB
m-769	153	27	to	to	ADP
m-769	153	28	the	the	DET
m-769	153	29	target	target	NOUN
m-769	153	30	group	group	NOUN
m-769	153	31	under	under	ADP
m-769	153	32	homomorphisms	homomorphism	NOUN
m-769	153	33	.	.	PUNCT
m-769	154	1	5	5	X
m-769	154	2	.	.	X
m-769	154	3	conclusion	conclusion	NOUN
m-769	154	4	this	this	DET
m-769	154	5	paper	paper	NOUN
m-769	154	6	establishes	establish	VERB
m-769	154	7	a	a	DET
m-769	154	8	profound	profound	ADJ
m-769	154	9	connection	connection	NOUN
m-769	154	10	between	between	ADP
m-769	154	11	the	the	DET
m-769	154	12	algebraic	algebraic	ADJ
m-769	154	13	and	and	CCONJ
m-769	154	14	topological	topological	ADJ
m-769	154	15	properties	property	NOUN
m-769	154	16	of	of	ADP
m-769	154	17	the	the	DET
m-769	154	18	enveloping	enveloping	NOUN
m-769	154	19	semigroupe	semigroupe	NOUN
m-769	154	20	associated	associate	VERB
m-769	154	21	with	with	ADP
m-769	154	22	transformation	transformation	NOUN
m-769	154	23	groups	group	NOUN
m-769	154	24	and	and	CCONJ
m-769	154	25	the	the	DET
m-769	154	26	proximal	proximal	ADJ
m-769	154	27	equivalence	equivalence	NOUN
m-769	154	28	relation	relation	NOUN
m-769	154	29	in	in	ADP
m-769	154	30	x.	x.	NOUN
m-769	154	31	the	the	DET
m-769	154	32	presence	presence	NOUN
m-769	154	33	of	of	ADP
m-769	154	34	a	a	DET
m-769	154	35	unique	unique	ADJ
m-769	154	36	minimal	minimal	ADJ
m-769	154	37	right	right	ADJ
m-769	154	38	ideal	ideal	NOUN
m-769	154	39	in	in	ADP
m-769	154	40	e	e	PROPN
m-769	154	41	is	be	AUX
m-769	154	42	shown	show	VERB
m-769	154	43	to	to	PART
m-769	154	44	be	be	AUX
m-769	154	45	a	a	DET
m-769	154	46	key	key	ADJ
m-769	154	47	factor	factor	NOUN
m-769	154	48	in	in	ADP
m-769	154	49	determining	determine	VERB
m-769	154	50	the	the	DET
m-769	154	51	nature	nature	NOUN
m-769	154	52	of	of	ADP
m-769	154	53	the	the	DET
m-769	154	54	proximal	proximal	ADJ
m-769	154	55	equivalence	equivalence	NOUN
m-769	154	56	relation	relation	NOUN
m-769	154	57	.	.	PUNCT
m-769	155	1	additionally	additionally	ADV
m-769	155	2	,	,	PUNCT
m-769	155	3	we	we	PRON
m-769	155	4	demonstrate	demonstrate	VERB
m-769	155	5	the	the	DET
m-769	155	6	applicability	applicability	NOUN
m-769	155	7	of	of	ADP
m-769	155	8	enveloping	envelop	VERB
m-769	155	9	semigroups	semigroup	NOUN
m-769	155	10	in	in	ADP
m-769	155	11	the	the	DET
m-769	155	12	study	study	NOUN
m-769	155	13	of	of	ADP
m-769	155	14	homomorphic	homomorphic	ADJ
m-769	155	15	images	image	NOUN
m-769	155	16	of	of	ADP
m-769	155	17	transformation	transformation	NOUN
m-769	155	18	groups	group	NOUN
m-769	155	19	.	.	PUNCT
m-769	156	1	these	these	DET
m-769	156	2	findings	finding	NOUN
m-769	156	3	contribute	contribute	VERB
m-769	156	4	to	to	ADP
m-769	156	5	a	a	DET
m-769	156	6	deeper	deep	ADJ
m-769	156	7	understanding	understanding	NOUN
m-769	156	8	of	of	ADP
m-769	156	9	the	the	DET
m-769	156	10	interplay	interplay	NOUN
m-769	156	11	between	between	ADP
m-769	156	12	algebraic	algebraic	ADJ
m-769	156	13	structures	structure	NOUN
m-769	156	14	and	and	CCONJ
m-769	156	15	topological	topological	ADJ
m-769	156	16	properties	property	NOUN
m-769	156	17	in	in	ADP
m-769	156	18	the	the	DET
m-769	156	19	context	context	NOUN
m-769	156	20	of	of	ADP
m-769	156	21	transformation	transformation	NOUN
m-769	156	22	groups	group	NOUN
m-769	156	23	with	with	ADP
m-769	156	24	compact	compact	ADJ
m-769	156	25	hausdorff	hausdorff	NOUN
m-769	156	26	phase	phase	NOUN
m-769	156	27	spaces	space	VERB
m-769	156	28	.	.	PUNCT
m-769	157	1	ijo	ijo	PROPN
m-769	157	2	journals	journal	NOUN
m-769	157	3	doi	doi	X
m-769	157	4	10.5281	10.5281	NUM
m-769	157	5	/	/	SYM
m-769	157	6	zenodo.10443958	zenodo.10443958	PROPN
m-769	157	7	volume	volume	NOUN
m-769	157	8	06	06	NUM
m-769	158	1	|	|	ADV
m-769	158	2	issue	issue	NOUN
m-769	158	3	12	12	NUM
m-769	158	4	|	|	CCONJ
m-769	158	5	december	december	PROPN
m-769	158	6	2023	2023	NUM
m-769	159	1	|	|	ADV
m-769	159	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-769	159	3	18	18	NUM
m-769	159	4	ijo	ijo	PROPN
m-769	159	5	international	international	PROPN
m-769	159	6	journal	journal	PROPN
m-769	159	7	of	of	ADP
m-769	159	8	mathematics	mathematics	PROPN
m-769	159	9	(	(	PUNCT
m-769	159	10	issn	issn	PROPN
m-769	159	11	:	:	PUNCT
m-769	159	12	2992	2992	NUM
m-769	159	13	-	-	SYM
m-769	159	14	4421	4421	NUM
m-769	159	15	)	)	PUNCT
m-769	160	1	michael	michael	PROPN
m-769	160	2	n.	n.	PROPN
m-769	160	3	john	john	PROPN
m-769	160	4	*	*	PROPN
m-769	160	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-769	160	6	volume	volume	NOUN
m-769	160	7	06	06	NUM
m-769	160	8	issue	issue	NOUN
m-769	160	9	12	12	NUM
m-769	160	10	||	||	NOUN
m-769	160	11	dec	dec	PROPN
m-769	160	12	.	.	PROPN
m-769	160	13	,	,	PUNCT
m-769	160	14	2023	2023	NUM
m-769	160	15	||	||	NOUN
m-769	161	1	algebraic	algebraic	ADJ
m-769	161	2	and	and	CCONJ
m-769	161	3	topological	topological	ADJ
m-769	161	4	analysis	analysis	NOUN
m-769	161	5	of	of	ADP
m-769	161	6	enveloping	envelop	VERB
m-769	161	7	semigroups	semigroup	NOUN
m-769	161	8	in	in	ADP
m-769	161	9	transformation	transformation	NOUN
m-769	161	10	groups	group	NOUN
m-769	161	11	:	:	PUNCT
m-769	161	12	proximal	proximal	ADJ
m-769	161	13	equivalence	equivalence	NOUN
m-769	161	14	and	and	CCONJ
m-769	161	15	homomorphic	homomorphic	ADJ
m-769	161	16	image	image	NOUN
m-769	161	17	6	6	NUM
m-769	161	18	.	.	PUNCT
m-769	161	19	corresponding	correspond	VERB
m-769	161	20	author	author	NOUN
m-769	161	21	michael	michael	PROPN
m-769	161	22	nsikan	nsikan	PROPN
m-769	161	23	john	john	PROPN
m-769	161	24	is	be	AUX
m-769	161	25	currently	currently	ADV
m-769	161	26	a	a	DET
m-769	161	27	phd	phd	NOUN
m-769	161	28	student	student	NOUN
m-769	161	29	of	of	ADP
m-769	161	30	mathematics	mathematics	PROPN
m-769	161	31	at	at	ADP
m-769	161	32	akwaibom	akwaibom	NOUN
m-769	161	33	state	state	NOUN
m-769	161	34	university	university	NOUN
m-769	161	35	.	.	PUNCT
m-769	162	1	michael	michael	PROPN
m-769	162	2	does	do	AUX
m-769	162	3	research	research	NOUN
m-769	162	4	in	in	ADP
m-769	162	5	algebra	algebra	NOUN
m-769	162	6	;	;	PUNCT
m-769	162	7	group	group	NOUN
m-769	162	8	theory	theory	NOUN
m-769	162	9	,	,	PUNCT
m-769	162	10	computational	computational	ADJ
m-769	162	11	group	group	NOUN
m-769	162	12	theory	theory	NOUN
m-769	162	13	,	,	PUNCT
m-769	162	14	algebraic	algebraic	ADJ
m-769	162	15	cryptography	cryptography	NOUN
m-769	162	16	,	,	PUNCT
m-769	162	17	number	number	NOUN
m-769	162	18	theory	theory	NOUN
m-769	162	19	,	,	PUNCT
m-769	162	20	combinatorics	combinatorics	PROPN
m-769	162	21	,	,	PUNCT
m-769	162	22	blockchain	blockchain	PROPN
m-769	162	23	technology	technology	PROPN
m-769	162	24	.	.	PUNCT
m-769	163	1	supervisor	supervisor	NOUN
m-769	163	2	:	:	PUNCT
m-769	163	3	otobong	otobong	PROPN
m-769	163	4	g.	g.	PROPN
m-769	163	5	udoaka	udoaka	ADV
m-769	163	6	for	for	ADP
m-769	163	7	more	more	ADJ
m-769	163	8	of	of	ADP
m-769	163	9	our	our	PRON
m-769	163	10	work	work	NOUN
m-769	163	11	,	,	PUNCT
m-769	163	12	please	please	INTJ
m-769	163	13	see	see	VERB
m-769	163	14	[	[	X
m-769	163	15	17]–[31	17]–[31	NOUN
m-769	163	16	]	]	X
m-769	163	17	references	reference	NOUN
m-769	163	18	[	[	X
m-769	163	19	1	1	NUM
m-769	163	20	]	]	X
m-769	163	21	bowen	bowen	PROPN
m-769	163	22	,	,	PUNCT
m-769	163	23	r.	r.	PROPN
m-769	163	24	(	(	PUNCT
m-769	163	25	1971	1971	NUM
m-769	163	26	)	)	PUNCT
m-769	163	27	.	.	PUNCT
m-769	164	1	proximal	proximal	ADJ
m-769	164	2	equivalence	equivalence	NOUN
m-769	164	3	in	in	ADP
m-769	164	4	topological	topological	ADJ
m-769	164	5	dynamics	dynamic	NOUN
m-769	164	6	.	.	PUNCT
m-769	165	1	transactions	transaction	NOUN
m-769	165	2	of	of	ADP
m-769	165	3	the	the	DET
m-769	165	4	american	american	PROPN
m-769	165	5	mathematical	mathematical	PROPN
m-769	165	6	society	society	NOUN
m-769	165	7	,	,	PUNCT
m-769	165	8	152(1	152(1	NUM
m-769	165	9	)	)	PUNCT
m-769	165	10	,	,	PUNCT
m-769	165	11	1	1	NUM
m-769	165	12	-	-	SYM
m-769	165	13	33	33	NUM
m-769	165	14	.	.	PUNCT
m-769	166	1	[	[	X
m-769	166	2	2	2	NUM
m-769	166	3	]	]	X
m-769	166	4	ellis	ellis	PROPN
m-769	166	5	,	,	PUNCT
m-769	166	6	r.	r.	PROPN
m-769	166	7	,	,	PUNCT
m-769	166	8	&	&	CCONJ
m-769	166	9	steprāns	steprāns	PROPN
m-769	166	10	,	,	PUNCT
m-769	166	11	j.	j.	PROPN
m-769	166	12	(	(	PUNCT
m-769	166	13	1976	1976	NUM
m-769	166	14	)	)	PUNCT
m-769	166	15	.	.	PUNCT
m-769	167	1	enveloping	envelop	VERB
m-769	167	2	semigroups	semigroup	NOUN
m-769	167	3	in	in	ADP
m-769	167	4	topological	topological	ADJ
m-769	167	5	transformation	transformation	NOUN
m-769	167	6	groups	group	NOUN
m-769	167	7	.	.	PUNCT
m-769	168	1	pacific	pacific	PROPN
m-769	168	2	journal	journal	PROPN
m-769	168	3	of	of	ADP
m-769	168	4	mathematics	mathematic	NOUN
m-769	168	5	,	,	PUNCT
m-769	168	6	65(1	65(1	NOUN
m-769	168	7	)	)	PUNCT
m-769	168	8	,	,	PUNCT
m-769	168	9	99	99	NUM
m-769	168	10	-	-	SYM
m-769	168	11	108	108	NUM
m-769	168	12	.	.	PUNCT
m-769	169	1	[	[	X
m-769	169	2	3	3	NUM
m-769	169	3	]	]	X
m-769	169	4	bergelson	bergelson	NOUN
m-769	169	5	,	,	PUNCT
m-769	169	6	v.	v.	PROPN
m-769	169	7	,	,	PUNCT
m-769	169	8	&	&	CCONJ
m-769	169	9	leibman	leibman	PROPN
m-769	169	10	,	,	PUNCT
m-769	169	11	a.	a.	NOUN
m-769	169	12	(	(	PUNCT
m-769	169	13	2005	2005	NUM
m-769	169	14	)	)	PUNCT
m-769	169	15	.	.	PUNCT
m-769	170	1	minimal	minimal	ADJ
m-769	170	2	right	right	ADJ
m-769	170	3	ideals	ideal	NOUN
m-769	170	4	in	in	ADP
m-769	170	5	semigroups	semigroup	NOUN
m-769	170	6	of	of	ADP
m-769	170	7	continuous	continuous	ADJ
m-769	170	8	maps	map	NOUN
m-769	170	9	.	.	PUNCT
m-769	171	1	ergodic	ergodic	ADJ
m-769	171	2	theory	theory	NOUN
m-769	171	3	and	and	CCONJ
m-769	171	4	dynamical	dynamical	ADJ
m-769	171	5	systems	system	NOUN
m-769	171	6	,	,	PUNCT
m-769	171	7	25(6	25(6	NOUN
m-769	171	8	)	)	PUNCT
m-769	171	9	,	,	PUNCT
m-769	171	10	1731	1731	NUM
m-769	171	11	-	-	SYM
m-769	171	12	1745	1745	NUM
m-769	171	13	.	.	PUNCT
m-769	172	1	[	[	X
m-769	172	2	4	4	NUM
m-769	172	3	]	]	X
m-769	172	4	hindman	hindman	PROPN
m-769	172	5	,	,	PUNCT
m-769	172	6	n.	n.	PROPN
m-769	172	7	,	,	PUNCT
m-769	172	8	&	&	CCONJ
m-769	172	9	strauss	strauss	PROPN
m-769	172	10	,	,	PUNCT
m-769	172	11	d.	d.	PROPN
m-769	172	12	(	(	PUNCT
m-769	172	13	1998	1998	NUM
m-769	172	14	)	)	PUNCT
m-769	172	15	.	.	PUNCT
m-769	173	1	homomorphisms	homomorphism	NOUN
m-769	173	2	and	and	CCONJ
m-769	173	3	enveloping	envelop	VERB
m-769	173	4	semigroups	semigroup	NOUN
m-769	173	5	of	of	ADP
m-769	173	6	transformation	transformation	NOUN
m-769	173	7	groups	group	NOUN
m-769	173	8	.	.	PUNCT
m-769	174	1	semigroup	semigroup	PROPN
m-769	174	2	forum	forum	PROPN
m-769	174	3	,	,	PUNCT
m-769	174	4	57(3	57(3	NUM
m-769	174	5	)	)	PUNCT
m-769	174	6	,	,	PUNCT
m-769	174	7	355	355	NUM
m-769	174	8	-	-	SYM
m-769	174	9	378	378	NUM
m-769	174	10	.	.	PUNCT
m-769	175	1	[	[	X
m-769	175	2	5	5	X
m-769	175	3	]	]	PUNCT
m-769	175	4	michael	michael	PROPN
m-769	175	5	n.	n.	PROPN
m-769	175	6	john	john	PROPN
m-769	175	7	&	&	CCONJ
m-769	175	8	udoaka	udoaka	PROPN
m-769	175	9	o.	o.	PROPN
m-769	175	10	g	g	PROPN
m-769	175	11	(	(	PUNCT
m-769	175	12	2023	2023	NUM
m-769	175	13	)	)	PUNCT
m-769	175	14	.	.	PUNCT
m-769	176	1	algorithm	algorithm	NOUN
m-769	176	2	and	and	CCONJ
m-769	176	3	cube	cube	NOUN
m-769	176	4	-	-	PUNCT
m-769	176	5	lattice	lattice	NOUN
m-769	176	6	-	-	PUNCT
m-769	176	7	based	base	VERB
m-769	176	8	cryptography	cryptography	NOUN
m-769	176	9	.	.	PUNCT
m-769	177	1	international	international	ADJ
m-769	177	2	journal	journal	PROPN
m-769	177	3	of	of	ADP
m-769	177	4	research	research	NOUN
m-769	177	5	publication	publication	NOUN
m-769	177	6	and	and	CCONJ
m-769	177	7	reviews	review	NOUN
m-769	177	8	,	,	PUNCT
m-769	177	9	vol	vol	NOUN
m-769	177	10	4	4	NUM
m-769	177	11	,	,	PUNCT
m-769	177	12	no	no	DET
m-769	177	13	10	10	NUM
m-769	177	14	,	,	PUNCT
m-769	177	15	pp	pp	ADJ
m-769	177	16	3312	3312	NUM
m-769	177	17	-	-	SYM
m-769	177	18	3315	3315	NUM
m-769	177	19	october	october	PROPN
m-769	177	20	2023	2023	NUM
m-769	177	21	.	.	PUNCT
m-769	178	1	doi	doi	NOUN
m-769	178	2	:	:	PUNCT
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m-769	178	4	/	/	SYM
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m-769	179	1	[	[	X
m-769	179	2	6	6	NUM
m-769	179	3	]	]	PUNCT
m-769	179	4	michael	michael	PROPN
m-769	179	5	n.	n.	PROPN
m-769	179	6	john	john	PROPN
m-769	179	7	,	,	PUNCT
m-769	179	8	udoaka	udoaka	ADV
m-769	179	9	o.	o.	PROPN
m-769	179	10	g.	g.	PROPN
m-769	179	11	,	,	PUNCT
m-769	179	12	"	"	PUNCT
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m-769	179	14	group	group	NOUN
m-769	179	15	theory	theory	NOUN
m-769	179	16	and	and	CCONJ
m-769	179	17	quantum	quantum	NOUN
m-769	179	18	-	-	PUNCT
m-769	179	19	era	era	NOUN
m-769	179	20	cryptography	cryptography	NOUN
m-769	179	21	"	"	PUNCT
m-769	179	22	,	,	PUNCT
m-769	179	23	international	international	ADJ
m-769	179	24	journal	journal	NOUN
m-769	179	25	of	of	ADP
m-769	179	26	scientific	scientific	ADJ
m-769	179	27	research	research	NOUN
m-769	179	28	in	in	ADP
m-769	179	29	science	science	NOUN
m-769	179	30	,	,	PUNCT
m-769	179	31	engineering	engineering	NOUN
m-769	179	32	and	and	CCONJ
m-769	179	33	technology	technology	NOUN
m-769	179	34	(	(	PUNCT
m-769	179	35	ijsrset	ijsrset	NOUN
m-769	179	36	)	)	PUNCT
m-769	179	37	,	,	PUNCT
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m-769	179	39	issn	issn	PROPN
m-769	179	40	:	:	PUNCT
m-769	179	41	2394	2394	NUM
m-769	179	42	-	-	SYM
m-769	179	43	4099	4099	NUM
m-769	179	44	,	,	PUNCT
m-769	179	45	print	print	NOUN
m-769	179	46	issn	issn	PROPN
m-769	179	47	:	:	PUNCT
m-769	179	48	2395	2395	NUM
m-769	179	49	-	-	SYM
m-769	179	50	1990	1990	NUM
m-769	179	51	,	,	PUNCT
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m-769	179	53	10	10	NUM
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m-769	179	56	.	.	PUNCT
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m-769	180	2	-	-	SYM
m-769	180	3	10	10	NUM
m-769	180	4	,	,	PUNCT
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m-769	180	6	-	-	PUNCT
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m-769	180	8	2023	2023	NUM
m-769	180	9	.	.	PUNCT
m-769	181	1	available	available	ADJ
m-769	181	2	at	at	ADP
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m-769	181	4	:	:	PUNCT
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m-769	181	7	7	7	NUM
m-769	181	8	]	]	PUNCT
m-769	181	9	michael	michael	PROPN
m-769	181	10	n.	n.	PROPN
m-769	181	11	john	john	PROPN
m-769	181	12	,	,	PUNCT
m-769	181	13	udoaka	udoaka	ADJ
m-769	181	14	,	,	PUNCT
m-769	181	15	otobong	otobong	PROPN
m-769	181	16	.	.	PUNCT
m-769	182	1	g.	g.	PROPN
m-769	182	2	,	,	PUNCT
m-769	182	3	alex	alex	PROPN
m-769	182	4	musa,"key	musa,"key	PROPN
m-769	182	5	agreement	agreement	PROPN
m-769	182	6	protocol	protocol	NOUN
m-769	182	7	using	use	VERB
m-769	182	8	conjugacy	conjugacy	ADJ
m-769	182	9	classes	class	NOUN
m-769	182	10	of	of	ADP
m-769	182	11	finitely	finitely	ADV
m-769	182	12	generated	generate	VERB
m-769	182	13	group	group	NOUN
m-769	182	14	”	"	PUNCT
m-769	182	15	,	,	PUNCT
m-769	182	16	international	international	ADJ
m-769	182	17	journal	journal	NOUN
m-769	182	18	of	of	ADP
m-769	182	19	scientific	scientific	ADJ
m-769	182	20	research	research	NOUN
m-769	182	21	in	in	ADP
m-769	182	22	science	science	NOUN
m-769	182	23	and	and	CCONJ
m-769	182	24	technology(ijsrst	technology(ijsrst	PROPN
m-769	182	25	)	)	PUNCT
m-769	182	26	,	,	PUNCT
m-769	182	27	volume	volume	NOUN
m-769	182	28	10	10	NUM
m-769	182	29	,	,	PUNCT
m-769	182	30	issue	issue	NOUN
m-769	182	31	6	6	NUM
m-769	182	32	,	,	PUNCT
m-769	182	33	pp52	pp52	PROPN
m-769	182	34	-	-	PUNCT
m-769	182	35	56	56	NUM
m-769	182	36	.	.	PUNCT
m-769	183	1	doi	doi	NOUN
m-769	183	2	:	:	PUNCT
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m-769	183	5	journals	journal	NOUN
m-769	183	6	doi	doi	NOUN
m-769	183	7	10.5281	10.5281	NUM
m-769	183	8	/	/	SYM
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m-769	183	11	06	06	NUM
m-769	184	1	|	|	ADV
m-769	184	2	issue	issue	NOUN
m-769	184	3	12	12	NUM
m-769	184	4	|	|	CCONJ
m-769	184	5	december	december	PROPN
m-769	184	6	2023	2023	NUM
m-769	185	1	|	|	ADV
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m-769	185	12	mathematics	mathematics	PROPN
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m-769	185	14	issn	issn	PROPN
m-769	185	15	:	:	PUNCT
m-769	185	16	2992	2992	NUM
m-769	185	17	-	-	SYM
m-769	185	18	4421	4421	NUM
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m-769	186	1	michael	michael	PROPN
m-769	186	2	n.	n.	PROPN
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m-769	186	7	06	06	NUM
m-769	186	8	issue	issue	NOUN
m-769	186	9	12	12	NUM
m-769	186	10	||	||	NOUN
m-769	186	11	dec	dec	PROPN
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m-769	186	15	||	||	NOUN
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m-769	187	7	semigroups	semigroup	NOUN
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m-769	187	9	transformation	transformation	NOUN
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m-769	187	13	equivalence	equivalence	NOUN
m-769	187	14	and	and	CCONJ
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m-769	187	22	john	john	PROPN
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m-769	187	25	,	,	PUNCT
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m-769	187	27	.	.	PUNCT
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m-769	188	2	,	,	PUNCT
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m-769	188	7	"	"	PUNCT
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m-769	188	9	-	-	PUNCT
m-769	188	10	curve	curve	NOUN
m-769	188	11	groups	group	NOUN
m-769	188	12	in	in	ADP
m-769	188	13	quantum	quantum	NOUN
m-769	188	14	-	-	PUNCT
m-769	188	15	era	era	NOUN
m-769	188	16	cryptography	cryptography	NOUN
m-769	188	17	”	"	PUNCT
m-769	188	18	,	,	PUNCT
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m-769	188	21	of	of	ADP
m-769	188	22	science	science	NOUN
m-769	188	23	and	and	CCONJ
m-769	188	24	technology	technology	NOUN
m-769	188	25	,	,	PUNCT
m-769	188	26	volume	volume	NOUN
m-769	188	27	1	1	NUM
m-769	188	28	,	,	PUNCT
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m-769	188	30	1	1	NUM
m-769	188	31	,	,	PUNCT
m-769	188	32	pp21	pp21	PROPN
m-769	188	33	-	-	PUNCT
m-769	188	34	24	24	NUM
m-769	188	35	.	.	PUNCT
m-769	189	1	doi	doi	NOUN
m-769	189	2	:	:	PUNCT
m-769	189	3	https://doi.org/10.5281/zenodo.10207536	https://doi.org/10.5281/zenodo.10207536	NOUN
m-769	189	4	[	[	X
m-769	189	5	9	9	NUM
m-769	189	6	]	]	X
m-769	189	7	michael	michael	PROPN
m-769	189	8	n	n	PROPN
m-769	189	9	john	john	PROPN
m-769	189	10	,	,	PUNCT
m-769	189	11	udoakaotobong	udoakaotobong	PROPN
m-769	189	12	g	g	PROPN
m-769	189	13	and	and	CCONJ
m-769	189	14	alex	alex	PROPN
m-769	189	15	musa	musa	PROPN
m-769	189	16	.	.	PUNCT
m-769	190	1	nilpotent	nilpotent	ADJ
m-769	190	2	groups	group	NOUN
m-769	190	3	in	in	ADP
m-769	190	4	cryptographic	cryptographic	ADJ
m-769	190	5	key	key	ADJ
m-769	190	6	exchange	exchange	NOUN
m-769	190	7	protocol	protocol	NOUN
m-769	190	8	for	for	ADP
m-769	190	9	n≥	n≥	PROPN
m-769	190	10	1	1	NUM
m-769	190	11	.	.	PUNCT
m-769	190	12	journal	journal	PROPN
m-769	190	13	of	of	ADP
m-769	190	14	mathematical	mathematical	ADJ
m-769	190	15	problems	problem	NOUN
m-769	190	16	,	,	PUNCT
m-769	190	17	equations	equation	NOUN
m-769	190	18	and	and	CCONJ
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m-769	190	20	.	.	PUNCT
m-769	191	1	2023	2023	NUM
m-769	191	2	;	;	PUNCT
m-769	191	3	4(2	4(2	NUM
m-769	191	4	):	):	PUNCT
m-769	191	5	32	32	NUM
m-769	191	6	-	-	SYM
m-769	191	7	34	34	NUM
m-769	191	8	.	.	PUNCT
m-769	191	9	doi	doi	NOUN
m-769	191	10	:	:	PUNCT
m-769	191	11	10.22271	10.22271	NUM
m-769	191	12	/	/	SYM
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m-769	191	14	[	[	NOUN
m-769	191	15	10	10	NUM
m-769	191	16	]	]	X
m-769	191	17	michael	michael	PROPN
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m-769	191	19	john	john	PROPN
m-769	191	20	,	,	PUNCT
m-769	191	21	udoakaotobong	udoakaotobong	PROPN
m-769	191	22	.	.	PUNCT
m-769	192	1	g.	g.	PROPN
m-769	192	2	,	,	PUNCT
m-769	192	3	&	&	CCONJ
m-769	192	4	alex	alex	PROPN
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m-769	192	6	.	.	PUNCT
m-769	193	1	(	(	PUNCT
m-769	193	2	2023	2023	NUM
m-769	193	3	)	)	PUNCT
m-769	193	4	.	.	PUNCT
m-769	194	1	symmetric	symmetric	PROPN
m-769	194	2	bilinear	bilinear	PROPN
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m-769	194	5	elliptic	elliptic	ADJ
m-769	194	6	curve	curve	NOUN
m-769	194	7	and	and	CCONJ
m-769	194	8	lie	lie	NOUN
m-769	194	9	algebra	algebra	NOUN
m-769	194	10	.	.	PUNCT
m-769	195	1	gph	gph	NOUN
m-769	195	2	international	international	ADJ
m-769	195	3	journal	journal	NOUN
m-769	195	4	of	of	ADP
m-769	195	5	mathematics	mathematic	NOUN
m-769	195	6	,	,	PUNCT
m-769	195	7	06(10	06(10	NOUN
m-769	195	8	)	)	PUNCT
m-769	195	9	,	,	PUNCT
m-769	196	1	01–15	01–15	PROPN
m-769	196	2	.	.	PUNCT
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m-769	197	1	[	[	PUNCT
m-769	197	2	11	11	NUM
m-769	197	3	]	]	X
m-769	197	4	john	john	PROPN
m-769	197	5	,	,	PUNCT
m-769	197	6	michael	michael	PROPN
m-769	197	7	n.	n.	PROPN
m-769	197	8	,	,	PUNCT
m-769	197	9	ozioma	ozioma	PROPN
m-769	197	10	,	,	PUNCT
m-769	197	11	o.	o.	PROPN
m-769	197	12	,	,	PUNCT
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m-769	197	14	,	,	PUNCT
m-769	197	15	p.	p.	PROPN
m-769	197	16	n.	n.	NOUN
m-769	197	17	,	,	PUNCT
m-769	197	18	egbogho	egbogho	PROPN
m-769	197	19	,	,	PUNCT
m-769	197	20	h.	h.	PROPN
m-769	197	21	e.	e.	PROPN
m-769	197	22	,	,	PUNCT
m-769	197	23	&	&	CCONJ
m-769	197	24	udoaka	udoaka	ADJ
m-769	197	25	,	,	PUNCT
m-769	197	26	o.	o.	PROPN
m-769	197	27	g.	g.	PROPN
m-769	197	28	(	(	PUNCT
m-769	197	29	2023	2023	NUM
m-769	197	30	)	)	PUNCT
m-769	197	31	.	.	PUNCT
m-769	198	1	lattices	lattice	NOUN
m-769	198	2	in	in	ADP
m-769	198	3	quantum	quantum	NOUN
m-769	198	4	-	-	PUNCT
m-769	198	5	era	era	NOUN
m-769	198	6	cryptography	cryptography	NOUN
m-769	198	7	.	.	PUNCT
m-769	199	1	international	international	ADJ
m-769	199	2	journal	journal	PROPN
m-769	199	3	of	of	ADP
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m-769	199	5	publication	publication	NOUN
m-769	199	6	and	and	CCONJ
m-769	199	7	reviews	review	NOUN
m-769	199	8	,	,	PUNCT
m-769	199	9	v	v	NOUN
m-769	199	10	,	,	PUNCT
m-769	199	11	4(11	4(11	NUM
m-769	199	12	)	)	PUNCT
m-769	199	13	,	,	PUNCT
m-769	199	14	2175–2179	2175–2179	NUM
m-769	199	15	.	.	PUNCT
m-769	200	1	https://doi.org/10.5281/zenodo.10207210	https://doi.org/10.5281/zenodo.10207210	NOUN
m-769	201	1	[	[	X
m-769	201	2	12	12	NUM
m-769	201	3	]	]	PUNCT
m-769	201	4	michael	michael	PROPN
m-769	201	5	n.	n.	PROPN
m-769	201	6	john	john	PROPN
m-769	201	7	,	,	PUNCT
m-769	201	8	ogoegbulemozioma	ogoegbulemozioma	NOUN
m-769	201	9	,	,	PUNCT
m-769	201	10	udoakaotobong	udoakaotobong	PROPN
m-769	201	11	.	.	PUNCT
m-769	202	1	g.	g.	PROPN
m-769	202	2	,	,	PUNCT
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m-769	202	4	o.	o.	PROPN
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m-769	202	7	&	&	CCONJ
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m-769	202	11	.	.	PUNCT
m-769	203	1	(	(	PUNCT
m-769	203	2	2023	2023	NUM
m-769	203	3	)	)	PUNCT
m-769	203	4	.	.	PUNCT
m-769	204	1	cryptographic	cryptographic	ADJ
m-769	204	2	encryption	encryption	NOUN
m-769	204	3	based	base	VERB
m-769	204	4	on	on	ADP
m-769	204	5	railfence	railfence	NOUN
m-769	204	6	permutation	permutation	NOUN
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m-769	204	8	.	.	PUNCT
m-769	205	1	gph	gph	VERB
m-769	205	2	international	international	ADJ
m-769	205	3	journal	journal	NOUN
m-769	205	4	of	of	ADP
m-769	205	5	mathematics	mathematic	NOUN
m-769	205	6	,	,	PUNCT
m-769	205	7	06(11	06(11	NUM
m-769	205	8	)	)	PUNCT
m-769	205	9	,	,	PUNCT
m-769	205	10	01–06	01–06	PROPN
m-769	205	11	.	.	PUNCT
m-769	206	1	https://doi.org/10.5281/zenodo.10207316	https://doi.org/10.5281/zenodo.10207316	VERB
m-769	207	1	[	[	X
m-769	207	2	13	13	NUM
m-769	207	3	]	]	PUNCT
m-769	207	4	michael	michael	PROPN
m-769	207	5	n.	n.	PROPN
m-769	207	6	john	john	PROPN
m-769	207	7	,	,	PUNCT
m-769	207	8	ogoegbulemozioma	ogoegbulemozioma	NOUN
m-769	207	9	,	,	PUNCT
m-769	207	10	obukohwo	obukohwo	PROPN
m-769	207	11	,	,	PUNCT
m-769	207	12	victor	victor	PROPN
m-769	207	13	,	,	PUNCT
m-769	207	14	&	&	CCONJ
m-769	207	15	henry	henry	PROPN
m-769	207	16	etarogheneegbogho	etarogheneegbogho	PROPN
m-769	207	17	.	.	PUNCT
m-769	208	1	(	(	PUNCT
m-769	208	2	2023	2023	NUM
m-769	208	3	)	)	PUNCT
m-769	208	4	.	.	PUNCT
m-769	209	1	number	number	NOUN
m-769	209	2	theory	theory	NOUN
m-769	209	3	in	in	ADP
m-769	209	4	rsa	rsa	NOUN
m-769	209	5	encryption	encryption	NOUN
m-769	209	6	systems	system	NOUN
m-769	209	7	.	.	PUNCT
m-769	210	1	gph	gph	VERB
m-769	210	2	international	international	ADJ
m-769	210	3	journal	journal	NOUN
m-769	210	4	of	of	ADP
m-769	210	5	mathematics	mathematic	NOUN
m-769	210	6	,	,	PUNCT
m-769	210	7	06(11	06(11	NUM
m-769	210	8	)	)	PUNCT
m-769	210	9	,	,	PUNCT
m-769	210	10	07–16	07–16	PROPN
m-769	210	11	.	.	PUNCT
m-769	211	1	https://doi.org/10.5281/zenodo.10207361	https://doi.org/10.5281/zenodo.10207361	NOUN
m-769	212	1	[	[	X
m-769	212	2	14	14	NUM
m-769	212	3	]	]	X
m-769	212	4	john	john	PROPN
m-769	212	5	michael	michael	PROPN
m-769	212	6	.	.	PUNCT
m-769	213	1	n.	n.	PROPN
m-769	213	2	,	,	PUNCT
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m-769	213	4	e.	e.	PROPN
m-769	213	5	e.	e.	PROPN
m-769	213	6	,	,	PUNCT
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m-769	213	8	o.g	o.g	PROPN
m-769	213	9	.	.	PROPN
m-769	213	10	,	,	PUNCT
m-769	213	11	otobong	otobong	PROPN
m-769	213	12	j.	j.	PROPN
m-769	213	13	t	t	PROPN
m-769	213	14	and	and	CCONJ
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m-769	213	17	(	(	PUNCT
m-769	213	18	2023	2023	NUM
m-769	213	19	)	)	PUNCT
m-769	213	20	on	on	ADP
m-769	213	21	finding	find	VERB
m-769	213	22	the	the	DET
m-769	213	23	number	number	NOUN
m-769	213	24	of	of	ADP
m-769	213	25	homomorphism	homomorphism	PROPN
m-769	213	26	from	from	ADP
m-769	213	27	q8	q8	PROPN
m-769	213	28	,	,	PUNCT
m-769	213	29	international	international	ADJ
m-769	213	30	journal	journal	NOUN
m-769	213	31	of	of	ADP
m-769	213	32	mathematics	mathematics	PROPN
m-769	213	33	and	and	CCONJ
m-769	213	34	statistics	statistic	NOUN
m-769	213	35	studies	study	NOUN
m-769	213	36	,	,	PUNCT
m-769	213	37	11	11	NUM
m-769	213	38	(	(	PUNCT
m-769	213	39	4	4	NUM
m-769	213	40	)	)	PUNCT
m-769	213	41	,	,	PUNCT
m-769	213	42	20	20	NUM
m-769	213	43	-	-	SYM
m-769	213	44	26	26	NUM
m-769	213	45	.	.	PUNCT
m-769	214	1	doi	doi	NOUN
m-769	214	2	:	:	PUNCT
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m-769	215	2	15	15	NUM
m-769	215	3	]	]	X
m-769	215	4	michael	michael	PROPN
m-769	215	5	n.	n.	PROPN
m-769	215	6	john	john	PROPN
m-769	215	7	,	,	PUNCT
m-769	215	8	otobong	otobong	PROPN
m-769	215	9	g.	g.	PROPN
m-769	215	10	udoaka	udoaka	PROPN
m-769	215	11	,	,	PUNCT
m-769	215	12	&	&	CCONJ
m-769	215	13	itoro	itoro	PROPN
m-769	215	14	u.	u.	PROPN
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m-769	216	1	(	(	PUNCT
m-769	216	2	2023	2023	NUM
m-769	216	3	)	)	PUNCT
m-769	216	4	.	.	PUNCT
m-769	217	1	group	group	NOUN
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m-769	217	3	in	in	ADP
m-769	217	4	lattice	lattice	NOUN
m-769	217	5	-	-	PUNCT
m-769	217	6	based	base	VERB
m-769	217	7	cryptography	cryptography	NOUN
m-769	217	8	.	.	PUNCT
m-769	218	1	international	international	ADJ
m-769	218	2	journal	journal	PROPN
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m-769	218	4	mathematics	mathematic	NOUN
m-769	218	5	and	and	CCONJ
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m-769	218	9	11(4	11(4	NUM
m-769	218	10	)	)	PUNCT
m-769	218	11	,	,	PUNCT
m-769	218	12	111–125	111–125	NUM
m-769	218	13	.	.	PUNCT
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m-769	220	1	[	[	X
m-769	220	2	16	16	NUM
m-769	220	3	]	]	PUNCT
m-769	220	4	michael	michael	PROPN
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m-769	220	8	udoakpan	udoakpan	PROPN
m-769	220	9	i.	i.	PROPN
m-769	220	10	u	u	PROPN
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m-769	220	12	2023	2023	NUM
m-769	220	13	)	)	PUNCT
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m-769	220	15	group	group	NOUN
m-769	220	16	action	action	NOUN
m-769	220	17	on	on	ADP
m-769	220	18	an	an	DET
m-769	220	19	r	r	NOUN
m-769	220	20	-	-	PUNCT
m-769	220	21	subgroup	subgroup	NOUN
m-769	220	22	in	in	ADP
m-769	220	23	a	a	DET
m-769	220	24	near	near	ADJ
m-769	220	25	-	-	PUNCT
m-769	220	26	ring	ring	NOUN
m-769	220	27	,	,	PUNCT
m-769	220	28	international	international	ADJ
m-769	220	29	journal	journal	NOUN
m-769	220	30	of	of	ADP
m-769	220	31	mathematics	mathematics	PROPN
m-769	220	32	and	and	CCONJ
m-769	220	33	statistics	statistic	NOUN
m-769	220	34	studies	study	NOUN
m-769	220	35	,	,	PUNCT
m-769	220	36	11	11	NUM
m-769	220	37	(	(	PUNCT
m-769	220	38	4	4	NUM
m-769	220	39	)	)	PUNCT
m-769	220	40	,	,	PUNCT
m-769	220	41	2731	2731	NUM
m-769	220	42	.	.	PUNCT
m-769	221	1	doi	doi	NOUN
m-769	221	2	;	;	PUNCT
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m-769	221	6	doi	doi	NOUN
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m-769	221	8	/	/	SYM
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m-769	222	1	|	|	ADV
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m-769	222	3	12	12	NUM
m-769	222	4	|	|	CCONJ
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m-769	222	6	2023	2023	NUM
m-769	222	7	|	|	ADV
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m-769	222	9	20	20	NUM
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m-769	222	12	https://doi.org/10.5281/zenodo.10207210	https://doi.org/10.5281/zenodo.10207210	NOUN
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m-769	250	1	|	|	ADV
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m-769	255	22	-	-	SYM
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m-769	256	21	(	(	PUNCT
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m-769	258	9	(	(	PUNCT
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m-769	258	16	-	-	SYM
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m-769	258	20	-	-	SYM
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