id	sid	tid	token	lemma	pos
m-1141	1	1	on	on	ADP
m-1141	1	2	stability	stability	NOUN
m-1141	1	3	in	in	ADP
m-1141	1	4	separative	separative	PROPN
m-1141	1	5	semigroup	semigroup	PROPN
m-1141	1	6	abstract	abstract	ADJ
m-1141	1	7	stability	stability	NOUN
m-1141	1	8	,	,	PUNCT
m-1141	1	9	as	as	SCONJ
m-1141	1	10	introduced	introduce	VERB
m-1141	1	11	by	by	ADP
m-1141	1	12	koch	koch	PROPN
m-1141	1	13	and	and	CCONJ
m-1141	1	14	wallace	wallace	PROPN
m-1141	1	15	(	(	PUNCT
m-1141	1	16	1956	1956	NUM
m-1141	1	17	)	)	PUNCT
m-1141	1	18	,	,	PUNCT
m-1141	1	19	has	have	AUX
m-1141	1	20	long	long	ADV
m-1141	1	21	stood	stand	VERB
m-1141	1	22	as	as	ADP
m-1141	1	23	a	a	DET
m-1141	1	24	central	central	ADJ
m-1141	1	25	notion	notion	NOUN
m-1141	1	26	in	in	ADP
m-1141	1	27	semigroup	semigroup	PROPN
m-1141	1	28	theory	theory	NOUN
m-1141	1	29	,	,	PUNCT
m-1141	1	30	ensuring	ensure	VERB
m-1141	1	31	that	that	SCONJ
m-1141	1	32	inclusions	inclusion	NOUN
m-1141	1	33	of	of	ADP
m-1141	1	34	principal	principal	ADJ
m-1141	1	35	ideals	ideal	NOUN
m-1141	1	36	collapse	collapse	VERB
m-1141	1	37	into	into	ADP
m-1141	1	38	equalities	equality	NOUN
m-1141	1	39	and	and	CCONJ
m-1141	1	40	that	that	SCONJ
m-1141	1	41	left	leave	VERB
m-1141	1	42	and	and	CCONJ
m-1141	1	43	right	right	ADJ
m-1141	1	44	structures	structure	NOUN
m-1141	1	45	align	align	VERB
m-1141	1	46	harmoniously	harmoniously	ADV
m-1141	1	47	.	.	PUNCT
m-1141	2	1	separativity	separativity	NOUN
m-1141	2	2	,	,	PUNCT
m-1141	2	3	on	on	ADP
m-1141	2	4	the	the	DET
m-1141	2	5	other	other	ADJ
m-1141	2	6	hand	hand	NOUN
m-1141	2	7	,	,	PUNCT
m-1141	2	8	generalises	generalise	VERB
m-1141	2	9	cancellativity	cancellativity	NOUN
m-1141	2	10	while	while	SCONJ
m-1141	2	11	retaining	retain	VERB
m-1141	2	12	algebraic	algebraic	ADJ
m-1141	2	13	regularity	regularity	NOUN
m-1141	2	14	,	,	PUNCT
m-1141	2	15	and	and	CCONJ
m-1141	2	16	burmistrovich	burmistrovich	PROPN
m-1141	2	17	's	's	PART
m-1141	2	18	decomposition	decomposition	NOUN
m-1141	2	19	theorem	theorem	NOUN
m-1141	2	20	revealed	reveal	VERB
m-1141	2	21	that	that	SCONJ
m-1141	2	22	every	every	DET
m-1141	2	23	separative	separative	NOUN
m-1141	2	24	semigroup	semigroup	NOUN
m-1141	2	25	can	can	AUX
m-1141	2	26	be	be	AUX
m-1141	2	27	expressed	express	VERB
m-1141	2	28	as	as	ADP
m-1141	2	29	a	a	DET
m-1141	2	30	semilattice	semilattice	NOUN
m-1141	2	31	of	of	ADP
m-1141	2	32	cancellative	cancellative	ADJ
m-1141	2	33	semigroups	semigroup	NOUN
m-1141	2	34	.	.	PUNCT
m-1141	3	1	yet	yet	ADV
m-1141	3	2	,	,	PUNCT
m-1141	3	3	whether	whether	SCONJ
m-1141	3	4	separative	separative	NOUN
m-1141	3	5	semigroups	semigroup	NOUN
m-1141	3	6	inherit	inherit	VERB
m-1141	3	7	stability	stability	NOUN
m-1141	3	8	in	in	ADP
m-1141	3	9	the	the	DET
m-1141	3	10	sense	sense	NOUN
m-1141	3	11	of	of	ADP
m-1141	3	12	koch	koch	PROPN
m-1141	3	13	and	and	CCONJ
m-1141	3	14	wallace	wallace	PROPN
m-1141	3	15	has	have	AUX
m-1141	3	16	remained	remain	VERB
m-1141	3	17	unresolved	unresolved	ADJ
m-1141	3	18	.	.	PUNCT
m-1141	4	1	in	in	ADP
m-1141	4	2	this	this	DET
m-1141	4	3	work	work	NOUN
m-1141	4	4	,	,	PUNCT
m-1141	4	5	we	we	PRON
m-1141	4	6	close	close	VERB
m-1141	4	7	this	this	DET
m-1141	4	8	gap	gap	NOUN
m-1141	4	9	:	:	PUNCT
m-1141	4	10	we	we	PRON
m-1141	4	11	prove	prove	VERB
m-1141	4	12	that	that	SCONJ
m-1141	4	13	semilattices	semilattice	NOUN
m-1141	4	14	of	of	ADP
m-1141	4	15	cancellative	cancellative	ADJ
m-1141	4	16	semigroups	semigroup	NOUN
m-1141	4	17	are	be	AUX
m-1141	4	18	stable	stable	ADJ
m-1141	4	19	,	,	PUNCT
m-1141	4	20	and	and	CCONJ
m-1141	4	21	hence	hence	ADV
m-1141	4	22	every	every	DET
m-1141	4	23	separative	separative	NOUN
m-1141	4	24	semigroup	semigroup	NOUN
m-1141	4	25	is	be	AUX
m-1141	4	26	inherently	inherently	ADV
m-1141	4	27	stable	stable	ADJ
m-1141	4	28	.	.	PUNCT
m-1141	5	1	this	this	DET
m-1141	5	2	result	result	NOUN
m-1141	5	3	elevates	elevate	VERB
m-1141	5	4	stability	stability	NOUN
m-1141	5	5	from	from	ADP
m-1141	5	6	a	a	DET
m-1141	5	7	supplementary	supplementary	ADJ
m-1141	5	8	condition	condition	NOUN
m-1141	5	9	to	to	ADP
m-1141	5	10	a	a	DET
m-1141	5	11	built	build	VERB
m-1141	5	12	-	-	PUNCT
m-1141	5	13	in	in	ADP
m-1141	5	14	feature	feature	NOUN
m-1141	5	15	of	of	ADP
m-1141	5	16	separative	separative	NOUN
m-1141	5	17	semigroups	semigroup	NOUN
m-1141	5	18	,	,	PUNCT
m-1141	5	19	o	o	NOUN
m-1141	5	20	�	�	NOUN
m-1141	5	21	ering	ere	VERB
m-1141	5	22	a	a	DET
m-1141	5	23	uni	uni	PROPN
m-1141	5	24	�	�	PROPN
m-1141	5	25	ed	ed	PROPN
m-1141	5	26	perspective	perspective	NOUN
m-1141	5	27	that	that	PRON
m-1141	5	28	strengthens	strengthen	VERB
m-1141	5	29	the	the	DET
m-1141	5	30	foundations	foundation	NOUN
m-1141	5	31	of	of	ADP
m-1141	5	32	semigroup	semigroup	PROPN
m-1141	5	33	theory	theory	NOUN
m-1141	5	34	and	and	CCONJ
m-1141	5	35	deepens	deepen	VERB
m-1141	5	36	its	its	PRON
m-1141	5	37	structural	structural	ADJ
m-1141	5	38	coherence	coherence	NOUN
m-1141	5	39	.	.	PUNCT
m-1141	6	1	keywords	keyword	NOUN
m-1141	6	2	:	:	PUNCT
m-1141	6	3	stability	stability	NOUN
m-1141	6	4	in	in	ADP
m-1141	6	5	semigroup	semigroup	PROPN
m-1141	6	6	;	;	PUNCT
m-1141	6	7	separative	separative	PROPN
m-1141	6	8	semigroup	semigroup	PROPN
m-1141	6	9	;	;	PUNCT
m-1141	6	10	cancellative	cancellative	ADJ
m-1141	6	11	semigroup	semigroup	NOUN
m-1141	6	12	;	;	PUNCT
m-1141	6	13	semillatice	semillatice	ADJ
m-1141	6	14	decomposition	decomposition	NOUN
m-1141	6	15	;	;	PUNCT
m-1141	6	16	and	and	CCONJ
m-1141	6	17	green	green	PROPN
m-1141	6	18	's	's	PART
m-1141	6	19	relations	relation	NOUN
m-1141	6	20	.	.	PUNCT
m-1141	7	1	otobong	otobong	PROPN
m-1141	7	2	j.	j.	PROPN
m-1141	7	3	tom∗	tom∗	PROPN
m-1141	7	4	fed	fed	PROPN
m-1141	7	5	.	.	PUNCT
m-1141	8	1	uni	uni	PROPN
m-1141	8	2	.	.	PUNCT
m-1141	8	3	tech	tech	PROPN
m-1141	8	4	.	.	PUNCT
m-1141	8	5	,	,	PUNCT
m-1141	8	6	ikot	ikot	PROPN
m-1141	8	7	abasi	abasi	PROPN
m-1141	8	8	&	&	CCONJ
m-1141	8	9	akwa	akwa	PROPN
m-1141	8	10	ibom	ibom	ADJ
m-1141	8	11	state	state	NOUN
m-1141	8	12	uni	uni	PROPN
m-1141	8	13	.	.	PROPN
m-1141	8	14	,	,	PUNCT
m-1141	8	15	ikot	ikot	PROPN
m-1141	8	16	akpaden	akpaden	VERB
m-1141	9	1	otobong	otobong	PROPN
m-1141	9	2	g.	g.	PROPN
m-1141	10	1	udoaka	udoaka	PROPN
m-1141	10	2	department	department	PROPN
m-1141	10	3	of	of	ADP
m-1141	10	4	mathematics	mathematic	NOUN
m-1141	10	5	,	,	PUNCT
m-1141	10	6	akwa	akwa	ADJ
m-1141	10	7	ibom	ibom	ADJ
m-1141	10	8	state	state	NOUN
m-1141	10	9	university	university	PROPN
m-1141	10	10	,	,	PUNCT
m-1141	10	11	ikot	ikot	PROPN
m-1141	10	12	akpaden	akpaden	VERB
m-1141	10	13	ijo	ijo	PROPN
m-1141	10	14	international	international	PROPN
m-1141	10	15	journal	journal	PROPN
m-1141	10	16	of	of	ADP
m-1141	10	17	mathematics	mathematics	PROPN
m-1141	10	18	(	(	PUNCT
m-1141	10	19	issn	issn	PROPN
m-1141	10	20	:	:	PUNCT
m-1141	10	21	2992	2992	NUM
m-1141	10	22	-	-	SYM
m-1141	10	23	4421	4421	NUM
m-1141	10	24	)	)	PUNCT
m-1141	10	25	volume	volume	NOUN
m-1141	10	26	08	08	NUM
m-1141	11	1	|	|	ADV
m-1141	11	2	issue	issue	NOUN
m-1141	11	3	9	9	NUM
m-1141	11	4	|	|	CCONJ
m-1141	11	5	september	september	PROPN
m-1141	11	6	2025	2025	NUM
m-1141	12	1	|	|	ADV
m-1141	12	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1141	12	3	1	1	NUM
m-1141	12	4	ekere	ekere	ADV
m-1141	12	5	s.	s.	PROPN
m-1141	12	6	udo	udo	PROPN
m-1141	12	7	�	�	PROPN
m-1141	12	8	a	a	DET
m-1141	12	9	department	department	NOUN
m-1141	12	10	of	of	ADP
m-1141	12	11	mathematics	mathematic	NOUN
m-1141	12	12	,	,	PUNCT
m-1141	12	13	akwa	akwa	ADJ
m-1141	12	14	ibom	ibom	ADJ
m-1141	12	15	state	state	NOUN
m-1141	12	16	university	university	PROPN
m-1141	12	17	,	,	PUNCT
m-1141	12	18	ikot	ikot	PROPN
m-1141	12	19	akpaden	akpaden	PROPN
m-1141	12	20	mailto	mailto	NOUN
m-1141	12	21	:	:	PUNCT
m-1141	13	1	tomdgreatest@gmail.com	tomdgreatest@gmail.com	X
m-1141	13	2	otobongawasi@aksu.edu.ng	otobongawasi@aksu.edu.ng	NOUN
m-1141	13	3	ekereudofia@yahoo.com	ekereudofia@yahoo.com	X
m-1141	13	4	1	1	NUM
m-1141	13	5	introduction	introduction	NOUN
m-1141	13	6	the	the	DET
m-1141	13	7	concept	concept	NOUN
m-1141	13	8	of	of	ADP
m-1141	13	9	stability	stability	NOUN
m-1141	13	10	in	in	ADP
m-1141	13	11	semigroups	semigroup	NOUN
m-1141	13	12	was	be	AUX
m-1141	13	13	formally	formally	ADV
m-1141	13	14	introduced	introduce	VERB
m-1141	13	15	by	by	ADP
m-1141	13	16	koch	koch	PROPN
m-1141	13	17	and	and	CCONJ
m-1141	13	18	wallace	wallace	PROPN
m-1141	14	1	[	[	X
m-1141	14	2	16	16	NUM
m-1141	14	3	]	]	PUNCT
m-1141	14	4	in	in	ADP
m-1141	14	5	their	their	PRON
m-1141	14	6	seminal	seminal	ADJ
m-1141	14	7	paper	paper	NOUN
m-1141	14	8	.	.	PUNCT
m-1141	15	1	a	a	DET
m-1141	15	2	semigroup	semigroup	NOUN
m-1141	15	3	is	be	AUX
m-1141	15	4	said	say	VERB
m-1141	15	5	to	to	PART
m-1141	15	6	be	be	AUX
m-1141	15	7	stable	stable	ADJ
m-1141	15	8	if	if	SCONJ
m-1141	15	9	for	for	SCONJ
m-1141	15	10	all	all	DET
m-1141	15	11	a	a	DET
m-1141	15	12	,	,	PUNCT
m-1141	15	13	b	b	X
m-1141	15	14	∈	∈	PROPN
m-1141	15	15	s	s	VERB
m-1141	15	16	:	:	PUNCT
m-1141	15	17	as	as	ADP
m-1141	15	18	⊆	⊆	NUM
m-1141	15	19	abs	ab	NOUN
m-1141	15	20	=	=	NOUN
m-1141	15	21	⇒	⇒	NOUN
m-1141	15	22	as	as	ADP
m-1141	15	23	=	=	NOUN
m-1141	15	24	abs	ab	NOUN
m-1141	15	25	,	,	PUNCT
m-1141	15	26	sa	sa	PROPN
m-1141	15	27	⊆	⊆	NUM
m-1141	15	28	sab	sab	ADJ
m-1141	15	29	=	=	NOUN
m-1141	15	30	⇒	⇒	NOUN
m-1141	15	31	sa	sa	X
m-1141	16	1	=	=	PUNCT
m-1141	16	2	sab	sab	PROPN
m-1141	16	3	.	.	PUNCT
m-1141	17	1	this	this	DET
m-1141	17	2	algebraic	algebraic	PROPN
m-1141	17	3	de	de	PROPN
m-1141	17	4	�	�	PROPN
m-1141	17	5	nition	nition	NOUN
m-1141	17	6	re	re	NOUN
m-1141	17	7	�	�	PROPN
m-1141	17	8	ects	ect	VERB
m-1141	17	9	a	a	DET
m-1141	17	10	form	form	NOUN
m-1141	17	11	of	of	ADP
m-1141	17	12	�	�	PROPN
m-1141	17	13	ideal	ideal	ADJ
m-1141	17	14	stability	stability	PROPN
m-1141	17	15	�	�	PROPN
m-1141	17	16	and	and	CCONJ
m-1141	17	17	has	have	AUX
m-1141	17	18	since	since	SCONJ
m-1141	17	19	become	become	VERB
m-1141	17	20	a	a	DET
m-1141	17	21	cornerstone	cornerstone	NOUN
m-1141	17	22	in	in	ADP
m-1141	17	23	the	the	DET
m-1141	17	24	theory	theory	NOUN
m-1141	17	25	of	of	ADP
m-1141	17	26	semigroups	semigroup	NOUN
m-1141	17	27	,	,	PUNCT
m-1141	17	28	inspiring	inspire	VERB
m-1141	17	29	further	further	ADJ
m-1141	17	30	structural	structural	ADJ
m-1141	17	31	investigations	investigation	NOUN
m-1141	17	32	[	[	X
m-1141	17	33	1	1	NUM
m-1141	17	34	,	,	PUNCT
m-1141	17	35	12	12	NUM
m-1141	17	36	]	]	PUNCT
m-1141	17	37	.	.	PUNCT
m-1141	18	1	stability	stability	NOUN
m-1141	18	2	is	be	AUX
m-1141	18	3	important	important	ADJ
m-1141	18	4	because	because	SCONJ
m-1141	18	5	it	it	PRON
m-1141	18	6	guarantees	guarantee	VERB
m-1141	18	7	that	that	SCONJ
m-1141	18	8	the	the	DET
m-1141	18	9	behaviour	behaviour	NOUN
m-1141	18	10	of	of	ADP
m-1141	18	11	principal	principal	ADJ
m-1141	18	12	ideals	ideal	NOUN
m-1141	18	13	remains	remain	VERB
m-1141	18	14	robust	robust	ADJ
m-1141	18	15	under	under	ADP
m-1141	18	16	multiplication	multiplication	NOUN
m-1141	18	17	.	.	PUNCT
m-1141	19	1	the	the	DET
m-1141	19	2	study	study	NOUN
m-1141	19	3	of	of	ADP
m-1141	19	4	stability	stability	NOUN
m-1141	19	5	has	have	AUX
m-1141	19	6	evolved	evolve	VERB
m-1141	19	7	,	,	PUNCT
m-1141	19	8	particularly	particularly	ADV
m-1141	19	9	through	through	ADP
m-1141	19	10	its	its	PRON
m-1141	19	11	connection	connection	NOUN
m-1141	19	12	with	with	ADP
m-1141	19	13	green	green	PROPN
m-1141	19	14	's	's	PART
m-1141	19	15	relations	relation	NOUN
m-1141	19	16	,	,	PUNCT
m-1141	19	17	which	which	PRON
m-1141	19	18	classify	classify	VERB
m-1141	19	19	semigroup	semigroup	ADJ
m-1141	19	20	elements	element	NOUN
m-1141	19	21	based	base	VERB
m-1141	19	22	on	on	ADP
m-1141	19	23	their	their	PRON
m-1141	19	24	divisibility	divisibility	NOUN
m-1141	19	25	properties	property	NOUN
m-1141	19	26	[	[	X
m-1141	19	27	16	16	NUM
m-1141	19	28	]	]	PUNCT
m-1141	19	29	.	.	PUNCT
m-1141	20	1	a	a	DET
m-1141	20	2	crucial	crucial	ADJ
m-1141	20	3	result	result	NOUN
m-1141	20	4	from	from	ADP
m-1141	20	5	koch	koch	PROPN
m-1141	20	6	and	and	CCONJ
m-1141	20	7	wallace	wallace	PROPN
m-1141	20	8	states	state	VERB
m-1141	20	9	that	that	SCONJ
m-1141	20	10	in	in	ADP
m-1141	20	11	a	a	DET
m-1141	20	12	stable	stable	ADJ
m-1141	20	13	semigroup	semigroup	NOUN
m-1141	20	14	,	,	PUNCT
m-1141	20	15	green	green	PROPN
m-1141	20	16	's	's	PART
m-1141	20	17	d−	d−	PROPN
m-1141	20	18	and	and	CCONJ
m-1141	20	19	j−	j−	PROPN
m-1141	20	20	equivalences	equivalence	VERB
m-1141	20	21	coincide	coincide	NOUN
m-1141	20	22	,	,	PUNCT
m-1141	20	23	implying	imply	VERB
m-1141	20	24	that	that	SCONJ
m-1141	20	25	left	leave	VERB
m-1141	20	26	and	and	CCONJ
m-1141	20	27	right	right	ADJ
m-1141	20	28	ideal	ideal	ADJ
m-1141	20	29	structures	structure	NOUN
m-1141	20	30	behave	behave	VERB
m-1141	20	31	symmetrically	symmetrically	ADV
m-1141	20	32	.	.	PUNCT
m-1141	21	1	anderson	anderson	PROPN
m-1141	21	2	,	,	PUNCT
m-1141	21	3	et	et	PROPN
m-1141	21	4	al	al	PROPN
m-1141	22	1	[	[	X
m-1141	22	2	11	11	NUM
m-1141	22	3	]	]	PUNCT
m-1141	22	4	further	far	ADV
m-1141	22	5	expanded	expand	VERB
m-1141	22	6	on	on	ADP
m-1141	22	7	this	this	DET
m-1141	22	8	idea	idea	NOUN
m-1141	22	9	by	by	ADP
m-1141	22	10	studying	study	VERB
m-1141	22	11	stability	stability	NOUN
m-1141	22	12	conditions	condition	NOUN
m-1141	22	13	in	in	ADP
m-1141	22	14	various	various	ADJ
m-1141	22	15	semigroup	semigroup	ADJ
m-1141	22	16	classes	class	NOUN
m-1141	22	17	,	,	PUNCT
m-1141	22	18	highlighting	highlight	VERB
m-1141	22	19	its	its	PRON
m-1141	22	20	role	role	NOUN
m-1141	22	21	in	in	ADP
m-1141	22	22	determining	determine	VERB
m-1141	22	23	structural	structural	ADJ
m-1141	22	24	simplicity	simplicity	NOUN
m-1141	22	25	and	and	CCONJ
m-1141	22	26	regularity	regularity	NOUN
m-1141	22	27	.	.	PUNCT
m-1141	23	1	the	the	DET
m-1141	23	2	structure	structure	NOUN
m-1141	23	3	of	of	ADP
m-1141	23	4	quasi	quasi	ADJ
m-1141	23	5	-	-	ADJ
m-1141	23	6	separative	separative	ADJ
m-1141	23	7	semigroup	semigroup	NOUN
m-1141	23	8	was	be	AUX
m-1141	23	9	introduced	introduce	VERB
m-1141	23	10	by	by	ADP
m-1141	23	11	drazin	drazin	PROPN
m-1141	24	1	[	[	X
m-1141	24	2	12	12	NUM
m-1141	24	3	]	]	PUNCT
m-1141	24	4	,	,	PUNCT
m-1141	24	5	where	where	SCONJ
m-1141	24	6	the	the	DET
m-1141	24	7	connections	connection	NOUN
m-1141	24	8	between	between	ADP
m-1141	24	9	it	it	PRON
m-1141	24	10	and	and	CCONJ
m-1141	24	11	other	other	ADJ
m-1141	24	12	semigroup	semigroup	PROPN
m-1141	24	13	properties	property	NOUN
m-1141	24	14	,	,	PUNCT
m-1141	24	15	such	such	ADJ
m-1141	24	16	as	as	ADP
m-1141	24	17	inversity	inversity	NOUN
m-1141	24	18	,	,	PUNCT
m-1141	24	19	regularity	regularity	NOUN
m-1141	24	20	,	,	PUNCT
m-1141	24	21	etc	etc	X
m-1141	24	22	,	,	PUNCT
m-1141	24	23	were	be	AUX
m-1141	24	24	established	establish	VERB
m-1141	24	25	.	.	PUNCT
m-1141	25	1	this	this	PRON
m-1141	25	2	was	be	AUX
m-1141	25	3	later	later	ADV
m-1141	25	4	extended	extend	VERB
m-1141	25	5	by	by	ADP
m-1141	25	6	krasilnikova	krasilnikova	PROPN
m-1141	25	7	and	and	CCONJ
m-1141	25	8	novikove	novikove	VERB
m-1141	25	9	[	[	X
m-1141	25	10	22	22	NUM
m-1141	25	11	]	]	PUNCT
m-1141	25	12	.	.	PUNCT
m-1141	26	1	parallel	parallel	ADJ
m-1141	26	2	to	to	ADP
m-1141	26	3	this	this	DET
m-1141	26	4	development	development	NOUN
m-1141	26	5	,	,	PUNCT
m-1141	26	6	the	the	DET
m-1141	26	7	notion	notion	NOUN
m-1141	26	8	of	of	ADP
m-1141	26	9	separativity	separativity	NOUN
m-1141	26	10	was	be	AUX
m-1141	26	11	introduced	introduce	VERB
m-1141	26	12	to	to	PART
m-1141	26	13	generalise	generalise	VERB
m-1141	26	14	cancellativity	cancellativity	NOUN
m-1141	26	15	while	while	SCONJ
m-1141	26	16	preserving	preserve	VERB
m-1141	26	17	a	a	DET
m-1141	26	18	degree	degree	NOUN
m-1141	26	19	of	of	ADP
m-1141	26	20	regularity	regularity	NOUN
m-1141	26	21	.	.	PUNCT
m-1141	27	1	east	east	NOUN
m-1141	27	2	and	and	CCONJ
m-1141	27	3	higgings	higging	NOUN
m-1141	27	4	[	[	X
m-1141	27	5	9	9	NUM
m-1141	27	6	]	]	PUNCT
m-1141	27	7	explored	explore	VERB
m-1141	27	8	green	green	PROPN
m-1141	27	9	's	's	PART
m-1141	27	10	relation	relation	NOUN
m-1141	27	11	in	in	ADP
m-1141	27	12	greater	great	ADJ
m-1141	27	13	depth	depth	NOUN
m-1141	27	14	,	,	PUNCT
m-1141	27	15	providing	provide	VERB
m-1141	27	16	a	a	DET
m-1141	27	17	more	more	ADJ
m-1141	27	18	re	re	ADJ
m-1141	27	19	�	�	PROPN
m-1141	27	20	ned	ned	ADJ
m-1141	27	21	classi	classi	PROPN
m-1141	27	22	�	�	PROPN
m-1141	27	23	cation	cation	NOUN
m-1141	27	24	of	of	ADP
m-1141	27	25	semigroup	semigroup	ADJ
m-1141	27	26	elements	element	NOUN
m-1141	27	27	'	'	PART
m-1141	27	28	stability	stability	NOUN
m-1141	27	29	constraints	constraint	NOUN
m-1141	27	30	.	.	PUNCT
m-1141	28	1	hewitt	hewitt	PROPN
m-1141	28	2	and	and	CCONJ
m-1141	28	3	zuckerman	zuckerman	NOUN
m-1141	29	1	[	[	X
m-1141	29	2	6	6	NUM
m-1141	29	3	]	]	PUNCT
m-1141	29	4	applied	apply	VERB
m-1141	29	5	separativity	separativity	NOUN
m-1141	29	6	properties	property	NOUN
m-1141	29	7	in	in	ADP
m-1141	29	8	their	their	PRON
m-1141	29	9	study	study	NOUN
m-1141	29	10	,	,	PUNCT
m-1141	29	11	titled	title	VERB
m-1141	29	12	�	�	NOUN
m-1141	29	13	l1−	l1−	NOUN
m-1141	29	14	algebra	algebra	NOUN
m-1141	29	15	of	of	ADP
m-1141	29	16	a	a	DET
m-1141	29	17	commutative	commutative	ADJ
m-1141	29	18	semigroup	semigroup	PROPN
m-1141	29	19	�	�	PROPN
m-1141	29	20	,	,	PUNCT
m-1141	29	21	where	where	SCONJ
m-1141	29	22	they	they	PRON
m-1141	29	23	introduce	introduce	VERB
m-1141	29	24	harmonic	harmonic	ADJ
m-1141	29	25	analy	analy	NOUN
m-1141	29	26	sis	sis	NOUN
m-1141	29	27	on	on	ADP
m-1141	29	28	discrete	discrete	ADJ
m-1141	29	29	commutative	commutative	ADJ
m-1141	29	30	semigroups	semigroup	NOUN
m-1141	29	31	.	.	PUNCT
m-1141	30	1	shourijeh	shourijeh	PROPN
m-1141	31	1	[	[	X
m-1141	31	2	2	2	NUM
m-1141	31	3	]	]	PUNCT
m-1141	31	4	established	establish	VERB
m-1141	31	5	the	the	DET
m-1141	31	6	commutativity	commutativity	NOUN
m-1141	31	7	of	of	ADP
m-1141	31	8	separative	separative	NOUN
m-1141	31	9	semigroup	semigroup	NOUN
m-1141	31	10	and	and	CCONJ
m-1141	31	11	discussed	discuss	VERB
m-1141	31	12	its	its	PRON
m-1141	31	13	left	left	ADJ
m-1141	31	14	presentation	presentation	NOUN
m-1141	31	15	.	.	PUNCT
m-1141	32	1	burmistrovich	burmistrovich	PROPN
m-1141	33	1	[	[	X
m-1141	33	2	7	7	NUM
m-1141	33	3	]	]	PUNCT
m-1141	33	4	provided	provide	VERB
m-1141	33	5	a	a	DET
m-1141	33	6	powerful	powerful	ADJ
m-1141	33	7	structural	structural	ADJ
m-1141	33	8	result	result	NOUN
m-1141	33	9	:	:	PUNCT
m-1141	33	10	a	a	DET
m-1141	33	11	semigroup	semigroup	NOUN
m-1141	33	12	is	be	AUX
m-1141	33	13	separative	separative	ADJ
m-1141	33	14	if	if	SCONJ
m-1141	33	15	and	and	CCONJ
m-1141	33	16	only	only	ADV
m-1141	33	17	if	if	SCONJ
m-1141	33	18	it	it	PRON
m-1141	33	19	is	be	AUX
m-1141	33	20	isomorphic	isomorphic	ADJ
m-1141	33	21	to	to	ADP
m-1141	33	22	a	a	DET
m-1141	33	23	semilattice	semilattice	NOUN
m-1141	33	24	of	of	ADP
m-1141	33	25	cancellative	cancellative	ADJ
m-1141	33	26	semigroups	semigroup	NOUN
m-1141	33	27	.	.	PUNCT
m-1141	34	1	this	this	PRON
m-1141	34	2	theorem	theorem	VERB
m-1141	34	3	places	place	NOUN
m-1141	34	4	separative	separative	NOUN
m-1141	34	5	semigroups	semigroup	NOUN
m-1141	34	6	in	in	ADP
m-1141	34	7	a	a	DET
m-1141	34	8	broader	broad	ADJ
m-1141	34	9	algebraic	algebraic	ADJ
m-1141	34	10	context	context	NOUN
m-1141	34	11	and	and	CCONJ
m-1141	34	12	has	have	AUX
m-1141	34	13	been	be	AUX
m-1141	34	14	extensively	extensively	ADV
m-1141	34	15	used	use	VERB
m-1141	34	16	in	in	ADP
m-1141	34	17	decomposition	decomposition	NOUN
m-1141	34	18	theory	theory	NOUN
m-1141	34	19	[	[	X
m-1141	34	20	13	13	NUM
m-1141	34	21	]	]	PUNCT
m-1141	34	22	.	.	PUNCT
m-1141	35	1	however	however	ADV
m-1141	35	2	,	,	PUNCT
m-1141	35	3	burmistrovich	burmistrovich	PROPN
m-1141	35	4	's	's	PART
m-1141	35	5	theorem	theorem	NOUN
m-1141	35	6	itself	itself	PRON
m-1141	35	7	does	do	AUX
m-1141	35	8	not	not	PART
m-1141	35	9	discuss	discuss	VERB
m-1141	35	10	stability	stability	NOUN
m-1141	35	11	,	,	PUNCT
m-1141	35	12	leaving	leave	VERB
m-1141	35	13	a	a	DET
m-1141	35	14	gap	gap	NOUN
m-1141	35	15	in	in	ADP
m-1141	35	16	the	the	DET
m-1141	35	17	understanding	understanding	NOUN
m-1141	35	18	of	of	ADP
m-1141	35	19	whether	whether	SCONJ
m-1141	35	20	separative	separative	PROPN
m-1141	35	21	semigroups	semigroup	NOUN
m-1141	35	22	inherit	inherit	VERB
m-1141	35	23	stability	stability	NOUN
m-1141	35	24	(	(	PUNCT
m-1141	35	25	in	in	ADP
m-1141	35	26	the	the	DET
m-1141	35	27	sense	sense	NOUN
m-1141	35	28	of	of	ADP
m-1141	35	29	[	[	X
m-1141	35	30	16	16	NUM
m-1141	35	31	]	]	PUNCT
m-1141	35	32	)	)	PUNCT
m-1141	35	33	.	.	PUNCT
m-1141	36	1	this	this	DET
m-1141	36	2	gap	gap	NOUN
m-1141	36	3	motivates	motivate	VERB
m-1141	36	4	the	the	DET
m-1141	36	5	present	present	ADJ
m-1141	36	6	study	study	NOUN
m-1141	36	7	.	.	PUNCT
m-1141	37	1	we	we	PRON
m-1141	37	2	aim	aim	VERB
m-1141	37	3	to	to	PART
m-1141	37	4	establish	establish	VERB
m-1141	37	5	that	that	SCONJ
m-1141	37	6	separative	separative	NOUN
m-1141	37	7	semigroups	semigroup	NOUN
m-1141	37	8	are	be	AUX
m-1141	37	9	indeed	indeed	ADV
m-1141	37	10	stable	stable	ADJ
m-1141	37	11	in	in	ADP
m-1141	37	12	the	the	DET
m-1141	37	13	sense	sense	NOUN
m-1141	37	14	of	of	ADP
m-1141	37	15	[	[	X
m-1141	37	16	19	19	NUM
m-1141	37	17	]	]	PUNCT
m-1141	37	18	.	.	PUNCT
m-1141	38	1	our	our	PRON
m-1141	38	2	approach	approach	NOUN
m-1141	38	3	hinges	hinge	VERB
m-1141	38	4	on	on	ADP
m-1141	38	5	re	re	VERB
m-1141	38	6	-	-	VERB
m-1141	38	7	examining	examine	VERB
m-1141	38	8	burmistrovich	burmistrovich	NOUN
m-1141	38	9	's	's	PART
m-1141	38	10	decomposition	decomposition	NOUN
m-1141	38	11	theorem	theorem	NOUN
m-1141	38	12	from	from	ADP
m-1141	38	13	the	the	DET
m-1141	38	14	perspective	perspective	NOUN
m-1141	38	15	of	of	ADP
m-1141	38	16	stability	stability	NOUN
m-1141	38	17	.	.	PUNCT
m-1141	39	1	we	we	PRON
m-1141	39	2	�	�	VERB
m-1141	39	3	rst	rst	PROPN
m-1141	39	4	establish	establish	VERB
m-1141	39	5	that	that	SCONJ
m-1141	39	6	a	a	DET
m-1141	39	7	semilattice	semilattice	NOUN
m-1141	39	8	of	of	ADP
m-1141	39	9	cancellative	cancellative	ADJ
m-1141	39	10	semigroups	semigroup	NOUN
m-1141	39	11	is	be	AUX
m-1141	39	12	stable	stable	ADJ
m-1141	39	13	�	�	X
m-1141	39	14	a	a	DET
m-1141	39	15	fact	fact	NOUN
m-1141	39	16	that	that	PRON
m-1141	39	17	has	have	AUX
m-1141	39	18	not	not	PART
m-1141	39	19	been	be	AUX
m-1141	39	20	explicitly	explicitly	ADV
m-1141	39	21	recorded	record	VERB
m-1141	39	22	in	in	ADP
m-1141	39	23	the	the	DET
m-1141	39	24	literature	literature	NOUN
m-1141	39	25	.	.	PUNCT
m-1141	40	1	since	since	SCONJ
m-1141	40	2	separative	separative	NOUN
m-1141	40	3	semigroups	semigroup	NOUN
m-1141	40	4	are	be	AUX
m-1141	40	5	precisely	precisely	ADV
m-1141	40	6	those	those	DET
m-1141	40	7	semigroups	semigroup	NOUN
m-1141	40	8	isomorphic	isomorphic	ADJ
m-1141	40	9	to	to	ADP
m-1141	40	10	such	such	ADJ
m-1141	40	11	semilattices	semilattice	NOUN
m-1141	40	12	,	,	PUNCT
m-1141	40	13	it	it	PRON
m-1141	40	14	follows	follow	VERB
m-1141	40	15	directly	directly	ADV
m-1141	40	16	that	that	SCONJ
m-1141	40	17	every	every	DET
m-1141	40	18	separative	separative	NOUN
m-1141	40	19	semigroup	semigroup	NOUN
m-1141	40	20	is	be	AUX
m-1141	40	21	stable	stable	ADJ
m-1141	40	22	.	.	PUNCT
m-1141	41	1	2	2	NUM
m-1141	41	2	preliminaries	preliminary	NOUN
m-1141	41	3	de	de	X
m-1141	41	4	�	�	NOUN
m-1141	41	5	nition	nition	NOUN
m-1141	41	6	2.1	2.1	NUM
m-1141	41	7	(	(	PUNCT
m-1141	41	8	groupoid	groupoid	PROPN
m-1141	41	9	)	)	PUNCT
m-1141	41	10	.	.	PUNCT
m-1141	42	1	let	let	VERB
m-1141	42	2	s	s	PRON
m-1141	42	3	be	be	AUX
m-1141	42	4	a	a	DET
m-1141	42	5	non	non	ADJ
m-1141	42	6	-	-	ADJ
m-1141	42	7	empty	empty	ADJ
m-1141	42	8	set	set	NOUN
m-1141	42	9	.	.	PUNCT
m-1141	43	1	let	let	VERB
m-1141	43	2	∗	∗	NOUN
m-1141	43	3	be	be	AUX
m-1141	43	4	an	an	DET
m-1141	43	5	operation	operation	NOUN
m-1141	43	6	such	such	ADJ
m-1141	43	7	that	that	SCONJ
m-1141	43	8	∗	∗	NOUN
m-1141	43	9	:	:	PUNCT
m-1141	44	1	s×s	s×s	PROPN
m-1141	44	2	→	→	SYM
m-1141	44	3	s	s	VERB
m-1141	44	4	be	be	AUX
m-1141	44	5	de	de	X
m-1141	44	6	�	�	NOUN
m-1141	44	7	ned	ne	VERB
m-1141	44	8	on	on	ADP
m-1141	44	9	s.	s.	PROPN
m-1141	44	10	then	then	ADV
m-1141	44	11	(	(	PUNCT
m-1141	44	12	s	s	X
m-1141	44	13	,	,	PUNCT
m-1141	44	14	∗	∗	NOUN
m-1141	44	15	)	)	PUNCT
m-1141	44	16	is	be	AUX
m-1141	44	17	called	call	VERB
m-1141	44	18	groupoid	groupoid	PROPN
m-1141	44	19	if	if	SCONJ
m-1141	44	20	for	for	ADP
m-1141	44	21	all	all	DET
m-1141	44	22	a	a	DET
m-1141	44	23	,	,	PUNCT
m-1141	44	24	b	b	X
m-1141	44	25	∈	∈	PROPN
m-1141	44	26	s	s	NOUN
m-1141	44	27	,	,	PUNCT
m-1141	44	28	a∗	a∗	PROPN
m-1141	44	29	b	b	PROPN
m-1141	44	30	∈	∈	PROPN
m-1141	44	31	s.	s.	PROPN
m-1141	44	32	de	de	PROPN
m-1141	44	33	�	�	PROPN
m-1141	44	34	nition	nition	NOUN
m-1141	44	35	2.2	2.2	NUM
m-1141	44	36	(	(	PUNCT
m-1141	44	37	semigroup	semigroup	NOUN
m-1141	44	38	)	)	PUNCT
m-1141	44	39	.	.	PUNCT
m-1141	45	1	a	a	DET
m-1141	45	2	groupoid	groupoid	PROPN
m-1141	45	3	is	be	AUX
m-1141	45	4	called	call	VERB
m-1141	45	5	semigroup	semigroup	ADJ
m-1141	45	6	if	if	SCONJ
m-1141	45	7	the	the	DET
m-1141	45	8	binary	binary	PROPN
m-1141	45	9	operation	operation	NOUN
m-1141	45	10	is	be	AUX
m-1141	45	11	associative	associative	ADJ
m-1141	45	12	(	(	PUNCT
m-1141	45	13	i.e.	i.e.	X
m-1141	45	14	,	,	PUNCT
m-1141	45	15	for	for	ADP
m-1141	45	16	all	all	DET
m-1141	45	17	a	a	DET
m-1141	45	18	,	,	PUNCT
m-1141	45	19	b	b	NOUN
m-1141	45	20	,	,	PUNCT
m-1141	45	21	c	c	PROPN
m-1141	45	22	∈	∈	PROPN
m-1141	45	23	s	s	X
m-1141	45	24	,	,	PUNCT
m-1141	45	25	we	we	PRON
m-1141	45	26	have	have	AUX
m-1141	45	27	(	(	PUNCT
m-1141	45	28	a	a	DET
m-1141	45	29	∗	∗	NOUN
m-1141	45	30	b	b	NOUN
m-1141	45	31	)	)	PUNCT
m-1141	45	32	∗	∗	NOUN
m-1141	45	33	c	c	NOUN
m-1141	45	34	=	=	PUNCT
m-1141	45	35	a	a	DET
m-1141	45	36	∗	∗	NOUN
m-1141	45	37	(	(	PUNCT
m-1141	45	38	b	b	NOUN
m-1141	45	39	∗	∗	NOUN
m-1141	45	40	c	c	NOUN
m-1141	45	41	)	)	PUNCT
m-1141	45	42	)	)	PUNCT
m-1141	45	43	.	.	PUNCT
m-1141	46	1	de	de	PROPN
m-1141	46	2	�	�	PROPN
m-1141	46	3	nition	nition	NOUN
m-1141	46	4	2.3	2.3	NUM
m-1141	46	5	(	(	PUNCT
m-1141	46	6	idempotent	idempotent	NOUN
m-1141	46	7	)	)	PUNCT
m-1141	46	8	.	.	PUNCT
m-1141	47	1	an	an	DET
m-1141	47	2	element	element	NOUN
m-1141	47	3	e	e	X
m-1141	47	4	∈	∈	NOUN
m-1141	47	5	s	s	PART
m-1141	47	6	is	be	AUX
m-1141	47	7	called	call	VERB
m-1141	47	8	an	an	DET
m-1141	47	9	idempotent	idempotent	ADJ
m-1141	47	10	element	element	NOUN
m-1141	47	11	if	if	SCONJ
m-1141	47	12	e2	e2	PROPN
m-1141	47	13	=	=	SYM
m-1141	47	14	e.	e.	PROPN
m-1141	47	15	ijo	ijo	PROPN
m-1141	47	16	international	international	PROPN
m-1141	47	17	journal	journal	PROPN
m-1141	47	18	of	of	ADP
m-1141	47	19	mathematics	mathematics	PROPN
m-1141	47	20	(	(	PUNCT
m-1141	47	21	issn	issn	PROPN
m-1141	47	22	:	:	PUNCT
m-1141	47	23	2992	2992	NUM
m-1141	47	24	-	-	SYM
m-1141	47	25	4421	4421	NUM
m-1141	47	26	)	)	PUNCT
m-1141	47	27	volume	volume	NOUN
m-1141	47	28	08	08	NUM
m-1141	48	1	|	|	ADV
m-1141	48	2	issue	issue	NOUN
m-1141	48	3	9	9	NUM
m-1141	48	4	|	|	CCONJ
m-1141	48	5	september	september	PROPN
m-1141	48	6	2025	2025	NUM
m-1141	49	1	|	|	ADV
m-1141	49	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1141	49	3	2	2	NUM
m-1141	49	4	de	de	X
m-1141	49	5	�	�	PROPN
m-1141	49	6	nition	nition	NOUN
m-1141	49	7	2.4	2.4	NUM
m-1141	49	8	(	(	PUNCT
m-1141	49	9	cancellative	cancellative	ADJ
m-1141	49	10	semigroup	semigroup	NOUN
m-1141	49	11	)	)	PUNCT
m-1141	49	12	.	.	PUNCT
m-1141	50	1	a	a	DET
m-1141	50	2	semigroup	semigroup	NOUN
m-1141	50	3	is	be	AUX
m-1141	50	4	called	call	VERB
m-1141	50	5	cancellative	cancellative	ADJ
m-1141	50	6	for	for	ADP
m-1141	50	7	all	all	DET
m-1141	50	8	a	a	DET
m-1141	50	9	,	,	PUNCT
m-1141	50	10	b	b	NOUN
m-1141	50	11	,	,	PUNCT
m-1141	50	12	c	c	PROPN
m-1141	50	13	∈	∈	PROPN
m-1141	50	14	s	s	PROPN
m-1141	50	15	,	,	PUNCT
m-1141	50	16	ab	ab	PROPN
m-1141	50	17	=	=	PUNCT
m-1141	50	18	ac	ac	PROPN
m-1141	50	19	=	=	AUX
m-1141	50	20	⇒	⇒	VERB
m-1141	50	21	a	a	DET
m-1141	50	22	=	=	PUNCT
m-1141	50	23	b(left	b(left	NOUN
m-1141	50	24	-	-	PUNCT
m-1141	50	25	cancellative	cancellative	NOUN
m-1141	50	26	)	)	PUNCT
m-1141	50	27	and	and	CCONJ
m-1141	51	1	ba	ba	NOUN
m-1141	51	2	=	=	PUNCT
m-1141	51	3	ca	can	AUX
m-1141	51	4	(	(	PUNCT
m-1141	51	5	right	right	ADV
m-1141	51	6	-	-	PUNCT
m-1141	51	7	cancellative	cancellative	ADJ
m-1141	51	8	)	)	PUNCT
m-1141	51	9	.	.	PUNCT
m-1141	52	1	for	for	ADP
m-1141	52	2	more	more	ADJ
m-1141	52	3	about	about	ADP
m-1141	52	4	this	this	PRON
m-1141	52	5	,	,	PUNCT
m-1141	52	6	the	the	DET
m-1141	52	7	reader	reader	NOUN
m-1141	52	8	is	be	AUX
m-1141	52	9	referred	refer	VERB
m-1141	52	10	to	to	ADP
m-1141	52	11	[	[	X
m-1141	52	12	1,4,5	1,4,5	NUM
m-1141	52	13	,	,	PUNCT
m-1141	52	14	7	7	NUM
m-1141	52	15	]	]	PUNCT
m-1141	52	16	.	.	PUNCT
m-1141	53	1	3	3	NUM
m-1141	53	2	separative	separative	PROPN
m-1141	53	3	semigroup	semigroup	PROPN
m-1141	53	4	de	de	PROPN
m-1141	53	5	�	�	PROPN
m-1141	53	6	nition	nition	NOUN
m-1141	53	7	3.1	3.1	NUM
m-1141	53	8	.	.	PUNCT
m-1141	54	1	a	a	DET
m-1141	54	2	semigroup	semigroup	NOUN
m-1141	54	3	s	s	VERB
m-1141	54	4	is	be	AUX
m-1141	54	5	called	call	VERB
m-1141	54	6	separative	separative	NOUN
m-1141	54	7	if	if	SCONJ
m-1141	54	8	and	and	CCONJ
m-1141	54	9	only	only	ADV
m-1141	54	10	if	if	SCONJ
m-1141	54	11	the	the	DET
m-1141	54	12	following	follow	VERB
m-1141	54	13	two	two	NUM
m-1141	54	14	conditions	condition	NOUN
m-1141	54	15	hold	hold	VERB
m-1141	54	16	for	for	ADP
m-1141	54	17	all	all	DET
m-1141	54	18	a	a	DET
m-1141	54	19	,	,	PUNCT
m-1141	54	20	b	b	X
m-1141	54	21	∈	∈	PROPN
m-1141	54	22	s	s	PART
m-1141	54	23	:	:	PUNCT
m-1141	54	24	�	�	PROPN
m-1141	54	25	a2	a2	PROPN
m-1141	54	26	=	=	PROPN
m-1141	54	27	ab	ab	PROPN
m-1141	54	28	and	and	CCONJ
m-1141	54	29	ba	ba	PROPN
m-1141	54	30	=	=	NOUN
m-1141	54	31	b2	b2	PROPN
m-1141	54	32	together	together	ADV
m-1141	54	33	imply	imply	VERB
m-1141	54	34	a	a	DET
m-1141	54	35	=	=	SYM
m-1141	54	36	b	b	NOUN
m-1141	54	37	,	,	PUNCT
m-1141	54	38	and	and	CCONJ
m-1141	54	39	�	�	PROPN
m-1141	54	40	a2	a2	PROPN
m-1141	54	41	=	=	SYM
m-1141	54	42	ba	ba	PROPN
m-1141	54	43	and	and	CCONJ
m-1141	54	44	ab	ab	PROPN
m-1141	54	45	=	=	PUNCT
m-1141	54	46	b2	b2	PROPN
m-1141	54	47	together	together	ADV
m-1141	54	48	imply	imply	VERB
m-1141	54	49	a	a	DET
m-1141	54	50	=	=	X
m-1141	54	51	b.	b.	NOUN
m-1141	55	1	the	the	DET
m-1141	55	2	reader	reader	NOUN
m-1141	55	3	can	can	AUX
m-1141	55	4	also	also	ADV
m-1141	55	5	see	see	VERB
m-1141	55	6	[	[	X
m-1141	55	7	15	15	NUM
m-1141	55	8	]	]	X
m-1141	55	9	de	de	X
m-1141	55	10	�	�	NOUN
m-1141	55	11	nition	nition	NOUN
m-1141	55	12	3.2	3.2	NUM
m-1141	55	13	(	(	PUNCT
m-1141	55	14	qusi	qusi	ADJ
m-1141	55	15	-	-	PUNCT
m-1141	55	16	separative	separative	NOUN
m-1141	55	17	)	)	PUNCT
m-1141	55	18	.	.	PUNCT
m-1141	56	1	a	a	DET
m-1141	56	2	semigroup	semigroup	NOUN
m-1141	56	3	is	be	AUX
m-1141	56	4	called	call	VERB
m-1141	56	5	quasi	quasi	PROPN
m-1141	56	6	�	�	PROPN
m-1141	56	7	separative	separative	NOUN
m-1141	56	8	if	if	SCONJ
m-1141	56	9	a2	a2	PROPN
m-1141	56	10	=	=	SYM
m-1141	56	11	ab	ab	PROPN
m-1141	56	12	=	=	PUNCT
m-1141	56	13	ba	ba	PROPN
m-1141	56	14	=	=	NOUN
m-1141	56	15	b2	b2	NOUN
m-1141	57	1	=	=	NOUN
m-1141	57	2	⇒	⇒	VERB
m-1141	57	3	a	a	DET
m-1141	57	4	=	=	SYM
m-1141	57	5	b	b	NOUN
m-1141	57	6	,	,	PUNCT
m-1141	57	7	[	[	X
m-1141	57	8	22	22	NUM
m-1141	57	9	]	]	PUNCT
m-1141	57	10	.	.	PUNCT
m-1141	58	1	on	on	ADP
m-1141	58	2	a	a	DET
m-1141	58	3	bright	bright	ADJ
m-1141	58	4	-	-	PUNCT
m-1141	58	5	line	line	NOUN
m-1141	58	6	,	,	PUNCT
m-1141	58	7	one	one	PRON
m-1141	58	8	can	can	AUX
m-1141	58	9	say	say	VERB
m-1141	58	10	that	that	SCONJ
m-1141	58	11	a	a	DET
m-1141	58	12	separative	separative	NOUN
m-1141	58	13	semigroup	semigroup	NOUN
m-1141	58	14	is	be	AUX
m-1141	58	15	a	a	DET
m-1141	58	16	quasi	quasi	NOUN
m-1141	58	17	-	-	NOUN
m-1141	58	18	separative	separative	ADJ
m-1141	58	19	.	.	PUNCT
m-1141	59	1	extracting	extract	VERB
m-1141	59	2	from	from	ADP
m-1141	59	3	[	[	X
m-1141	59	4	20	20	NUM
m-1141	59	5	]	]	PUNCT
m-1141	59	6	we	we	PROPN
m-1141	59	7	de	de	PROPN
m-1141	59	8	�	�	PROPN
m-1141	59	9	ne	ne	PROPN
m-1141	59	10	a	a	DET
m-1141	59	11	weakly	weakly	ADJ
m-1141	59	12	separative	separative	NOUN
m-1141	59	13	semigroup	semigroup	NOUN
m-1141	59	14	when	when	SCONJ
m-1141	59	15	we	we	PRON
m-1141	59	16	have	have	VERB
m-1141	59	17	asa	asa	PROPN
m-1141	59	18	=	=	PUNCT
m-1141	59	19	asb	asb	PROPN
m-1141	59	20	=	=	PROPN
m-1141	59	21	bsa	bsa	PROPN
m-1141	59	22	=	=	PROPN
m-1141	59	23	bsb	bsb	PROPN
m-1141	59	24	only	only	ADV
m-1141	59	25	if	if	SCONJ
m-1141	59	26	a	a	DET
m-1141	59	27	=	=	SYM
m-1141	59	28	b	b	NOUN
m-1141	59	29	for	for	ADP
m-1141	59	30	all	all	DET
m-1141	59	31	a	a	DET
m-1141	59	32	,	,	PUNCT
m-1141	59	33	b	b	NOUN
m-1141	59	34	,	,	PUNCT
m-1141	59	35	s	s	PROPN
m-1141	59	36	∈	∈	PROPN
m-1141	59	37	s.	s.	PROPN
m-1141	59	38	theorem	theorem	VERB
m-1141	59	39	3.1	3.1	NUM
m-1141	59	40	(	(	PUNCT
m-1141	59	41	proposition	proposition	NOUN
m-1141	59	42	1	1	NUM
m-1141	59	43	of	of	ADP
m-1141	59	44	[	[	X
m-1141	59	45	12	12	NUM
m-1141	59	46	]	]	PUNCT
m-1141	59	47	)	)	PUNCT
m-1141	59	48	.	.	PUNCT
m-1141	60	1	if	if	SCONJ
m-1141	60	2	s	s	NOUN
m-1141	60	3	is	be	AUX
m-1141	60	4	any	any	DET
m-1141	60	5	quasi	quasi	ADJ
m-1141	60	6	-	-	ADJ
m-1141	60	7	separative	separative	ADJ
m-1141	60	8	semigroup	semigroup	NOUN
m-1141	60	9	,	,	PUNCT
m-1141	60	10	then	then	ADV
m-1141	60	11	,	,	PUNCT
m-1141	60	12	for	for	ADP
m-1141	60	13	all	all	DET
m-1141	60	14	a	a	DET
m-1141	60	15	,	,	PUNCT
m-1141	60	16	b	b	X
m-1141	60	17	∈	∈	NOUN
m-1141	60	18	s	s	VERB
m-1141	60	19	we	we	PRON
m-1141	60	20	have	have	VERB
m-1141	60	21	a2	a2	PROPN
m-1141	60	22	=	=	SYM
m-1141	60	23	ab	ab	PROPN
m-1141	60	24	=	=	PROPN
m-1141	60	25	b2	b2	PROPN
m-1141	60	26	if	if	SCONJ
m-1141	60	27	and	and	CCONJ
m-1141	60	28	only	only	ADV
m-1141	60	29	if	if	SCONJ
m-1141	60	30	a	a	DET
m-1141	60	31	=	=	X
m-1141	60	32	b.	b.	PROPN
m-1141	60	33	and	and	CCONJ
m-1141	60	34	the	the	DET
m-1141	60	35	converse	converse	NOUN
m-1141	60	36	also	also	ADV
m-1141	60	37	holds	hold	VERB
m-1141	60	38	.	.	PUNCT
m-1141	61	1	proof	proof	NOUN
m-1141	61	2	.	.	PUNCT
m-1141	62	1	if	if	SCONJ
m-1141	62	2	a2	a2	PROPN
m-1141	62	3	=	=	SYM
m-1141	62	4	ab	ab	PROPN
m-1141	62	5	=	=	PUNCT
m-1141	62	6	b2	b2	PROPN
m-1141	62	7	,	,	PUNCT
m-1141	62	8	then	then	ADV
m-1141	62	9	we	we	PRON
m-1141	62	10	have	have	VERB
m-1141	62	11	(	(	PUNCT
m-1141	62	12	ab)2	ab)2	PROPN
m-1141	62	13	=	=	SYM
m-1141	62	14	a2b2	a2b2	PROPN
m-1141	62	15	=	=	SYM
m-1141	62	16	(	(	PUNCT
m-1141	62	17	aa)(bb	aa)(bb	ADJ
m-1141	62	18	)	)	PUNCT
m-1141	62	19	=	=	SYM
m-1141	63	1	a2b2	a2b2	PROPN
m-1141	63	2	,	,	PUNCT
m-1141	63	3	and	and	CCONJ
m-1141	63	4	(	(	PUNCT
m-1141	63	5	ab)(ba	ab)(ba	X
m-1141	63	6	)	)	PUNCT
m-1141	63	7	=	=	SYM
m-1141	63	8	(	(	PUNCT
m-1141	63	9	aa)a(a	aa)a(a	PROPN
m-1141	63	10	)	)	PUNCT
m-1141	63	11	=	=	SYM
m-1141	63	12	a4	a4	PROPN
m-1141	63	13	,	,	PUNCT
m-1141	63	14	(	(	PUNCT
m-1141	63	15	ba)(ab	ba)(ab	NOUN
m-1141	63	16	)	)	PUNCT
m-1141	63	17	=	=	SYM
m-1141	63	18	(	(	PUNCT
m-1141	63	19	bb)b(b	bb)b(b	X
m-1141	63	20	)	)	PUNCT
m-1141	63	21	=	=	SYM
m-1141	63	22	b4	b4	NOUN
m-1141	63	23	.	.	PUNCT
m-1141	64	1	also	also	ADV
m-1141	64	2	,	,	PUNCT
m-1141	64	3	(	(	PUNCT
m-1141	64	4	ba)2	ba)2	NOUN
m-1141	64	5	=	=	SYM
m-1141	64	6	b(ab)a	b(ab)a	PROPN
m-1141	64	7	=	=	SYM
m-1141	64	8	(	(	PUNCT
m-1141	64	9	bb)(aa	bb)(aa	PROPN
m-1141	64	10	)	)	PUNCT
m-1141	64	11	=	=	SYM
m-1141	64	12	b2a2	b2a2	PROPN
m-1141	64	13	,	,	PUNCT
m-1141	64	14	and	and	CCONJ
m-1141	64	15	a(ab	a(ab	NOUN
m-1141	64	16	)	)	PUNCT
m-1141	64	17	=	=	SYM
m-1141	64	18	a(aa)a	a(aa)a	PROPN
m-1141	64	19	=	=	NOUN
m-1141	64	20	a3	a3	NOUN
m-1141	64	21	,	,	PUNCT
m-1141	64	22	b(ba	b(ba	X
m-1141	64	23	)	)	PUNCT
m-1141	64	24	=	=	SYM
m-1141	65	1	b(bb)b	b(bb)b	NOUN
m-1141	65	2	=	=	SYM
m-1141	65	3	b3	b3	PROPN
m-1141	65	4	.	.	PUNCT
m-1141	66	1	therefore	therefore	ADV
m-1141	66	2	,	,	PUNCT
m-1141	66	3	(	(	PUNCT
m-1141	66	4	ab)(ab	ab)(ab	PROPN
m-1141	66	5	)	)	PUNCT
m-1141	66	6	=	=	SYM
m-1141	66	7	(	(	PUNCT
m-1141	66	8	ba)(ba	ba)(ba	X
m-1141	66	9	)	)	PUNCT
m-1141	66	10	=	=	SYM
m-1141	66	11	(	(	PUNCT
m-1141	66	12	ab)(ba	ab)(ba	NOUN
m-1141	66	13	)	)	PUNCT
m-1141	66	14	,	,	PUNCT
m-1141	66	15	thus	thus	ADV
m-1141	66	16	,	,	PUNCT
m-1141	66	17	by	by	ADP
m-1141	66	18	quasi	quasi	NOUN
m-1141	66	19	-	-	NOUN
m-1141	66	20	separativity	separativity	NOUN
m-1141	66	21	,	,	PUNCT
m-1141	66	22	we	we	PRON
m-1141	66	23	have	have	VERB
m-1141	66	24	ab	ab	PROPN
m-1141	66	25	=	=	SYM
m-1141	66	26	ba	ba	PROPN
m-1141	66	27	.	.	PUNCT
m-1141	67	1	so	so	ADV
m-1141	67	2	we	we	PRON
m-1141	67	3	obtain	obtain	VERB
m-1141	67	4	a2	a2	PROPN
m-1141	67	5	=	=	SYM
m-1141	67	6	ab	ab	PROPN
m-1141	67	7	=	=	PUNCT
m-1141	67	8	ba	ba	PROPN
m-1141	67	9	=	=	PUNCT
m-1141	67	10	b2	b2	PROPN
m-1141	67	11	,	,	PUNCT
m-1141	67	12	and	and	CCONJ
m-1141	67	13	by	by	ADP
m-1141	67	14	quasi	quasi	NOUN
m-1141	67	15	-	-	NOUN
m-1141	67	16	separativity	separativity	NOUN
m-1141	67	17	again	again	ADV
m-1141	67	18	we	we	PRON
m-1141	67	19	have	have	VERB
m-1141	67	20	a	a	DET
m-1141	67	21	=	=	PROPN
m-1141	67	22	b.	b.	PROPN
m-1141	67	23	ijo	ijo	PROPN
m-1141	67	24	international	international	PROPN
m-1141	67	25	journal	journal	PROPN
m-1141	67	26	of	of	ADP
m-1141	67	27	mathematics	mathematics	PROPN
m-1141	67	28	(	(	PUNCT
m-1141	67	29	issn	issn	PROPN
m-1141	67	30	:	:	PUNCT
m-1141	67	31	2992	2992	NUM
m-1141	67	32	-	-	SYM
m-1141	67	33	4421	4421	NUM
m-1141	67	34	)	)	PUNCT
m-1141	67	35	volume	volume	NOUN
m-1141	67	36	08	08	NUM
m-1141	68	1	|	|	ADV
m-1141	68	2	issue	issue	NOUN
m-1141	68	3	9	9	NUM
m-1141	68	4	|	|	CCONJ
m-1141	68	5	september	september	PROPN
m-1141	68	6	2025	2025	NUM
m-1141	68	7	|	|	ADV
m-1141	68	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1141	68	9	3	3	NUM
m-1141	68	10	4	4	NUM
m-1141	68	11	semilattice	semilattice	NOUN
m-1141	68	12	of	of	ADP
m-1141	68	13	a	a	DET
m-1141	68	14	semigroup	semigroup	PROPN
m-1141	68	15	de	de	PROPN
m-1141	68	16	�	�	NOUN
m-1141	68	17	nition	nition	NOUN
m-1141	68	18	4.1	4.1	NUM
m-1141	68	19	(	(	PUNCT
m-1141	68	20	semilattice	semilattice	NOUN
m-1141	68	21	)	)	PUNCT
m-1141	68	22	.	.	PUNCT
m-1141	69	1	a	a	DET
m-1141	69	2	semilattice	semilattice	NOUN
m-1141	69	3	is	be	AUX
m-1141	69	4	a	a	DET
m-1141	69	5	commutative	commutative	ADJ
m-1141	69	6	idempotent	idempotent	NOUN
m-1141	69	7	semigroup	semigroup	NOUN
m-1141	69	8	.	.	PUNCT
m-1141	70	1	that	that	PRON
m-1141	70	2	is	is	ADV
m-1141	70	3	,	,	PUNCT
m-1141	70	4	a	a	DET
m-1141	70	5	set	set	NOUN
m-1141	70	6	y	y	PROPN
m-1141	70	7	with	with	ADP
m-1141	70	8	a	a	DET
m-1141	70	9	binary	binary	ADJ
m-1141	70	10	operation	operation	NOUN
m-1141	70	11	∧	∧	NOUN
m-1141	70	12	:	:	PUNCT
m-1141	70	13	y	y	PROPN
m-1141	70	14	×	×	NOUN
m-1141	70	15	y	y	PROPN
m-1141	70	16	→	→	PUNCT
m-1141	70	17	y	y	PROPN
m-1141	70	18	such	such	ADJ
m-1141	70	19	that	that	SCONJ
m-1141	70	20	,	,	PUNCT
m-1141	70	21	for	for	ADP
m-1141	70	22	all	all	DET
m-1141	70	23	e	e	NOUN
m-1141	70	24	,	,	PUNCT
m-1141	70	25	f	f	X
m-1141	70	26	,	,	PUNCT
m-1141	70	27	g	g	PROPN
m-1141	70	28	∈	∈	PROPN
m-1141	70	29	y	y	NOUN
m-1141	70	30	:	:	PUNCT
m-1141	70	31	1	1	X
m-1141	70	32	.	.	PUNCT
m-1141	70	33	(	(	PUNCT
m-1141	70	34	e	e	X
m-1141	70	35	∧	∧	PROPN
m-1141	70	36	f	f	NOUN
m-1141	70	37	)	)	PUNCT
m-1141	70	38	∧	∧	NOUN
m-1141	70	39	g	g	NOUN
m-1141	70	40	=	=	SYM
m-1141	70	41	e	e	X
m-1141	70	42	∧	∧	PROPN
m-1141	70	43	(	(	PUNCT
m-1141	70	44	f	f	PROPN
m-1141	70	45	∧	∧	PROPN
m-1141	70	46	g	g	PROPN
m-1141	70	47	)	)	PUNCT
m-1141	70	48	(	(	PUNCT
m-1141	70	49	associativity	associativity	NOUN
m-1141	70	50	)	)	PUNCT
m-1141	70	51	,	,	PUNCT
m-1141	70	52	2	2	X
m-1141	70	53	.	.	X
m-1141	70	54	e	e	X
m-1141	70	55	∧	∧	NOUN
m-1141	70	56	f	f	PROPN
m-1141	70	57	=	=	SYM
m-1141	70	58	f	f	PROPN
m-1141	70	59	∧	∧	PROPN
m-1141	70	60	e	e	X
m-1141	70	61	(	(	PUNCT
m-1141	70	62	commutativity	commutativity	NOUN
m-1141	70	63	)	)	PUNCT
m-1141	70	64	,	,	PUNCT
m-1141	70	65	3	3	X
m-1141	70	66	.	.	X
m-1141	70	67	e	e	X
m-1141	70	68	∧	∧	NOUN
m-1141	70	69	e	e	NOUN
m-1141	70	70	=	=	SYM
m-1141	70	71	e	e	X
m-1141	70	72	(	(	PUNCT
m-1141	70	73	idempotency	idempotency	NOUN
m-1141	70	74	)	)	PUNCT
m-1141	70	75	.	.	PUNCT
m-1141	71	1	interpretation	interpretation	NOUN
m-1141	71	2	:	:	PUNCT
m-1141	71	3	�	�	PROPN
m-1141	71	4	the	the	DET
m-1141	71	5	operation	operation	NOUN
m-1141	71	6	∧	∧	PROPN
m-1141	71	7	can	can	AUX
m-1141	71	8	be	be	AUX
m-1141	71	9	thought	think	VERB
m-1141	71	10	of	of	ADP
m-1141	71	11	as	as	ADP
m-1141	71	12	a	a	DET
m-1141	71	13	�	�	PROPN
m-1141	71	14	meet	meet	NOUN
m-1141	71	15	�	�	PROPN
m-1141	71	16	(	(	PUNCT
m-1141	71	17	greatest	greatest	ADV
m-1141	71	18	lower	lower	ADV
m-1141	71	19	bound	bind	VERB
m-1141	71	20	)	)	PUNCT
m-1141	71	21	in	in	ADP
m-1141	71	22	an	an	DET
m-1141	71	23	ordered	order	VERB
m-1141	71	24	set	set	NOUN
m-1141	71	25	.	.	PUNCT
m-1141	72	1	�	�	PROPN
m-1141	72	2	the	the	DET
m-1141	72	3	partial	partial	ADJ
m-1141	72	4	order	order	NOUN
m-1141	72	5	associated	associate	VERB
m-1141	72	6	with	with	ADP
m-1141	72	7	a	a	DET
m-1141	72	8	semilattice	semilattice	NOUN
m-1141	72	9	is	be	AUX
m-1141	72	10	given	give	VERB
m-1141	72	11	by	by	ADP
m-1141	72	12	e	e	PROPN
m-1141	72	13	≤	≤	PROPN
m-1141	72	14	f	f	X
m-1141	72	15	⇐	⇐	ADJ
m-1141	72	16	⇒	⇒	NOUN
m-1141	72	17	e	e	X
m-1141	72	18	∧	∧	PROPN
m-1141	72	19	f	f	PROPN
m-1141	72	20	=	=	PROPN
m-1141	72	21	e.	e.	PROPN
m-1141	72	22	�	�	PROPN
m-1141	72	23	thus	thus	ADV
m-1141	72	24	,	,	PUNCT
m-1141	72	25	a	a	DET
m-1141	72	26	semilattice	semilattice	NOUN
m-1141	72	27	is	be	AUX
m-1141	72	28	both	both	CCONJ
m-1141	72	29	an	an	DET
m-1141	72	30	algebraic	algebraic	ADJ
m-1141	72	31	structure	structure	NOUN
m-1141	72	32	(	(	PUNCT
m-1141	72	33	a	a	DET
m-1141	72	34	special	special	ADJ
m-1141	72	35	semigroup	semigroup	NOUN
m-1141	72	36	)	)	PUNCT
m-1141	72	37	and	and	CCONJ
m-1141	72	38	an	an	DET
m-1141	72	39	order	order	NOUN
m-1141	72	40	-	-	PUNCT
m-1141	72	41	theoretic	theoretic	NOUN
m-1141	72	42	one	one	NOUN
m-1141	72	43	(	(	PUNCT
m-1141	72	44	a	a	DET
m-1141	72	45	meet	meet	ADJ
m-1141	72	46	-	-	PUNCT
m-1141	72	47	semilattice	semilattice	NOUN
m-1141	72	48	)	)	PUNCT
m-1141	72	49	in	in	ADP
m-1141	72	50	a	a	DET
m-1141	72	51	poset	poset	NOUN
m-1141	72	52	.	.	PUNCT
m-1141	73	1	de	de	PROPN
m-1141	73	2	�	�	PROPN
m-1141	73	3	nition	nition	NOUN
m-1141	73	4	4.2	4.2	NUM
m-1141	73	5	(	(	PUNCT
m-1141	73	6	semilattice	semilattice	NOUN
m-1141	73	7	of	of	ADP
m-1141	73	8	semigroups	semigroup	NOUN
m-1141	73	9	)	)	PUNCT
m-1141	73	10	.	.	PUNCT
m-1141	74	1	let	let	VERB
m-1141	74	2	y	y	PRON
m-1141	74	3	be	be	AUX
m-1141	74	4	a	a	DET
m-1141	74	5	semilattice	semilattice	NOUN
m-1141	74	6	with	with	ADP
m-1141	74	7	operation	operation	NOUN
m-1141	74	8	∧.	∧.	NOUN
m-1141	74	9	a	a	DET
m-1141	74	10	semilattice	semilattice	NOUN
m-1141	74	11	of	of	ADP
m-1141	74	12	semigroups	semigroup	NOUN
m-1141	74	13	is	be	AUX
m-1141	74	14	a	a	DET
m-1141	74	15	semigroup	semigroup	NOUN
m-1141	74	16	s	s	VERB
m-1141	74	17	together	together	ADV
m-1141	74	18	with	with	ADP
m-1141	74	19	a	a	DET
m-1141	74	20	decomposition	decomposition	NOUN
m-1141	74	21	s	s	PART
m-1141	74	22	=	=	PUNCT
m-1141	74	23	⋃	⋃	NOUN
m-1141	74	24	e∈y	e∈y	NOUN
m-1141	74	25	se	se	NOUN
m-1141	74	26	,	,	PUNCT
m-1141	74	27	where	where	SCONJ
m-1141	74	28	each	each	DET
m-1141	74	29	se	se	PROPN
m-1141	74	30	is	be	AUX
m-1141	74	31	a	a	DET
m-1141	74	32	subsemigroup	subsemigroup	NOUN
m-1141	74	33	of	of	ADP
m-1141	74	34	s	s	PROPN
m-1141	74	35	,	,	PUNCT
m-1141	74	36	such	such	ADJ
m-1141	74	37	that	that	SCONJ
m-1141	74	38	for	for	ADP
m-1141	74	39	all	all	DET
m-1141	74	40	e	e	NOUN
m-1141	75	1	,	,	PUNCT
m-1141	75	2	f	f	PROPN
m-1141	75	3	∈	∈	PROPN
m-1141	75	4	y	y	PROPN
m-1141	75	5	:	:	PUNCT
m-1141	75	6	se	se	X
m-1141	75	7	·	·	PUNCT
m-1141	75	8	sf	sf	PROPN
m-1141	75	9	⊆	⊆	NUM
m-1141	75	10	se∧f	se∧f	NOUN
m-1141	75	11	.	.	PUNCT
m-1141	76	1	that	that	PRON
m-1141	76	2	is	is	ADV
m-1141	76	3	,	,	PUNCT
m-1141	76	4	the	the	DET
m-1141	76	5	product	product	NOUN
m-1141	76	6	of	of	ADP
m-1141	76	7	an	an	DET
m-1141	76	8	element	element	NOUN
m-1141	76	9	from	from	ADP
m-1141	76	10	se	se	PROPN
m-1141	76	11	and	and	CCONJ
m-1141	76	12	an	an	DET
m-1141	76	13	element	element	NOUN
m-1141	76	14	from	from	ADP
m-1141	76	15	sf	sf	PRON
m-1141	76	16	always	always	ADV
m-1141	76	17	lies	lie	VERB
m-1141	76	18	in	in	ADP
m-1141	76	19	the	the	DET
m-1141	76	20	same	same	ADJ
m-1141	76	21	component	component	NOUN
m-1141	76	22	corresponding	correspond	VERB
m-1141	76	23	to	to	ADP
m-1141	76	24	the	the	DET
m-1141	76	25	meet	meet	NOUN
m-1141	76	26	e	e	PROPN
m-1141	76	27	∧	∧	PROPN
m-1141	76	28	f	f	PROPN
m-1141	76	29	,	,	PUNCT
m-1141	76	30	see	see	VERB
m-1141	76	31	�	�	PROPN
m-1141	76	32	gure	gure	NOUN
m-1141	76	33	1	1	NUM
m-1141	76	34	below	below	ADV
m-1141	76	35	.	.	PUNCT
m-1141	77	1	se	se	PROPN
m-1141	78	1	sf	sf	PROPN
m-1141	78	2	se∧f	se∧f	VERB
m-1141	78	3	a	a	DET
m-1141	78	4	∈	∈	PROPN
m-1141	78	5	se	se	X
m-1141	78	6	b	b	PROPN
m-1141	78	7	∈	∈	PROPN
m-1141	78	8	sf	sf	PROPN
m-1141	78	9	ab	ab	PROPN
m-1141	78	10	∈	∈	PROPN
m-1141	78	11	se∧f	se∧f	NOUN
m-1141	78	12	figure	figure	NOUN
m-1141	78	13	1	1	NUM
m-1141	78	14	:	:	PUNCT
m-1141	78	15	product	product	NOUN
m-1141	78	16	in	in	ADP
m-1141	78	17	a	a	DET
m-1141	78	18	semilattice	semilattice	NOUN
m-1141	78	19	of	of	ADP
m-1141	78	20	semigroups	semigroup	NOUN
m-1141	78	21	:	:	PUNCT
m-1141	78	22	a	a	DET
m-1141	78	23	∈	∈	PROPN
m-1141	78	24	se	se	X
m-1141	78	25	,	,	PUNCT
m-1141	78	26	b	b	X
m-1141	78	27	∈	∈	NOUN
m-1141	78	28	sf	sf	X
m-1141	78	29	multiply	multiply	VERB
m-1141	78	30	into	into	ADP
m-1141	78	31	se∧f	se∧f	PROPN
m-1141	78	32	.	.	PUNCT
m-1141	79	1	remarks	remark	VERB
m-1141	79	2	:	:	PUNCT
m-1141	79	3	ijo	ijo	PROPN
m-1141	79	4	international	international	PROPN
m-1141	79	5	journal	journal	PROPN
m-1141	79	6	of	of	ADP
m-1141	79	7	mathematics	mathematics	PROPN
m-1141	79	8	(	(	PUNCT
m-1141	79	9	issn	issn	PROPN
m-1141	79	10	:	:	PUNCT
m-1141	79	11	2992	2992	NUM
m-1141	79	12	-	-	SYM
m-1141	79	13	4421	4421	NUM
m-1141	79	14	)	)	PUNCT
m-1141	79	15	volume	volume	NOUN
m-1141	79	16	08	08	NUM
m-1141	80	1	|	|	ADV
m-1141	80	2	issue	issue	NOUN
m-1141	80	3	9	9	NUM
m-1141	80	4	|	|	CCONJ
m-1141	80	5	september	september	PROPN
m-1141	80	6	2025	2025	NUM
m-1141	80	7	|	|	ADV
m-1141	80	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1141	80	9	4	4	NUM
m-1141	80	10	�	�	PROPN
m-1141	80	11	each	each	DET
m-1141	80	12	component	component	NOUN
m-1141	80	13	se	se	PROPN
m-1141	80	14	is	be	AUX
m-1141	80	15	itself	itself	PRON
m-1141	80	16	a	a	DET
m-1141	80	17	semigroup	semigroup	NOUN
m-1141	80	18	.	.	PUNCT
m-1141	81	1	�	�	PROPN
m-1141	82	1	the	the	DET
m-1141	82	2	semilattice	semilattice	NOUN
m-1141	82	3	y	y	PROPN
m-1141	82	4	controls	control	VERB
m-1141	82	5	how	how	SCONJ
m-1141	82	6	the	the	DET
m-1141	82	7	components	component	NOUN
m-1141	82	8	interact	interact	VERB
m-1141	82	9	with	with	ADP
m-1141	82	10	each	each	DET
m-1141	82	11	other	other	ADJ
m-1141	82	12	:	:	PUNCT
m-1141	82	13	multiplication	multiplication	NOUN
m-1141	82	14	�	�	PROPN
m-1141	82	15	descends	descend	VERB
m-1141	82	16	�	�	PROPN
m-1141	82	17	to	to	ADP
m-1141	82	18	the	the	DET
m-1141	82	19	meet	meet	NOUN
m-1141	82	20	in	in	ADP
m-1141	82	21	y	y	PROPN
m-1141	82	22	.	.	PUNCT
m-1141	83	1	�	�	PROPN
m-1141	83	2	if	if	SCONJ
m-1141	83	3	all	all	DET
m-1141	83	4	components	component	NOUN
m-1141	83	5	se	se	X
m-1141	83	6	are	be	AUX
m-1141	83	7	cancellative	cancellative	ADJ
m-1141	83	8	semigroups	semigroup	NOUN
m-1141	83	9	,	,	PUNCT
m-1141	83	10	then	then	ADV
m-1141	83	11	s	s	VERB
m-1141	83	12	is	be	AUX
m-1141	83	13	called	call	VERB
m-1141	83	14	a	a	DET
m-1141	83	15	semilattice	semilattice	NOUN
m-1141	83	16	of	of	ADP
m-1141	83	17	cancellative	cancellative	ADJ
m-1141	83	18	semigroups	semigroup	NOUN
m-1141	83	19	.	.	PUNCT
m-1141	84	1	de	de	PROPN
m-1141	84	2	�	�	PROPN
m-1141	84	3	nition	nition	NOUN
m-1141	84	4	4.3	4.3	NUM
m-1141	84	5	(	(	PUNCT
m-1141	84	6	green	green	PROPN
m-1141	84	7	's	's	PART
m-1141	84	8	relations	relation	NOUN
m-1141	84	9	)	)	PUNCT
m-1141	84	10	.	.	PUNCT
m-1141	85	1	let	let	VERB
m-1141	85	2	s	s	PRON
m-1141	85	3	be	be	AUX
m-1141	85	4	a	a	DET
m-1141	85	5	semigroup	semigroup	NOUN
m-1141	85	6	.	.	PUNCT
m-1141	86	1	green	green	PROPN
m-1141	86	2	's	's	PART
m-1141	86	3	relations	relation	NOUN
m-1141	86	4	are	be	AUX
m-1141	86	5	the	the	DET
m-1141	86	6	equivalence	equivalence	NOUN
m-1141	86	7	relations	relation	NOUN
m-1141	86	8	l	l	PROPN
m-1141	86	9	,	,	PUNCT
m-1141	86	10	r	r	PROPN
m-1141	86	11	,	,	PUNCT
m-1141	86	12	j	j	PROPN
m-1141	86	13	,	,	PUNCT
m-1141	86	14	h	h	PROPN
m-1141	86	15	,	,	PUNCT
m-1141	86	16	d	d	PROPN
m-1141	86	17	de	de	X
m-1141	86	18	�	�	PROPN
m-1141	86	19	ned	ne	VERB
m-1141	86	20	on	on	ADP
m-1141	86	21	s	s	PRON
m-1141	86	22	as	as	SCONJ
m-1141	86	23	follows	follow	VERB
m-1141	86	24	:	:	PUNCT
m-1141	86	25	�	�	PROPN
m-1141	86	26	the	the	DET
m-1141	86	27	l	l	NOUN
m-1141	86	28	-	-	NOUN
m-1141	86	29	relation	relation	NOUN
m-1141	86	30	:	:	PUNCT
m-1141	86	31	for	for	ADP
m-1141	86	32	a	a	DET
m-1141	86	33	,	,	PUNCT
m-1141	86	34	b	b	PROPN
m-1141	86	35	∈	∈	PROPN
m-1141	86	36	s	s	PROPN
m-1141	86	37	,	,	PUNCT
m-1141	86	38	al	al	PROPN
m-1141	86	39	b	b	PROPN
m-1141	86	40	⇐	⇐	PROPN
m-1141	86	41	⇒	⇒	PROPN
m-1141	86	42	s1a	s1a	PUNCT
m-1141	86	43	=	=	SYM
m-1141	86	44	s1b	s1b	PROPN
m-1141	86	45	,	,	PUNCT
m-1141	86	46	that	that	ADV
m-1141	86	47	is	is	ADV
m-1141	86	48	,	,	PUNCT
m-1141	86	49	a	a	PRON
m-1141	86	50	and	and	CCONJ
m-1141	86	51	b	b	NOUN
m-1141	86	52	generate	generate	VERB
m-1141	86	53	the	the	DET
m-1141	86	54	same	same	ADJ
m-1141	86	55	principal	principal	NOUN
m-1141	86	56	left	leave	VERB
m-1141	86	57	ideal	ideal	ADJ
m-1141	86	58	.	.	PUNCT
m-1141	87	1	here	here	ADV
m-1141	87	2	s1	s1	PROPN
m-1141	87	3	denotes	denote	VERB
m-1141	87	4	s	s	NOUN
m-1141	87	5	with	with	ADP
m-1141	87	6	identity	identity	NOUN
m-1141	87	7	adjoined	adjoin	VERB
m-1141	87	8	if	if	SCONJ
m-1141	87	9	necessary	necessary	ADJ
m-1141	87	10	.	.	PUNCT
m-1141	88	1	�	�	PROPN
m-1141	88	2	the	the	DET
m-1141	88	3	r	r	NOUN
m-1141	88	4	-	-	PUNCT
m-1141	88	5	relation	relation	NOUN
m-1141	88	6	:	:	PUNCT
m-1141	88	7	for	for	ADP
m-1141	88	8	a	a	DET
m-1141	88	9	,	,	PUNCT
m-1141	88	10	b	b	PROPN
m-1141	88	11	∈	∈	PROPN
m-1141	88	12	s	s	PROPN
m-1141	88	13	,	,	PUNCT
m-1141	88	14	a	a	DET
m-1141	88	15	r	r	NOUN
m-1141	88	16	b	b	NOUN
m-1141	88	17	⇐	⇐	ADJ
m-1141	88	18	⇒	⇒	PROPN
m-1141	88	19	as1	as1	PROPN
m-1141	88	20	=	=	SYM
m-1141	88	21	bs1	bs1	PROPN
m-1141	88	22	,	,	PUNCT
m-1141	88	23	that	that	ADV
m-1141	88	24	is	is	ADV
m-1141	88	25	,	,	PUNCT
m-1141	88	26	a	a	PRON
m-1141	88	27	and	and	CCONJ
m-1141	88	28	b	b	NOUN
m-1141	88	29	generate	generate	VERB
m-1141	88	30	the	the	DET
m-1141	88	31	same	same	ADJ
m-1141	88	32	principal	principal	ADJ
m-1141	88	33	right	right	ADJ
m-1141	88	34	ideal	ideal	NOUN
m-1141	88	35	.	.	PUNCT
m-1141	89	1	�	�	PROPN
m-1141	89	2	the	the	DET
m-1141	89	3	j	j	PROPN
m-1141	89	4	-relation	-relation	PROPN
m-1141	89	5	:	:	PUNCT
m-1141	89	6	for	for	ADP
m-1141	89	7	a	a	DET
m-1141	89	8	,	,	PUNCT
m-1141	89	9	b	b	PROPN
m-1141	89	10	∈	∈	PROPN
m-1141	89	11	s	s	PROPN
m-1141	89	12	,	,	PUNCT
m-1141	89	13	a	a	DET
m-1141	89	14	j	j	PROPN
m-1141	89	15	b	b	SYM
m-1141	89	16	⇐	⇐	ADJ
m-1141	89	17	⇒	⇒	PROPN
m-1141	89	18	s1as1	s1as1	NOUN
m-1141	89	19	=	=	SYM
m-1141	89	20	s1bs1	s1bs1	NOUN
m-1141	89	21	,	,	PUNCT
m-1141	89	22	that	that	ADV
m-1141	89	23	is	is	ADV
m-1141	89	24	,	,	PUNCT
m-1141	89	25	a	a	PRON
m-1141	89	26	and	and	CCONJ
m-1141	89	27	b	b	NOUN
m-1141	89	28	generate	generate	VERB
m-1141	89	29	the	the	DET
m-1141	89	30	same	same	ADJ
m-1141	89	31	two	two	NUM
m-1141	89	32	-	-	PUNCT
m-1141	89	33	sided	sided	ADJ
m-1141	89	34	ideal	ideal	NOUN
m-1141	89	35	.	.	PUNCT
m-1141	90	1	�	�	PROPN
m-1141	90	2	the	the	DET
m-1141	90	3	h	h	NOUN
m-1141	90	4	-	-	PUNCT
m-1141	90	5	relation	relation	NOUN
m-1141	90	6	:	:	PUNCT
m-1141	91	1	h	h	NOUN
m-1141	91	2	=	=	SYM
m-1141	91	3	l	l	NOUN
m-1141	91	4	∩r	∩r	NOUN
m-1141	91	5	,	,	PUNCT
m-1141	91	6	that	that	ADV
m-1141	91	7	is	is	ADV
m-1141	91	8	,	,	PUNCT
m-1141	91	9	a	a	DET
m-1141	91	10	h	h	NOUN
m-1141	91	11	b	b	NOUN
m-1141	91	12	if	if	SCONJ
m-1141	92	1	and	and	CCONJ
m-1141	92	2	only	only	ADV
m-1141	92	3	if	if	SCONJ
m-1141	92	4	al	al	PROPN
m-1141	92	5	b	b	PROPN
m-1141	92	6	and	and	CCONJ
m-1141	92	7	ar	ar	PROPN
m-1141	92	8	b.	b.	PROPN
m-1141	92	9	�	�	PROPN
m-1141	92	10	the	the	DET
m-1141	92	11	d	d	NOUN
m-1141	92	12	-	-	NOUN
m-1141	92	13	relation	relation	NOUN
m-1141	92	14	:	:	PUNCT
m-1141	93	1	d	d	X
m-1141	93	2	=	=	PUNCT
m-1141	93	3	l	l	NOUN
m-1141	93	4	◦	◦	NOUN
m-1141	93	5	r	r	NOUN
m-1141	93	6	=	=	SYM
m-1141	93	7	r	r	NOUN
m-1141	93	8	◦	◦	NOUN
m-1141	93	9	l	l	NOUN
m-1141	93	10	,	,	PUNCT
m-1141	93	11	that	that	ADV
m-1141	93	12	is	is	ADV
m-1141	93	13	,	,	PUNCT
m-1141	93	14	a	a	DET
m-1141	93	15	d	d	X
m-1141	93	16	b	b	NOUN
m-1141	93	17	if	if	SCONJ
m-1141	93	18	and	and	CCONJ
m-1141	93	19	only	only	ADV
m-1141	93	20	if	if	SCONJ
m-1141	93	21	there	there	PRON
m-1141	93	22	exists	exist	VERB
m-1141	93	23	c	c	NOUN
m-1141	93	24	∈	∈	PROPN
m-1141	93	25	s	s	VERB
m-1141	93	26	such	such	ADJ
m-1141	93	27	that	that	SCONJ
m-1141	93	28	a	a	DET
m-1141	93	29	l	l	NOUN
m-1141	93	30	c	c	NOUN
m-1141	93	31	and	and	CCONJ
m-1141	93	32	c	c	PROPN
m-1141	93	33	r	r	PROPN
m-1141	93	34	b.	b.	PROPN
m-1141	93	35	de	de	PROPN
m-1141	93	36	�	�	PROPN
m-1141	93	37	nition	nition	NOUN
m-1141	93	38	4.4	4.4	NUM
m-1141	93	39	(	(	PUNCT
m-1141	93	40	associated	associate	VERB
m-1141	93	41	preorders	preorder	NOUN
m-1141	93	42	)	)	PUNCT
m-1141	93	43	.	.	PUNCT
m-1141	94	1	besides	besides	SCONJ
m-1141	94	2	the	the	DET
m-1141	94	3	equivalence	equivalence	NOUN
m-1141	94	4	relations	relation	NOUN
m-1141	94	5	,	,	PUNCT
m-1141	94	6	one	one	PRON
m-1141	94	7	often	often	ADV
m-1141	94	8	considers	consider	VERB
m-1141	94	9	the	the	DET
m-1141	94	10	following	follow	VERB
m-1141	94	11	preorders	preorder	NOUN
m-1141	94	12	:	:	PUNCT
m-1141	94	13	�	�	PROPN
m-1141	94	14	the	the	DET
m-1141	94	15	≤l	≤l	NOUN
m-1141	94	16	-	-	PUNCT
m-1141	94	17	relation	relation	NOUN
m-1141	94	18	:	:	PUNCT
m-1141	94	19	for	for	ADP
m-1141	94	20	a	a	DET
m-1141	94	21	,	,	PUNCT
m-1141	94	22	b	b	PROPN
m-1141	94	23	∈	∈	PROPN
m-1141	94	24	s	s	PROPN
m-1141	94	25	,	,	PUNCT
m-1141	94	26	a	a	DET
m-1141	94	27	≤l	≤l	PROPN
m-1141	94	28	b	b	SYM
m-1141	94	29	⇐	⇐	PROPN
m-1141	94	30	⇒	⇒	NOUN
m-1141	94	31	s1a	s1a	VERB
m-1141	94	32	⊆	⊆	NUM
m-1141	94	33	s1b	s1b	PROPN
m-1141	94	34	.	.	PUNCT
m-1141	94	35	�	�	PROPN
m-1141	94	36	the	the	DET
m-1141	94	37	≤r	≤r	PROPN
m-1141	94	38	-	-	NOUN
m-1141	94	39	relation	relation	NOUN
m-1141	94	40	:	:	PUNCT
m-1141	94	41	for	for	ADP
m-1141	94	42	a	a	DET
m-1141	94	43	,	,	PUNCT
m-1141	94	44	b	b	PROPN
m-1141	94	45	∈	∈	PROPN
m-1141	94	46	s	s	PROPN
m-1141	94	47	,	,	PUNCT
m-1141	94	48	a	a	DET
m-1141	94	49	≤r	≤r	PROPN
m-1141	94	50	b	b	X
m-1141	94	51	⇐	⇐	ADJ
m-1141	94	52	⇒	⇒	PROPN
m-1141	94	53	as1	as1	NOUN
m-1141	94	54	⊆	⊆	NUM
m-1141	94	55	bs1	bs1	NOUN
m-1141	94	56	.	.	PUNCT
m-1141	95	1	ijo	ijo	PROPN
m-1141	95	2	international	international	PROPN
m-1141	95	3	journal	journal	PROPN
m-1141	95	4	of	of	ADP
m-1141	95	5	mathematics	mathematics	PROPN
m-1141	95	6	(	(	PUNCT
m-1141	95	7	issn	issn	PROPN
m-1141	95	8	:	:	PUNCT
m-1141	95	9	2992	2992	NUM
m-1141	95	10	-	-	SYM
m-1141	95	11	4421	4421	NUM
m-1141	95	12	)	)	PUNCT
m-1141	95	13	volume	volume	NOUN
m-1141	95	14	08	08	NUM
m-1141	96	1	|	|	ADV
m-1141	96	2	issue	issue	NOUN
m-1141	96	3	9	9	NUM
m-1141	96	4	|	|	CCONJ
m-1141	96	5	september	september	PROPN
m-1141	96	6	2025	2025	NUM
m-1141	96	7	|	|	ADV
m-1141	96	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1141	96	9	5	5	NUM
m-1141	96	10	�	�	PROPN
m-1141	96	11	the	the	DET
m-1141	96	12	≤j	≤j	NOUN
m-1141	96	13	-relation	-relation	NOUN
m-1141	96	14	:	:	PUNCT
m-1141	96	15	for	for	ADP
m-1141	96	16	a	a	DET
m-1141	96	17	,	,	PUNCT
m-1141	96	18	b	b	PROPN
m-1141	96	19	∈	∈	PROPN
m-1141	96	20	s	s	PROPN
m-1141	96	21	,	,	PUNCT
m-1141	96	22	a	a	DET
m-1141	96	23	≤j	≤j	NOUN
m-1141	96	24	b	b	NOUN
m-1141	96	25	⇐	⇐	ADJ
m-1141	96	26	⇒	⇒	PROPN
m-1141	96	27	s1as1	s1as1	NOUN
m-1141	96	28	⊆	⊆	NUM
m-1141	96	29	s1bs1	s1bs1	NOUN
m-1141	96	30	.	.	PUNCT
m-1141	97	1	for	for	ADP
m-1141	97	2	more	more	ADJ
m-1141	97	3	clearity	clearity	NOUN
m-1141	97	4	,	,	PUNCT
m-1141	97	5	reader	reader	NOUN
m-1141	97	6	should	should	AUX
m-1141	97	7	see	see	VERB
m-1141	97	8	[	[	X
m-1141	97	9	3	3	NUM
m-1141	97	10	,	,	PUNCT
m-1141	97	11	8	8	NUM
m-1141	97	12	,	,	PUNCT
m-1141	97	13	10,17,19	10,17,19	NUM
m-1141	97	14	�	�	NOUN
m-1141	97	15	21	21	NUM
m-1141	97	16	]	]	PUNCT
m-1141	97	17	theorem	theorem	VERB
m-1141	97	18	4.1	4.1	NUM
m-1141	97	19	.	.	PUNCT
m-1141	98	1	let	let	VERB
m-1141	98	2	s	s	PRON
m-1141	98	3	be	be	AUX
m-1141	98	4	a	a	DET
m-1141	98	5	semigroup	semigroup	NOUN
m-1141	98	6	.	.	PUNCT
m-1141	99	1	then	then	ADV
m-1141	99	2	d	d	X
m-1141	99	3	=	=	PUNCT
m-1141	99	4	j	j	PROPN
m-1141	99	5	if	if	SCONJ
m-1141	99	6	and	and	CCONJ
m-1141	99	7	only	only	ADV
m-1141	99	8	if	if	SCONJ
m-1141	99	9	for	for	ADP
m-1141	99	10	every	every	DET
m-1141	99	11	a	a	PROPN
m-1141	99	12	,	,	PUNCT
m-1141	99	13	b	b	PROPN
m-1141	99	14	∈	∈	PROPN
m-1141	99	15	s	s	NOUN
m-1141	99	16	,	,	PUNCT
m-1141	99	17	sas	sas	X
m-1141	99	18	=	=	SYM
m-1141	99	19	sbs	sbs	NOUN
m-1141	99	20	⇒	⇒	VERB
m-1141	99	21	∃x	∃x	PROPN
m-1141	99	22	∈	∈	PROPN
m-1141	99	23	s	s	VERB
m-1141	99	24	such	such	ADJ
m-1141	99	25	that	that	SCONJ
m-1141	99	26	alxr	alxr	PROPN
m-1141	99	27	b.	b.	PROPN
m-1141	99	28	proof	proof	PROPN
m-1141	99	29	.	.	PUNCT
m-1141	100	1	(	(	PUNCT
m-1141	100	2	⇒	⇒	NOUN
m-1141	100	3	):	):	PUNCT
m-1141	100	4	suppose	suppose	VERB
m-1141	100	5	d	d	X
m-1141	100	6	=	=	SYM
m-1141	100	7	j	j	PROPN
m-1141	100	8	.	.	PUNCT
m-1141	101	1	let	let	VERB
m-1141	101	2	a	a	DET
m-1141	101	3	,	,	PUNCT
m-1141	101	4	b	b	X
m-1141	101	5	∈	∈	NOUN
m-1141	101	6	s	s	X
m-1141	101	7	with	with	ADP
m-1141	101	8	sas	sas	PROPN
m-1141	101	9	=	=	SYM
m-1141	101	10	sbs	sbs	PROPN
m-1141	101	11	,	,	PUNCT
m-1141	101	12	i.e.	i.e.	X
m-1141	101	13	aj	aj	PROPN
m-1141	101	14	b.	b.	PROPN
m-1141	101	15	since	since	SCONJ
m-1141	101	16	d	d	PROPN
m-1141	101	17	=	=	SYM
m-1141	101	18	j	j	PROPN
m-1141	101	19	,	,	PUNCT
m-1141	101	20	this	this	PRON
m-1141	101	21	implies	imply	VERB
m-1141	101	22	ad	ad	NOUN
m-1141	101	23	b.	b.	PROPN
m-1141	101	24	by	by	ADP
m-1141	101	25	the	the	DET
m-1141	101	26	de	de	PROPN
m-1141	101	27	�	�	PROPN
m-1141	101	28	nition	nition	NOUN
m-1141	101	29	of	of	ADP
m-1141	101	30	d	d	PROPN
m-1141	101	31	,	,	PUNCT
m-1141	101	32	there	there	PRON
m-1141	101	33	exists	exist	VERB
m-1141	101	34	x	x	X
m-1141	101	35	∈	∈	PROPN
m-1141	101	36	s	s	VERB
m-1141	101	37	such	such	ADJ
m-1141	101	38	that	that	SCONJ
m-1141	101	39	alx	alx	PROPN
m-1141	101	40	and	and	CCONJ
m-1141	101	41	xr	xr	PROPN
m-1141	101	42	b.	b.	PROPN
m-1141	102	1	(	(	PUNCT
m-1141	102	2	⇐	⇐	PROPN
m-1141	102	3	):	):	PUNCT
m-1141	102	4	suppose	suppose	VERB
m-1141	102	5	the	the	DET
m-1141	102	6	stated	state	VERB
m-1141	102	7	condition	condition	NOUN
m-1141	102	8	holds	hold	VERB
m-1141	102	9	.	.	PUNCT
m-1141	103	1	take	take	VERB
m-1141	103	2	any	any	DET
m-1141	103	3	a	a	NOUN
m-1141	103	4	,	,	PUNCT
m-1141	103	5	b	b	X
m-1141	103	6	∈	∈	NOUN
m-1141	103	7	s	s	VERB
m-1141	103	8	with	with	ADP
m-1141	103	9	aj	aj	PROPN
m-1141	103	10	b	b	PROPN
m-1141	103	11	,	,	PUNCT
m-1141	103	12	i.e.	i.e.	X
m-1141	103	13	sas	sas	PROPN
m-1141	103	14	=	=	SYM
m-1141	103	15	sbs	sbs	NOUN
m-1141	103	16	.	.	PUNCT
m-1141	104	1	by	by	ADP
m-1141	104	2	hypothesis	hypothesis	NOUN
m-1141	104	3	,	,	PUNCT
m-1141	104	4	there	there	PRON
m-1141	104	5	exists	exist	VERB
m-1141	104	6	x	x	X
m-1141	104	7	∈	∈	PROPN
m-1141	104	8	s	s	PART
m-1141	104	9	with	with	ADP
m-1141	104	10	alxr	alxr	PROPN
m-1141	104	11	b.	b.	PROPN
m-1141	104	12	hence	hence	ADV
m-1141	104	13	ad	ad	PROPN
m-1141	104	14	b.	b.	PROPN
m-1141	104	15	since	since	SCONJ
m-1141	104	16	a	a	DET
m-1141	104	17	,	,	PUNCT
m-1141	104	18	b	b	X
m-1141	104	19	∈	∈	NOUN
m-1141	104	20	s	s	VERB
m-1141	104	21	were	be	AUX
m-1141	104	22	arbitrary	arbitrary	ADJ
m-1141	104	23	,	,	PUNCT
m-1141	104	24	this	this	PRON
m-1141	104	25	shows	show	VERB
m-1141	104	26	j	j	PROPN
m-1141	104	27	⊆	⊆	NUM
m-1141	104	28	d.	d.	PROPN
m-1141	104	29	but	but	CCONJ
m-1141	104	30	in	in	ADP
m-1141	104	31	any	any	DET
m-1141	104	32	semigroup	semigroup	NOUN
m-1141	104	33	it	it	PRON
m-1141	104	34	is	be	AUX
m-1141	104	35	always	always	ADV
m-1141	104	36	true	true	ADJ
m-1141	104	37	that	that	SCONJ
m-1141	104	38	d	d	PROPN
m-1141	104	39	⊆	⊆	NUM
m-1141	104	40	j	j	NOUN
m-1141	104	41	(	(	PUNCT
m-1141	104	42	see	see	VERB
m-1141	104	43	[	[	X
m-1141	104	44	1	1	NUM
m-1141	104	45	,	,	PUNCT
m-1141	104	46	18,19	18,19	NUM
m-1141	104	47	]	]	PUNCT
m-1141	104	48	)	)	PUNCT
m-1141	104	49	.	.	PUNCT
m-1141	105	1	therefore	therefore	ADV
m-1141	105	2	,	,	PUNCT
m-1141	105	3	d	d	PROPN
m-1141	105	4	=	=	SYM
m-1141	105	5	j	j	PROPN
m-1141	105	6	.	.	PUNCT
m-1141	105	7	5	5	NUM
m-1141	105	8	results	result	NOUN
m-1141	105	9	de	de	NOUN
m-1141	105	10	�	�	NOUN
m-1141	105	11	nition	nition	NOUN
m-1141	105	12	5.1	5.1	NUM
m-1141	105	13	(	(	PUNCT
m-1141	105	14	koch	koch	PROPN
m-1141	105	15	�	�	PROPN
m-1141	105	16	wallace	wallace	PROPN
m-1141	105	17	stability	stability	NOUN
m-1141	105	18	[	[	X
m-1141	105	19	16	16	NUM
m-1141	105	20	]	]	PUNCT
m-1141	105	21	)	)	PUNCT
m-1141	105	22	.	.	PUNCT
m-1141	106	1	a	a	DET
m-1141	106	2	semigroup	semigroup	NOUN
m-1141	106	3	s	s	VERB
m-1141	106	4	is	be	AUX
m-1141	106	5	called	call	VERB
m-1141	106	6	stable	stable	ADJ
m-1141	106	7	if	if	SCONJ
m-1141	106	8	for	for	ADP
m-1141	106	9	all	all	DET
m-1141	106	10	a	a	PRON
m-1141	106	11	,	,	PUNCT
m-1141	106	12	b	b	X
m-1141	106	13	∈	∈	PROPN
m-1141	106	14	s	s	PART
m-1141	106	15	:	:	PUNCT
m-1141	106	16	1	1	NUM
m-1141	106	17	.	.	PUNCT
m-1141	106	18	(	(	PUNCT
m-1141	106	19	right	right	ADJ
m-1141	106	20	stability	stability	NOUN
m-1141	106	21	):	):	PUNCT
m-1141	106	22	as	as	ADP
m-1141	106	23	⊆	⊆	NUM
m-1141	106	24	abs	ab	NOUN
m-1141	106	25	=	=	NOUN
m-1141	106	26	⇒	⇒	NOUN
m-1141	106	27	as	as	ADP
m-1141	106	28	=	=	NOUN
m-1141	106	29	abs	ab	NOUN
m-1141	106	30	,	,	PUNCT
m-1141	106	31	and	and	CCONJ
m-1141	106	32	2	2	X
m-1141	106	33	.	.	PUNCT
m-1141	107	1	(	(	PUNCT
m-1141	107	2	left	leave	VERB
m-1141	107	3	stability	stability	NOUN
m-1141	107	4	):	):	PUNCT
m-1141	107	5	sa	sa	PROPN
m-1141	107	6	⊆	⊆	NUM
m-1141	107	7	sab	sab	ADJ
m-1141	107	8	=	=	NOUN
m-1141	107	9	⇒	⇒	NOUN
m-1141	107	10	sa	sa	X
m-1141	108	1	=	=	PUNCT
m-1141	108	2	sab	sab	PROPN
m-1141	108	3	.	.	PUNCT
m-1141	109	1	this	this	DET
m-1141	109	2	de	de	PROPN
m-1141	109	3	�	�	PROPN
m-1141	109	4	nition	nition	NOUN
m-1141	109	5	was	be	AUX
m-1141	109	6	also	also	ADV
m-1141	109	7	given	give	VERB
m-1141	109	8	by	by	ADP
m-1141	109	9	east	east	NOUN
m-1141	109	10	using	use	VERB
m-1141	109	11	green	green	PROPN
m-1141	109	12	's	's	PART
m-1141	109	13	relation	relation	NOUN
m-1141	109	14	in	in	ADP
m-1141	109	15	[	[	X
m-1141	109	16	9	9	NUM
m-1141	109	17	]	]	PUNCT
m-1141	109	18	.	.	PUNCT
m-1141	110	1	equivalently	equivalently	ADV
m-1141	110	2	,	,	PUNCT
m-1141	110	3	if	if	SCONJ
m-1141	110	4	a	a	DET
m-1141	110	5	≤j	≤j	PROPN
m-1141	110	6	ab	ab	PROPN
m-1141	110	7	,	,	PUNCT
m-1141	110	8	then	then	ADV
m-1141	110	9	arab	arab	PROPN
m-1141	110	10	,	,	PUNCT
m-1141	110	11	and	and	CCONJ
m-1141	110	12	if	if	SCONJ
m-1141	110	13	a	a	DET
m-1141	110	14	≤j	≤j	PROPN
m-1141	110	15	ba	ba	PROPN
m-1141	110	16	,	,	PUNCT
m-1141	110	17	then	then	ADV
m-1141	110	18	alba	alba	PROPN
m-1141	110	19	.	.	PUNCT
m-1141	111	1	it	it	PRON
m-1141	111	2	is	be	AUX
m-1141	111	3	now	now	ADV
m-1141	111	4	necessary	necessary	ADJ
m-1141	111	5	to	to	PART
m-1141	111	6	establish	establish	VERB
m-1141	111	7	some	some	DET
m-1141	111	8	useful	useful	ADJ
m-1141	111	9	relationship	relationship	NOUN
m-1141	111	10	between	between	ADP
m-1141	111	11	stable	stable	ADJ
m-1141	111	12	and	and	CCONJ
m-1141	111	13	separative	separative	NOUN
m-1141	111	14	semigroups	semigroup	NOUN
m-1141	111	15	.	.	PUNCT
m-1141	112	1	this	this	PRON
m-1141	112	2	is	be	AUX
m-1141	112	3	achieved	achieve	VERB
m-1141	112	4	through	through	ADP
m-1141	112	5	the	the	DET
m-1141	112	6	following	follow	VERB
m-1141	112	7	propositions	proposition	NOUN
m-1141	112	8	and	and	CCONJ
m-1141	112	9	theorems	theorem	NOUN
m-1141	112	10	.	.	PUNCT
m-1141	113	1	proposition	proposition	NOUN
m-1141	113	2	5.1	5.1	NUM
m-1141	113	3	.	.	PUNCT
m-1141	114	1	let	let	VERB
m-1141	114	2	s	s	PRON
m-1141	114	3	=	=	PUNCT
m-1141	114	4	⋃	⋃	NOUN
m-1141	114	5	e∈y	e∈y	NOUN
m-1141	114	6	se	se	X
m-1141	114	7	be	be	AUX
m-1141	114	8	a	a	DET
m-1141	114	9	semilattice	semilattice	NOUN
m-1141	114	10	(	(	PUNCT
m-1141	114	11	with	with	ADP
m-1141	114	12	meet	meet	VERB
m-1141	114	13	∧	∧	PROPN
m-1141	114	14	)	)	PUNCT
m-1141	114	15	of	of	ADP
m-1141	114	16	cancellative	cancellative	ADJ
m-1141	114	17	semigroups	semigroup	NOUN
m-1141	114	18	se	se	X
m-1141	114	19	,	,	PUNCT
m-1141	114	20	so	so	SCONJ
m-1141	114	21	that	that	PRON
m-1141	114	22	sesf	sesf	VERB
m-1141	114	23	⊆	⊆	NUM
m-1141	114	24	s	s	NOUN
m-1141	114	25	e∧f	e∧f	NOUN
m-1141	114	26	for	for	ADP
m-1141	114	27	all	all	DET
m-1141	114	28	e	e	NOUN
m-1141	114	29	,	,	PUNCT
m-1141	115	1	f	f	PROPN
m-1141	115	2	∈	∈	PROPN
m-1141	115	3	y.	y.	NOUN
m-1141	116	1	then	then	ADV
m-1141	116	2	s	s	VERB
m-1141	116	3	is	be	AUX
m-1141	116	4	separative	separative	NOUN
m-1141	116	5	.	.	PUNCT
m-1141	117	1	proof	proof	NOUN
m-1141	117	2	.	.	PUNCT
m-1141	118	1	let	let	VERB
m-1141	118	2	a	a	DET
m-1141	118	3	,	,	PUNCT
m-1141	118	4	b	b	X
m-1141	118	5	∈	∈	PROPN
m-1141	118	6	s	s	NOUN
m-1141	118	7	,	,	PUNCT
m-1141	118	8	suppose	suppose	VERB
m-1141	118	9	a2	a2	PROPN
m-1141	118	10	=	=	PROPN
m-1141	118	11	ab	ab	PROPN
m-1141	118	12	and	and	CCONJ
m-1141	118	13	ba	ba	PROPN
m-1141	118	14	=	=	PUNCT
m-1141	118	15	b2	b2	PROPN
m-1141	118	16	.	.	PUNCT
m-1141	119	1	let	let	VERB
m-1141	119	2	a	a	DET
m-1141	119	3	∈	∈	PROPN
m-1141	119	4	se	se	X
m-1141	119	5	,	,	PUNCT
m-1141	119	6	b	b	X
m-1141	119	7	∈	∈	PROPN
m-1141	119	8	sf	sf	NOUN
m-1141	119	9	.	.	PUNCT
m-1141	120	1	since	since	SCONJ
m-1141	120	2	a	a	DET
m-1141	120	3	2	2	NUM
m-1141	120	4	∈	∈	NOUN
m-1141	120	5	se	se	X
m-1141	120	6	but	but	CCONJ
m-1141	120	7	ab	ab	PROPN
m-1141	120	8	∈	∈	PROPN
m-1141	120	9	se∧f	se∧f	PROPN
m-1141	120	10	,	,	PUNCT
m-1141	120	11	the	the	DET
m-1141	120	12	equality	equality	NOUN
m-1141	120	13	a	a	DET
m-1141	120	14	2	2	NUM
m-1141	120	15	=	=	SYM
m-1141	120	16	ab	ab	PROPN
m-1141	120	17	forces	force	NOUN
m-1141	120	18	e	e	NOUN
m-1141	120	19	=	=	PUNCT
m-1141	120	20	e∧	e∧	PROPN
m-1141	120	21	f	f	NOUN
m-1141	120	22	and	and	CCONJ
m-1141	120	23	so	so	ADV
m-1141	120	24	e	e	PROPN
m-1141	120	25	≤	≤	NUM
m-1141	120	26	f	f	X
m-1141	120	27	.	.	PUNCT
m-1141	121	1	likewise	likewise	ADV
m-1141	121	2	,	,	PUNCT
m-1141	121	3	b2	b2	NOUN
m-1141	121	4	∈	∈	PROPN
m-1141	121	5	sf	sf	PROPN
m-1141	121	6	and	and	CCONJ
m-1141	121	7	ba	ba	PROPN
m-1141	121	8	∈	∈	PROPN
m-1141	121	9	se∧f	se∧f	VERB
m-1141	121	10	then	then	ADV
m-1141	121	11	b2	b2	PROPN
m-1141	121	12	=	=	SYM
m-1141	121	13	ba	ba	PROPN
m-1141	121	14	forces	force	NOUN
m-1141	121	15	f	f	PROPN
m-1141	121	16	≤	≤	PROPN
m-1141	121	17	e.	e.	PROPN
m-1141	121	18	hence	hence	PROPN
m-1141	121	19	,	,	PUNCT
m-1141	121	20	e	e	PROPN
m-1141	121	21	=	=	SYM
m-1141	121	22	f	f	PROPN
m-1141	121	23	,	,	PUNCT
m-1141	121	24	and	and	CCONJ
m-1141	121	25	so	so	ADV
m-1141	121	26	a	a	PRON
m-1141	121	27	,	,	PUNCT
m-1141	122	1	b	b	DET
m-1141	122	2	lie	lie	NOUN
m-1141	122	3	in	in	ADP
m-1141	122	4	the	the	DET
m-1141	122	5	same	same	ADJ
m-1141	122	6	cancellative	cancellative	ADJ
m-1141	122	7	component	component	NOUN
m-1141	122	8	se	se	X
m-1141	122	9	.	.	PROPN
m-1141	122	10	from	from	ADP
m-1141	122	11	a2	a2	PROPN
m-1141	122	12	=	=	SYM
m-1141	122	13	ab	ab	PROPN
m-1141	122	14	,	,	PUNCT
m-1141	122	15	we	we	PRON
m-1141	122	16	have	have	VERB
m-1141	122	17	aa	aa	NOUN
m-1141	122	18	=	=	PUNCT
m-1141	122	19	ab	ab	PROPN
m-1141	122	20	=	=	NOUN
m-1141	122	21	⇒	⇒	VERB
m-1141	122	22	a	a	DET
m-1141	122	23	=	=	X
m-1141	122	24	b.	b.	PROPN
m-1141	122	25	hus	hus	PROPN
m-1141	122	26	,	,	PUNCT
m-1141	122	27	the	the	DET
m-1141	122	28	�	�	PROPN
m-1141	122	29	rst	rst	PROPN
m-1141	122	30	separative	separative	PROPN
m-1141	122	31	implication	implication	NOUN
m-1141	122	32	holds	hold	VERB
m-1141	122	33	.	.	PUNCT
m-1141	123	1	the	the	DET
m-1141	123	2	second	second	ADJ
m-1141	123	3	(	(	PUNCT
m-1141	123	4	symmetric	symmetric	ADJ
m-1141	123	5	)	)	PUNCT
m-1141	123	6	implication	implication	NOUN
m-1141	123	7	is	be	AUX
m-1141	123	8	identical	identical	ADJ
m-1141	123	9	in	in	ADP
m-1141	123	10	structure	structure	NOUN
m-1141	123	11	.	.	PUNCT
m-1141	124	1	therefore	therefore	ADV
m-1141	124	2	,	,	PUNCT
m-1141	124	3	s	s	PROPN
m-1141	124	4	satis	satis	NOUN
m-1141	124	5	�	�	PROPN
m-1141	124	6	es	es	ADP
m-1141	124	7	the	the	DET
m-1141	124	8	separativity	separativity	NOUN
m-1141	124	9	conditions	condition	NOUN
m-1141	124	10	and	and	CCONJ
m-1141	124	11	so	so	ADV
m-1141	124	12	is	be	AUX
m-1141	124	13	separative	separative	NOUN
m-1141	124	14	.	.	PUNCT
m-1141	125	1	theorem	theorem	VERB
m-1141	125	2	5.1	5.1	NUM
m-1141	125	3	.	.	PUNCT
m-1141	126	1	let	let	VERB
m-1141	126	2	s	s	PRON
m-1141	126	3	=	=	PUNCT
m-1141	126	4	⋃	⋃	NOUN
m-1141	126	5	e∈y	e∈y	NOUN
m-1141	126	6	se	se	X
m-1141	126	7	be	be	AUX
m-1141	126	8	a	a	DET
m-1141	126	9	semilattice	semilattice	NOUN
m-1141	126	10	(	(	PUNCT
m-1141	126	11	index	index	NOUN
m-1141	126	12	set	set	VERB
m-1141	126	13	y	y	PROPN
m-1141	126	14	with	with	ADP
m-1141	126	15	meet	meet	NOUN
m-1141	126	16	∧	∧	PROPN
m-1141	126	17	)	)	PUNCT
m-1141	126	18	of	of	ADP
m-1141	126	19	cancellative	cancellative	ADJ
m-1141	126	20	semigroups	semigroup	NOUN
m-1141	127	1	se	se	X
m-1141	127	2	.	.	PUNCT
m-1141	128	1	if	if	SCONJ
m-1141	128	2	a	a	DET
m-1141	128	3	∈	∈	PROPN
m-1141	128	4	se	se	X
m-1141	128	5	and	and	CCONJ
m-1141	128	6	b	b	X
m-1141	128	7	∈	∈	PROPN
m-1141	128	8	sf	sf	NOUN
m-1141	128	9	then	then	ADV
m-1141	128	10	ab	ab	PROPN
m-1141	128	11	∈	∈	PROPN
m-1141	128	12	se∧f	se∧f	VERB
m-1141	128	13	.	.	PUNCT
m-1141	129	1	then	then	ADV
m-1141	129	2	s	s	VERB
m-1141	129	3	is	be	AUX
m-1141	129	4	stable	stable	ADJ
m-1141	129	5	,	,	PUNCT
m-1141	129	6	i.e.	i.e.	X
m-1141	129	7	for	for	ADP
m-1141	129	8	all	all	DET
m-1141	129	9	a	a	DET
m-1141	129	10	,	,	PUNCT
m-1141	129	11	b	b	X
m-1141	129	12	∈	∈	PROPN
m-1141	129	13	s	s	PART
m-1141	129	14	:	:	PUNCT
m-1141	129	15	ijo	ijo	PROPN
m-1141	129	16	international	international	PROPN
m-1141	129	17	journal	journal	PROPN
m-1141	129	18	of	of	ADP
m-1141	129	19	mathematics	mathematics	PROPN
m-1141	129	20	(	(	PUNCT
m-1141	129	21	issn	issn	PROPN
m-1141	129	22	:	:	PUNCT
m-1141	129	23	2992	2992	NUM
m-1141	129	24	-	-	SYM
m-1141	129	25	4421	4421	NUM
m-1141	129	26	)	)	PUNCT
m-1141	130	1	volume	volume	NOUN
m-1141	130	2	08	08	NUM
m-1141	131	1	|	|	ADV
m-1141	131	2	issue	issue	NOUN
m-1141	131	3	9	9	NUM
m-1141	131	4	|	|	CCONJ
m-1141	131	5	september	september	PROPN
m-1141	131	6	2025	2025	NUM
m-1141	132	1	|	|	ADV
m-1141	132	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1141	132	3	6	6	NUM
m-1141	132	4	1	1	NUM
m-1141	132	5	.	.	PUNCT
m-1141	133	1	if	if	SCONJ
m-1141	133	2	sa	sa	PROPN
m-1141	133	3	⊆	⊆	NUM
m-1141	133	4	s(ab	s(ab	NOUN
m-1141	133	5	)	)	PUNCT
m-1141	133	6	then	then	ADV
m-1141	133	7	sa	sa	PROPN
m-1141	133	8	=	=	PUNCT
m-1141	133	9	s(ab	s(ab	PROPN
m-1141	133	10	)	)	PUNCT
m-1141	133	11	.	.	PUNCT
m-1141	134	1	2	2	X
m-1141	134	2	.	.	X
m-1141	134	3	if	if	SCONJ
m-1141	134	4	as	as	ADP
m-1141	134	5	⊆	⊆	NUM
m-1141	134	6	(	(	PUNCT
m-1141	134	7	ab)s	ab)s	PROPN
m-1141	134	8	then	then	ADV
m-1141	134	9	as	as	ADP
m-1141	134	10	=	=	PRON
m-1141	134	11	(	(	PUNCT
m-1141	134	12	ab)s	ab)s	PROPN
m-1141	134	13	.	.	PUNCT
m-1141	135	1	proof	proof	NOUN
m-1141	135	2	.	.	PUNCT
m-1141	136	1	write	write	VERB
m-1141	136	2	principal	principal	NOUN
m-1141	136	3	left	leave	VERB
m-1141	136	4	ideals	ideal	NOUN
m-1141	136	5	by	by	ADP
m-1141	136	6	sa	sa	PROPN
m-1141	136	7	=	=	SYM
m-1141	136	8	{	{	PUNCT
m-1141	136	9	xa	xa	PROPN
m-1141	136	10	:	:	PUNCT
m-1141	136	11	x	x	X
m-1141	136	12	∈	∈	PROPN
m-1141	136	13	s	s	PART
m-1141	136	14	}	}	PUNCT
m-1141	136	15	and	and	CCONJ
m-1141	136	16	right	right	ADJ
m-1141	136	17	ideals	ideal	NOUN
m-1141	136	18	by	by	ADP
m-1141	136	19	as	as	ADP
m-1141	136	20	=	=	VERB
m-1141	136	21	{	{	PUNCT
m-1141	136	22	ax	ax	NOUN
m-1141	136	23	:	:	PUNCT
m-1141	136	24	x	x	PUNCT
m-1141	136	25	∈	∈	PROPN
m-1141	136	26	s	s	PART
m-1141	136	27	}	}	PUNCT
m-1141	136	28	.	.	PUNCT
m-1141	137	1	let	let	VERB
m-1141	137	2	a	a	DET
m-1141	137	3	∈	∈	PROPN
m-1141	137	4	se	se	X
m-1141	137	5	and	and	CCONJ
m-1141	137	6	b	b	X
m-1141	137	7	∈	∈	PROPN
m-1141	137	8	sf	sf	X
m-1141	137	9	.	.	PUNCT
m-1141	138	1	(	(	PUNCT
m-1141	138	2	1	1	X
m-1141	138	3	)	)	PUNCT
m-1141	138	4	assume	assume	VERB
m-1141	138	5	sa	sa	PROPN
m-1141	138	6	⊆	⊆	NUM
m-1141	138	7	s(ab	s(ab	NOUN
m-1141	138	8	)	)	PUNCT
m-1141	138	9	.	.	PUNCT
m-1141	139	1	in	in	ADP
m-1141	139	2	particular	particular	ADJ
m-1141	139	3	a	a	DET
m-1141	139	4	∈	∈	PROPN
m-1141	139	5	s(ab	s(ab	NOUN
m-1141	139	6	)	)	PUNCT
m-1141	139	7	,	,	PUNCT
m-1141	139	8	so	so	CCONJ
m-1141	139	9	there	there	PRON
m-1141	139	10	exists	exist	VERB
m-1141	139	11	t	t	PROPN
m-1141	139	12	∈	∈	PROPN
m-1141	139	13	s	s	PART
m-1141	139	14	with	with	ADP
m-1141	139	15	a	a	DET
m-1141	139	16	=	=	NOUN
m-1141	139	17	t(ab	t(ab	PROPN
m-1141	139	18	)	)	PUNCT
m-1141	139	19	.	.	PUNCT
m-1141	140	1	writing	write	VERB
m-1141	140	2	t	t	PROPN
m-1141	140	3	∈	∈	PROPN
m-1141	140	4	sg	sg	PROPN
m-1141	140	5	,	,	PUNCT
m-1141	140	6	the	the	DET
m-1141	140	7	product	product	NOUN
m-1141	140	8	t(ab	t(ab	AUX
m-1141	140	9	)	)	PUNCT
m-1141	140	10	∈	∈	PROPN
m-1141	140	11	sg∧e∧f	sg∧e∧f	PROPN
m-1141	140	12	,	,	PUNCT
m-1141	140	13	while	while	SCONJ
m-1141	140	14	a	a	DET
m-1141	140	15	∈	∈	PROPN
m-1141	140	16	se	se	X
m-1141	140	17	.	.	PUNCT
m-1141	141	1	equality	equality	NOUN
m-1141	141	2	of	of	ADP
m-1141	141	3	components	component	NOUN
m-1141	141	4	forces	force	NOUN
m-1141	141	5	e	e	NOUN
m-1141	141	6	=	=	SYM
m-1141	141	7	g	g	PROPN
m-1141	141	8	∧	∧	PROPN
m-1141	141	9	e	e	PROPN
m-1141	141	10	∧	∧	PROPN
m-1141	141	11	f	f	PROPN
m-1141	141	12	,	,	PUNCT
m-1141	141	13	hence	hence	ADV
m-1141	141	14	e	e	X
m-1141	141	15	≤	≤	ADJ
m-1141	141	16	f	f	NOUN
m-1141	141	17	and	and	CCONJ
m-1141	141	18	so	so	ADV
m-1141	141	19	e	e	PROPN
m-1141	141	20	∧	∧	PROPN
m-1141	141	21	f	f	PROPN
m-1141	141	22	=	=	PROPN
m-1141	141	23	e.	e.	PROPN
m-1141	141	24	thus	thus	ADV
m-1141	141	25	ab	ab	PROPN
m-1141	141	26	∈	∈	PROPN
m-1141	141	27	se	se	PROPN
m-1141	141	28	.	.	PROPN
m-1141	142	1	for	for	ADP
m-1141	142	2	any	any	DET
m-1141	142	3	h	h	NOUN
m-1141	142	4	∈	∈	PROPN
m-1141	142	5	y	y	PROPN
m-1141	142	6	,	,	PUNCT
m-1141	142	7	the	the	DET
m-1141	142	8	map	map	NOUN
m-1141	142	9	φh	φh	VERB
m-1141	142	10	:	:	PUNCT
m-1141	142	11	sh∧ea→	sh∧ea→	PROPN
m-1141	142	12	sh∧e(ab	sh∧e(ab	PROPN
m-1141	142	13	)	)	PUNCT
m-1141	142	14	de	de	X
m-1141	142	15	�	�	NOUN
m-1141	142	16	ned	ne	VERB
m-1141	142	17	by	by	ADP
m-1141	142	18	φh(xa	φh(xa	PROPN
m-1141	142	19	)	)	PUNCT
m-1141	142	20	=	=	PUNCT
m-1141	143	1	(	(	PUNCT
m-1141	143	2	xa)b	xa)b	PROPN
m-1141	143	3	is	be	AUX
m-1141	143	4	bijective	bijective	ADJ
m-1141	143	5	by	by	ADP
m-1141	143	6	cancellativity	cancellativity	NOUN
m-1141	143	7	.	.	PUNCT
m-1141	144	1	taking	take	VERB
m-1141	144	2	unions	union	NOUN
m-1141	144	3	over	over	ADP
m-1141	144	4	h	h	NOUN
m-1141	144	5	yields	yield	NOUN
m-1141	144	6	sa	sa	PROPN
m-1141	144	7	=	=	PUNCT
m-1141	144	8	s(ab	s(ab	PROPN
m-1141	144	9	)	)	PUNCT
m-1141	144	10	.	.	PUNCT
m-1141	145	1	(	(	PUNCT
m-1141	145	2	2	2	X
m-1141	145	3	)	)	PUNCT
m-1141	145	4	symmetrically	symmetrically	ADV
m-1141	145	5	,	,	PUNCT
m-1141	145	6	assume	assume	VERB
m-1141	145	7	as	as	ADP
m-1141	145	8	⊆	⊆	NUM
m-1141	145	9	(	(	PUNCT
m-1141	145	10	ab)s	ab)s	PROPN
m-1141	145	11	.	.	PUNCT
m-1141	146	1	then	then	ADV
m-1141	146	2	a	a	DET
m-1141	146	3	=	=	X
m-1141	146	4	(	(	PUNCT
m-1141	146	5	ab)u	ab)u	NOUN
m-1141	146	6	for	for	ADP
m-1141	146	7	some	some	DET
m-1141	146	8	u	u	NOUN
m-1141	146	9	∈	∈	PROPN
m-1141	146	10	s.	s.	PROPN
m-1141	146	11	this	this	DET
m-1141	146	12	forces	force	NOUN
m-1141	146	13	e	e	NOUN
m-1141	146	14	≤	≤	NUM
m-1141	146	15	f	f	PROPN
m-1141	146	16	and	and	CCONJ
m-1141	146	17	ab	ab	PROPN
m-1141	146	18	∈	∈	PROPN
m-1141	146	19	se	se	PROPN
m-1141	146	20	.	.	PUNCT
m-1141	147	1	for	for	ADP
m-1141	147	2	each	each	DET
m-1141	147	3	h	h	NOUN
m-1141	147	4	,	,	PUNCT
m-1141	147	5	the	the	DET
m-1141	147	6	map	map	NOUN
m-1141	147	7	ψh	ψh	ADP
m-1141	147	8	:	:	PUNCT
m-1141	147	9	ase∧h	ase∧h	ADV
m-1141	147	10	→	→	SYM
m-1141	147	11	(	(	PUNCT
m-1141	147	12	ab)se∧h	ab)se∧h	PROPN
m-1141	147	13	,	,	PUNCT
m-1141	147	14	ψh(ax	ψh(ax	PROPN
m-1141	147	15	)	)	PUNCT
m-1141	148	1	=	=	PUNCT
m-1141	148	2	(	(	PUNCT
m-1141	148	3	ab)x	ab)x	NOUN
m-1141	148	4	,	,	PUNCT
m-1141	148	5	is	be	AUX
m-1141	148	6	bijective	bijective	ADJ
m-1141	148	7	.	.	PUNCT
m-1141	149	1	taking	take	VERB
m-1141	149	2	unions	union	NOUN
m-1141	149	3	gives	give	VERB
m-1141	149	4	as	as	ADP
m-1141	149	5	=	=	ADJ
m-1141	149	6	(	(	PUNCT
m-1141	149	7	ab)s	ab)s	PROPN
m-1141	149	8	.	.	PUNCT
m-1141	150	1	hence	hence	ADV
m-1141	150	2	s	s	VERB
m-1141	150	3	is	be	AUX
m-1141	150	4	stable	stable	ADJ
m-1141	150	5	.	.	PUNCT
m-1141	151	1	remark	remark	VERB
m-1141	151	2	5.1	5.1	NUM
m-1141	151	3	.	.	PUNCT
m-1141	152	1	the	the	DET
m-1141	152	2	proof	proof	NOUN
m-1141	152	3	uses	use	VERB
m-1141	152	4	(	(	PUNCT
m-1141	152	5	i	i	NOUN
m-1141	152	6	)	)	PUNCT
m-1141	152	7	semilattice	semilattice	NOUN
m-1141	152	8	decomposition	decomposition	NOUN
m-1141	152	9	of	of	ADP
m-1141	152	10	separative	separative	NOUN
m-1141	152	11	semigroups	semigroup	NOUN
m-1141	152	12	,	,	PUNCT
m-1141	152	13	and	and	CCONJ
m-1141	152	14	(	(	PUNCT
m-1141	152	15	ii	ii	NOUN
m-1141	152	16	)	)	PUNCT
m-1141	152	17	cancellativity	cancellativity	NOUN
m-1141	152	18	in	in	ADP
m-1141	152	19	each	each	DET
m-1141	152	20	component	component	NOUN
m-1141	152	21	,	,	PUNCT
m-1141	152	22	ensuring	ensure	VERB
m-1141	152	23	injectivity	injectivity	NOUN
m-1141	152	24	of	of	ADP
m-1141	152	25	the	the	DET
m-1141	152	26	maps	map	NOUN
m-1141	152	27	xa	xa	PROPN
m-1141	152	28	7→	7→	NUM
m-1141	152	29	(	(	PUNCT
m-1141	152	30	xa)b	xa)b	PROPN
m-1141	152	31	and	and	CCONJ
m-1141	152	32	ax	ax	NOUN
m-1141	152	33	7→	7→	NUM
m-1141	152	34	(	(	PUNCT
m-1141	152	35	ab)x	ab)x	PROPN
m-1141	152	36	.	.	PUNCT
m-1141	153	1	these	these	PRON
m-1141	153	2	yield	yield	NOUN
m-1141	153	3	stability	stability	NOUN
m-1141	153	4	(	(	PUNCT
m-1141	153	5	in	in	ADP
m-1141	153	6	the	the	DET
m-1141	153	7	sense	sense	NOUN
m-1141	153	8	of	of	ADP
m-1141	153	9	[	[	X
m-1141	153	10	16	16	NUM
m-1141	153	11	]	]	SYM
m-1141	153	12	)	)	PUNCT
m-1141	153	13	.	.	PUNCT
m-1141	154	1	theorem	theorem	VERB
m-1141	154	2	5.2	5.2	NUM
m-1141	154	3	.	.	PUNCT
m-1141	155	1	(	(	PUNCT
m-1141	155	2	burmistrovich	burmistrovich	PRON
m-1141	155	3	's	's	PART
m-1141	155	4	theorem	theorem	ADJ
m-1141	155	5	[	[	X
m-1141	155	6	10	10	NUM
m-1141	155	7	]	]	NUM
m-1141	155	8	)	)	PUNCT
m-1141	155	9	.	.	PUNCT
m-1141	156	1	a	a	DET
m-1141	156	2	semigroup	semigroup	NOUN
m-1141	156	3	s	s	VERB
m-1141	156	4	is	be	AUX
m-1141	156	5	separative	separative	ADJ
m-1141	156	6	if	if	SCONJ
m-1141	156	7	and	and	CCONJ
m-1141	156	8	only	only	ADV
m-1141	156	9	if	if	SCONJ
m-1141	156	10	it	it	PRON
m-1141	156	11	is	be	AUX
m-1141	156	12	isomorphic	isomorphic	ADJ
m-1141	156	13	to	to	ADP
m-1141	156	14	a	a	DET
m-1141	156	15	semilattice	semilattice	NOUN
m-1141	156	16	of	of	ADP
m-1141	156	17	cancellative	cancellative	ADJ
m-1141	156	18	semigroups	semigroup	NOUN
m-1141	156	19	.	.	PUNCT
m-1141	157	1	proof	proof	NOUN
m-1141	157	2	.	.	PUNCT
m-1141	158	1	see	see	VERB
m-1141	158	2	[	[	X
m-1141	158	3	7	7	NUM
m-1141	158	4	]	]	SYM
m-1141	158	5	6	6	NUM
m-1141	158	6	conclusion	conclusion	NOUN
m-1141	158	7	in	in	ADP
m-1141	158	8	this	this	DET
m-1141	158	9	work	work	NOUN
m-1141	158	10	,	,	PUNCT
m-1141	158	11	we	we	PRON
m-1141	158	12	have	have	AUX
m-1141	158	13	established	establish	VERB
m-1141	158	14	that	that	SCONJ
m-1141	158	15	every	every	DET
m-1141	158	16	separative	separative	NOUN
m-1141	158	17	semigroup	semigroup	NOUN
m-1141	158	18	is	be	AUX
m-1141	158	19	stable	stable	ADJ
m-1141	158	20	.	.	PUNCT
m-1141	159	1	the	the	DET
m-1141	159	2	argument	argument	NOUN
m-1141	159	3	relies	rely	VERB
m-1141	159	4	on	on	ADP
m-1141	159	5	two	two	NUM
m-1141	159	6	fundamental	fundamental	ADJ
m-1141	159	7	ingredients	ingredient	NOUN
m-1141	159	8	:	:	PUNCT
m-1141	159	9	�	�	PROPN
m-1141	159	10	rst	rst	PROPN
m-1141	159	11	,	,	PUNCT
m-1141	159	12	the	the	DET
m-1141	159	13	structural	structural	ADJ
m-1141	159	14	theorem	theorem	NOUN
m-1141	159	15	of	of	ADP
m-1141	159	16	burmistrovich	burmistrovich	PROPN
m-1141	159	17	(	(	PUNCT
m-1141	159	18	theorem	theorem	VERB
m-1141	159	19	5.2	5.2	NUM
m-1141	159	20	)	)	PUNCT
m-1141	159	21	,	,	PUNCT
m-1141	159	22	which	which	PRON
m-1141	159	23	asserts	assert	VERB
m-1141	159	24	that	that	SCONJ
m-1141	159	25	every	every	DET
m-1141	159	26	separative	separative	NOUN
m-1141	159	27	semigroup	semigroup	NOUN
m-1141	159	28	is	be	AUX
m-1141	159	29	isomorphic	isomorphic	ADJ
m-1141	159	30	to	to	ADP
m-1141	159	31	a	a	DET
m-1141	159	32	semilattice	semilattice	NOUN
m-1141	159	33	of	of	ADP
m-1141	159	34	cancellative	cancellative	ADJ
m-1141	159	35	semigroups	semigroup	NOUN
m-1141	159	36	;	;	PUNCT
m-1141	159	37	second	second	X
m-1141	159	38	,	,	PUNCT
m-1141	159	39	the	the	DET
m-1141	159	40	fact	fact	NOUN
m-1141	159	41	that	that	SCONJ
m-1141	159	42	semilattices	semilattice	NOUN
m-1141	159	43	of	of	ADP
m-1141	159	44	cancellative	cancellative	ADJ
m-1141	159	45	semigroups	semigroup	NOUN
m-1141	159	46	preserve	preserve	VERB
m-1141	159	47	the	the	DET
m-1141	159	48	stability	stability	NOUN
m-1141	159	49	conditions	condition	NOUN
m-1141	159	50	(	(	PUNCT
m-1141	159	51	theorem	theorem	VERB
m-1141	159	52	5.1	5.1	NUM
m-1141	159	53	)	)	PUNCT
m-1141	159	54	.	.	PUNCT
m-1141	160	1	by	by	ADP
m-1141	160	2	combining	combine	VERB
m-1141	160	3	these	these	DET
m-1141	160	4	facts	fact	NOUN
m-1141	160	5	along	along	ADP
m-1141	160	6	with	with	ADP
m-1141	160	7	proposition	proposition	NOUN
m-1141	160	8	5.1	5.1	NUM
m-1141	160	9	,	,	PUNCT
m-1141	160	10	we	we	PRON
m-1141	160	11	conclude	conclude	VERB
m-1141	160	12	that	that	SCONJ
m-1141	160	13	the	the	DET
m-1141	160	14	separative	separative	NOUN
m-1141	160	15	property	property	NOUN
m-1141	160	16	not	not	PART
m-1141	160	17	only	only	ADV
m-1141	160	18	encodes	encode	NOUN
m-1141	160	19	a	a	DET
m-1141	160	20	re	re	NOUN
m-1141	160	21	�	�	NOUN
m-1141	160	22	ned	ned	ADJ
m-1141	160	23	form	form	NOUN
m-1141	160	24	of	of	ADP
m-1141	160	25	cancellativity	cancellativity	NOUN
m-1141	160	26	but	but	CCONJ
m-1141	160	27	also	also	ADV
m-1141	160	28	guarantees	guarantee	VERB
m-1141	160	29	algebraic	algebraic	ADJ
m-1141	160	30	stability	stability	NOUN
m-1141	160	31	in	in	ADP
m-1141	160	32	the	the	DET
m-1141	160	33	sense	sense	NOUN
m-1141	160	34	of	of	ADP
m-1141	160	35	koch	koch	PROPN
m-1141	160	36	and	and	CCONJ
m-1141	160	37	wallace	wallace	NOUN
m-1141	160	38	.	.	PUNCT
m-1141	161	1	this	this	DET
m-1141	161	2	result	result	NOUN
m-1141	161	3	strengthens	strengthen	VERB
m-1141	161	4	the	the	DET
m-1141	161	5	conceptual	conceptual	ADJ
m-1141	161	6	link	link	NOUN
m-1141	161	7	between	between	ADP
m-1141	161	8	structural	structural	ADJ
m-1141	161	9	decomposition	decomposition	NOUN
m-1141	161	10	and	and	CCONJ
m-1141	161	11	stability	stability	NOUN
m-1141	161	12	theory	theory	NOUN
m-1141	161	13	in	in	ADP
m-1141	161	14	semigroup	semigroup	PROPN
m-1141	161	15	theory	theory	NOUN
m-1141	161	16	.	.	PUNCT
m-1141	162	1	it	it	PRON
m-1141	162	2	shows	show	VERB
m-1141	162	3	that	that	SCONJ
m-1141	162	4	separativity	separativity	NOUN
m-1141	162	5	,	,	PUNCT
m-1141	162	6	originally	originally	ADV
m-1141	162	7	introduced	introduce	VERB
m-1141	162	8	to	to	PART
m-1141	162	9	capture	capture	VERB
m-1141	162	10	subtle	subtle	ADJ
m-1141	162	11	cancellation	cancellation	NOUN
m-1141	162	12	phenomena	phenomenon	NOUN
m-1141	162	13	,	,	PUNCT
m-1141	162	14	naturally	naturally	ADV
m-1141	162	15	enforces	enforce	VERB
m-1141	162	16	stability	stability	NOUN
m-1141	162	17	(	(	PUNCT
m-1141	162	18	in	in	ADP
m-1141	162	19	the	the	DET
m-1141	162	20	sense	sense	NOUN
m-1141	162	21	of	of	ADP
m-1141	162	22	koch	koch	PROPN
m-1141	162	23	and	and	CCONJ
m-1141	162	24	wallace	wallace	PROPN
m-1141	162	25	)	)	PUNCT
m-1141	162	26	through	through	ADP
m-1141	162	27	its	its	PRON
m-1141	162	28	semilattice	semilattice	NOUN
m-1141	162	29	decomposition	decomposition	NOUN
m-1141	162	30	.	.	PUNCT
m-1141	163	1	consequently	consequently	ADV
m-1141	163	2	,	,	PUNCT
m-1141	163	3	the	the	DET
m-1141	163	4	class	class	NOUN
m-1141	163	5	of	of	ADP
m-1141	163	6	separative	separative	NOUN
m-1141	163	7	semigroups	semigroup	NOUN
m-1141	163	8	provides	provide	VERB
m-1141	163	9	a	a	DET
m-1141	163	10	robust	robust	ADJ
m-1141	163	11	and	and	CCONJ
m-1141	163	12	stable	stable	ADJ
m-1141	163	13	framework	framework	NOUN
m-1141	163	14	for	for	ADP
m-1141	163	15	further	further	ADJ
m-1141	163	16	exploration	exploration	NOUN
m-1141	163	17	in	in	ADP
m-1141	163	18	the	the	DET
m-1141	163	19	algebraic	algebraic	ADJ
m-1141	163	20	theory	theory	NOUN
m-1141	163	21	of	of	ADP
m-1141	163	22	semigroups	semigroup	NOUN
m-1141	163	23	.	.	PUNCT
m-1141	164	1	declaration	declaration	NOUN
m-1141	164	2	of	of	ADP
m-1141	164	3	interest	interest	NOUN
m-1141	164	4	:	:	PUNCT
m-1141	164	5	1	1	X
m-1141	164	6	.	.	X
m-1141	164	7	funding	funding	NOUN
m-1141	164	8	:	:	PUNCT
m-1141	164	9	not	not	PART
m-1141	164	10	applicable	applicable	ADJ
m-1141	164	11	2	2	NUM
m-1141	164	12	.	.	PUNCT
m-1141	165	1	informed	informed	ADJ
m-1141	165	2	consent	consent	NOUN
m-1141	165	3	statement	statement	NOUN
m-1141	165	4	:	:	PUNCT
m-1141	165	5	not	not	PART
m-1141	165	6	applicable	applicable	ADJ
m-1141	165	7	3	3	NUM
m-1141	165	8	.	.	PUNCT
m-1141	165	9	data	datum	NOUN
m-1141	165	10	availability	availability	NOUN
m-1141	165	11	:	:	PUNCT
m-1141	165	12	not	not	PART
m-1141	165	13	applicable	applicable	ADJ
m-1141	165	14	4	4	NUM
m-1141	165	15	.	.	PUNCT
m-1141	166	1	con	con	PROPN
m-1141	166	2	�	�	PROPN
m-1141	166	3	ict	ict	PROPN
m-1141	166	4	of	of	ADP
m-1141	166	5	interest	interest	NOUN
m-1141	166	6	statement	statement	NOUN
m-1141	166	7	:	:	PUNCT
m-1141	166	8	no	no	DET
m-1141	166	9	con	con	PROPN
m-1141	166	10	�	�	PROPN
m-1141	166	11	ict	ict	PROPN
m-1141	166	12	of	of	ADP
m-1141	166	13	interest	interest	NOUN
m-1141	166	14	.	.	PUNCT
m-1141	167	1	ijo	ijo	PROPN
m-1141	167	2	international	international	PROPN
m-1141	167	3	journal	journal	PROPN
m-1141	167	4	of	of	ADP
m-1141	167	5	mathematics	mathematics	PROPN
m-1141	167	6	(	(	PUNCT
m-1141	167	7	issn	issn	PROPN
m-1141	167	8	:	:	PUNCT
m-1141	167	9	2992	2992	NUM
m-1141	167	10	-	-	SYM
m-1141	167	11	4421	4421	NUM
m-1141	167	12	)	)	PUNCT
m-1141	167	13	volume	volume	NOUN
m-1141	167	14	08	08	NUM
m-1141	168	1	|	|	ADV
m-1141	168	2	issue	issue	NOUN
m-1141	168	3	9	9	NUM
m-1141	168	4	|	|	CCONJ
m-1141	168	5	september	september	PROPN
m-1141	168	6	2025	2025	NUM
m-1141	168	7	|	|	ADV
m-1141	168	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1141	168	9	7	7	NUM
m-1141	168	10	references	reference	NOUN
m-1141	168	11	[	[	X
m-1141	168	12	1	1	NUM
m-1141	168	13	]	]	PUNCT
m-1141	168	14	a.	a.	NOUN
m-1141	168	15	h.	h.	PROPN
m-1141	168	16	cli	cli	PROPN
m-1141	168	17	�	�	PROPN
m-1141	168	18	ord	ord	PROPN
m-1141	168	19	and	and	CCONJ
m-1141	168	20	g.	g.	PROPN
m-1141	168	21	b.	b.	PROPN
m-1141	168	22	preston	preston	PROPN
m-1141	168	23	,	,	PUNCT
m-1141	168	24	the	the	DET
m-1141	168	25	algebraic	algebraic	ADJ
m-1141	168	26	theory	theory	NOUN
m-1141	168	27	of	of	ADP
m-1141	168	28	semigroups	semigroup	NOUN
m-1141	168	29	,	,	PUNCT
m-1141	168	30	american	american	PROPN
m-1141	168	31	mathematical	mathematical	ADJ
m-1141	168	32	society	society	NOUN
m-1141	168	33	(	(	PUNCT
m-1141	168	34	1961	1961	NUM
m-1141	168	35	)	)	PUNCT
m-1141	168	36	.	.	PUNCT
m-1141	169	1	[	[	X
m-1141	169	2	2	2	NUM
m-1141	169	3	]	]	X
m-1141	169	4	b.	b.	PROPN
m-1141	169	5	t.	t.	PROPN
m-1141	169	6	shourijeh	shourijeh	PROPN
m-1141	169	7	,	,	PUNCT
m-1141	169	8	c∗−algebras	c∗−algebra	NOUN
m-1141	169	9	of	of	ADP
m-1141	169	10	some	some	DET
m-1141	169	11	semigroups	semigroup	NOUN
m-1141	169	12	,	,	PUNCT
m-1141	169	13	honam	honam	PROPN
m-1141	169	14	mathematical	mathematical	PROPN
m-1141	169	15	j.	j.	PROPN
m-1141	169	16	vol	vol	PROPN
m-1141	169	17	.	.	PROPN
m-1141	170	1	26	26	NUM
m-1141	170	2	,	,	PUNCT
m-1141	170	3	no	no	INTJ
m-1141	170	4	.	.	NOUN
m-1141	170	5	4	4	X
m-1141	170	6	.	.	X
m-1141	171	1	pp	pp	ADJ
m-1141	171	2	.	.	PUNCT
m-1141	172	1	483	483	NUM
m-1141	172	2	�	�	PROPN
m-1141	172	3	507	507	NUM
m-1141	172	4	,	,	PUNCT
m-1141	172	5	2004	2004	NUM
m-1141	172	6	.	.	PUNCT
m-1141	173	1	[	[	X
m-1141	173	2	3	3	X
m-1141	173	3	]	]	X
m-1141	173	4	c.	c.	PROPN
m-1141	173	5	hollings	holling	NOUN
m-1141	173	6	,	,	PUNCT
m-1141	173	7	mathematics	mathematic	NOUN
m-1141	173	8	across	across	ADP
m-1141	173	9	the	the	DET
m-1141	173	10	iron	iron	NOUN
m-1141	173	11	curtain	curtain	NOUN
m-1141	173	12	:	:	PUNCT
m-1141	173	13	a	a	DET
m-1141	173	14	history	history	NOUN
m-1141	173	15	of	of	ADP
m-1141	173	16	soviet	soviet	ADJ
m-1141	173	17	mathematics	mathematic	NOUN
m-1141	173	18	,	,	PUNCT
m-1141	173	19	american	american	PROPN
m-1141	173	20	mathematical	mathematical	PROPN
m-1141	173	21	society	society	NOUN
m-1141	173	22	,	,	PUNCT
m-1141	173	23	2014	2014	NUM
m-1141	173	24	.	.	PUNCT
m-1141	174	1	[	[	X
m-1141	174	2	4	4	X
m-1141	174	3	]	]	X
m-1141	174	4	d.	d.	PROPN
m-1141	174	5	d.	d.	PROPN
m-1141	174	6	miller	miller	PROPN
m-1141	174	7	and	and	CCONJ
m-1141	174	8	cli	cli	PROPN
m-1141	174	9	�	�	PROPN
m-1141	174	10	ord	ord	PROPN
m-1141	174	11	,	,	PUNCT
m-1141	174	12	regular	regular	ADJ
m-1141	174	13	classes	class	NOUN
m-1141	174	14	in	in	ADP
m-1141	174	15	semigroups	semigroup	NOUN
m-1141	174	16	,	,	PUNCT
m-1141	174	17	tran	tran	PROPN
m-1141	174	18	.	.	PUNCT
m-1141	175	1	amer	amer	PROPN
m-1141	175	2	.	.	PUNCT
m-1141	175	3	math	math	PROPN
m-1141	175	4	.	.	PUNCT
m-1141	176	1	soc	soc	PROPN
m-1141	176	2	.	.	PUNCT
m-1141	177	1	vol	vol	NOUN
m-1141	177	2	.	.	PROPN
m-1141	178	1	82	82	NUM
m-1141	178	2	,	,	PUNCT
m-1141	178	3	1956	1956	NUM
m-1141	178	4	.	.	PUNCT
m-1141	179	1	[	[	X
m-1141	179	2	5	5	X
m-1141	179	3	]	]	PUNCT
m-1141	179	4	d.	d.	PROPN
m-1141	179	5	rees	rees	PROPN
m-1141	179	6	,	,	PUNCT
m-1141	179	7	on	on	ADP
m-1141	179	8	semi	semi	NOUN
m-1141	179	9	-	-	NOUN
m-1141	179	10	groups	group	NOUN
m-1141	179	11	,	,	PUNCT
m-1141	179	12	mathematical	mathematical	ADJ
m-1141	179	13	proceedings	proceeding	NOUN
m-1141	179	14	of	of	ADP
m-1141	179	15	the	the	DET
m-1141	179	16	cambridge	cambridge	PROPN
m-1141	179	17	philosophical	philosophical	ADJ
m-1141	179	18	society	society	NOUN
m-1141	179	19	,	,	PUNCT
m-1141	179	20	vol	vol	NOUN
m-1141	179	21	.	.	PROPN
m-1141	180	1	36	36	NUM
m-1141	180	2	,	,	PUNCT
m-1141	180	3	no	no	INTJ
m-1141	180	4	.	.	NOUN
m-1141	180	5	4	4	NUM
m-1141	180	6	,	,	PUNCT
m-1141	180	7	pp	pp	ADJ
m-1141	180	8	.	.	PUNCT
m-1141	181	1	387	387	NUM
m-1141	181	2	�	�	PROPN
m-1141	181	3	400	400	NUM
m-1141	181	4	1940	1940	NUM
m-1141	181	5	.	.	PUNCT
m-1141	182	1	[	[	X
m-1141	182	2	6	6	NUM
m-1141	182	3	]	]	PUNCT
m-1141	182	4	e.	e.	PROPN
m-1141	182	5	hewitt	hewitt	PROPN
m-1141	182	6	,	,	PUNCT
m-1141	182	7	h.	h.	PROPN
m-1141	182	8	s.	s.	PROPN
m-1141	182	9	zuckerman	zuckerman	PROPN
m-1141	182	10	,	,	PUNCT
m-1141	182	11	the	the	DET
m-1141	182	12	l1˘algebra	l1˘algebra	NOUN
m-1141	182	13	of	of	ADP
m-1141	182	14	commutative	commutative	ADJ
m-1141	182	15	semigroup	semigroup	PROPN
m-1141	182	16	,	,	PUNCT
m-1141	182	17	trans	trans	PROPN
m-1141	182	18	.	.	PROPN
m-1141	183	1	amer	amer	PROPN
m-1141	183	2	.	.	PUNCT
m-1141	183	3	math	math	PROPN
m-1141	183	4	.	.	PUNCT
m-1141	184	1	soc	soc	PROPN
m-1141	184	2	.	.	PUNCT
m-1141	185	1	,	,	PUNCT
m-1141	185	2	83	83	NUM
m-1141	185	3	,	,	PUNCT
m-1141	185	4	pp	pp	ADJ
m-1141	185	5	.	.	PUNCT
m-1141	186	1	70	70	NUM
m-1141	186	2	-	-	SYM
m-1141	186	3	97	97	NUM
m-1141	186	4	,	,	PUNCT
m-1141	186	5	1956	1956	NUM
m-1141	186	6	.	.	PUNCT
m-1141	187	1	[	[	X
m-1141	187	2	7	7	X
m-1141	187	3	]	]	X
m-1141	187	4	i.	i.	PROPN
m-1141	187	5	e.	e.	PROPN
m-1141	187	6	burmistrovitch	burmistrovitch	PROPN
m-1141	187	7	,	,	PUNCT
m-1141	187	8	commutative	commutative	ADJ
m-1141	187	9	bands	band	NOUN
m-1141	187	10	of	of	ADP
m-1141	187	11	cancellative	cancellative	ADJ
m-1141	187	12	semigroups	semigroup	NOUN
m-1141	187	13	,	,	PUNCT
m-1141	187	14	siberian	siberian	ADJ
m-1141	187	15	mat	mat	NOUN
m-1141	187	16	.	.	PUNCT
m-1141	188	1	z.	z.	PROPN
m-1141	188	2	vol	vol	NOUN
m-1141	188	3	.	.	PROPN
m-1141	189	1	6	6	NUM
m-1141	189	2	,	,	PUNCT
m-1141	189	3	pp	pp	ADJ
m-1141	189	4	.	.	PUNCT
m-1141	190	1	284	284	NUM
m-1141	190	2	�	�	PROPN
m-1141	190	3	299	299	NUM
m-1141	190	4	,	,	PUNCT
m-1141	190	5	1965	1965	NUM
m-1141	190	6	.	.	PUNCT
m-1141	191	1	[	[	X
m-1141	191	2	8	8	NUM
m-1141	191	3	]	]	X
m-1141	191	4	j.a	j.a	PROPN
m-1141	191	5	.	.	PROPN
m-1141	191	6	green	green	PROPN
m-1141	191	7	,	,	PUNCT
m-1141	191	8	on	on	ADP
m-1141	191	9	the	the	DET
m-1141	191	10	structure	structure	NOUN
m-1141	191	11	of	of	ADP
m-1141	191	12	semigroups	semigroup	NOUN
m-1141	191	13	,	,	PUNCT
m-1141	191	14	ann	ann	PROPN
m-1141	191	15	.	.	PROPN
m-1141	191	16	of	of	ADP
m-1141	191	17	math	math	NOUN
m-1141	191	18	.	.	PUNCT
m-1141	192	1	,	,	PUNCT
m-1141	192	2	pp	pp	PROPN
m-1141	192	3	.	.	PUNCT
m-1141	193	1	163-	163-	NUM
m-1141	193	2	�	�	SYM
m-1141	193	3	172	172	NUM
m-1141	193	4	,	,	PUNCT
m-1141	193	5	1951	1951	NUM
m-1141	193	6	.	.	PUNCT
m-1141	194	1	[	[	X
m-1141	194	2	9	9	NUM
m-1141	194	3	]	]	PUNCT
m-1141	194	4	j.	j.	PROPN
m-1141	194	5	east	east	PROPN
m-1141	194	6	and	and	CCONJ
m-1141	194	7	p.	p.	PROPN
m-1141	194	8	m.	m.	PROPN
m-1141	194	9	higgins	higgins	PROPN
m-1141	194	10	,	,	PUNCT
m-1141	194	11	green	green	PROPN
m-1141	194	12	's	's	PART
m-1141	194	13	relations	relation	NOUN
m-1141	194	14	and	and	CCONJ
m-1141	194	15	stability	stability	NOUN
m-1141	194	16	for	for	ADP
m-1141	194	17	subsemigroups	subsemigroup	NOUN
m-1141	194	18	,	,	PUNCT
m-1141	194	19	semigroup	semigroup	PROPN
m-1141	194	20	forum	forum	PROPN
m-1141	194	21	,	,	PUNCT
m-1141	194	22	vol	vol	NOUN
m-1141	194	23	.	.	PROPN
m-1141	195	1	101	101	NUM
m-1141	195	2	,	,	PUNCT
m-1141	195	3	no	no	INTJ
m-1141	195	4	.	.	NOUN
m-1141	195	5	1	1	NUM
m-1141	195	6	,	,	PUNCT
m-1141	195	7	pp	pp	ADJ
m-1141	195	8	.	.	PUNCT
m-1141	196	1	77	77	NUM
m-1141	196	2	�	�	NOUN
m-1141	196	3	86	86	NUM
m-1141	196	4	,	,	PUNCT
m-1141	196	5	2020	2020	NUM
m-1141	196	6	.	.	PUNCT
m-1141	197	1	[	[	X
m-1141	197	2	10	10	NUM
m-1141	197	3	]	]	X
m-1141	197	4	o.	o.	PROPN
m-1141	197	5	g.	g.	PROPN
m-1141	197	6	udoaka	udoaka	ADV
m-1141	197	7	,	,	PUNCT
m-1141	197	8	generators	generator	NOUN
m-1141	197	9	and	and	CCONJ
m-1141	197	10	inner	inner	ADJ
m-1141	197	11	automorphism	automorphism	NOUN
m-1141	197	12	,	,	PUNCT
m-1141	197	13	the	the	DET
m-1141	197	14	colloquium	colloquium	NOUN
m-1141	197	15	-	-	PUNCT
m-1141	197	16	a	a	DET
m-1141	197	17	multidisciplinary	multidisciplinary	ADJ
m-1141	197	18	thematc	thematc	NOUN
m-1141	197	19	policy	policy	NOUN
m-1141	197	20	journal	journal	PROPN
m-1141	197	21	vol	vol	NOUN
m-1141	197	22	.	.	PROPN
m-1141	197	23	10	10	NUM
m-1141	197	24	,	,	PUNCT
m-1141	197	25	no	no	INTJ
m-1141	197	26	.	.	NOUN
m-1141	197	27	1	1	NUM
m-1141	197	28	,	,	PUNCT
m-1141	197	29	pp	pp	ADJ
m-1141	197	30	.	.	PUNCT
m-1141	198	1	102	102	NUM
m-1141	198	2	�	�	NOUN
m-1141	198	3	111	111	NUM
m-1141	198	4	,	,	PUNCT
m-1141	198	5	2022	2022	NUM
m-1141	198	6	.	.	PUNCT
m-1141	199	1	[	[	X
m-1141	199	2	11	11	NUM
m-1141	199	3	]	]	PUNCT
m-1141	199	4	l.	l.	PROPN
m-1141	199	5	w.	w.	PROPN
m-1141	199	6	anderson	anderson	PROPN
m-1141	199	7	,	,	PUNCT
m-1141	199	8	r.	r.	PROPN
m-1141	199	9	p.	p.	PROPN
m-1141	199	10	hunter	hunter	PROPN
m-1141	199	11	and	and	CCONJ
m-1141	199	12	r.	r.	PROPN
m-1141	199	13	j.	j.	PROPN
m-1141	199	14	koch	koch	PROPN
m-1141	199	15	,	,	PUNCT
m-1141	199	16	some	some	PRON
m-1141	199	17	results	result	VERB
m-1141	199	18	on	on	ADP
m-1141	199	19	stability	stability	NOUN
m-1141	199	20	in	in	ADP
m-1141	199	21	semigroups	semigroup	NOUN
m-1141	199	22	.	.	PUNCT
m-1141	200	1	ams	am	NOUN
m-1141	200	2	journal	journal	PROPN
m-1141	200	3	,	,	PUNCT
m-1141	200	4	pp	pp	PROPN
m-1141	200	5	.	.	PUNCT
m-1141	201	1	521	521	NUM
m-1141	201	2	�	�	PROPN
m-1141	201	3	529	529	NUM
m-1141	201	4	,	,	PUNCT
m-1141	201	5	1965	1965	NUM
m-1141	201	6	.	.	PUNCT
m-1141	202	1	[	[	X
m-1141	202	2	12	12	NUM
m-1141	202	3	]	]	PUNCT
m-1141	202	4	m.	m.	NOUN
m-1141	202	5	p.	p.	PROPN
m-1141	202	6	drazin	drazin	PROPN
m-1141	202	7	,	,	PUNCT
m-1141	202	8	a	a	DET
m-1141	202	9	partial	partial	ADJ
m-1141	202	10	order	order	NOUN
m-1141	202	11	in	in	ADP
m-1141	202	12	completely	completely	ADV
m-1141	202	13	regular	regular	ADJ
m-1141	202	14	semigroups	semigroup	NOUN
m-1141	202	15	,	,	PUNCT
m-1141	202	16	journal	journal	NOUN
m-1141	202	17	of	of	ADP
m-1141	202	18	algebra	algebra	PROPN
m-1141	202	19	,	,	PUNCT
m-1141	202	20	vol	vol	NOUN
m-1141	202	21	.	.	PROPN
m-1141	202	22	98	98	NUM
m-1141	202	23	,	,	PUNCT
m-1141	202	24	pp	pp	ADJ
m-1141	202	25	.	.	PUNCT
m-1141	202	26	362	362	NUM
m-1141	202	27	�	�	PROPN
m-1141	202	28	374	374	NUM
m-1141	202	29	,	,	PUNCT
m-1141	202	30	1986	1986	NUM
m-1141	202	31	.	.	PUNCT
m-1141	203	1	[	[	X
m-1141	203	2	13	13	NUM
m-1141	203	3	]	]	PUNCT
m-1141	203	4	m.	m.	NOUN
m-1141	203	5	petrich	petrich	NOUN
m-1141	203	6	,	,	PUNCT
m-1141	203	7	introduction	introduction	NOUN
m-1141	203	8	to	to	ADP
m-1141	203	9	semigroups	semigroup	NOUN
m-1141	203	10	,	,	PUNCT
m-1141	203	11	charles	charles	PROPN
m-1141	203	12	babbage	babbage	PROPN
m-1141	203	13	research	research	PROPN
m-1141	203	14	centre	centre	NOUN
m-1141	203	15	,	,	PUNCT
m-1141	203	16	1984	1984	NUM
m-1141	203	17	.	.	PUNCT
m-1141	204	1	[	[	X
m-1141	204	2	14	14	NUM
m-1141	204	3	]	]	X
m-1141	204	4	n.	n.	PROPN
m-1141	204	5	kumar	kumar	PROPN
m-1141	204	6	and	and	CCONJ
m-1141	204	7	b.	b.	PROPN
m-1141	204	8	kumar	kumar	PROPN
m-1141	204	9	,	,	PUNCT
m-1141	204	10	some	some	DET
m-1141	204	11	fundamental	fundamental	ADJ
m-1141	204	12	properties	property	NOUN
m-1141	204	13	of	of	ADP
m-1141	204	14	semigroups	semigroup	NOUN
m-1141	204	15	and	and	CCONJ
m-1141	204	16	their	their	PRON
m-1141	204	17	classi	classi	ADJ
m-1141	204	18	�	�	PROPN
m-1141	204	19	cations	cation	NOUN
m-1141	204	20	,	,	PUNCT
m-1141	204	21	internal	internal	ADJ
m-1141	204	22	journal	journal	NOUN
m-1141	204	23	of	of	ADP
m-1141	204	24	mathematics	mathematics	NOUN
m-1141	204	25	trends	trend	NOUN
m-1141	204	26	and	and	CCONJ
m-1141	204	27	technology	technology	NOUN
m-1141	204	28	,	,	PUNCT
m-1141	204	29	vol	vol	NOUN
m-1141	204	30	.	.	PUNCT
m-1141	204	31	70	70	NUM
m-1141	204	32	issue	issue	NOUN
m-1141	204	33	9	9	NUM
m-1141	204	34	,	,	PUNCT
m-1141	204	35	pp	pp	ADJ
m-1141	204	36	.	.	PUNCT
m-1141	204	37	8	8	NUM
m-1141	204	38	�	�	NOUN
m-1141	204	39	11	11	NUM
m-1141	204	40	,	,	PUNCT
m-1141	204	41	2024	2024	NUM
m-1141	204	42	.	.	PUNCT
m-1141	205	1	[	[	X
m-1141	205	2	15	15	NUM
m-1141	205	3	]	]	X
m-1141	205	4	r.	r.	PROPN
m-1141	205	5	thakur	thakur	PROPN
m-1141	205	6	,	,	PUNCT
m-1141	205	7	a	a	DET
m-1141	205	8	short	short	ADJ
m-1141	205	9	note	note	NOUN
m-1141	205	10	on	on	ADP
m-1141	205	11	completely	completely	ADV
m-1141	205	12	regular	regular	ADJ
m-1141	205	13	semigroup	semigroup	NOUN
m-1141	205	14	,	,	PUNCT
m-1141	205	15	international	international	ADJ
m-1141	205	16	journal	journal	NOUN
m-1141	205	17	of	of	ADP
m-1141	205	18	creative	creative	ADJ
m-1141	205	19	research	research	NOUN
m-1141	205	20	thoughts	thought	NOUN
m-1141	205	21	,	,	PUNCT
m-1141	205	22	vol	vol	NOUN
m-1141	205	23	.	.	PROPN
m-1141	205	24	11	11	NUM
m-1141	205	25	,	,	PUNCT
m-1141	205	26	issue	issue	NOUN
m-1141	205	27	5	5	NUM
m-1141	205	28	,	,	PUNCT
m-1141	205	29	2023	2023	NUM
m-1141	205	30	.	.	PUNCT
m-1141	206	1	[	[	X
m-1141	206	2	16	16	NUM
m-1141	206	3	]	]	PUNCT
m-1141	206	4	r.	r.	PROPN
m-1141	206	5	j.	j.	PROPN
m-1141	206	6	koch	koch	PROPN
m-1141	206	7	and	and	CCONJ
m-1141	206	8	a.	a.	PROPN
m-1141	206	9	d.	d.	PROPN
m-1141	206	10	wallace	wallace	PROPN
m-1141	206	11	,	,	PUNCT
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m-1141	206	14	semigroups	semigroup	NOUN
m-1141	206	15	,	,	PUNCT
m-1141	206	16	duke	duke	PROPN
m-1141	206	17	math	math	PROPN
m-1141	206	18	.	.	PUNCT
m-1141	207	1	j.	j.	PROPN
m-1141	207	2	,	,	PUNCT
m-1141	207	3	vol	vol	NOUN
m-1141	207	4	.	.	PROPN
m-1141	207	5	24	24	NUM
m-1141	207	6	,	,	PUNCT
m-1141	207	7	pp	pp	ADJ
m-1141	207	8	.	.	PUNCT
m-1141	207	9	193	193	NUM
m-1141	207	10	�	�	PROPN
m-1141	207	11	196	196	NUM
m-1141	207	12	,	,	PUNCT
m-1141	207	13	1957	1957	NUM
m-1141	207	14	.	.	PUNCT
m-1141	208	1	[	[	X
m-1141	208	2	17	17	NUM
m-1141	208	3	]	]	X
m-1141	208	4	r.	r.	PROPN
m-1141	208	5	u.	u.	PROPN
m-1141	208	6	ndubisi	ndubisi	PROPN
m-1141	208	7	,	,	PUNCT
m-1141	208	8	and	and	CCONJ
m-1141	208	9	o.	o.	PROPN
m-1141	208	10	g.	g.	PROPN
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m-1141	208	12	,	,	PUNCT
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m-1141	208	14	left	left	ADJ
m-1141	208	15	restriction	restriction	NOUN
m-1141	208	16	semigroups	semigroup	NOUN
m-1141	208	17	,	,	PUNCT
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m-1141	208	19	journal	journal	NOUN
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m-1141	208	21	algebra	algebra	PROPN
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m-1141	208	26	.	.	PUNCT
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m-1141	210	1	1	1	NUM
m-1141	210	2	,	,	PUNCT
m-1141	210	3	pp	pp	ADJ
m-1141	210	4	.	.	PUNCT
m-1141	211	1	59	59	NUM
m-1141	211	2	�	�	PROPN
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m-1141	211	4	,	,	PUNCT
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m-1141	212	7	1083	1083	NUM
m-1141	212	8	.	.	PUNCT
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m-1141	213	2	international	international	PROPN
m-1141	213	3	journal	journal	PROPN
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m-1141	214	1	|	|	ADV
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m-1141	214	4	|	|	CCONJ
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m-1141	214	6	2025	2025	NUM
m-1141	215	1	|	|	ADV
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m-1141	215	7	o.	o.	PROPN
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m-1141	216	7	theory	theory	NOUN
m-1141	216	8	,	,	PUNCT
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m-1141	216	10	.	.	PROPN
m-1141	216	11	9	9	NUM
m-1141	216	12	,	,	PUNCT
m-1141	216	13	no	no	INTJ
m-1141	216	14	.	.	NOUN
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m-1141	216	16	,	,	PUNCT
m-1141	216	17	pp	pp	ADJ
m-1141	216	18	.	.	PUNCT
m-1141	217	1	90	90	NUM
m-1141	217	2	�	�	NOUN
m-1141	217	3	100	100	NUM
m-1141	217	4	,	,	PUNCT
m-1141	217	5	2023	2023	NUM
m-1141	217	6	.	.	PUNCT
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m-1141	218	2	.	.	PUNCT
m-1141	219	1	d.o.i	d.o.i	NOUN
m-1141	219	2	:	:	PUNCT
m-1141	219	3	10.56201	10.56201	NUM
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m-1141	220	3	]	]	X
m-1141	220	4	o.	o.	PROPN
m-1141	220	5	g.	g.	PROPN
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m-1141	220	18	in	in	ADP
m-1141	220	19	quasi	quasi	ADJ
m-1141	220	20	-	-	ADJ
m-1141	220	21	idempotent	idempotent	ADJ
m-1141	220	22	generated	generate	VERB
m-1141	220	23	semigroup	semigroup	PROPN
m-1141	220	24	,	,	PUNCT
m-1141	220	25	international	international	ADJ
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m-1141	220	32	,	,	PUNCT
m-1141	220	33	vol	vol	NOUN
m-1141	220	34	.	.	PROPN
m-1141	220	35	8	8	NUM
m-1141	220	36	,	,	PUNCT
m-1141	220	37	no	no	INTJ
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m-1141	220	40	,	,	PUNCT
m-1141	220	41	pp	pp	ADJ
m-1141	220	42	.	.	PUNCT
m-1141	221	1	2456	2456	NUM
m-1141	221	2	�	�	NOUN
m-1141	221	3	3315	3315	NUM
m-1141	221	4	,	,	PUNCT
m-1141	221	5	2023	2023	NUM
m-1141	221	6	.	.	PUNCT
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m-1141	222	3	]	]	PUNCT
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m-1141	222	5	d.	d.	PROPN
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m-1141	222	14	partial	partial	ADJ
m-1141	222	15	order	order	NOUN
m-1141	222	16	relation	relation	NOUN
m-1141	222	17	on	on	ADP
m-1141	222	18	semiprime	semiprime	NOUN
m-1141	222	19	rings	ring	NOUN
m-1141	222	20	and	and	CCONJ
m-1141	222	21	on	on	ADP
m-1141	222	22	semigroups	semigroup	NOUN
m-1141	222	23	,	,	PUNCT
m-1141	222	24	semigroup	semigroup	PROPN
m-1141	222	25	forum	forum	PROPN
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m-1141	222	27	.	.	PROPN
m-1141	223	1	16	16	NUM
m-1141	223	2	,	,	PUNCT
m-1141	223	3	pp	pp	ADJ
m-1141	223	4	.	.	PUNCT
m-1141	224	1	133	133	NUM
m-1141	224	2	�	�	NOUN
m-1141	224	3	140	140	NUM
m-1141	224	4	,	,	PUNCT
m-1141	224	5	1978	1978	NUM
m-1141	224	6	.	.	PUNCT
m-1141	225	1	[	[	X
m-1141	225	2	21	21	NUM
m-1141	225	3	]	]	PUNCT
m-1141	225	4	x.	x.	NOUN
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m-1141	225	6	,	,	PUNCT
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m-1141	225	8	the	the	DET
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m-1141	225	10	of	of	ADP
m-1141	225	11	semigroups	semigroup	NOUN
m-1141	225	12	whose	whose	DET
m-1141	225	13	regular	regular	ADJ
m-1141	225	14	elements	element	NOUN
m-1141	225	15	are	be	AUX
m-1141	225	16	completely	completely	ADV
m-1141	225	17	regular	regular	ADJ
m-1141	225	18	.	.	PUNCT
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m-1141	225	20	.	.	PUNCT
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m-1141	226	2	.	.	PUNCT
m-1141	227	1	[	[	X
m-1141	227	2	22	22	NUM
m-1141	227	3	]	]	PUNCT
m-1141	227	4	on	on	ADP
m-1141	227	5	quasi	quasi	ADJ
m-1141	227	6	-	-	ADJ
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m-1141	227	9	.	.	PUNCT
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m-1141	229	1	v.	v.	PROPN
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m-1141	229	3	,	,	PUNCT
m-1141	229	4	arxiv	arxiv	PROPN
m-1141	229	5	:	:	PUNCT
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m-1141	230	1	[	[	X
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m-1141	231	5	mathematics	mathematics	PROPN
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m-1141	231	7	issn	issn	PROPN
m-1141	231	8	:	:	PUNCT
m-1141	231	9	2992	2992	NUM
m-1141	231	10	-	-	SYM
m-1141	231	11	4421	4421	NUM
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m-1141	231	13	volume	volume	NOUN
m-1141	231	14	08	08	NUM
m-1141	232	1	|	|	ADV
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m-1141	232	4	|	|	CCONJ
m-1141	232	5	september	september	PROPN
m-1141	232	6	2025	2025	NUM
m-1141	232	7	|	|	ADV
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m-1141	232	9	9	9	NUM
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m-1141	232	16	a	a	DET
m-1141	232	17	semigroup	semigroup	ADJ
m-1141	232	18	results	result	NOUN
m-1141	232	19	conclusion	conclusion	VERB