id	sid	tid	token	lemma	pos
ejpam-5514	1	1	european	european	PROPN
ejpam-5514	1	2	journal	journal	PROPN
ejpam-5514	1	3	of	of	ADP
ejpam-5514	1	4	pure	pure	ADJ
ejpam-5514	1	5	and	and	CCONJ
ejpam-5514	1	6	applied	applied	ADJ
ejpam-5514	1	7	mathematics	mathematic	NOUN
ejpam-5514	1	8	2025	2025	NUM
ejpam-5514	1	9	,	,	PUNCT
ejpam-5514	1	10	vol	vol	NOUN
ejpam-5514	1	11	.	.	PROPN
ejpam-5514	1	12	18	18	NUM
ejpam-5514	1	13	,	,	PUNCT
ejpam-5514	1	14	issue	issue	NOUN
ejpam-5514	1	15	1	1	NUM
ejpam-5514	1	16	,	,	PUNCT
ejpam-5514	1	17	article	article	NOUN
ejpam-5514	1	18	number	number	NOUN
ejpam-5514	1	19	5514	5514	NUM
ejpam-5514	1	20	issn	issn	PROPN
ejpam-5514	1	21	1307	1307	NUM
ejpam-5514	1	22	-	-	SYM
ejpam-5514	1	23	5543	5543	NUM
ejpam-5514	1	24	–	–	PUNCT
ejpam-5514	1	25	ejpam.com	ejpam.com	X
ejpam-5514	1	26	published	publish	VERB
ejpam-5514	1	27	by	by	ADP
ejpam-5514	1	28	new	new	PROPN
ejpam-5514	1	29	york	york	PROPN
ejpam-5514	1	30	business	business	PROPN
ejpam-5514	1	31	global	global	ADJ
ejpam-5514	1	32	exploring	explore	VERB
ejpam-5514	1	33	ideals	ideal	NOUN
ejpam-5514	1	34	:	:	PUNCT
ejpam-5514	1	35	an	an	DET
ejpam-5514	1	36	analysis	analysis	NOUN
ejpam-5514	1	37	through	through	ADP
ejpam-5514	1	38	rough	rough	ADJ
ejpam-5514	1	39	set	set	NOUN
ejpam-5514	1	40	theory	theory	NOUN
ejpam-5514	1	41	with	with	ADP
ejpam-5514	1	42	examples	example	NOUN
ejpam-5514	1	43	anand	anand	PROPN
ejpam-5514	1	44	prakash1	prakash1	PROPN
ejpam-5514	1	45	,	,	PUNCT
ejpam-5514	1	46	,	,	PUNCT
ejpam-5514	1	47	rahul	rahul	PROPN
ejpam-5514	1	48	shukla2∗	shukla2∗	PROPN
ejpam-5514	1	49	1	1	NUM
ejpam-5514	1	50	department	department	NOUN
ejpam-5514	1	51	of	of	ADP
ejpam-5514	1	52	mathematics	mathematic	NOUN
ejpam-5514	1	53	,	,	PUNCT
ejpam-5514	1	54	shri	shri	PROPN
ejpam-5514	1	55	vaishnav	vaishnav	PROPN
ejpam-5514	1	56	vidyapeeth	vidyapeeth	PROPN
ejpam-5514	1	57	vishwavidyalaya	vishwavidyalaya	PROPN
ejpam-5514	1	58	,	,	PUNCT
ejpam-5514	1	59	indore	indore	PROPN
ejpam-5514	1	60	,	,	PUNCT
ejpam-5514	1	61	mp	mp	PROPN
ejpam-5514	1	62	,	,	PUNCT
ejpam-5514	1	63	india	india	PROPN
ejpam-5514	1	64	,	,	PUNCT
ejpam-5514	1	65	pin-453111	pin-453111	NOUN
ejpam-5514	1	66	2	2	NUM
ejpam-5514	1	67	department	department	NOUN
ejpam-5514	1	68	of	of	ADP
ejpam-5514	1	69	mathematical	mathematical	ADJ
ejpam-5514	1	70	sciences	sciences	PROPN
ejpam-5514	1	71	and	and	CCONJ
ejpam-5514	1	72	computing	computing	NOUN
ejpam-5514	1	73	,	,	PUNCT
ejpam-5514	1	74	walter	walter	PROPN
ejpam-5514	1	75	sisulu	sisulu	PROPN
ejpam-5514	1	76	university	university	PROPN
ejpam-5514	1	77	,	,	PUNCT
ejpam-5514	1	78	mthatha	mthatha	NOUN
ejpam-5514	1	79	5117	5117	NUM
ejpam-5514	1	80	,	,	PUNCT
ejpam-5514	1	81	south	south	PROPN
ejpam-5514	1	82	africa	africa	PROPN
ejpam-5514	1	83	abstract	abstract	PROPN
ejpam-5514	1	84	.	.	PUNCT
ejpam-5514	2	1	rough	rough	ADJ
ejpam-5514	2	2	set	set	PROPN
ejpam-5514	2	3	theory	theory	NOUN
ejpam-5514	2	4	,	,	PUNCT
ejpam-5514	2	5	a	a	DET
ejpam-5514	2	6	powerful	powerful	ADJ
ejpam-5514	2	7	mathematical	mathematical	ADJ
ejpam-5514	2	8	framework	framework	NOUN
ejpam-5514	2	9	,	,	PUNCT
ejpam-5514	2	10	excels	excel	VERB
ejpam-5514	2	11	in	in	ADP
ejpam-5514	2	12	addressing	address	VERB
ejpam-5514	2	13	uncertainty	uncertainty	NOUN
ejpam-5514	2	14	and	and	CCONJ
ejpam-5514	2	15	imprecision	imprecision	NOUN
ejpam-5514	2	16	.	.	PUNCT
ejpam-5514	3	1	in	in	ADP
ejpam-5514	3	2	this	this	DET
ejpam-5514	3	3	study	study	NOUN
ejpam-5514	3	4	,	,	PUNCT
ejpam-5514	3	5	we	we	PRON
ejpam-5514	3	6	explore	explore	VERB
ejpam-5514	3	7	the	the	DET
ejpam-5514	3	8	analysis	analysis	NOUN
ejpam-5514	3	9	and	and	CCONJ
ejpam-5514	3	10	provide	provide	VERB
ejpam-5514	3	11	examples	example	NOUN
ejpam-5514	3	12	of	of	ADP
ejpam-5514	3	13	ideals	ideal	NOUN
ejpam-5514	3	14	and	and	CCONJ
ejpam-5514	3	15	prime	prime	ADJ
ejpam-5514	3	16	ideals	ideal	NOUN
ejpam-5514	3	17	within	within	ADP
ejpam-5514	3	18	imprecise	imprecise	ADJ
ejpam-5514	3	19	scenarios	scenario	NOUN
ejpam-5514	3	20	,	,	PUNCT
ejpam-5514	3	21	utilizing	utilize	VERB
ejpam-5514	3	22	rough	rough	ADJ
ejpam-5514	3	23	set	set	NOUN
ejpam-5514	3	24	theory	theory	NOUN
ejpam-5514	3	25	to	to	PART
ejpam-5514	3	26	quantify	quantify	VERB
ejpam-5514	3	27	and	and	CCONJ
ejpam-5514	3	28	navigate	navigate	VERB
ejpam-5514	3	29	the	the	DET
ejpam-5514	3	30	inherent	inherent	ADJ
ejpam-5514	3	31	imprecision	imprecision	NOUN
ejpam-5514	3	32	and	and	CCONJ
ejpam-5514	3	33	roughness	roughness	NOUN
ejpam-5514	3	34	within	within	ADP
ejpam-5514	3	35	these	these	DET
ejpam-5514	3	36	algebraic	algebraic	ADJ
ejpam-5514	3	37	structures	structure	NOUN
ejpam-5514	3	38	.	.	PUNCT
ejpam-5514	4	1	this	this	DET
ejpam-5514	4	2	research	research	NOUN
ejpam-5514	4	3	contributes	contribute	VERB
ejpam-5514	4	4	to	to	ADP
ejpam-5514	4	5	our	our	PRON
ejpam-5514	4	6	comprehension	comprehension	NOUN
ejpam-5514	4	7	of	of	ADP
ejpam-5514	4	8	how	how	SCONJ
ejpam-5514	4	9	rough	rough	ADJ
ejpam-5514	4	10	set	set	NOUN
ejpam-5514	4	11	theory	theory	NOUN
ejpam-5514	4	12	effectively	effectively	ADV
ejpam-5514	4	13	manages	manage	VERB
ejpam-5514	4	14	imprecision	imprecision	NOUN
ejpam-5514	4	15	in	in	ADP
ejpam-5514	4	16	algebraic	algebraic	ADJ
ejpam-5514	4	17	contexts	contexts	NOUN
ejpam-5514	4	18	.	.	PUNCT
ejpam-5514	5	1	2020	2020	NUM
ejpam-5514	5	2	mathematics	mathematic	NOUN
ejpam-5514	5	3	subject	subject	NOUN
ejpam-5514	5	4	classifications	classification	NOUN
ejpam-5514	5	5	:	:	PUNCT
ejpam-5514	5	6	ams	am	NOUN
ejpam-5514	5	7	classification	classification	NOUN
ejpam-5514	5	8	codes	code	VERB
ejpam-5514	5	9	key	key	ADJ
ejpam-5514	5	10	words	word	NOUN
ejpam-5514	5	11	and	and	CCONJ
ejpam-5514	5	12	phrases	phrase	NOUN
ejpam-5514	5	13	:	:	PUNCT
ejpam-5514	5	14	rough	rough	ADJ
ejpam-5514	5	15	set	set	NOUN
ejpam-5514	5	16	,	,	PUNCT
ejpam-5514	5	17	ideals	ideal	NOUN
ejpam-5514	5	18	,	,	PUNCT
ejpam-5514	5	19	upper	upper	ADJ
ejpam-5514	5	20	and	and	CCONJ
ejpam-5514	5	21	lower	low	ADJ
ejpam-5514	5	22	approximation	approximation	NOUN
ejpam-5514	5	23	1	1	NUM
ejpam-5514	5	24	.	.	PUNCT
ejpam-5514	6	1	introduction	introduction	NOUN
ejpam-5514	6	2	real	real	ADJ
ejpam-5514	6	3	-	-	PUNCT
ejpam-5514	6	4	world	world	NOUN
ejpam-5514	6	5	problems	problem	NOUN
ejpam-5514	6	6	are	be	AUX
ejpam-5514	6	7	inherently	inherently	ADV
ejpam-5514	6	8	plagued	plague	VERB
ejpam-5514	6	9	by	by	ADP
ejpam-5514	6	10	imprecision	imprecision	NOUN
ejpam-5514	6	11	,	,	PUNCT
ejpam-5514	6	12	necessitating	necessitate	VERB
ejpam-5514	6	13	the	the	DET
ejpam-5514	6	14	employment	employment	NOUN
ejpam-5514	6	15	of	of	ADP
ejpam-5514	6	16	mathematical	mathematical	ADJ
ejpam-5514	6	17	tools	tool	NOUN
ejpam-5514	6	18	capable	capable	ADJ
ejpam-5514	6	19	of	of	ADP
ejpam-5514	6	20	handling	handle	VERB
ejpam-5514	6	21	uncertainty	uncertainty	NOUN
ejpam-5514	6	22	.	.	PUNCT
ejpam-5514	7	1	various	various	ADJ
ejpam-5514	7	2	mathematical	mathematical	ADJ
ejpam-5514	7	3	theories	theory	NOUN
ejpam-5514	7	4	,	,	PUNCT
ejpam-5514	7	5	including	include	VERB
ejpam-5514	7	6	fuzzy	fuzzy	ADJ
ejpam-5514	7	7	set	set	NOUN
ejpam-5514	7	8	theory	theory	NOUN
ejpam-5514	7	9	,	,	PUNCT
ejpam-5514	7	10	soft	soft	ADJ
ejpam-5514	7	11	set	set	NOUN
ejpam-5514	7	12	theory	theory	NOUN
ejpam-5514	7	13	,	,	PUNCT
ejpam-5514	7	14	and	and	CCONJ
ejpam-5514	7	15	rough	rough	ADJ
ejpam-5514	7	16	set	set	NOUN
ejpam-5514	7	17	theory	theory	NOUN
ejpam-5514	7	18	,	,	PUNCT
ejpam-5514	7	19	have	have	AUX
ejpam-5514	7	20	emerged	emerge	VERB
ejpam-5514	7	21	to	to	PART
ejpam-5514	7	22	address	address	VERB
ejpam-5514	7	23	these	these	DET
ejpam-5514	7	24	imprecise	imprecise	ADJ
ejpam-5514	7	25	scenarios	scenario	NOUN
ejpam-5514	7	26	.	.	PUNCT
ejpam-5514	8	1	among	among	ADP
ejpam-5514	8	2	these	these	PRON
ejpam-5514	8	3	,	,	PUNCT
ejpam-5514	8	4	rough	rough	ADJ
ejpam-5514	8	5	set	set	NOUN
ejpam-5514	8	6	theory	theory	NOUN
ejpam-5514	8	7	stands	stand	VERB
ejpam-5514	8	8	out	out	ADP
ejpam-5514	8	9	due	due	ADP
ejpam-5514	8	10	to	to	ADP
ejpam-5514	8	11	its	its	PRON
ejpam-5514	8	12	unique	unique	ADJ
ejpam-5514	8	13	feature	feature	NOUN
ejpam-5514	8	14	of	of	ADP
ejpam-5514	8	15	requiring	require	VERB
ejpam-5514	8	16	only	only	ADV
ejpam-5514	8	17	the	the	DET
ejpam-5514	8	18	dataset	dataset	NOUN
ejpam-5514	8	19	itself	itself	PRON
ejpam-5514	8	20	,	,	PUNCT
ejpam-5514	8	21	without	without	ADP
ejpam-5514	8	22	any	any	DET
ejpam-5514	8	23	prior	prior	ADJ
ejpam-5514	8	24	information	information	NOUN
ejpam-5514	8	25	,	,	PUNCT
ejpam-5514	8	26	to	to	PART
ejpam-5514	8	27	analyze	analyze	VERB
ejpam-5514	8	28	uncertainty	uncertainty	NOUN
ejpam-5514	8	29	.	.	PUNCT
ejpam-5514	9	1	in	in	ADP
ejpam-5514	9	2	the	the	DET
ejpam-5514	9	3	realm	realm	NOUN
ejpam-5514	9	4	of	of	ADP
ejpam-5514	9	5	mathematics	mathematic	NOUN
ejpam-5514	9	6	and	and	CCONJ
ejpam-5514	9	7	its	its	PRON
ejpam-5514	9	8	practical	practical	ADJ
ejpam-5514	9	9	applications	application	NOUN
ejpam-5514	9	10	,	,	PUNCT
ejpam-5514	9	11	the	the	DET
ejpam-5514	9	12	study	study	NOUN
ejpam-5514	9	13	of	of	ADP
ejpam-5514	9	14	ideals	ideal	NOUN
ejpam-5514	9	15	and	and	CCONJ
ejpam-5514	9	16	prime	prime	ADJ
ejpam-5514	9	17	ideals	ideal	NOUN
ejpam-5514	9	18	assumes	assume	VERB
ejpam-5514	9	19	a	a	DET
ejpam-5514	9	20	foundational	foundational	ADJ
ejpam-5514	9	21	role	role	NOUN
ejpam-5514	9	22	in	in	ADP
ejpam-5514	9	23	understanding	understand	VERB
ejpam-5514	9	24	algebraic	algebraic	ADJ
ejpam-5514	9	25	structures	structure	NOUN
ejpam-5514	9	26	.	.	PUNCT
ejpam-5514	10	1	these	these	DET
ejpam-5514	10	2	concepts	concept	NOUN
ejpam-5514	10	3	,	,	PUNCT
ejpam-5514	10	4	originating	originate	VERB
ejpam-5514	10	5	from	from	ADP
ejpam-5514	10	6	abstract	abstract	ADJ
ejpam-5514	10	7	algebra	algebra	NOUN
ejpam-5514	10	8	,	,	PUNCT
ejpam-5514	10	9	possess	possess	VERB
ejpam-5514	10	10	far	far	ADV
ejpam-5514	10	11	-	-	PUNCT
ejpam-5514	10	12	reaching	reach	VERB
ejpam-5514	10	13	implications	implication	NOUN
ejpam-5514	10	14	across	across	ADP
ejpam-5514	10	15	various	various	ADJ
ejpam-5514	10	16	domains	domain	NOUN
ejpam-5514	10	17	,	,	PUNCT
ejpam-5514	10	18	including	include	VERB
ejpam-5514	10	19	number	number	NOUN
ejpam-5514	10	20	theory	theory	NOUN
ejpam-5514	10	21	,	,	PUNCT
ejpam-5514	10	22	ring	ring	NOUN
ejpam-5514	10	23	theory	theory	NOUN
ejpam-5514	10	24	,	,	PUNCT
ejpam-5514	10	25	and	and	CCONJ
ejpam-5514	10	26	algebraic	algebraic	ADJ
ejpam-5514	10	27	geometry	geometry	NOUN
ejpam-5514	10	28	.	.	PUNCT
ejpam-5514	11	1	however	however	ADV
ejpam-5514	11	2	,	,	PUNCT
ejpam-5514	11	3	when	when	SCONJ
ejpam-5514	11	4	confronted	confront	VERB
ejpam-5514	11	5	with	with	ADP
ejpam-5514	11	6	real	real	ADJ
ejpam-5514	11	7	-	-	PUNCT
ejpam-5514	11	8	world	world	NOUN
ejpam-5514	11	9	data	datum	NOUN
ejpam-5514	11	10	and	and	CCONJ
ejpam-5514	11	11	imprecise	imprecise	ADJ
ejpam-5514	11	12	information	information	NOUN
ejpam-5514	11	13	,	,	PUNCT
ejpam-5514	11	14	traditional	traditional	ADJ
ejpam-5514	11	15	algebraic	algebraic	ADJ
ejpam-5514	11	16	methods	method	NOUN
ejpam-5514	11	17	may	may	AUX
ejpam-5514	11	18	fall	fall	VERB
ejpam-5514	11	19	short	short	ADV
ejpam-5514	11	20	in	in	ADP
ejpam-5514	11	21	providing	provide	VERB
ejpam-5514	11	22	meaningful	meaningful	ADJ
ejpam-5514	11	23	insights	insight	NOUN
ejpam-5514	11	24	.	.	PUNCT
ejpam-5514	12	1	this	this	PRON
ejpam-5514	12	2	is	be	AUX
ejpam-5514	12	3	where	where	SCONJ
ejpam-5514	12	4	rough	rough	ADJ
ejpam-5514	12	5	set	set	NOUN
ejpam-5514	12	6	theory	theory	NOUN
ejpam-5514	12	7	,	,	PUNCT
ejpam-5514	12	8	a	a	DET
ejpam-5514	12	9	branch	branch	NOUN
ejpam-5514	12	10	of	of	ADP
ejpam-5514	12	11	mathematics	mathematic	NOUN
ejpam-5514	12	12	tailored	tailor	VERB
ejpam-5514	12	13	for	for	ADP
ejpam-5514	12	14	handling	handle	VERB
ejpam-5514	12	15	uncertainty	uncertainty	NOUN
ejpam-5514	12	16	and	and	CCONJ
ejpam-5514	12	17	imprecision	imprecision	NOUN
ejpam-5514	12	18	,	,	PUNCT
ejpam-5514	12	19	comes	come	VERB
ejpam-5514	12	20	into	into	ADP
ejpam-5514	12	21	play	play	NOUN
ejpam-5514	12	22	.	.	PUNCT
ejpam-5514	13	1	rough	rough	ADJ
ejpam-5514	13	2	set	set	PROPN
ejpam-5514	13	3	theory	theory	NOUN
ejpam-5514	13	4	,	,	PUNCT
ejpam-5514	13	5	introduced	introduce	VERB
ejpam-5514	13	6	by	by	ADP
ejpam-5514	13	7	polish	polish	ADJ
ejpam-5514	13	8	mathematician	mathematician	ADJ
ejpam-5514	13	9	zdzis	zdzis	PROPN
ejpam-5514	13	10	law	law	NOUN
ejpam-5514	13	11	pawlak	pawlak	ADJ
ejpam-5514	13	12	in	in	ADP
ejpam-5514	13	13	the	the	DET
ejpam-5514	13	14	1980s	1980s	NUM
ejpam-5514	13	15	,	,	PUNCT
ejpam-5514	13	16	provides	provide	VERB
ejpam-5514	13	17	a	a	DET
ejpam-5514	13	18	mathematical	mathematical	ADJ
ejpam-5514	13	19	framework	framework	NOUN
ejpam-5514	13	20	designed	design	VERB
ejpam-5514	13	21	specifically	specifically	ADV
ejpam-5514	13	22	to	to	PART
ejpam-5514	13	23	grapple	grapple	VERB
ejpam-5514	13	24	with	with	ADP
ejpam-5514	13	25	imprecision	imprecision	NOUN
ejpam-5514	13	26	and	and	CCONJ
ejpam-5514	13	27	∗corresponding	∗corresponde	VERB
ejpam-5514	13	28	author	author	NOUN
ejpam-5514	13	29	.	.	PUNCT
ejpam-5514	14	1	doi	doi	NOUN
ejpam-5514	14	2	:	:	PUNCT
ejpam-5514	14	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5514	https://doi.org/10.29020/nybg.ejpam.v18i1.5514	NOUN
ejpam-5514	14	4	email	email	NOUN
ejpam-5514	14	5	addresses	address	NOUN
ejpam-5514	14	6	:	:	PUNCT
ejpam-5514	14	7	anand.p.pal@gmail.com	anand.p.pal@gmail.com	PROPN
ejpam-5514	14	8	(	(	PUNCT
ejpam-5514	14	9	a.	a.	PROPN
ejpam-5514	14	10	prakash	prakash	PROPN
ejpam-5514	14	11	)	)	PUNCT
ejpam-5514	14	12	,	,	PUNCT
ejpam-5514	14	13	rshukla@wsu.ac.za	rshukla@wsu.ac.za	NOUN
ejpam-5514	14	14	(	(	PUNCT
ejpam-5514	14	15	r.	r.	NOUN
ejpam-5514	14	16	shukla	shukla	PROPN
ejpam-5514	14	17	)	)	PUNCT
ejpam-5514	14	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5514	14	19	1	1	NUM
ejpam-5514	14	20	copyright	copyright	NOUN
ejpam-5514	14	21	:	:	PUNCT
ejpam-5514	15	1	©	©	PROPN
ejpam-5514	15	2	2025	2025	NUM
ejpam-5514	15	3	the	the	DET
ejpam-5514	15	4	author(s	author(s	NOUN
ejpam-5514	15	5	)	)	PUNCT
ejpam-5514	15	6	.	.	PUNCT
ejpam-5514	16	1	(	(	PUNCT
ejpam-5514	16	2	cc	cc	NOUN
ejpam-5514	16	3	by	by	ADP
ejpam-5514	16	4	-	-	PUNCT
ejpam-5514	16	5	nc	nc	PROPN
ejpam-5514	16	6	4.0	4.0	NUM
ejpam-5514	16	7	)	)	PUNCT
ejpam-5514	16	8	a.	a.	NOUN
ejpam-5514	16	9	prakash	prakash	PROPN
ejpam-5514	16	10	,	,	PUNCT
ejpam-5514	16	11	r.	r.	PROPN
ejpam-5514	16	12	shukla	shukla	PROPN
ejpam-5514	16	13	/	/	SYM
ejpam-5514	16	14	eur	eur	PROPN
ejpam-5514	16	15	.	.	PUNCT
ejpam-5514	17	1	j.	j.	PROPN
ejpam-5514	17	2	pure	pure	PROPN
ejpam-5514	17	3	appl	appl	PROPN
ejpam-5514	17	4	.	.	PROPN
ejpam-5514	17	5	math	math	PROPN
ejpam-5514	17	6	,	,	PUNCT
ejpam-5514	17	7	18	18	NUM
ejpam-5514	17	8	(	(	PUNCT
ejpam-5514	17	9	1	1	NUM
ejpam-5514	17	10	)	)	PUNCT
ejpam-5514	17	11	(	(	PUNCT
ejpam-5514	17	12	2025	2025	NUM
ejpam-5514	17	13	)	)	PUNCT
ejpam-5514	17	14	,	,	PUNCT
ejpam-5514	17	15	5514	5514	NUM
ejpam-5514	17	16	2	2	NUM
ejpam-5514	17	17	of	of	ADP
ejpam-5514	17	18	9	9	NUM
ejpam-5514	17	19	uncertainty	uncertainty	NOUN
ejpam-5514	17	20	in	in	ADP
ejpam-5514	17	21	data	datum	NOUN
ejpam-5514	17	22	[	[	X
ejpam-5514	17	23	14	14	NUM
ejpam-5514	17	24	]	]	PUNCT
ejpam-5514	17	25	.	.	PUNCT
ejpam-5514	18	1	at	at	ADP
ejpam-5514	18	2	its	its	PRON
ejpam-5514	18	3	core	core	NOUN
ejpam-5514	18	4	,	,	PUNCT
ejpam-5514	18	5	rough	rough	ADJ
ejpam-5514	18	6	set	set	NOUN
ejpam-5514	18	7	theory	theory	NOUN
ejpam-5514	18	8	deals	deal	VERB
ejpam-5514	18	9	with	with	ADP
ejpam-5514	18	10	approximations	approximation	NOUN
ejpam-5514	18	11	and	and	CCONJ
ejpam-5514	18	12	object	object	VERB
ejpam-5514	18	13	classification	classification	NOUN
ejpam-5514	18	14	based	base	VERB
ejpam-5514	18	15	on	on	ADP
ejpam-5514	18	16	available	available	ADJ
ejpam-5514	18	17	information	information	NOUN
ejpam-5514	18	18	.	.	PUNCT
ejpam-5514	19	1	the	the	DET
ejpam-5514	19	2	central	central	ADJ
ejpam-5514	19	3	concept	concept	NOUN
ejpam-5514	19	4	in	in	ADP
ejpam-5514	19	5	rough	rough	ADJ
ejpam-5514	19	6	set	set	NOUN
ejpam-5514	19	7	theory	theory	NOUN
ejpam-5514	19	8	is	be	AUX
ejpam-5514	19	9	the	the	DET
ejpam-5514	19	10	notion	notion	NOUN
ejpam-5514	19	11	of	of	ADP
ejpam-5514	19	12	a	a	DET
ejpam-5514	19	13	rough	rough	ADJ
ejpam-5514	19	14	set	set	NOUN
ejpam-5514	19	15	,	,	PUNCT
ejpam-5514	19	16	characterized	characterize	VERB
ejpam-5514	19	17	by	by	ADP
ejpam-5514	19	18	two	two	NUM
ejpam-5514	19	19	sets	set	NOUN
ejpam-5514	19	20	:	:	PUNCT
ejpam-5514	19	21	the	the	DET
ejpam-5514	19	22	lower	low	ADJ
ejpam-5514	19	23	approximation	approximation	NOUN
ejpam-5514	19	24	(	(	PUNCT
ejpam-5514	19	25	containing	contain	VERB
ejpam-5514	19	26	certain	certain	ADJ
ejpam-5514	19	27	information	information	NOUN
ejpam-5514	19	28	)	)	PUNCT
ejpam-5514	19	29	and	and	CCONJ
ejpam-5514	19	30	the	the	DET
ejpam-5514	19	31	upper	upper	ADJ
ejpam-5514	19	32	approximation	approximation	NOUN
ejpam-5514	19	33	(	(	PUNCT
ejpam-5514	19	34	containing	contain	VERB
ejpam-5514	19	35	potential	potential	ADJ
ejpam-5514	19	36	information	information	NOUN
ejpam-5514	19	37	)	)	PUNCT
ejpam-5514	19	38	.	.	PUNCT
ejpam-5514	20	1	these	these	DET
ejpam-5514	20	2	sets	set	NOUN
ejpam-5514	20	3	are	be	AUX
ejpam-5514	20	4	used	use	VERB
ejpam-5514	20	5	to	to	PART
ejpam-5514	20	6	define	define	VERB
ejpam-5514	20	7	boundaries	boundary	NOUN
ejpam-5514	20	8	and	and	CCONJ
ejpam-5514	20	9	make	make	VERB
ejpam-5514	20	10	decisions	decision	NOUN
ejpam-5514	20	11	when	when	SCONJ
ejpam-5514	20	12	faced	face	VERB
ejpam-5514	20	13	with	with	ADP
ejpam-5514	20	14	imprecise	imprecise	ADJ
ejpam-5514	20	15	data	datum	NOUN
ejpam-5514	20	16	.	.	PUNCT
ejpam-5514	21	1	developments	development	NOUN
ejpam-5514	21	2	in	in	ADP
ejpam-5514	21	3	rough	rough	ADJ
ejpam-5514	21	4	set	set	NOUN
ejpam-5514	21	5	theory	theory	NOUN
ejpam-5514	21	6	,	,	PUNCT
ejpam-5514	21	7	spanning	span	VERB
ejpam-5514	21	8	advancements	advancement	NOUN
ejpam-5514	21	9	in	in	ADP
ejpam-5514	21	10	pure	pure	ADJ
ejpam-5514	21	11	mathematics	mathematic	NOUN
ejpam-5514	21	12	,	,	PUNCT
ejpam-5514	21	13	the	the	DET
ejpam-5514	21	14	establishment	establishment	NOUN
ejpam-5514	21	15	of	of	ADP
ejpam-5514	21	16	algebraic	algebraic	ADJ
ejpam-5514	21	17	foundations	foundation	NOUN
ejpam-5514	21	18	,	,	PUNCT
ejpam-5514	21	19	and	and	CCONJ
ejpam-5514	21	20	practical	practical	ADJ
ejpam-5514	21	21	applications	application	NOUN
ejpam-5514	21	22	,	,	PUNCT
ejpam-5514	21	23	have	have	AUX
ejpam-5514	21	24	been	be	AUX
ejpam-5514	21	25	rapidly	rapidly	ADV
ejpam-5514	21	26	progressing	progress	VERB
ejpam-5514	21	27	since	since	SCONJ
ejpam-5514	21	28	its	its	PRON
ejpam-5514	21	29	inception	inception	NOUN
ejpam-5514	21	30	.	.	PUNCT
ejpam-5514	22	1	pawlak	pawlak	ADJ
ejpam-5514	22	2	’s	’s	PART
ejpam-5514	22	3	seminal	seminal	ADJ
ejpam-5514	22	4	work	work	NOUN
ejpam-5514	22	5	[	[	X
ejpam-5514	22	6	16	16	NUM
ejpam-5514	22	7	]	]	PUNCT
ejpam-5514	22	8	laid	lay	VERB
ejpam-5514	22	9	the	the	DET
ejpam-5514	22	10	primary	primary	ADJ
ejpam-5514	22	11	algebraic	algebraic	ADJ
ejpam-5514	22	12	groundwork	groundwork	NOUN
ejpam-5514	22	13	for	for	ADP
ejpam-5514	22	14	rough	rough	ADJ
ejpam-5514	22	15	sets	set	NOUN
ejpam-5514	22	16	,	,	PUNCT
ejpam-5514	22	17	culminating	culminate	VERB
ejpam-5514	22	18	in	in	ADP
ejpam-5514	22	19	the	the	DET
ejpam-5514	22	20	exploration	exploration	NOUN
ejpam-5514	22	21	of	of	ADP
ejpam-5514	22	22	various	various	ADJ
ejpam-5514	22	23	algebraic	algebraic	ADJ
ejpam-5514	22	24	properties	property	NOUN
ejpam-5514	22	25	and	and	CCONJ
ejpam-5514	22	26	theories	theory	NOUN
ejpam-5514	22	27	in	in	ADP
ejpam-5514	22	28	this	this	DET
ejpam-5514	22	29	field	field	NOUN
ejpam-5514	23	1	[	[	X
ejpam-5514	23	2	6	6	NUM
ejpam-5514	23	3	]	]	PUNCT
ejpam-5514	23	4	.	.	PUNCT
ejpam-5514	24	1	this	this	DET
ejpam-5514	24	2	paper	paper	NOUN
ejpam-5514	24	3	references	reference	VERB
ejpam-5514	24	4	several	several	ADJ
ejpam-5514	24	5	related	relate	VERB
ejpam-5514	24	6	works	work	NOUN
ejpam-5514	24	7	to	to	PART
ejpam-5514	24	8	underscore	underscore	VERB
ejpam-5514	24	9	the	the	DET
ejpam-5514	24	10	significance	significance	NOUN
ejpam-5514	24	11	of	of	ADP
ejpam-5514	24	12	rough	rough	ADJ
ejpam-5514	24	13	set	set	NOUN
ejpam-5514	24	14	theory	theory	NOUN
ejpam-5514	24	15	.	.	PUNCT
ejpam-5514	25	1	pawlak	pawlak	PROPN
ejpam-5514	25	2	himself	himself	PRON
ejpam-5514	25	3	emphasized	emphasize	VERB
ejpam-5514	25	4	the	the	DET
ejpam-5514	25	5	importance	importance	NOUN
ejpam-5514	25	6	of	of	ADP
ejpam-5514	25	7	rough	rough	ADJ
ejpam-5514	25	8	sets	set	NOUN
ejpam-5514	25	9	[	[	X
ejpam-5514	25	10	15	15	NUM
ejpam-5514	25	11	]	]	PUNCT
ejpam-5514	25	12	.	.	PUNCT
ejpam-5514	26	1	concepts	concept	NOUN
ejpam-5514	26	2	such	such	ADJ
ejpam-5514	26	3	as	as	ADP
ejpam-5514	26	4	rough	rough	ADJ
ejpam-5514	26	5	groups	group	NOUN
ejpam-5514	26	6	,	,	PUNCT
ejpam-5514	26	7	rough	rough	ADJ
ejpam-5514	26	8	subgroups	subgroup	NOUN
ejpam-5514	26	9	,	,	PUNCT
ejpam-5514	26	10	and	and	CCONJ
ejpam-5514	26	11	their	their	PRON
ejpam-5514	26	12	properties	property	NOUN
ejpam-5514	26	13	have	have	AUX
ejpam-5514	26	14	been	be	AUX
ejpam-5514	26	15	thoroughly	thoroughly	ADV
ejpam-5514	26	16	investigated	investigate	VERB
ejpam-5514	26	17	in	in	ADP
ejpam-5514	26	18	[	[	X
ejpam-5514	26	19	12	12	NUM
ejpam-5514	26	20	]	]	PUNCT
ejpam-5514	26	21	.	.	PUNCT
ejpam-5514	27	1	q.	q.	PROPN
ejpam-5514	27	2	xiao	xiao	PROPN
ejpam-5514	27	3	and	and	CCONJ
ejpam-5514	27	4	z.	z.	PROPN
ejpam-5514	27	5	zhang	zhang	PROPN
ejpam-5514	27	6	introduced	introduce	VERB
ejpam-5514	27	7	the	the	DET
ejpam-5514	27	8	concepts	concept	NOUN
ejpam-5514	27	9	of	of	ADP
ejpam-5514	27	10	rough	rough	ADJ
ejpam-5514	27	11	prime	prime	ADJ
ejpam-5514	27	12	ideals	ideal	NOUN
ejpam-5514	27	13	and	and	CCONJ
ejpam-5514	27	14	rough	rough	ADJ
ejpam-5514	27	15	fuzzy	fuzzy	ADJ
ejpam-5514	27	16	prime	prime	ADJ
ejpam-5514	27	17	ideals	ideal	NOUN
ejpam-5514	27	18	[	[	X
ejpam-5514	27	19	20	20	NUM
ejpam-5514	27	20	]	]	PUNCT
ejpam-5514	27	21	.	.	PUNCT
ejpam-5514	28	1	fuzzy	fuzzy	ADJ
ejpam-5514	28	2	ideals	ideal	NOUN
ejpam-5514	28	3	within	within	ADP
ejpam-5514	28	4	a	a	DET
ejpam-5514	28	5	ring	ring	NOUN
ejpam-5514	28	6	have	have	AUX
ejpam-5514	28	7	been	be	AUX
ejpam-5514	28	8	meticulously	meticulously	ADV
ejpam-5514	28	9	explored	explore	VERB
ejpam-5514	28	10	by	by	ADP
ejpam-5514	28	11	t.	t.	PROPN
ejpam-5514	28	12	mukherjee	mukherjee	PROPN
ejpam-5514	28	13	and	and	CCONJ
ejpam-5514	28	14	m.	m.	PROPN
ejpam-5514	28	15	sen	sen	PROPN
ejpam-5514	29	1	[	[	X
ejpam-5514	29	2	13	13	NUM
ejpam-5514	29	3	]	]	PUNCT
ejpam-5514	29	4	.	.	PUNCT
ejpam-5514	30	1	b.	b.	PROPN
ejpam-5514	30	2	davvaz	davvaz	PROPN
ejpam-5514	30	3	explores	explore	NOUN
ejpam-5514	30	4	into	into	ADP
ejpam-5514	30	5	roughness	roughness	NOUN
ejpam-5514	30	6	based	base	VERB
ejpam-5514	30	7	on	on	ADP
ejpam-5514	30	8	fuzzy	fuzzy	ADJ
ejpam-5514	30	9	ideals	ideal	NOUN
ejpam-5514	30	10	and	and	CCONJ
ejpam-5514	30	11	roughness	roughness	NOUN
ejpam-5514	30	12	within	within	ADP
ejpam-5514	30	13	rings	ring	NOUN
ejpam-5514	30	14	[	[	X
ejpam-5514	30	15	7	7	NUM
ejpam-5514	30	16	,	,	PUNCT
ejpam-5514	30	17	8	8	NUM
ejpam-5514	30	18	]	]	PUNCT
ejpam-5514	30	19	.	.	PUNCT
ejpam-5514	31	1	kuroki	kuroki	PROPN
ejpam-5514	31	2	has	have	AUX
ejpam-5514	31	3	conducted	conduct	VERB
ejpam-5514	31	4	research	research	NOUN
ejpam-5514	31	5	on	on	ADP
ejpam-5514	31	6	rough	rough	ADJ
ejpam-5514	31	7	ideals	ideal	NOUN
ejpam-5514	31	8	in	in	ADP
ejpam-5514	31	9	semigroups	semigroup	NOUN
ejpam-5514	31	10	[	[	X
ejpam-5514	31	11	11	11	NUM
ejpam-5514	31	12	]	]	PUNCT
ejpam-5514	31	13	.	.	PUNCT
ejpam-5514	32	1	a.k	a.k	PROPN
ejpam-5514	32	2	.	.	PROPN
ejpam-5514	32	3	sinha	sinha	PROPN
ejpam-5514	32	4	and	and	CCONJ
ejpam-5514	32	5	a.	a.	PROPN
ejpam-5514	32	6	prakash	prakash	PROPN
ejpam-5514	32	7	have	have	AUX
ejpam-5514	32	8	meticulously	meticulously	ADV
ejpam-5514	32	9	examined	examine	VERB
ejpam-5514	32	10	the	the	DET
ejpam-5514	32	11	algebraic	algebraic	ADJ
ejpam-5514	32	12	properties	property	NOUN
ejpam-5514	32	13	of	of	ADP
ejpam-5514	32	14	rough	rough	ADJ
ejpam-5514	32	15	set	set	NOUN
ejpam-5514	32	16	theory	theory	NOUN
ejpam-5514	32	17	[	[	X
ejpam-5514	32	18	18	18	NUM
ejpam-5514	32	19	,	,	PUNCT
ejpam-5514	32	20	19	19	NUM
ejpam-5514	32	21	]	]	PUNCT
ejpam-5514	32	22	.	.	PUNCT
ejpam-5514	33	1	recent	recent	ADJ
ejpam-5514	33	2	extensions	extension	NOUN
ejpam-5514	33	3	of	of	ADP
ejpam-5514	33	4	rough	rough	ADJ
ejpam-5514	33	5	set	set	NOUN
ejpam-5514	33	6	theory	theory	NOUN
ejpam-5514	33	7	can	can	AUX
ejpam-5514	33	8	be	be	AUX
ejpam-5514	33	9	found	find	VERB
ejpam-5514	33	10	in	in	ADP
ejpam-5514	33	11	[	[	X
ejpam-5514	33	12	10	10	NUM
ejpam-5514	33	13	,	,	PUNCT
ejpam-5514	33	14	17	17	NUM
ejpam-5514	33	15	,	,	PUNCT
ejpam-5514	33	16	21	21	NUM
ejpam-5514	33	17	]	]	PUNCT
ejpam-5514	33	18	.	.	PUNCT
ejpam-5514	34	1	the	the	DET
ejpam-5514	34	2	article	article	NOUN
ejpam-5514	34	3	by	by	ADP
ejpam-5514	34	4	hosny	hosny	PROPN
ejpam-5514	34	5	et	et	PROPN
ejpam-5514	34	6	al	al	PROPN
ejpam-5514	34	7	.	.	PUNCT
ejpam-5514	35	1	[	[	X
ejpam-5514	35	2	9	9	NUM
ejpam-5514	35	3	]	]	PUNCT
ejpam-5514	35	4	explores	explore	VERB
ejpam-5514	35	5	how	how	SCONJ
ejpam-5514	35	6	rough	rough	ADJ
ejpam-5514	35	7	set	set	NOUN
ejpam-5514	35	8	theory	theory	NOUN
ejpam-5514	35	9	can	can	AUX
ejpam-5514	35	10	be	be	AUX
ejpam-5514	35	11	enhanced	enhance	VERB
ejpam-5514	35	12	using	use	VERB
ejpam-5514	35	13	ideals	ideal	NOUN
ejpam-5514	35	14	and	and	CCONJ
ejpam-5514	35	15	maximal	maximal	ADJ
ejpam-5514	35	16	right	right	ADJ
ejpam-5514	35	17	neighborhoods	neighborhood	NOUN
ejpam-5514	35	18	.	.	PUNCT
ejpam-5514	36	1	the	the	DET
ejpam-5514	36	2	study	study	NOUN
ejpam-5514	36	3	focuses	focus	VERB
ejpam-5514	36	4	on	on	ADP
ejpam-5514	36	5	minimizing	minimize	VERB
ejpam-5514	36	6	boundary	boundary	ADJ
ejpam-5514	36	7	regions	region	NOUN
ejpam-5514	36	8	and	and	CCONJ
ejpam-5514	36	9	increasing	increase	VERB
ejpam-5514	36	10	classification	classification	NOUN
ejpam-5514	36	11	accuracy	accuracy	NOUN
ejpam-5514	36	12	,	,	PUNCT
ejpam-5514	36	13	particularly	particularly	ADV
ejpam-5514	36	14	in	in	ADP
ejpam-5514	36	15	medical	medical	ADJ
ejpam-5514	36	16	cases	case	NOUN
ejpam-5514	36	17	such	such	ADJ
ejpam-5514	36	18	as	as	ADP
ejpam-5514	36	19	covid-19	covid-19	PROPN
ejpam-5514	36	20	and	and	CCONJ
ejpam-5514	36	21	heart	heart	NOUN
ejpam-5514	36	22	disease	disease	NOUN
ejpam-5514	36	23	diagnosis	diagnosis	NOUN
ejpam-5514	36	24	.	.	PUNCT
ejpam-5514	37	1	similarly	similarly	ADV
ejpam-5514	37	2	,	,	PUNCT
ejpam-5514	37	3	the	the	DET
ejpam-5514	37	4	work	work	NOUN
ejpam-5514	37	5	by	by	ADP
ejpam-5514	37	6	al	al	PROPN
ejpam-5514	37	7	-	-	PUNCT
ejpam-5514	37	8	shami	shami	PROPN
ejpam-5514	37	9	et	et	PROPN
ejpam-5514	37	10	al	al	PROPN
ejpam-5514	37	11	.	.	PUNCT
ejpam-5514	38	1	[	[	X
ejpam-5514	38	2	5	5	NUM
ejpam-5514	38	3	]	]	PUNCT
ejpam-5514	38	4	introduces	introduce	VERB
ejpam-5514	38	5	an	an	DET
ejpam-5514	38	6	advanced	advanced	ADJ
ejpam-5514	38	7	extension	extension	NOUN
ejpam-5514	38	8	of	of	ADP
ejpam-5514	38	9	rough	rough	ADJ
ejpam-5514	38	10	set	set	NOUN
ejpam-5514	38	11	theory	theory	NOUN
ejpam-5514	38	12	by	by	ADP
ejpam-5514	38	13	incorporating	incorporate	VERB
ejpam-5514	38	14	ik	ik	NOUN
ejpam-5514	38	15	-	-	NOUN
ejpam-5514	38	16	neighborhoods	neighborhood	NOUN
ejpam-5514	38	17	,	,	PUNCT
ejpam-5514	38	18	which	which	PRON
ejpam-5514	38	19	combine	combine	VERB
ejpam-5514	38	20	neighborhood	neighborhood	NOUN
ejpam-5514	38	21	systems	system	NOUN
ejpam-5514	38	22	and	and	CCONJ
ejpam-5514	38	23	ideal	ideal	ADJ
ejpam-5514	38	24	structures	structure	NOUN
ejpam-5514	38	25	.	.	PUNCT
ejpam-5514	39	1	this	this	DET
ejpam-5514	39	2	approach	approach	NOUN
ejpam-5514	39	3	enhances	enhance	VERB
ejpam-5514	39	4	decision	decision	NOUN
ejpam-5514	39	5	-	-	PUNCT
ejpam-5514	39	6	making	make	VERB
ejpam-5514	39	7	accuracy	accuracy	NOUN
ejpam-5514	39	8	and	and	CCONJ
ejpam-5514	39	9	is	be	AUX
ejpam-5514	39	10	applied	apply	VERB
ejpam-5514	39	11	to	to	PART
ejpam-5514	39	12	analyze	analyze	VERB
ejpam-5514	39	13	data	datum	NOUN
ejpam-5514	39	14	related	relate	VERB
ejpam-5514	39	15	to	to	ADP
ejpam-5514	39	16	chikungunya	chikungunya	NOUN
ejpam-5514	39	17	disease	disease	NOUN
ejpam-5514	39	18	.	.	PUNCT
ejpam-5514	40	1	their	their	PRON
ejpam-5514	40	2	results	result	NOUN
ejpam-5514	40	3	highlight	highlight	VERB
ejpam-5514	40	4	the	the	DET
ejpam-5514	40	5	effectiveness	effectiveness	NOUN
ejpam-5514	40	6	of	of	ADP
ejpam-5514	40	7	these	these	DET
ejpam-5514	40	8	models	model	NOUN
ejpam-5514	40	9	in	in	ADP
ejpam-5514	40	10	reducing	reduce	VERB
ejpam-5514	40	11	uncertainty	uncertainty	NOUN
ejpam-5514	40	12	,	,	PUNCT
ejpam-5514	40	13	aligning	align	VERB
ejpam-5514	40	14	with	with	ADP
ejpam-5514	40	15	the	the	DET
ejpam-5514	40	16	current	current	ADJ
ejpam-5514	40	17	article	article	NOUN
ejpam-5514	40	18	’s	’s	PART
ejpam-5514	40	19	focus	focus	NOUN
ejpam-5514	40	20	on	on	ADP
ejpam-5514	40	21	the	the	DET
ejpam-5514	40	22	algebraic	algebraic	ADJ
ejpam-5514	40	23	foundations	foundation	NOUN
ejpam-5514	40	24	of	of	ADP
ejpam-5514	40	25	ideals	ideal	NOUN
ejpam-5514	40	26	within	within	ADP
ejpam-5514	40	27	rough	rough	ADJ
ejpam-5514	40	28	set	set	NOUN
ejpam-5514	40	29	theory	theory	NOUN
ejpam-5514	40	30	.	.	PUNCT
ejpam-5514	41	1	further	far	ADV
ejpam-5514	41	2	,	,	PUNCT
ejpam-5514	41	3	al	al	PROPN
ejpam-5514	41	4	-	-	PUNCT
ejpam-5514	41	5	shami	shami	PROPN
ejpam-5514	41	6	et	et	PROPN
ejpam-5514	41	7	al	al	PROPN
ejpam-5514	41	8	.	.	PUNCT
ejpam-5514	42	1	[	[	X
ejpam-5514	42	2	1	1	X
ejpam-5514	42	3	]	]	X
ejpam-5514	42	4	develop	develop	VERB
ejpam-5514	42	5	a	a	DET
ejpam-5514	42	6	novel	novel	ADJ
ejpam-5514	42	7	decision	decision	NOUN
ejpam-5514	42	8	-	-	PUNCT
ejpam-5514	42	9	making	make	VERB
ejpam-5514	42	10	framework	framework	NOUN
ejpam-5514	42	11	that	that	PRON
ejpam-5514	42	12	uses	use	VERB
ejpam-5514	42	13	rough	rough	ADJ
ejpam-5514	42	14	set	set	NOUN
ejpam-5514	42	15	theory	theory	NOUN
ejpam-5514	42	16	with	with	ADP
ejpam-5514	42	17	new	new	ADJ
ejpam-5514	42	18	approximation	approximation	NOUN
ejpam-5514	42	19	models	model	NOUN
ejpam-5514	42	20	based	base	VERB
ejpam-5514	42	21	on	on	ADP
ejpam-5514	42	22	basic	basic	ADJ
ejpam-5514	42	23	-	-	PUNCT
ejpam-5514	42	24	minimal	minimal	ADJ
ejpam-5514	42	25	neighborhoods	neighborhood	NOUN
ejpam-5514	42	26	.	.	PUNCT
ejpam-5514	43	1	their	their	PRON
ejpam-5514	43	2	application	application	NOUN
ejpam-5514	43	3	to	to	ADP
ejpam-5514	43	4	heart	heart	NOUN
ejpam-5514	43	5	failure	failure	NOUN
ejpam-5514	43	6	diagnosis	diagnosis	NOUN
ejpam-5514	43	7	achieves	achieve	VERB
ejpam-5514	43	8	100	100	NUM
ejpam-5514	43	9	%	%	NOUN
ejpam-5514	43	10	accuracy	accuracy	NOUN
ejpam-5514	43	11	,	,	PUNCT
ejpam-5514	43	12	significantly	significantly	ADV
ejpam-5514	43	13	improving	improve	VERB
ejpam-5514	43	14	upon	upon	SCONJ
ejpam-5514	43	15	previous	previous	ADJ
ejpam-5514	43	16	methods	method	NOUN
ejpam-5514	43	17	.	.	PUNCT
ejpam-5514	44	1	this	this	DET
ejpam-5514	44	2	work	work	NOUN
ejpam-5514	44	3	complements	complement	VERB
ejpam-5514	44	4	the	the	DET
ejpam-5514	44	5	present	present	ADJ
ejpam-5514	44	6	study	study	NOUN
ejpam-5514	44	7	by	by	ADP
ejpam-5514	44	8	expanding	expand	VERB
ejpam-5514	44	9	the	the	DET
ejpam-5514	44	10	connection	connection	NOUN
ejpam-5514	44	11	between	between	ADP
ejpam-5514	44	12	rough	rough	ADJ
ejpam-5514	44	13	set	set	NOUN
ejpam-5514	44	14	theory	theory	NOUN
ejpam-5514	44	15	and	and	CCONJ
ejpam-5514	44	16	topological	topological	ADJ
ejpam-5514	44	17	structures	structure	NOUN
ejpam-5514	44	18	,	,	PUNCT
ejpam-5514	44	19	reinforcing	reinforce	VERB
ejpam-5514	44	20	the	the	DET
ejpam-5514	44	21	practical	practical	ADJ
ejpam-5514	44	22	utility	utility	NOUN
ejpam-5514	44	23	of	of	ADP
ejpam-5514	44	24	ideals	ideal	NOUN
ejpam-5514	44	25	-	-	PUNCT
ejpam-5514	44	26	based	base	VERB
ejpam-5514	44	27	methods	method	NOUN
ejpam-5514	44	28	in	in	ADP
ejpam-5514	44	29	medical	medical	ADJ
ejpam-5514	44	30	applications	application	NOUN
ejpam-5514	44	31	.	.	PUNCT
ejpam-5514	45	1	these	these	DET
ejpam-5514	45	2	recent	recent	ADJ
ejpam-5514	45	3	developments	development	NOUN
ejpam-5514	45	4	,	,	PUNCT
ejpam-5514	45	5	alongside	alongside	ADP
ejpam-5514	45	6	the	the	DET
ejpam-5514	45	7	current	current	ADJ
ejpam-5514	45	8	article	article	NOUN
ejpam-5514	45	9	,	,	PUNCT
ejpam-5514	45	10	underscore	underscore	VERB
ejpam-5514	45	11	the	the	DET
ejpam-5514	45	12	potential	potential	NOUN
ejpam-5514	45	13	of	of	ADP
ejpam-5514	45	14	ideals	ideal	NOUN
ejpam-5514	45	15	-	-	PUNCT
ejpam-5514	45	16	based	base	VERB
ejpam-5514	45	17	rough	rough	ADJ
ejpam-5514	45	18	set	set	NOUN
ejpam-5514	45	19	theory	theory	NOUN
ejpam-5514	45	20	in	in	ADP
ejpam-5514	45	21	enhancing	enhance	VERB
ejpam-5514	45	22	decision	decision	NOUN
ejpam-5514	45	23	-	-	PUNCT
ejpam-5514	45	24	making	make	VERB
ejpam-5514	45	25	accuracy	accuracy	NOUN
ejpam-5514	45	26	and	and	CCONJ
ejpam-5514	45	27	addressing	address	VERB
ejpam-5514	45	28	uncertainties	uncertainty	NOUN
ejpam-5514	45	29	[	[	X
ejpam-5514	45	30	3	3	X
ejpam-5514	45	31	]	]	PUNCT
ejpam-5514	45	32	in	in	ADP
ejpam-5514	45	33	various	various	ADJ
ejpam-5514	45	34	fields	field	NOUN
ejpam-5514	45	35	,	,	PUNCT
ejpam-5514	45	36	especially	especially	ADV
ejpam-5514	45	37	in	in	ADP
ejpam-5514	45	38	medical	medical	ADJ
ejpam-5514	45	39	data	datum	NOUN
ejpam-5514	45	40	analysis	analysis	NOUN
ejpam-5514	45	41	.	.	PUNCT
ejpam-5514	46	1	recent	recent	ADJ
ejpam-5514	46	2	advancements	advancement	NOUN
ejpam-5514	46	3	in	in	ADP
ejpam-5514	46	4	rough	rough	ADJ
ejpam-5514	46	5	set	set	NOUN
ejpam-5514	46	6	theory	theory	NOUN
ejpam-5514	46	7	,	,	PUNCT
ejpam-5514	46	8	including	include	VERB
ejpam-5514	46	9	concepts	concept	NOUN
ejpam-5514	46	10	like	like	ADP
ejpam-5514	46	11	somewhere	somewhere	ADV
ejpam-5514	46	12	dense	dense	ADJ
ejpam-5514	46	13	sets	set	NOUN
ejpam-5514	46	14	[	[	X
ejpam-5514	46	15	2	2	NUM
ejpam-5514	46	16	]	]	PUNCT
ejpam-5514	46	17	,	,	PUNCT
ejpam-5514	46	18	containment	containment	NOUN
ejpam-5514	46	19	neighborhoods	neighborhood	NOUN
ejpam-5514	46	20	,	,	PUNCT
ejpam-5514	46	21	and	and	CCONJ
ejpam-5514	46	22	supra	supra	ADJ
ejpam-5514	46	23	-	-	PUNCT
ejpam-5514	46	24	topological	topological	ADJ
ejpam-5514	46	25	frameworks	framework	NOUN
ejpam-5514	46	26	[	[	X
ejpam-5514	46	27	4	4	NUM
ejpam-5514	46	28	]	]	PUNCT
ejpam-5514	46	29	,	,	PUNCT
ejpam-5514	46	30	have	have	AUX
ejpam-5514	46	31	improved	improve	VERB
ejpam-5514	46	32	accuracy	accuracy	NOUN
ejpam-5514	46	33	and	and	CCONJ
ejpam-5514	46	34	decision	decision	NOUN
ejpam-5514	46	35	-	-	PUNCT
ejpam-5514	46	36	making	making	NOUN
ejpam-5514	46	37	,	,	PUNCT
ejpam-5514	46	38	particularly	particularly	ADV
ejpam-5514	46	39	in	in	ADP
ejpam-5514	46	40	medical	medical	ADJ
ejpam-5514	46	41	diagnostics	diagnostic	NOUN
ejpam-5514	46	42	[	[	X
ejpam-5514	46	43	1	1	NUM
ejpam-5514	46	44	]	]	PUNCT
ejpam-5514	46	45	.	.	PUNCT
ejpam-5514	47	1	future	future	ADJ
ejpam-5514	47	2	research	research	NOUN
ejpam-5514	47	3	could	could	AUX
ejpam-5514	47	4	explore	explore	VERB
ejpam-5514	47	5	these	these	DET
ejpam-5514	47	6	enhanced	enhance	VERB
ejpam-5514	47	7	rough	rough	ADJ
ejpam-5514	47	8	set	set	NOUN
ejpam-5514	47	9	models	model	NOUN
ejpam-5514	47	10	in	in	ADP
ejpam-5514	47	11	complex	complex	ADJ
ejpam-5514	47	12	applications	application	NOUN
ejpam-5514	47	13	,	,	PUNCT
ejpam-5514	47	14	such	such	ADJ
ejpam-5514	47	15	as	as	ADP
ejpam-5514	47	16	broader	broad	ADJ
ejpam-5514	47	17	clinical	clinical	ADJ
ejpam-5514	47	18	contexts	context	NOUN
ejpam-5514	47	19	and	and	CCONJ
ejpam-5514	47	20	other	other	ADJ
ejpam-5514	47	21	data	data	NOUN
ejpam-5514	47	22	-	-	PUNCT
ejpam-5514	47	23	driven	drive	VERB
ejpam-5514	47	24	environments	environment	NOUN
ejpam-5514	47	25	,	,	PUNCT
ejpam-5514	47	26	to	to	PART
ejpam-5514	47	27	strengthen	strengthen	VERB
ejpam-5514	47	28	decision	decision	NOUN
ejpam-5514	47	29	frameworks	framework	NOUN
ejpam-5514	47	30	under	under	ADP
ejpam-5514	47	31	uncertain	uncertain	ADJ
ejpam-5514	47	32	data	datum	NOUN
ejpam-5514	47	33	conditions	condition	NOUN
ejpam-5514	47	34	.	.	PUNCT
ejpam-5514	48	1	a.	a.	PROPN
ejpam-5514	48	2	prakash	prakash	PROPN
ejpam-5514	48	3	,	,	PUNCT
ejpam-5514	48	4	r.	r.	PROPN
ejpam-5514	48	5	shukla	shukla	PROPN
ejpam-5514	48	6	/	/	SYM
ejpam-5514	48	7	eur	eur	PROPN
ejpam-5514	48	8	.	.	PUNCT
ejpam-5514	49	1	j.	j.	PROPN
ejpam-5514	49	2	pure	pure	PROPN
ejpam-5514	49	3	appl	appl	PROPN
ejpam-5514	49	4	.	.	PROPN
ejpam-5514	49	5	math	math	PROPN
ejpam-5514	49	6	,	,	PUNCT
ejpam-5514	49	7	18	18	NUM
ejpam-5514	49	8	(	(	PUNCT
ejpam-5514	49	9	1	1	NUM
ejpam-5514	49	10	)	)	PUNCT
ejpam-5514	49	11	(	(	PUNCT
ejpam-5514	49	12	2025	2025	NUM
ejpam-5514	49	13	)	)	PUNCT
ejpam-5514	49	14	,	,	PUNCT
ejpam-5514	49	15	5514	5514	NUM
ejpam-5514	49	16	3	3	NUM
ejpam-5514	49	17	of	of	ADP
ejpam-5514	49	18	9	9	NUM
ejpam-5514	49	19	this	this	DET
ejpam-5514	49	20	introduction	introduction	NOUN
ejpam-5514	49	21	lays	lay	VERB
ejpam-5514	49	22	the	the	DET
ejpam-5514	49	23	groundwork	groundwork	NOUN
ejpam-5514	49	24	for	for	ADP
ejpam-5514	49	25	a	a	DET
ejpam-5514	49	26	comprehensive	comprehensive	ADJ
ejpam-5514	49	27	investigation	investigation	NOUN
ejpam-5514	49	28	into	into	ADP
ejpam-5514	49	29	the	the	DET
ejpam-5514	49	30	application	application	NOUN
ejpam-5514	49	31	of	of	ADP
ejpam-5514	49	32	rough	rough	ADJ
ejpam-5514	49	33	set	set	NOUN
ejpam-5514	49	34	theory	theory	NOUN
ejpam-5514	49	35	to	to	PART
ejpam-5514	49	36	analyze	analyze	VERB
ejpam-5514	49	37	ideals	ideal	NOUN
ejpam-5514	49	38	and	and	CCONJ
ejpam-5514	49	39	prime	prime	ADJ
ejpam-5514	49	40	ideals	ideal	NOUN
ejpam-5514	49	41	within	within	ADP
ejpam-5514	49	42	imprecise	imprecise	ADJ
ejpam-5514	49	43	scenarios	scenario	NOUN
ejpam-5514	49	44	.	.	PUNCT
ejpam-5514	50	1	it	it	PRON
ejpam-5514	50	2	underscores	underscore	VERB
ejpam-5514	50	3	the	the	DET
ejpam-5514	50	4	necessity	necessity	NOUN
ejpam-5514	50	5	of	of	ADP
ejpam-5514	50	6	such	such	DET
ejpam-5514	50	7	an	an	DET
ejpam-5514	50	8	approach	approach	NOUN
ejpam-5514	50	9	in	in	ADP
ejpam-5514	50	10	addressing	address	VERB
ejpam-5514	50	11	real	real	ADJ
ejpam-5514	50	12	-	-	PUNCT
ejpam-5514	50	13	world	world	NOUN
ejpam-5514	50	14	problems	problem	NOUN
ejpam-5514	50	15	characterized	characterize	VERB
ejpam-5514	50	16	by	by	ADP
ejpam-5514	50	17	uncertainty	uncertainty	NOUN
ejpam-5514	50	18	and	and	CCONJ
ejpam-5514	50	19	imprecision	imprecision	NOUN
ejpam-5514	50	20	.	.	PUNCT
ejpam-5514	51	1	2	2	X
ejpam-5514	51	2	.	.	X
ejpam-5514	51	3	preliminaries	preliminary	NOUN
ejpam-5514	51	4	in	in	ADP
ejpam-5514	51	5	this	this	DET
ejpam-5514	51	6	section	section	NOUN
ejpam-5514	51	7	,	,	PUNCT
ejpam-5514	51	8	we	we	PRON
ejpam-5514	51	9	present	present	VERB
ejpam-5514	51	10	a	a	DET
ejpam-5514	51	11	foundational	foundational	ADJ
ejpam-5514	51	12	understanding	understanding	NOUN
ejpam-5514	51	13	of	of	ADP
ejpam-5514	51	14	rough	rough	ADJ
ejpam-5514	51	15	set	set	NOUN
ejpam-5514	51	16	theory	theory	NOUN
ejpam-5514	51	17	with	with	ADP
ejpam-5514	51	18	key	key	ADJ
ejpam-5514	51	19	algebraic	algebraic	ADJ
ejpam-5514	51	20	properties	property	NOUN
ejpam-5514	51	21	.	.	PUNCT
ejpam-5514	52	1	let	let	VERB
ejpam-5514	52	2	u	u	PRON
ejpam-5514	52	3	represent	represent	VERB
ejpam-5514	52	4	a	a	DET
ejpam-5514	52	5	universal	universal	ADJ
ejpam-5514	52	6	set	set	NOUN
ejpam-5514	52	7	,	,	PUNCT
ejpam-5514	52	8	and	and	CCONJ
ejpam-5514	52	9	let	let	VERB
ejpam-5514	52	10	θ	θ	PROPN
ejpam-5514	52	11	denote	denote	VERB
ejpam-5514	52	12	an	an	DET
ejpam-5514	52	13	equivalence	equivalence	NOUN
ejpam-5514	52	14	relation	relation	NOUN
ejpam-5514	52	15	on	on	ADP
ejpam-5514	52	16	u	u	PROPN
ejpam-5514	52	17	.	.	PUNCT
ejpam-5514	53	1	the	the	DET
ejpam-5514	53	2	equivalence	equivalence	NOUN
ejpam-5514	53	3	class	class	NOUN
ejpam-5514	53	4	of	of	ADP
ejpam-5514	53	5	an	an	DET
ejpam-5514	53	6	element	element	NOUN
ejpam-5514	53	7	x	x	SYM
ejpam-5514	53	8	∈	∈	NOUN
ejpam-5514	53	9	u	u	NOUN
ejpam-5514	53	10	is	be	AUX
ejpam-5514	53	11	defined	define	VERB
ejpam-5514	53	12	as	as	ADP
ejpam-5514	53	13	the	the	DET
ejpam-5514	53	14	set	set	NOUN
ejpam-5514	53	15	of	of	ADP
ejpam-5514	53	16	elements	element	NOUN
ejpam-5514	53	17	related	relate	VERB
ejpam-5514	53	18	to	to	ADP
ejpam-5514	53	19	x	x	PUNCT
ejpam-5514	53	20	in	in	ADP
ejpam-5514	53	21	u	u	NOUN
ejpam-5514	53	22	,	,	PUNCT
ejpam-5514	53	23	denoted	denote	VERB
ejpam-5514	53	24	as	as	ADP
ejpam-5514	53	25	[	[	X
ejpam-5514	53	26	x]θ	x]θ	X
ejpam-5514	53	27	.	.	PUNCT
ejpam-5514	54	1	the	the	DET
ejpam-5514	54	2	pair	pair	NOUN
ejpam-5514	54	3	(	(	PUNCT
ejpam-5514	54	4	u	u	NOUN
ejpam-5514	54	5	,	,	PUNCT
ejpam-5514	54	6	θ	θ	PROPN
ejpam-5514	54	7	)	)	PUNCT
ejpam-5514	54	8	,	,	PUNCT
ejpam-5514	54	9	where	where	SCONJ
ejpam-5514	54	10	u	u	NOUN
ejpam-5514	54	11	is	be	AUX
ejpam-5514	54	12	non	non	ADJ
ejpam-5514	54	13	-	-	ADJ
ejpam-5514	54	14	empty	empty	ADJ
ejpam-5514	54	15	,	,	PUNCT
ejpam-5514	54	16	and	and	CCONJ
ejpam-5514	54	17	θ	θ	PROPN
ejpam-5514	54	18	is	be	AUX
ejpam-5514	54	19	an	an	DET
ejpam-5514	54	20	equivalence	equivalence	NOUN
ejpam-5514	54	21	relation	relation	NOUN
ejpam-5514	54	22	on	on	ADP
ejpam-5514	54	23	u	u	NOUN
ejpam-5514	54	24	,	,	PUNCT
ejpam-5514	54	25	constitutes	constitute	VERB
ejpam-5514	54	26	an	an	DET
ejpam-5514	54	27	approximation	approximation	NOUN
ejpam-5514	54	28	space	space	NOUN
ejpam-5514	54	29	.	.	PUNCT
ejpam-5514	55	1	in	in	ADP
ejpam-5514	55	2	this	this	DET
ejpam-5514	55	3	context	context	NOUN
ejpam-5514	55	4	,	,	PUNCT
ejpam-5514	55	5	rough	rough	ADJ
ejpam-5514	55	6	approximation	approximation	NOUN
ejpam-5514	55	7	[	[	X
ejpam-5514	55	8	16	16	NUM
ejpam-5514	55	9	]	]	PUNCT
ejpam-5514	55	10	in	in	ADP
ejpam-5514	55	11	(	(	PUNCT
ejpam-5514	55	12	u	u	NOUN
ejpam-5514	55	13	,	,	PUNCT
ejpam-5514	55	14	θ	θ	NOUN
ejpam-5514	55	15	)	)	PUNCT
ejpam-5514	55	16	is	be	AUX
ejpam-5514	55	17	represented	represent	VERB
ejpam-5514	55	18	by	by	ADP
ejpam-5514	55	19	a	a	DET
ejpam-5514	55	20	mapping	mapping	NOUN
ejpam-5514	55	21	apr	apr	NOUN
ejpam-5514	55	22	:	:	PUNCT
ejpam-5514	56	1	p	p	X
ejpam-5514	56	2	(	(	PUNCT
ejpam-5514	56	3	u	u	NOUN
ejpam-5514	56	4	)	)	PUNCT
ejpam-5514	56	5	→	→	SYM
ejpam-5514	56	6	p	p	X
ejpam-5514	56	7	(	(	PUNCT
ejpam-5514	56	8	u	u	NOUN
ejpam-5514	56	9	)	)	PUNCT
ejpam-5514	56	10	×	×	NOUN
ejpam-5514	56	11	p	p	X
ejpam-5514	56	12	(	(	PUNCT
ejpam-5514	56	13	u	u	NOUN
ejpam-5514	56	14	)	)	PUNCT
ejpam-5514	56	15	,	,	PUNCT
ejpam-5514	56	16	defined	define	VERB
ejpam-5514	56	17	for	for	ADP
ejpam-5514	56	18	every	every	DET
ejpam-5514	56	19	x	x	SYM
ejpam-5514	56	20	∈	∈	PROPN
ejpam-5514	56	21	p	p	X
ejpam-5514	56	22	(	(	PUNCT
ejpam-5514	56	23	u	u	NOUN
ejpam-5514	56	24	)	)	PUNCT
ejpam-5514	56	25	as	as	SCONJ
ejpam-5514	56	26	follows	follow	VERB
ejpam-5514	56	27	:	:	PUNCT
ejpam-5514	56	28	apr(x	apr(x	X
ejpam-5514	56	29	)	)	PUNCT
ejpam-5514	56	30	=	=	SYM
ejpam-5514	56	31	(	(	PUNCT
ejpam-5514	56	32	θ−(x	θ−(x	PROPN
ejpam-5514	56	33	)	)	PUNCT
ejpam-5514	56	34	,	,	PUNCT
ejpam-5514	56	35	θ−(x	θ−(x	PROPN
ejpam-5514	56	36	)	)	PUNCT
ejpam-5514	56	37	)	)	PUNCT
ejpam-5514	56	38	,	,	PUNCT
ejpam-5514	56	39	where	where	SCONJ
ejpam-5514	56	40	θ−(x	θ−(x	NOUN
ejpam-5514	56	41	)	)	PUNCT
ejpam-5514	56	42	=	=	PRON
ejpam-5514	57	1	{	{	PUNCT
ejpam-5514	57	2	x	x	PUNCT
ejpam-5514	57	3	∈	∈	PROPN
ejpam-5514	57	4	u	u	NOUN
ejpam-5514	57	5	|	|	NOUN
ejpam-5514	58	1	[	[	X
ejpam-5514	58	2	x]θ	x]θ	NOUN
ejpam-5514	58	3	⊆	⊆	NUM
ejpam-5514	58	4	x	x	SYM
ejpam-5514	58	5	}	}	PUNCT
ejpam-5514	58	6	and	and	CCONJ
ejpam-5514	58	7	θ−(x	θ−(x	PROPN
ejpam-5514	58	8	)	)	PUNCT
ejpam-5514	59	1	=	=	PRON
ejpam-5514	59	2	{	{	PUNCT
ejpam-5514	59	3	x	x	PUNCT
ejpam-5514	59	4	∈	∈	PROPN
ejpam-5514	59	5	u	u	NOUN
ejpam-5514	60	1	|	|	NOUN
ejpam-5514	61	1	[	[	X
ejpam-5514	61	2	x]θ	x]θ	X
ejpam-5514	61	3	∩	∩	NOUN
ejpam-5514	61	4	x	x	SYM
ejpam-5514	61	5	̸=	̸=	PROPN
ejpam-5514	61	6	∅	∅	NOUN
ejpam-5514	61	7	}	}	PUNCT
ejpam-5514	61	8	.	.	PUNCT
ejpam-5514	62	1	these	these	DET
ejpam-5514	62	2	sets	set	NOUN
ejpam-5514	62	3	,	,	PUNCT
ejpam-5514	62	4	θ−(x	θ−(x	PROPN
ejpam-5514	62	5	)	)	PUNCT
ejpam-5514	62	6	and	and	CCONJ
ejpam-5514	62	7	θ−(x	θ−(x	PROPN
ejpam-5514	62	8	)	)	PUNCT
ejpam-5514	62	9	,	,	PUNCT
ejpam-5514	62	10	are	be	AUX
ejpam-5514	62	11	known	know	VERB
ejpam-5514	62	12	as	as	ADP
ejpam-5514	62	13	the	the	DET
ejpam-5514	62	14	lower	low	ADJ
ejpam-5514	62	15	and	and	CCONJ
ejpam-5514	62	16	upper	upper	ADJ
ejpam-5514	62	17	approximations	approximation	NOUN
ejpam-5514	62	18	of	of	ADP
ejpam-5514	62	19	set	set	NOUN
ejpam-5514	62	20	x	x	SYM
ejpam-5514	62	21	within	within	ADP
ejpam-5514	62	22	(	(	PUNCT
ejpam-5514	62	23	u	u	NOUN
ejpam-5514	62	24	,	,	PUNCT
ejpam-5514	62	25	θ	θ	NOUN
ejpam-5514	62	26	)	)	PUNCT
ejpam-5514	62	27	,	,	PUNCT
ejpam-5514	62	28	respectively	respectively	ADV
ejpam-5514	62	29	.	.	PUNCT
ejpam-5514	63	1	the	the	DET
ejpam-5514	63	2	accuracy	accuracy	NOUN
ejpam-5514	63	3	of	of	ADP
ejpam-5514	63	4	rough	rough	ADJ
ejpam-5514	63	5	set	set	NOUN
ejpam-5514	63	6	can	can	AUX
ejpam-5514	63	7	be	be	AUX
ejpam-5514	63	8	measured	measure	VERB
ejpam-5514	63	9	as	as	ADP
ejpam-5514	63	10	[	[	X
ejpam-5514	63	11	16	16	NUM
ejpam-5514	63	12	]	]	X
ejpam-5514	63	13	:	:	PUNCT
ejpam-5514	63	14	αθ(x	αθ(x	NUM
ejpam-5514	63	15	)	)	PUNCT
ejpam-5514	63	16	=	=	SYM
ejpam-5514	63	17	card	card	NOUN
ejpam-5514	63	18	(	(	PUNCT
ejpam-5514	63	19	θ−(x	θ−(x	PROPN
ejpam-5514	63	20	)	)	PUNCT
ejpam-5514	63	21	)	)	PUNCT
ejpam-5514	63	22	card	card	NOUN
ejpam-5514	63	23	(	(	PUNCT
ejpam-5514	63	24	θ−x	θ−x	PROPN
ejpam-5514	63	25	)	)	PUNCT
ejpam-5514	63	26	,	,	PUNCT
ejpam-5514	63	27	where	where	SCONJ
ejpam-5514	63	28	x	x	X
ejpam-5514	63	29	̸=	̸=	PROPN
ejpam-5514	63	30	∅.	∅.	ADP
ejpam-5514	63	31	the	the	DET
ejpam-5514	63	32	accuracy	accuracy	NOUN
ejpam-5514	63	33	measure	measure	NOUN
ejpam-5514	63	34	αθ(x	αθ(x	PUNCT
ejpam-5514	63	35	)	)	PUNCT
ejpam-5514	63	36	quantifies	quantify	VERB
ejpam-5514	63	37	the	the	DET
ejpam-5514	63	38	degree	degree	NOUN
ejpam-5514	63	39	of	of	ADP
ejpam-5514	63	40	completeness	completeness	NOUN
ejpam-5514	63	41	of	of	ADP
ejpam-5514	63	42	our	our	PRON
ejpam-5514	63	43	knowledge	knowledge	NOUN
ejpam-5514	63	44	about	about	ADP
ejpam-5514	63	45	the	the	DET
ejpam-5514	63	46	set	set	NOUN
ejpam-5514	63	47	x.	x.	NOUN
ejpam-5514	63	48	it	it	PRON
ejpam-5514	63	49	holds	hold	VERB
ejpam-5514	63	50	that	that	SCONJ
ejpam-5514	63	51	0	0	NUM
ejpam-5514	63	52	≤	≤	NOUN
ejpam-5514	63	53	αθ(x	αθ(x	PUNCT
ejpam-5514	63	54	)	)	PUNCT
ejpam-5514	63	55	≤	≤	NUM
ejpam-5514	63	56	1	1	NUM
ejpam-5514	63	57	,	,	PUNCT
ejpam-5514	63	58	and	and	CCONJ
ejpam-5514	63	59	when	when	SCONJ
ejpam-5514	63	60	αθ(x	αθ(x	NUM
ejpam-5514	63	61	)	)	PUNCT
ejpam-5514	63	62	=	=	SYM
ejpam-5514	63	63	1	1	NUM
ejpam-5514	63	64	,	,	PUNCT
ejpam-5514	63	65	the	the	DET
ejpam-5514	63	66	θ	θ	NOUN
ejpam-5514	63	67	-	-	PUNCT
ejpam-5514	63	68	borderline	borderline	ADJ
ejpam-5514	63	69	region	region	NOUN
ejpam-5514	63	70	of	of	ADP
ejpam-5514	63	71	x	x	PUNCT
ejpam-5514	63	72	is	be	AUX
ejpam-5514	63	73	empty	empty	ADJ
ejpam-5514	63	74	.	.	PUNCT
ejpam-5514	64	1	the	the	DET
ejpam-5514	64	2	θ	θ	NOUN
ejpam-5514	64	3	-	-	NOUN
ejpam-5514	64	4	roughness	roughness	NOUN
ejpam-5514	64	5	of	of	ADP
ejpam-5514	64	6	x	x	PUNCT
ejpam-5514	65	1	[	[	X
ejpam-5514	65	2	16	16	NUM
ejpam-5514	65	3	]	]	PUNCT
ejpam-5514	65	4	is	be	AUX
ejpam-5514	65	5	defined	define	VERB
ejpam-5514	65	6	as	as	ADP
ejpam-5514	65	7	:	:	PUNCT
ejpam-5514	65	8	ρθ(x	ρθ(x	NUM
ejpam-5514	65	9	)	)	PUNCT
ejpam-5514	66	1	=	=	SYM
ejpam-5514	66	2	1	1	NUM
ejpam-5514	66	3	−	−	NOUN
ejpam-5514	66	4	αθ(x	αθ(x	PUNCT
ejpam-5514	66	5	)	)	PUNCT
ejpam-5514	66	6	=	=	SYM
ejpam-5514	66	7	1	1	NUM
ejpam-5514	66	8	−	−	NOUN
ejpam-5514	66	9	card	card	NOUN
ejpam-5514	66	10	(	(	PUNCT
ejpam-5514	66	11	θ−(x	θ−(x	PROPN
ejpam-5514	66	12	)	)	PUNCT
ejpam-5514	66	13	)	)	PUNCT
ejpam-5514	66	14	card	card	NOUN
ejpam-5514	66	15	(	(	PUNCT
ejpam-5514	66	16	θ−(x	θ−(x	PROPN
ejpam-5514	66	17	)	)	PUNCT
ejpam-5514	66	18	)	)	PUNCT
ejpam-5514	66	19	.	.	PUNCT
ejpam-5514	67	1	it	it	PRON
ejpam-5514	67	2	’s	’	VERB
ejpam-5514	67	3	worth	worth	ADJ
ejpam-5514	67	4	noting	note	VERB
ejpam-5514	67	5	that	that	SCONJ
ejpam-5514	67	6	lower	low	ADJ
ejpam-5514	67	7	roughness	roughness	NOUN
ejpam-5514	67	8	of	of	ADP
ejpam-5514	67	9	a	a	DET
ejpam-5514	67	10	subset	subset	NOUN
ejpam-5514	67	11	indicates	indicate	VERB
ejpam-5514	67	12	a	a	DET
ejpam-5514	67	13	better	well	ADJ
ejpam-5514	67	14	approximation	approximation	NOUN
ejpam-5514	67	15	.	.	PUNCT
ejpam-5514	68	1	in	in	ADP
ejpam-5514	68	2	this	this	DET
ejpam-5514	68	3	paper	paper	NOUN
ejpam-5514	68	4	,	,	PUNCT
ejpam-5514	68	5	we	we	PRON
ejpam-5514	68	6	consider	consider	VERB
ejpam-5514	68	7	r	r	NOUN
ejpam-5514	68	8	as	as	ADP
ejpam-5514	68	9	a	a	DET
ejpam-5514	68	10	ring	ring	NOUN
ejpam-5514	68	11	with	with	ADP
ejpam-5514	68	12	standard	standard	ADJ
ejpam-5514	68	13	operations	operation	NOUN
ejpam-5514	68	14	.	.	PUNCT
ejpam-5514	69	1	let	let	VERB
ejpam-5514	69	2	i	i	PRON
ejpam-5514	69	3	represent	represent	VERB
ejpam-5514	69	4	an	an	DET
ejpam-5514	69	5	ideal	ideal	NOUN
ejpam-5514	69	6	of	of	ADP
ejpam-5514	69	7	r	r	NOUN
ejpam-5514	69	8	,	,	PUNCT
ejpam-5514	69	9	and	and	CCONJ
ejpam-5514	69	10	x	x	X
ejpam-5514	69	11	(	(	PUNCT
ejpam-5514	69	12	where	where	SCONJ
ejpam-5514	69	13	x	x	PUNCT
ejpam-5514	69	14	̸=	̸=	PROPN
ejpam-5514	69	15	emptyset	emptyset	NOUN
ejpam-5514	69	16	)	)	PUNCT
ejpam-5514	69	17	be	be	AUX
ejpam-5514	69	18	a	a	DET
ejpam-5514	69	19	subset	subset	NOUN
ejpam-5514	69	20	of	of	ADP
ejpam-5514	69	21	r.	r.	PROPN
ejpam-5514	69	22	we	we	PRON
ejpam-5514	69	23	define	define	VERB
ejpam-5514	69	24	i−(x	i−(x	PROPN
ejpam-5514	69	25	)	)	PUNCT
ejpam-5514	69	26	and	and	CCONJ
ejpam-5514	69	27	i−(x	i−(x	PROPN
ejpam-5514	69	28	)	)	PUNCT
ejpam-5514	69	29	as	as	SCONJ
ejpam-5514	69	30	follows	follow	VERB
ejpam-5514	69	31	:	:	PUNCT
ejpam-5514	69	32	i−(x	i−(x	PROPN
ejpam-5514	69	33	)	)	PUNCT
ejpam-5514	70	1	=	=	PUNCT
ejpam-5514	70	2	{	{	PUNCT
ejpam-5514	70	3	x	x	PUNCT
ejpam-5514	70	4	∈	∈	PROPN
ejpam-5514	70	5	r	r	NOUN
ejpam-5514	70	6	|	|	NOUN
ejpam-5514	70	7	(	(	PUNCT
ejpam-5514	70	8	x	x	PROPN
ejpam-5514	70	9	+	+	PUNCT
ejpam-5514	70	10	i	i	NOUN
ejpam-5514	70	11	)	)	PUNCT
ejpam-5514	70	12	⊆	⊆	NUM
ejpam-5514	70	13	x	x	SYM
ejpam-5514	70	14	}	}	PUNCT
ejpam-5514	70	15	i−(x	i−(x	PROPN
ejpam-5514	70	16	)	)	PUNCT
ejpam-5514	70	17	=	=	PUNCT
ejpam-5514	71	1	{	{	PUNCT
ejpam-5514	71	2	x	x	PUNCT
ejpam-5514	71	3	∈	∈	PROPN
ejpam-5514	71	4	r	r	NOUN
ejpam-5514	71	5	|	|	NOUN
ejpam-5514	71	6	(	(	PUNCT
ejpam-5514	71	7	x	x	PROPN
ejpam-5514	71	8	+	+	PUNCT
ejpam-5514	71	9	i	i	NOUN
ejpam-5514	71	10	)	)	PUNCT
ejpam-5514	71	11	∩x	∩x	X
ejpam-5514	72	1	̸=	̸=	NOUN
ejpam-5514	72	2	∅	∅	NOUN
ejpam-5514	72	3	}	}	PUNCT
ejpam-5514	72	4	,	,	PUNCT
ejpam-5514	72	5	representing	represent	VERB
ejpam-5514	72	6	the	the	DET
ejpam-5514	72	7	lower	low	ADJ
ejpam-5514	72	8	and	and	CCONJ
ejpam-5514	72	9	upper	upper	ADJ
ejpam-5514	72	10	approximations	approximation	NOUN
ejpam-5514	72	11	of	of	ADP
ejpam-5514	72	12	set	set	NOUN
ejpam-5514	72	13	x	x	SYM
ejpam-5514	72	14	concerning	concern	VERB
ejpam-5514	72	15	the	the	DET
ejpam-5514	72	16	ideal	ideal	NOUN
ejpam-5514	72	17	i	i	PROPN
ejpam-5514	72	18	,	,	PUNCT
ejpam-5514	72	19	respectively	respectively	ADV
ejpam-5514	72	20	.	.	PUNCT
ejpam-5514	73	1	additionally	additionally	ADV
ejpam-5514	73	2	,	,	PUNCT
ejpam-5514	73	3	we	we	PRON
ejpam-5514	73	4	introduce	introduce	VERB
ejpam-5514	73	5	notations	notation	NOUN
ejpam-5514	73	6	for	for	ADP
ejpam-5514	73	7	r	r	NOUN
ejpam-5514	73	8	as	as	ADP
ejpam-5514	73	9	a	a	DET
ejpam-5514	73	10	ring	ring	NOUN
ejpam-5514	73	11	and	and	CCONJ
ejpam-5514	73	12	ρ	ρ	NOUN
ejpam-5514	73	13	as	as	ADP
ejpam-5514	73	14	a	a	DET
ejpam-5514	73	15	congruence	congruence	NOUN
ejpam-5514	73	16	relation	relation	NOUN
ejpam-5514	73	17	on	on	ADP
ejpam-5514	73	18	r	r	NOUN
ejpam-5514	73	19	:	:	PUNCT
ejpam-5514	73	20	ρ−(a	ρ−(a	ADV
ejpam-5514	73	21	)	)	PUNCT
ejpam-5514	73	22	=	=	PRON
ejpam-5514	74	1	{	{	PUNCT
ejpam-5514	74	2	x	x	PUNCT
ejpam-5514	74	3	∈	∈	PROPN
ejpam-5514	74	4	r	r	NOUN
ejpam-5514	74	5	|	|	NOUN
ejpam-5514	75	1	[	[	X
ejpam-5514	75	2	x]ρ	x]ρ	PROPN
ejpam-5514	75	3	⊆	⊆	NUM
ejpam-5514	75	4	a	a	DET
ejpam-5514	75	5	}	}	PUNCT
ejpam-5514	75	6	a.	a.	NOUN
ejpam-5514	75	7	prakash	prakash	PROPN
ejpam-5514	75	8	,	,	PUNCT
ejpam-5514	75	9	r.	r.	PROPN
ejpam-5514	75	10	shukla	shukla	PROPN
ejpam-5514	75	11	/	/	SYM
ejpam-5514	75	12	eur	eur	PROPN
ejpam-5514	75	13	.	.	PUNCT
ejpam-5514	76	1	j.	j.	PROPN
ejpam-5514	76	2	pure	pure	PROPN
ejpam-5514	76	3	appl	appl	PROPN
ejpam-5514	76	4	.	.	PROPN
ejpam-5514	76	5	math	math	PROPN
ejpam-5514	76	6	,	,	PUNCT
ejpam-5514	76	7	18	18	NUM
ejpam-5514	76	8	(	(	PUNCT
ejpam-5514	76	9	1	1	NUM
ejpam-5514	76	10	)	)	PUNCT
ejpam-5514	76	11	(	(	PUNCT
ejpam-5514	76	12	2025	2025	NUM
ejpam-5514	76	13	)	)	PUNCT
ejpam-5514	76	14	,	,	PUNCT
ejpam-5514	76	15	5514	5514	NUM
ejpam-5514	76	16	4	4	NUM
ejpam-5514	76	17	of	of	ADP
ejpam-5514	76	18	9	9	NUM
ejpam-5514	76	19	ρ−(a	ρ−(a	ADV
ejpam-5514	76	20	)	)	PUNCT
ejpam-5514	76	21	=	=	SYM
ejpam-5514	76	22	{	{	PUNCT
ejpam-5514	76	23	x	x	PUNCT
ejpam-5514	76	24	∈	∈	PROPN
ejpam-5514	76	25	r	r	NOUN
ejpam-5514	77	1	|	|	NOUN
ejpam-5514	78	1	[	[	X
ejpam-5514	78	2	x]ρ	x]ρ	PRON
ejpam-5514	78	3	∩a	∩a	PROPN
ejpam-5514	78	4	̸=	̸=	PROPN
ejpam-5514	78	5	∅	∅	NOUN
ejpam-5514	78	6	}	}	PUNCT
ejpam-5514	78	7	a	a	DET
ejpam-5514	78	8	congruence	congruence	NOUN
ejpam-5514	78	9	relation	relation	NOUN
ejpam-5514	78	10	on	on	ADP
ejpam-5514	78	11	r	r	NOUN
ejpam-5514	78	12	is	be	AUX
ejpam-5514	78	13	termed	term	VERB
ejpam-5514	78	14	complete	complete	ADJ
ejpam-5514	78	15	if	if	SCONJ
ejpam-5514	78	16	[	[	X
ejpam-5514	78	17	a]ρ[b]ρ	a]ρ[b]ρ	X
ejpam-5514	78	18	=	=	PUNCT
ejpam-5514	79	1	[	[	X
ejpam-5514	79	2	ab]ρ	ab]ρ	NOUN
ejpam-5514	79	3	for	for	ADP
ejpam-5514	79	4	any	any	DET
ejpam-5514	79	5	a	a	DET
ejpam-5514	79	6	,	,	PUNCT
ejpam-5514	79	7	b	b	X
ejpam-5514	79	8	∈	∈	PROPN
ejpam-5514	79	9	r.	r.	NOUN
ejpam-5514	79	10	we	we	PRON
ejpam-5514	79	11	also	also	ADV
ejpam-5514	79	12	present	present	VERB
ejpam-5514	79	13	key	key	ADJ
ejpam-5514	79	14	propositions	proposition	NOUN
ejpam-5514	79	15	and	and	CCONJ
ejpam-5514	79	16	theorems	theorem	NOUN
ejpam-5514	79	17	relevant	relevant	ADJ
ejpam-5514	79	18	to	to	ADP
ejpam-5514	79	19	our	our	PRON
ejpam-5514	79	20	study	study	NOUN
ejpam-5514	79	21	:	:	PUNCT
ejpam-5514	79	22	proposition	proposition	NOUN
ejpam-5514	79	23	1	1	NUM
ejpam-5514	79	24	.	.	PUNCT
ejpam-5514	80	1	(	(	PUNCT
ejpam-5514	80	2	[	[	X
ejpam-5514	80	3	7	7	NUM
ejpam-5514	80	4	]	]	PUNCT
ejpam-5514	80	5	,	,	PUNCT
ejpam-5514	80	6	proposition	proposition	NOUN
ejpam-5514	80	7	3.4	3.4	NUM
ejpam-5514	80	8	)	)	PUNCT
ejpam-5514	80	9	let	let	VERB
ejpam-5514	80	10	i	i	PRON
ejpam-5514	80	11	be	be	AUX
ejpam-5514	80	12	an	an	DET
ejpam-5514	80	13	ideal	ideal	NOUN
ejpam-5514	80	14	of	of	ADP
ejpam-5514	80	15	r	r	NOUN
ejpam-5514	80	16	,	,	PUNCT
ejpam-5514	80	17	and	and	CCONJ
ejpam-5514	80	18	a	a	DET
ejpam-5514	80	19	,	,	PUNCT
ejpam-5514	80	20	b	b	X
ejpam-5514	80	21	non	non	ADJ
ejpam-5514	80	22	-	-	ADJ
ejpam-5514	80	23	empty	empty	ADJ
ejpam-5514	80	24	subsets	subset	NOUN
ejpam-5514	80	25	of	of	ADP
ejpam-5514	80	26	r.	r.	PROPN
ejpam-5514	80	27	then	then	ADV
ejpam-5514	80	28	i−(a	i−(a	ADV
ejpam-5514	80	29	)	)	PUNCT
ejpam-5514	80	30	·	·	PUNCT
ejpam-5514	81	1	i−(b	i−(b	NOUN
ejpam-5514	81	2	)	)	PUNCT
ejpam-5514	81	3	=	=	PUNCT
ejpam-5514	81	4	i−(a	i−(a	ADV
ejpam-5514	81	5	·	·	SYM
ejpam-5514	81	6	b	b	NOUN
ejpam-5514	81	7	)	)	PUNCT
ejpam-5514	81	8	.	.	PUNCT
ejpam-5514	82	1	proposition	proposition	NOUN
ejpam-5514	82	2	2	2	NUM
ejpam-5514	82	3	.	.	PUNCT
ejpam-5514	83	1	(	(	PUNCT
ejpam-5514	83	2	[	[	X
ejpam-5514	83	3	7	7	NUM
ejpam-5514	83	4	]	]	PUNCT
ejpam-5514	83	5	,	,	PUNCT
ejpam-5514	83	6	proposition	proposition	NOUN
ejpam-5514	83	7	3.5	3.5	NUM
ejpam-5514	83	8	)	)	PUNCT
ejpam-5514	83	9	let	let	VERB
ejpam-5514	83	10	i	i	PRON
ejpam-5514	83	11	be	be	AUX
ejpam-5514	83	12	an	an	DET
ejpam-5514	83	13	ideal	ideal	NOUN
ejpam-5514	83	14	of	of	ADP
ejpam-5514	83	15	r	r	NOUN
ejpam-5514	83	16	,	,	PUNCT
ejpam-5514	83	17	and	and	CCONJ
ejpam-5514	83	18	a	a	DET
ejpam-5514	83	19	,	,	PUNCT
ejpam-5514	83	20	b	b	X
ejpam-5514	83	21	non	non	ADJ
ejpam-5514	83	22	-	-	ADJ
ejpam-5514	83	23	empty	empty	ADJ
ejpam-5514	83	24	subsets	subset	NOUN
ejpam-5514	83	25	of	of	ADP
ejpam-5514	83	26	r.	r.	PROPN
ejpam-5514	83	27	then	then	ADV
ejpam-5514	83	28	i−(a	i−(a	ADV
ejpam-5514	83	29	)	)	PUNCT
ejpam-5514	83	30	·	·	PUNCT
ejpam-5514	84	1	i−(b	i−(b	NOUN
ejpam-5514	84	2	)	)	PUNCT
ejpam-5514	84	3	=	=	PUNCT
ejpam-5514	84	4	i−(a	i−(a	ADV
ejpam-5514	84	5	·	·	SYM
ejpam-5514	84	6	b	b	NOUN
ejpam-5514	84	7	)	)	PUNCT
ejpam-5514	84	8	.	.	PUNCT
ejpam-5514	85	1	theorem	theorem	NOUN
ejpam-5514	85	2	1	1	NUM
ejpam-5514	85	3	.	.	PUNCT
ejpam-5514	86	1	(	(	PUNCT
ejpam-5514	86	2	theorem	theorem	VERB
ejpam-5514	86	3	3.12	3.12	NUM
ejpam-5514	86	4	,	,	PUNCT
ejpam-5514	86	5	[	[	X
ejpam-5514	86	6	7	7	NUM
ejpam-5514	86	7	]	]	PUNCT
ejpam-5514	86	8	)	)	PUNCT
ejpam-5514	86	9	let	let	VERB
ejpam-5514	86	10	i	i	PRON
ejpam-5514	86	11	and	and	CCONJ
ejpam-5514	86	12	j	j	PROPN
ejpam-5514	86	13	be	be	VERB
ejpam-5514	86	14	two	two	NUM
ejpam-5514	86	15	ideals	ideal	NOUN
ejpam-5514	86	16	of	of	ADP
ejpam-5514	86	17	the	the	DET
ejpam-5514	86	18	ring	ring	NOUN
ejpam-5514	86	19	r.	r.	PROPN
ejpam-5514	86	20	then	then	ADV
ejpam-5514	86	21	i−(j	i−(j	NOUN
ejpam-5514	86	22	)	)	PUNCT
ejpam-5514	86	23	is	be	AUX
ejpam-5514	86	24	an	an	DET
ejpam-5514	86	25	ideal	ideal	NOUN
ejpam-5514	86	26	of	of	ADP
ejpam-5514	86	27	r.	r.	PROPN
ejpam-5514	86	28	theorem	theorem	PROPN
ejpam-5514	86	29	2	2	NUM
ejpam-5514	86	30	.	.	PUNCT
ejpam-5514	86	31	(	(	PUNCT
ejpam-5514	86	32	theorem	theorem	NOUN
ejpam-5514	86	33	3.13	3.13	NUM
ejpam-5514	86	34	,	,	PUNCT
ejpam-5514	86	35	[	[	X
ejpam-5514	86	36	7	7	NUM
ejpam-5514	86	37	]	]	PUNCT
ejpam-5514	86	38	)	)	PUNCT
ejpam-5514	86	39	let	let	VERB
ejpam-5514	86	40	i	i	PRON
ejpam-5514	86	41	and	and	CCONJ
ejpam-5514	86	42	j	j	PROPN
ejpam-5514	86	43	be	be	VERB
ejpam-5514	86	44	two	two	NUM
ejpam-5514	86	45	ideals	ideal	NOUN
ejpam-5514	86	46	of	of	ADP
ejpam-5514	86	47	the	the	DET
ejpam-5514	86	48	ring	ring	NOUN
ejpam-5514	86	49	r.	r.	PROPN
ejpam-5514	86	50	then	then	ADV
ejpam-5514	86	51	i−(j	i−(j	NOUN
ejpam-5514	86	52	)	)	PUNCT
ejpam-5514	86	53	is	be	AUX
ejpam-5514	86	54	an	an	DET
ejpam-5514	86	55	ideal	ideal	NOUN
ejpam-5514	86	56	of	of	ADP
ejpam-5514	86	57	r.	r.	NOUN
ejpam-5514	86	58	these	these	DET
ejpam-5514	86	59	propositions	proposition	NOUN
ejpam-5514	86	60	and	and	CCONJ
ejpam-5514	86	61	theorems	theorem	NOUN
ejpam-5514	86	62	establish	establish	VERB
ejpam-5514	86	63	essential	essential	ADJ
ejpam-5514	86	64	properties	property	NOUN
ejpam-5514	86	65	and	and	CCONJ
ejpam-5514	86	66	relationships	relationship	NOUN
ejpam-5514	86	67	within	within	ADP
ejpam-5514	86	68	our	our	PRON
ejpam-5514	86	69	research	research	NOUN
ejpam-5514	86	70	domain	domain	NOUN
ejpam-5514	86	71	.	.	PUNCT
ejpam-5514	87	1	3	3	X
ejpam-5514	87	2	.	.	X
ejpam-5514	87	3	rough	rough	ADJ
ejpam-5514	87	4	set	set	VERB
ejpam-5514	87	5	analysis	analysis	NOUN
ejpam-5514	87	6	of	of	ADP
ejpam-5514	87	7	ideals	ideal	NOUN
ejpam-5514	87	8	:	:	PUNCT
ejpam-5514	87	9	theory	theory	NOUN
ejpam-5514	87	10	,	,	PUNCT
ejpam-5514	87	11	applications	application	NOUN
ejpam-5514	87	12	,	,	PUNCT
ejpam-5514	87	13	and	and	CCONJ
ejpam-5514	87	14	illustrated	illustrate	VERB
ejpam-5514	87	15	examples	example	NOUN
ejpam-5514	87	16	in	in	ADP
ejpam-5514	87	17	this	this	DET
ejpam-5514	87	18	section	section	NOUN
ejpam-5514	87	19	,	,	PUNCT
ejpam-5514	87	20	we	we	PRON
ejpam-5514	87	21	explore	explore	VERB
ejpam-5514	87	22	the	the	DET
ejpam-5514	87	23	fundamental	fundamental	ADJ
ejpam-5514	87	24	theorems	theorem	NOUN
ejpam-5514	87	25	related	relate	VERB
ejpam-5514	87	26	to	to	ADP
ejpam-5514	87	27	ideals	ideal	NOUN
ejpam-5514	87	28	,	,	PUNCT
ejpam-5514	87	29	with	with	ADP
ejpam-5514	87	30	a	a	DET
ejpam-5514	87	31	specific	specific	ADJ
ejpam-5514	87	32	focus	focus	NOUN
ejpam-5514	87	33	on	on	ADP
ejpam-5514	87	34	upper	upper	ADJ
ejpam-5514	87	35	and	and	CCONJ
ejpam-5514	87	36	lower	low	ADJ
ejpam-5514	87	37	approximations	approximation	NOUN
ejpam-5514	87	38	.	.	PUNCT
ejpam-5514	88	1	to	to	PART
ejpam-5514	88	2	aid	aid	VERB
ejpam-5514	88	3	in	in	ADP
ejpam-5514	88	4	comprehension	comprehension	NOUN
ejpam-5514	88	5	,	,	PUNCT
ejpam-5514	88	6	we	we	PRON
ejpam-5514	88	7	provide	provide	VERB
ejpam-5514	88	8	concrete	concrete	ADJ
ejpam-5514	88	9	examples	example	NOUN
ejpam-5514	88	10	that	that	PRON
ejpam-5514	88	11	illustrate	illustrate	VERB
ejpam-5514	88	12	these	these	DET
ejpam-5514	88	13	theorems	theorem	NOUN
ejpam-5514	88	14	.	.	PUNCT
ejpam-5514	89	1	we	we	PRON
ejpam-5514	89	2	also	also	ADV
ejpam-5514	89	3	explore	explore	VERB
ejpam-5514	89	4	the	the	DET
ejpam-5514	89	5	concept	concept	NOUN
ejpam-5514	89	6	of	of	ADP
ejpam-5514	89	7	rough	rough	ADJ
ejpam-5514	89	8	ideals	ideal	NOUN
ejpam-5514	89	9	,	,	PUNCT
ejpam-5514	89	10	offering	offer	VERB
ejpam-5514	89	11	a	a	DET
ejpam-5514	89	12	precise	precise	ADJ
ejpam-5514	89	13	definition	definition	NOUN
ejpam-5514	89	14	to	to	PART
ejpam-5514	89	15	facilitate	facilitate	VERB
ejpam-5514	89	16	understanding	understanding	NOUN
ejpam-5514	89	17	.	.	PUNCT
ejpam-5514	90	1	further	far	ADV
ejpam-5514	90	2	,	,	PUNCT
ejpam-5514	90	3	we	we	PRON
ejpam-5514	90	4	present	present	VERB
ejpam-5514	90	5	additional	additional	ADJ
ejpam-5514	90	6	theorems	theorem	NOUN
ejpam-5514	90	7	and	and	CCONJ
ejpam-5514	90	8	illustrative	illustrative	ADJ
ejpam-5514	90	9	examples	example	NOUN
ejpam-5514	90	10	to	to	PART
ejpam-5514	90	11	enhance	enhance	VERB
ejpam-5514	90	12	the	the	DET
ejpam-5514	90	13	practical	practical	ADJ
ejpam-5514	90	14	application	application	NOUN
ejpam-5514	90	15	of	of	ADP
ejpam-5514	90	16	these	these	DET
ejpam-5514	90	17	concepts	concept	NOUN
ejpam-5514	90	18	.	.	PUNCT
ejpam-5514	91	1	theorem	theorem	NOUN
ejpam-5514	91	2	3	3	X
ejpam-5514	91	3	.	.	PUNCT
ejpam-5514	92	1	let	let	VERB
ejpam-5514	92	2	ρ	ρ	NOUN
ejpam-5514	92	3	be	be	AUX
ejpam-5514	92	4	a	a	DET
ejpam-5514	92	5	congruence	congruence	NOUN
ejpam-5514	92	6	relation	relation	NOUN
ejpam-5514	92	7	on	on	ADP
ejpam-5514	92	8	a	a	DET
ejpam-5514	92	9	ring	ring	NOUN
ejpam-5514	92	10	r.	r.	PROPN
ejpam-5514	92	11	suppose	suppose	VERB
ejpam-5514	92	12	a	a	PRON
ejpam-5514	92	13	is	be	AUX
ejpam-5514	92	14	a	a	DET
ejpam-5514	92	15	subset	subset	NOUN
ejpam-5514	92	16	of	of	ADP
ejpam-5514	92	17	ring	ring	NOUN
ejpam-5514	92	18	r	r	NOUN
ejpam-5514	92	19	,	,	PUNCT
ejpam-5514	92	20	and	and	CCONJ
ejpam-5514	92	21	if	if	SCONJ
ejpam-5514	92	22	a	a	PRON
ejpam-5514	92	23	is	be	AUX
ejpam-5514	92	24	an	an	DET
ejpam-5514	92	25	ideal	ideal	NOUN
ejpam-5514	92	26	of	of	ADP
ejpam-5514	92	27	ring	ring	NOUN
ejpam-5514	92	28	r	r	NOUN
ejpam-5514	92	29	,	,	PUNCT
ejpam-5514	92	30	then	then	ADV
ejpam-5514	92	31	the	the	DET
ejpam-5514	92	32	upper	upper	ADJ
ejpam-5514	92	33	approximation	approximation	NOUN
ejpam-5514	92	34	of	of	ADP
ejpam-5514	92	35	a	a	PRON
ejpam-5514	92	36	,	,	PUNCT
ejpam-5514	92	37	denoted	denote	VERB
ejpam-5514	92	38	as	as	ADP
ejpam-5514	92	39	ρ−(a	ρ−(a	ADV
ejpam-5514	92	40	)	)	PUNCT
ejpam-5514	92	41	,	,	PUNCT
ejpam-5514	92	42	is	be	AUX
ejpam-5514	92	43	also	also	ADV
ejpam-5514	92	44	an	an	DET
ejpam-5514	92	45	ideal	ideal	NOUN
ejpam-5514	92	46	of	of	ADP
ejpam-5514	92	47	ring	ring	PROPN
ejpam-5514	92	48	r.	r.	PROPN
ejpam-5514	92	49	proof	proof	PROPN
ejpam-5514	92	50	.	.	PUNCT
ejpam-5514	93	1	assume	assume	VERB
ejpam-5514	93	2	that	that	SCONJ
ejpam-5514	93	3	a	a	PRON
ejpam-5514	93	4	is	be	AUX
ejpam-5514	93	5	an	an	DET
ejpam-5514	93	6	ideal	ideal	NOUN
ejpam-5514	93	7	of	of	ADP
ejpam-5514	93	8	ring	ring	NOUN
ejpam-5514	93	9	r	r	NOUN
ejpam-5514	93	10	,	,	PUNCT
ejpam-5514	93	11	which	which	PRON
ejpam-5514	93	12	implies	imply	VERB
ejpam-5514	93	13	that	that	SCONJ
ejpam-5514	93	14	ra	ra	PROPN
ejpam-5514	93	15	⊆	⊆	NUM
ejpam-5514	93	16	a	a	PRON
ejpam-5514	93	17	and	and	CCONJ
ejpam-5514	93	18	ar	ar	NOUN
ejpam-5514	93	19	⊆	⊆	NUM
ejpam-5514	93	20	a.	a.	NOUN
ejpam-5514	93	21	note	note	NOUN
ejpam-5514	93	22	that	that	SCONJ
ejpam-5514	93	23	ρ−(r	ρ−(r	PROPN
ejpam-5514	93	24	)	)	PUNCT
ejpam-5514	94	1	=	=	VERB
ejpam-5514	94	2	r.	r.	NOUN
ejpam-5514	94	3	by	by	ADP
ejpam-5514	94	4	proposition	proposition	NOUN
ejpam-5514	94	5	1	1	NUM
ejpam-5514	94	6	,	,	PUNCT
ejpam-5514	94	7	we	we	PRON
ejpam-5514	94	8	have	have	VERB
ejpam-5514	94	9	:	:	PUNCT
ejpam-5514	94	10	rρ−(a	rρ−(a	ADV
ejpam-5514	94	11	)	)	PUNCT
ejpam-5514	94	12	=	=	PUNCT
ejpam-5514	94	13	ρ−(r)ρ−(a	ρ−(r)ρ−(a	NOUN
ejpam-5514	94	14	)	)	PUNCT
ejpam-5514	94	15	⊆	⊆	NUM
ejpam-5514	94	16	ρ−(ra	ρ−(ra	NOUN
ejpam-5514	94	17	)	)	PUNCT
ejpam-5514	94	18	⊆	⊆	NUM
ejpam-5514	94	19	ρ−(a	ρ−(a	ADV
ejpam-5514	94	20	)	)	PUNCT
ejpam-5514	94	21	ρ−(a)r	ρ−(a)r	NOUN
ejpam-5514	94	22	=	=	SYM
ejpam-5514	94	23	ρ−(a)ρ−(r	ρ−(a)ρ−(r	NOUN
ejpam-5514	94	24	)	)	PUNCT
ejpam-5514	94	25	⊆	⊆	NUM
ejpam-5514	94	26	ρ−(ar	ρ−(ar	NOUN
ejpam-5514	94	27	)	)	PUNCT
ejpam-5514	94	28	⊆	⊆	NUM
ejpam-5514	94	29	ρ−(a	ρ−(a	ADV
ejpam-5514	94	30	)	)	PUNCT
ejpam-5514	94	31	this	this	PRON
ejpam-5514	94	32	implies	imply	VERB
ejpam-5514	94	33	that	that	SCONJ
ejpam-5514	94	34	ρ−(a	ρ−(a	ADV
ejpam-5514	94	35	)	)	PUNCT
ejpam-5514	94	36	is	be	AUX
ejpam-5514	94	37	an	an	DET
ejpam-5514	94	38	ideal	ideal	NOUN
ejpam-5514	94	39	of	of	ADP
ejpam-5514	94	40	r	r	NOUN
ejpam-5514	94	41	,	,	PUNCT
ejpam-5514	94	42	making	make	VERB
ejpam-5514	94	43	it	it	PRON
ejpam-5514	94	44	an	an	DET
ejpam-5514	94	45	upper	upper	ADJ
ejpam-5514	94	46	rough	rough	ADJ
ejpam-5514	94	47	ideal	ideal	NOUN
ejpam-5514	94	48	of	of	ADP
ejpam-5514	94	49	ring	ring	PROPN
ejpam-5514	94	50	r.	r.	PROPN
ejpam-5514	94	51	it	it	PRON
ejpam-5514	94	52	’s	’	VERB
ejpam-5514	94	53	important	important	ADJ
ejpam-5514	94	54	to	to	PART
ejpam-5514	94	55	note	note	VERB
ejpam-5514	94	56	that	that	SCONJ
ejpam-5514	94	57	the	the	DET
ejpam-5514	94	58	converse	converse	NOUN
ejpam-5514	94	59	of	of	ADP
ejpam-5514	94	60	this	this	DET
ejpam-5514	94	61	theorem	theorem	NOUN
ejpam-5514	94	62	does	do	AUX
ejpam-5514	94	63	not	not	PART
ejpam-5514	94	64	hold	hold	VERB
ejpam-5514	94	65	in	in	ADP
ejpam-5514	94	66	general	general	ADJ
ejpam-5514	94	67	.	.	PUNCT
ejpam-5514	95	1	a.	a.	PROPN
ejpam-5514	95	2	prakash	prakash	PROPN
ejpam-5514	95	3	,	,	PUNCT
ejpam-5514	95	4	r.	r.	PROPN
ejpam-5514	95	5	shukla	shukla	PROPN
ejpam-5514	95	6	/	/	SYM
ejpam-5514	95	7	eur	eur	PROPN
ejpam-5514	95	8	.	.	PUNCT
ejpam-5514	96	1	j.	j.	PROPN
ejpam-5514	96	2	pure	pure	PROPN
ejpam-5514	96	3	appl	appl	PROPN
ejpam-5514	96	4	.	.	PROPN
ejpam-5514	96	5	math	math	PROPN
ejpam-5514	96	6	,	,	PUNCT
ejpam-5514	96	7	18	18	NUM
ejpam-5514	96	8	(	(	PUNCT
ejpam-5514	96	9	1	1	NUM
ejpam-5514	96	10	)	)	PUNCT
ejpam-5514	96	11	(	(	PUNCT
ejpam-5514	96	12	2025	2025	NUM
ejpam-5514	96	13	)	)	PUNCT
ejpam-5514	96	14	,	,	PUNCT
ejpam-5514	96	15	5514	5514	NUM
ejpam-5514	96	16	5	5	NUM
ejpam-5514	96	17	of	of	ADP
ejpam-5514	96	18	9	9	NUM
ejpam-5514	96	19	example	example	NOUN
ejpam-5514	96	20	1	1	NUM
ejpam-5514	96	21	.	.	PUNCT
ejpam-5514	97	1	let	let	VERB
ejpam-5514	97	2	’s	’s	NOUN
ejpam-5514	97	3	consider	consider	VERB
ejpam-5514	97	4	r	r	NOUN
ejpam-5514	97	5	=	=	SYM
ejpam-5514	97	6	z12	z12	NUM
ejpam-5514	97	7	,	,	PUNCT
ejpam-5514	97	8	i	i	PRON
ejpam-5514	97	9	=	=	PUNCT
ejpam-5514	97	10	{	{	PUNCT
ejpam-5514	97	11	0	0	NUM
ejpam-5514	97	12	,	,	PUNCT
ejpam-5514	97	13	6	6	NUM
ejpam-5514	97	14	}	}	PUNCT
ejpam-5514	97	15	as	as	ADP
ejpam-5514	97	16	a	a	DET
ejpam-5514	97	17	congruence	congruence	NOUN
ejpam-5514	97	18	relation	relation	NOUN
ejpam-5514	97	19	on	on	ADP
ejpam-5514	97	20	z12	z12	PROPN
ejpam-5514	97	21	,	,	PUNCT
ejpam-5514	97	22	and	and	CCONJ
ejpam-5514	97	23	a	a	PRON
ejpam-5514	97	24	=	=	X
ejpam-5514	97	25	{	{	PUNCT
ejpam-5514	97	26	0	0	NUM
ejpam-5514	97	27	,	,	PUNCT
ejpam-5514	97	28	4	4	NUM
ejpam-5514	97	29	,	,	PUNCT
ejpam-5514	97	30	8	8	NUM
ejpam-5514	97	31	}	}	PUNCT
ejpam-5514	97	32	as	as	ADP
ejpam-5514	97	33	a	a	DET
ejpam-5514	97	34	subset	subset	NOUN
ejpam-5514	97	35	of	of	ADP
ejpam-5514	97	36	z12	z12	PROPN
ejpam-5514	97	37	.	.	PUNCT
ejpam-5514	98	1	in	in	ADP
ejpam-5514	98	2	this	this	DET
ejpam-5514	98	3	case	case	NOUN
ejpam-5514	98	4	,	,	PUNCT
ejpam-5514	98	5	i−(a	i−(a	ADV
ejpam-5514	98	6	)	)	PUNCT
ejpam-5514	98	7	=	=	SYM
ejpam-5514	98	8	{	{	PUNCT
ejpam-5514	98	9	0	0	NUM
ejpam-5514	98	10	,	,	PUNCT
ejpam-5514	98	11	2	2	NUM
ejpam-5514	98	12	,	,	PUNCT
ejpam-5514	98	13	4	4	NUM
ejpam-5514	98	14	,	,	PUNCT
ejpam-5514	98	15	6	6	NUM
ejpam-5514	98	16	,	,	PUNCT
ejpam-5514	98	17	8	8	NUM
ejpam-5514	98	18	,	,	PUNCT
ejpam-5514	98	19	10	10	NUM
ejpam-5514	98	20	}	}	PUNCT
ejpam-5514	98	21	is	be	AUX
ejpam-5514	98	22	an	an	DET
ejpam-5514	98	23	ideal	ideal	NOUN
ejpam-5514	98	24	of	of	ADP
ejpam-5514	98	25	z12	z12	PROPN
ejpam-5514	98	26	.	.	PUNCT
ejpam-5514	99	1	however	however	ADV
ejpam-5514	99	2	,	,	PUNCT
ejpam-5514	99	3	the	the	DET
ejpam-5514	99	4	lower	low	ADJ
ejpam-5514	99	5	approximation	approximation	NOUN
ejpam-5514	99	6	is	be	AUX
ejpam-5514	99	7	an	an	DET
ejpam-5514	99	8	empty	empty	ADJ
ejpam-5514	99	9	set	set	NOUN
ejpam-5514	99	10	,	,	PUNCT
ejpam-5514	99	11	i.e.	i.e.	X
ejpam-5514	99	12	,	,	PUNCT
ejpam-5514	99	13	i−(a	i−(a	ADV
ejpam-5514	99	14	)	)	PUNCT
ejpam-5514	99	15	=	=	PUNCT
ejpam-5514	99	16	∅.	∅.	ADP
ejpam-5514	99	17	now	now	ADV
ejpam-5514	99	18	,	,	PUNCT
ejpam-5514	99	19	consider	consider	VERB
ejpam-5514	99	20	another	another	DET
ejpam-5514	99	21	subset	subset	NOUN
ejpam-5514	99	22	,	,	PUNCT
ejpam-5514	99	23	b	b	X
ejpam-5514	99	24	=	=	SYM
ejpam-5514	99	25	{	{	PUNCT
ejpam-5514	99	26	0	0	NUM
ejpam-5514	99	27	,	,	PUNCT
ejpam-5514	99	28	2	2	NUM
ejpam-5514	99	29	,	,	PUNCT
ejpam-5514	99	30	4	4	NUM
ejpam-5514	99	31	,	,	PUNCT
ejpam-5514	99	32	6	6	NUM
ejpam-5514	99	33	,	,	PUNCT
ejpam-5514	99	34	8	8	NUM
ejpam-5514	99	35	,	,	PUNCT
ejpam-5514	99	36	10	10	NUM
ejpam-5514	99	37	}	}	PUNCT
ejpam-5514	99	38	in	in	ADP
ejpam-5514	99	39	z12	z12	PROPN
ejpam-5514	99	40	.	.	PUNCT
ejpam-5514	100	1	here	here	ADV
ejpam-5514	100	2	,	,	PUNCT
ejpam-5514	100	3	i−(b	i−(b	PROPN
ejpam-5514	100	4	)	)	PUNCT
ejpam-5514	100	5	=	=	PRON
ejpam-5514	100	6	{	{	PUNCT
ejpam-5514	100	7	0	0	NUM
ejpam-5514	100	8	,	,	PUNCT
ejpam-5514	100	9	2	2	NUM
ejpam-5514	100	10	,	,	PUNCT
ejpam-5514	100	11	4	4	NUM
ejpam-5514	100	12	,	,	PUNCT
ejpam-5514	100	13	6	6	NUM
ejpam-5514	100	14	,	,	PUNCT
ejpam-5514	100	15	8	8	NUM
ejpam-5514	100	16	,	,	PUNCT
ejpam-5514	100	17	10	10	NUM
ejpam-5514	100	18	}	}	PUNCT
ejpam-5514	100	19	is	be	AUX
ejpam-5514	100	20	also	also	ADV
ejpam-5514	100	21	an	an	DET
ejpam-5514	100	22	ideal	ideal	NOUN
ejpam-5514	100	23	of	of	ADP
ejpam-5514	100	24	z12	z12	PROPN
ejpam-5514	100	25	,	,	PUNCT
ejpam-5514	100	26	and	and	CCONJ
ejpam-5514	100	27	its	its	PRON
ejpam-5514	100	28	lower	low	ADJ
ejpam-5514	100	29	approximation	approximation	NOUN
ejpam-5514	100	30	is	be	AUX
ejpam-5514	100	31	i−(b	i−(b	NOUN
ejpam-5514	100	32	)	)	PUNCT
ejpam-5514	100	33	=	=	PRON
ejpam-5514	101	1	{	{	PUNCT
ejpam-5514	101	2	0	0	NUM
ejpam-5514	101	3	,	,	PUNCT
ejpam-5514	101	4	6	6	NUM
ejpam-5514	101	5	}	}	PUNCT
ejpam-5514	101	6	.	.	PUNCT
ejpam-5514	102	1	the	the	DET
ejpam-5514	102	2	roughness	roughness	NOUN
ejpam-5514	102	3	ρ(b	ρ(b	PROPN
ejpam-5514	102	4	)	)	PUNCT
ejpam-5514	102	5	=	=	SYM
ejpam-5514	102	6	0.67	0.67	NUM
ejpam-5514	102	7	,	,	PUNCT
ejpam-5514	102	8	indicating	indicate	VERB
ejpam-5514	102	9	the	the	DET
ejpam-5514	102	10	degree	degree	NOUN
ejpam-5514	102	11	of	of	ADP
ejpam-5514	102	12	completeness	completeness	NOUN
ejpam-5514	102	13	of	of	ADP
ejpam-5514	102	14	our	our	PRON
ejpam-5514	102	15	knowledge	knowledge	NOUN
ejpam-5514	102	16	about	about	ADP
ejpam-5514	102	17	set	set	PROPN
ejpam-5514	102	18	b.	b.	PROPN
ejpam-5514	102	19	theorem	theorem	PROPN
ejpam-5514	102	20	4	4	NUM
ejpam-5514	102	21	.	.	PUNCT
ejpam-5514	103	1	let	let	VERB
ejpam-5514	103	2	ρ	ρ	NOUN
ejpam-5514	103	3	be	be	AUX
ejpam-5514	103	4	a	a	DET
ejpam-5514	103	5	complete	complete	ADJ
ejpam-5514	103	6	congruence	congruence	NOUN
ejpam-5514	103	7	relation	relation	NOUN
ejpam-5514	103	8	on	on	ADP
ejpam-5514	103	9	a	a	DET
ejpam-5514	103	10	ring	ring	NOUN
ejpam-5514	103	11	r	r	NOUN
ejpam-5514	103	12	,	,	PUNCT
ejpam-5514	103	13	and	and	CCONJ
ejpam-5514	103	14	a	a	PRON
ejpam-5514	103	15	be	be	AUX
ejpam-5514	103	16	an	an	DET
ejpam-5514	103	17	ideal	ideal	NOUN
ejpam-5514	103	18	of	of	ADP
ejpam-5514	103	19	r.	r.	PROPN
ejpam-5514	103	20	if	if	SCONJ
ejpam-5514	103	21	the	the	DET
ejpam-5514	103	22	lower	low	ADJ
ejpam-5514	103	23	approximation	approximation	NOUN
ejpam-5514	103	24	ρ−(a	ρ−(a	ADV
ejpam-5514	103	25	)	)	PUNCT
ejpam-5514	103	26	is	be	AUX
ejpam-5514	103	27	non	non	ADJ
ejpam-5514	103	28	-	-	ADJ
ejpam-5514	103	29	empty	empty	ADJ
ejpam-5514	103	30	,	,	PUNCT
ejpam-5514	103	31	then	then	ADV
ejpam-5514	103	32	it	it	PRON
ejpam-5514	103	33	is	be	AUX
ejpam-5514	103	34	also	also	ADV
ejpam-5514	103	35	an	an	DET
ejpam-5514	103	36	ideal	ideal	NOUN
ejpam-5514	103	37	of	of	ADP
ejpam-5514	103	38	ring	ring	PROPN
ejpam-5514	103	39	r.	r.	PROPN
ejpam-5514	103	40	proof	proof	NOUN
ejpam-5514	103	41	.	.	PUNCT
ejpam-5514	104	1	consider	consider	VERB
ejpam-5514	104	2	a	a	PRON
ejpam-5514	104	3	as	as	ADP
ejpam-5514	104	4	an	an	DET
ejpam-5514	104	5	ideal	ideal	NOUN
ejpam-5514	104	6	of	of	ADP
ejpam-5514	104	7	ring	ring	NOUN
ejpam-5514	104	8	r	r	NOUN
ejpam-5514	104	9	,	,	PUNCT
ejpam-5514	104	10	implying	imply	VERB
ejpam-5514	104	11	ra	ra	PROPN
ejpam-5514	104	12	⊆	⊆	NUM
ejpam-5514	104	13	a	a	PRON
ejpam-5514	104	14	and	and	CCONJ
ejpam-5514	104	15	ar	ar	NOUN
ejpam-5514	104	16	⊆	⊆	NUM
ejpam-5514	104	17	a.	a.	NOUN
ejpam-5514	104	18	note	note	NOUN
ejpam-5514	104	19	that	that	SCONJ
ejpam-5514	104	20	ρ−(r	ρ−(r	PROPN
ejpam-5514	104	21	)	)	PUNCT
ejpam-5514	104	22	=	=	VERB
ejpam-5514	104	23	r.	r.	NOUN
ejpam-5514	104	24	by	by	ADP
ejpam-5514	104	25	proposition	proposition	NOUN
ejpam-5514	104	26	2	2	NUM
ejpam-5514	104	27	,	,	PUNCT
ejpam-5514	104	28	we	we	PRON
ejpam-5514	104	29	have	have	AUX
ejpam-5514	104	30	:	:	PUNCT
ejpam-5514	104	31	rρ−(a	rρ−(a	ADV
ejpam-5514	104	32	)	)	PUNCT
ejpam-5514	104	33	=	=	PUNCT
ejpam-5514	104	34	ρ−(r)ρ−(a	ρ−(r)ρ−(a	NOUN
ejpam-5514	104	35	)	)	PUNCT
ejpam-5514	104	36	⊆	⊆	NUM
ejpam-5514	104	37	ρ−(ra	ρ−(ra	NOUN
ejpam-5514	104	38	)	)	PUNCT
ejpam-5514	104	39	⊆	⊆	NUM
ejpam-5514	104	40	ρ−(a	ρ−(a	ADV
ejpam-5514	104	41	)	)	PUNCT
ejpam-5514	104	42	this	this	PRON
ejpam-5514	104	43	indicates	indicate	VERB
ejpam-5514	104	44	that	that	SCONJ
ejpam-5514	104	45	ρ−(a	ρ−(a	ADV
ejpam-5514	104	46	)	)	PUNCT
ejpam-5514	104	47	is	be	AUX
ejpam-5514	104	48	an	an	DET
ejpam-5514	104	49	ideal	ideal	NOUN
ejpam-5514	104	50	of	of	ADP
ejpam-5514	104	51	r	r	NOUN
ejpam-5514	104	52	,	,	PUNCT
ejpam-5514	104	53	and	and	CCONJ
ejpam-5514	104	54	it	it	PRON
ejpam-5514	104	55	represents	represent	VERB
ejpam-5514	104	56	a	a	DET
ejpam-5514	104	57	lower	low	ADJ
ejpam-5514	104	58	rough	rough	ADJ
ejpam-5514	104	59	ideal	ideal	NOUN
ejpam-5514	104	60	.	.	PUNCT
ejpam-5514	105	1	definition	definition	NOUN
ejpam-5514	105	2	1	1	NUM
ejpam-5514	105	3	.	.	PUNCT
ejpam-5514	105	4	suppose	suppose	VERB
ejpam-5514	105	5	ρ	ρ	PROPN
ejpam-5514	105	6	is	be	AUX
ejpam-5514	105	7	a	a	DET
ejpam-5514	105	8	congruence	congruence	NOUN
ejpam-5514	105	9	relation	relation	NOUN
ejpam-5514	105	10	on	on	ADP
ejpam-5514	105	11	a	a	DET
ejpam-5514	105	12	ring	ring	NOUN
ejpam-5514	105	13	r	r	NOUN
ejpam-5514	105	14	,	,	PUNCT
ejpam-5514	105	15	and	and	CCONJ
ejpam-5514	105	16	a	a	DET
ejpam-5514	105	17	⊆	⊆	NUM
ejpam-5514	105	18	r.	r.	NOUN
ejpam-5514	105	19	we	we	PRON
ejpam-5514	105	20	define	define	VERB
ejpam-5514	105	21	ρ(a	ρ(a	PROPN
ejpam-5514	105	22	)	)	PUNCT
ejpam-5514	106	1	=	=	SYM
ejpam-5514	106	2	(	(	PUNCT
ejpam-5514	106	3	ρ−(a	ρ−(a	ADV
ejpam-5514	106	4	)	)	PUNCT
ejpam-5514	106	5	,	,	PUNCT
ejpam-5514	106	6	ρ−(a	ρ−(a	ADV
ejpam-5514	106	7	)	)	PUNCT
ejpam-5514	106	8	)	)	PUNCT
ejpam-5514	107	1	as	as	ADP
ejpam-5514	107	2	a	a	DET
ejpam-5514	107	3	rough	rough	ADJ
ejpam-5514	107	4	ideal	ideal	NOUN
ejpam-5514	107	5	of	of	ADP
ejpam-5514	107	6	r	r	NOUN
ejpam-5514	107	7	if	if	SCONJ
ejpam-5514	107	8	both	both	CCONJ
ejpam-5514	107	9	the	the	DET
ejpam-5514	107	10	lower	low	ADJ
ejpam-5514	107	11	approximation	approximation	NOUN
ejpam-5514	107	12	ρ−(a	ρ−(a	ADV
ejpam-5514	107	13	)	)	PUNCT
ejpam-5514	107	14	and	and	CCONJ
ejpam-5514	107	15	the	the	DET
ejpam-5514	107	16	upper	upper	ADJ
ejpam-5514	107	17	approximation	approximation	NOUN
ejpam-5514	107	18	ρ−(a	ρ−(a	ADV
ejpam-5514	107	19	)	)	PUNCT
ejpam-5514	107	20	are	be	AUX
ejpam-5514	107	21	ideals	ideal	NOUN
ejpam-5514	107	22	of	of	ADP
ejpam-5514	107	23	the	the	DET
ejpam-5514	107	24	ring	ring	PROPN
ejpam-5514	107	25	r.	r.	PROPN
ejpam-5514	107	26	example	example	NOUN
ejpam-5514	108	1	2	2	NUM
ejpam-5514	108	2	.	.	PUNCT
ejpam-5514	108	3	let	let	VERB
ejpam-5514	108	4	’s	’s	PRON
ejpam-5514	108	5	take	take	VERB
ejpam-5514	108	6	r	r	NOUN
ejpam-5514	108	7	=	=	SYM
ejpam-5514	108	8	z12	z12	NUM
ejpam-5514	108	9	,	,	PUNCT
ejpam-5514	108	10	i	i	PRON
ejpam-5514	108	11	=	=	PUNCT
ejpam-5514	108	12	{	{	PUNCT
ejpam-5514	108	13	0	0	NUM
ejpam-5514	108	14	,	,	PUNCT
ejpam-5514	108	15	4	4	NUM
ejpam-5514	108	16	}	}	PUNCT
ejpam-5514	108	17	as	as	ADP
ejpam-5514	108	18	an	an	DET
ejpam-5514	108	19	ideal	ideal	NOUN
ejpam-5514	108	20	of	of	ADP
ejpam-5514	108	21	r	r	NOUN
ejpam-5514	108	22	,	,	PUNCT
ejpam-5514	108	23	and	and	CCONJ
ejpam-5514	108	24	a	a	PRON
ejpam-5514	108	25	=	=	X
ejpam-5514	108	26	{	{	PUNCT
ejpam-5514	108	27	0	0	NUM
ejpam-5514	108	28	,	,	PUNCT
ejpam-5514	108	29	4	4	NUM
ejpam-5514	108	30	,	,	PUNCT
ejpam-5514	108	31	6	6	NUM
ejpam-5514	108	32	}	}	PUNCT
ejpam-5514	108	33	as	as	ADP
ejpam-5514	108	34	a	a	DET
ejpam-5514	108	35	subset	subset	NOUN
ejpam-5514	108	36	of	of	ADP
ejpam-5514	108	37	r.	r.	PROPN
ejpam-5514	108	38	in	in	ADP
ejpam-5514	108	39	this	this	DET
ejpam-5514	108	40	case	case	NOUN
ejpam-5514	108	41	,	,	PUNCT
ejpam-5514	108	42	i−(a	i−(a	ADV
ejpam-5514	108	43	)	)	PUNCT
ejpam-5514	108	44	=	=	SYM
ejpam-5514	108	45	{	{	PUNCT
ejpam-5514	108	46	0	0	NUM
ejpam-5514	108	47	,	,	PUNCT
ejpam-5514	108	48	4	4	NUM
ejpam-5514	108	49	}	}	PUNCT
ejpam-5514	108	50	and	and	CCONJ
ejpam-5514	108	51	i−(a	i−(a	ADV
ejpam-5514	108	52	)	)	PUNCT
ejpam-5514	108	53	=	=	PRON
ejpam-5514	108	54	{	{	PUNCT
ejpam-5514	108	55	0	0	NUM
ejpam-5514	108	56	,	,	PUNCT
ejpam-5514	108	57	2	2	NUM
ejpam-5514	108	58	,	,	PUNCT
ejpam-5514	108	59	4	4	NUM
ejpam-5514	108	60	,	,	PUNCT
ejpam-5514	108	61	6	6	NUM
ejpam-5514	108	62	}	}	PUNCT
ejpam-5514	108	63	are	be	AUX
ejpam-5514	108	64	the	the	DET
ejpam-5514	108	65	lower	low	ADJ
ejpam-5514	108	66	and	and	CCONJ
ejpam-5514	108	67	upper	upper	ADJ
ejpam-5514	108	68	approximations	approximation	NOUN
ejpam-5514	108	69	of	of	ADP
ejpam-5514	108	70	a	a	DET
ejpam-5514	108	71	concerning	concern	VERB
ejpam-5514	108	72	the	the	DET
ejpam-5514	108	73	congruence	congruence	PROPN
ejpam-5514	108	74	relation	relation	PROPN
ejpam-5514	108	75	i.	i.	PROPN
ejpam-5514	108	76	since	since	SCONJ
ejpam-5514	108	77	both	both	PRON
ejpam-5514	108	78	i−(a	i−(a	ADV
ejpam-5514	108	79	)	)	PUNCT
ejpam-5514	108	80	and	and	CCONJ
ejpam-5514	108	81	i−(a	i−(a	ADV
ejpam-5514	108	82	)	)	PUNCT
ejpam-5514	108	83	are	be	AUX
ejpam-5514	108	84	ideals	ideal	NOUN
ejpam-5514	108	85	of	of	ADP
ejpam-5514	108	86	r	r	NOUN
ejpam-5514	108	87	,	,	PUNCT
ejpam-5514	108	88	this	this	PRON
ejpam-5514	108	89	implies	imply	VERB
ejpam-5514	108	90	that	that	SCONJ
ejpam-5514	108	91	i(a	i(a	PROPN
ejpam-5514	108	92	)	)	PUNCT
ejpam-5514	108	93	=	=	PUNCT
ejpam-5514	108	94	(	(	PUNCT
ejpam-5514	108	95	i−(a	i−(a	ADV
ejpam-5514	108	96	)	)	PUNCT
ejpam-5514	108	97	,	,	PUNCT
ejpam-5514	108	98	i−(a	i−(a	ADV
ejpam-5514	108	99	)	)	PUNCT
ejpam-5514	108	100	)	)	PUNCT
ejpam-5514	108	101	is	be	AUX
ejpam-5514	108	102	a	a	DET
ejpam-5514	108	103	rough	rough	ADJ
ejpam-5514	108	104	ideal	ideal	NOUN
ejpam-5514	108	105	of	of	ADP
ejpam-5514	108	106	z12	z12	PROPN
ejpam-5514	108	107	.	.	PUNCT
ejpam-5514	109	1	the	the	DET
ejpam-5514	109	2	roughness	roughness	NOUN
ejpam-5514	109	3	of	of	ADP
ejpam-5514	109	4	a	a	PRON
ejpam-5514	109	5	is	be	AUX
ejpam-5514	109	6	calculated	calculate	VERB
ejpam-5514	109	7	as	as	ADP
ejpam-5514	109	8	ρ(a	ρ(a	PROPN
ejpam-5514	109	9	)	)	PUNCT
ejpam-5514	109	10	=	=	SYM
ejpam-5514	109	11	0.5	0.5	NUM
ejpam-5514	109	12	.	.	PUNCT
ejpam-5514	110	1	theorem	theorem	VERB
ejpam-5514	110	2	5	5	NUM
ejpam-5514	110	3	.	.	PUNCT
ejpam-5514	111	1	if	if	SCONJ
ejpam-5514	111	2	ρ	ρ	PROPN
ejpam-5514	111	3	and	and	CCONJ
ejpam-5514	111	4	λ	λ	PROPN
ejpam-5514	111	5	are	be	AUX
ejpam-5514	111	6	congruence	congruence	NOUN
ejpam-5514	111	7	relations	relation	NOUN
ejpam-5514	111	8	on	on	ADP
ejpam-5514	111	9	a	a	DET
ejpam-5514	111	10	ring	ring	NOUN
ejpam-5514	111	11	r	r	NOUN
ejpam-5514	111	12	,	,	PUNCT
ejpam-5514	111	13	and	and	CCONJ
ejpam-5514	111	14	ρ	ρ	PROPN
ejpam-5514	111	15	⊂	⊂	PROPN
ejpam-5514	111	16	λ	λ	PROPN
ejpam-5514	111	17	,	,	PUNCT
ejpam-5514	111	18	and	and	CCONJ
ejpam-5514	111	19	if	if	SCONJ
ejpam-5514	111	20	a	a	PRON
ejpam-5514	111	21	is	be	AUX
ejpam-5514	111	22	a	a	DET
ejpam-5514	111	23	non	non	ADJ
ejpam-5514	111	24	-	-	ADJ
ejpam-5514	111	25	empty	empty	ADJ
ejpam-5514	111	26	subset	subset	NOUN
ejpam-5514	111	27	of	of	ADP
ejpam-5514	111	28	r	r	NOUN
ejpam-5514	111	29	,	,	PUNCT
ejpam-5514	111	30	then	then	ADV
ejpam-5514	111	31	(	(	PUNCT
ejpam-5514	111	32	ρ	ρ	PROPN
ejpam-5514	111	33	∩	∩	X
ejpam-5514	111	34	λ)−(a	λ)−(a	NUM
ejpam-5514	111	35	)	)	PUNCT
ejpam-5514	111	36	=	=	SYM
ejpam-5514	111	37	ρ−(a	ρ−(a	ADV
ejpam-5514	111	38	)	)	PUNCT
ejpam-5514	111	39	∩	∩	ADJ
ejpam-5514	111	40	λ−(a	λ−(a	NOUN
ejpam-5514	111	41	)	)	PUNCT
ejpam-5514	111	42	.	.	PUNCT
ejpam-5514	112	1	proof	proof	NOUN
ejpam-5514	112	2	.	.	PUNCT
ejpam-5514	113	1	since	since	SCONJ
ejpam-5514	113	2	ρ	ρ	PROPN
ejpam-5514	113	3	and	and	CCONJ
ejpam-5514	113	4	λ	λ	PROPN
ejpam-5514	113	5	are	be	AUX
ejpam-5514	113	6	congruence	congruence	NOUN
ejpam-5514	113	7	relations	relation	NOUN
ejpam-5514	113	8	on	on	ADP
ejpam-5514	113	9	a	a	DET
ejpam-5514	113	10	ring	ring	NOUN
ejpam-5514	113	11	r	r	NOUN
ejpam-5514	113	12	,	,	PUNCT
ejpam-5514	113	13	this	this	PRON
ejpam-5514	113	14	implies	imply	VERB
ejpam-5514	113	15	that	that	SCONJ
ejpam-5514	113	16	ρ∩λ	ρ∩λ	PROPN
ejpam-5514	113	17	is	be	AUX
ejpam-5514	113	18	also	also	ADV
ejpam-5514	113	19	a	a	DET
ejpam-5514	113	20	congruence	congruence	NOUN
ejpam-5514	113	21	relation	relation	NOUN
ejpam-5514	113	22	on	on	ADP
ejpam-5514	113	23	ring	ring	PROPN
ejpam-5514	113	24	r.	r.	PROPN
ejpam-5514	113	25	for	for	ADP
ejpam-5514	113	26	any	any	DET
ejpam-5514	113	27	c	c	PROPN
ejpam-5514	113	28	∈	∈	PROPN
ejpam-5514	113	29	(	(	PUNCT
ejpam-5514	113	30	ρ	ρ	PROPN
ejpam-5514	113	31	∩	∩	PROPN
ejpam-5514	113	32	λ)−(a	λ)−(a	NUM
ejpam-5514	113	33	)	)	PUNCT
ejpam-5514	113	34	,	,	PUNCT
ejpam-5514	113	35	we	we	PRON
ejpam-5514	113	36	can	can	AUX
ejpam-5514	113	37	deduce	deduce	VERB
ejpam-5514	113	38	that	that	DET
ejpam-5514	113	39	c	c	PROPN
ejpam-5514	113	40	∈	∈	PROPN
ejpam-5514	113	41	ρ−(a	ρ−(a	ADV
ejpam-5514	113	42	)	)	PUNCT
ejpam-5514	113	43	and	and	CCONJ
ejpam-5514	113	44	c	c	NOUN
ejpam-5514	113	45	∈	∈	PROPN
ejpam-5514	113	46	λ−(a	λ−(a	NOUN
ejpam-5514	113	47	)	)	PUNCT
ejpam-5514	113	48	.	.	PUNCT
ejpam-5514	114	1	this	this	PRON
ejpam-5514	114	2	establishes	establish	VERB
ejpam-5514	114	3	that	that	SCONJ
ejpam-5514	114	4	(	(	PUNCT
ejpam-5514	114	5	ρ	ρ	PROPN
ejpam-5514	114	6	∩	∩	X
ejpam-5514	114	7	λ)−(a	λ)−(a	NUM
ejpam-5514	114	8	)	)	PUNCT
ejpam-5514	114	9	⊆	⊆	NUM
ejpam-5514	114	10	ρ−(a	ρ−(a	ADV
ejpam-5514	114	11	)	)	PUNCT
ejpam-5514	114	12	∩	∩	ADJ
ejpam-5514	114	13	λ−(a	λ−(a	NOUN
ejpam-5514	114	14	)	)	PUNCT
ejpam-5514	114	15	.	.	PUNCT
ejpam-5514	115	1	conversely	conversely	ADV
ejpam-5514	115	2	,	,	PUNCT
ejpam-5514	115	3	for	for	ADP
ejpam-5514	115	4	c	c	PROPN
ejpam-5514	115	5	∈	∈	PROPN
ejpam-5514	115	6	ρ−(a	ρ−(a	ADV
ejpam-5514	115	7	)	)	PUNCT
ejpam-5514	115	8	∩	∩	ADJ
ejpam-5514	115	9	λ−(a	λ−(a	NOUN
ejpam-5514	115	10	)	)	PUNCT
ejpam-5514	115	11	,	,	PUNCT
ejpam-5514	115	12	we	we	PRON
ejpam-5514	115	13	can	can	AUX
ejpam-5514	115	14	show	show	VERB
ejpam-5514	115	15	that	that	SCONJ
ejpam-5514	115	16	c	c	PROPN
ejpam-5514	115	17	∈	∈	PROPN
ejpam-5514	115	18	(	(	PUNCT
ejpam-5514	115	19	ρ	ρ	PROPN
ejpam-5514	115	20	∩	∩	PROPN
ejpam-5514	115	21	λ)−(a	λ)−(a	NUM
ejpam-5514	115	22	)	)	PUNCT
ejpam-5514	115	23	,	,	PUNCT
ejpam-5514	115	24	which	which	PRON
ejpam-5514	115	25	leads	lead	VERB
ejpam-5514	115	26	to	to	ADP
ejpam-5514	115	27	the	the	DET
ejpam-5514	115	28	conclusion	conclusion	NOUN
ejpam-5514	115	29	that	that	SCONJ
ejpam-5514	115	30	ρ−(a	ρ−(a	ADV
ejpam-5514	115	31	)	)	PUNCT
ejpam-5514	115	32	∩	∩	ADJ
ejpam-5514	115	33	λ−(a	λ−(a	NOUN
ejpam-5514	115	34	)	)	PUNCT
ejpam-5514	115	35	⊆	⊆	NUM
ejpam-5514	115	36	(	(	PUNCT
ejpam-5514	115	37	ρ	ρ	NOUN
ejpam-5514	115	38	∩	∩	X
ejpam-5514	115	39	λ)−(a	λ)−(a	NUM
ejpam-5514	115	40	)	)	PUNCT
ejpam-5514	115	41	.	.	PUNCT
ejpam-5514	116	1	hence	hence	ADV
ejpam-5514	116	2	,	,	PUNCT
ejpam-5514	116	3	we	we	PRON
ejpam-5514	116	4	have	have	AUX
ejpam-5514	116	5	proven	prove	VERB
ejpam-5514	116	6	(	(	PUNCT
ejpam-5514	116	7	ρ	ρ	PROPN
ejpam-5514	116	8	∩	∩	X
ejpam-5514	116	9	λ)−(a	λ)−(a	NUM
ejpam-5514	116	10	)	)	PUNCT
ejpam-5514	116	11	=	=	SYM
ejpam-5514	116	12	ρ−(a	ρ−(a	ADV
ejpam-5514	116	13	)	)	PUNCT
ejpam-5514	116	14	∩	∩	ADJ
ejpam-5514	116	15	λ−(a	λ−(a	NOUN
ejpam-5514	116	16	)	)	PUNCT
ejpam-5514	116	17	.	.	PUNCT
ejpam-5514	117	1	a.	a.	PROPN
ejpam-5514	117	2	prakash	prakash	PROPN
ejpam-5514	117	3	,	,	PUNCT
ejpam-5514	117	4	r.	r.	PROPN
ejpam-5514	117	5	shukla	shukla	PROPN
ejpam-5514	117	6	/	/	SYM
ejpam-5514	117	7	eur	eur	PROPN
ejpam-5514	117	8	.	.	PUNCT
ejpam-5514	118	1	j.	j.	PROPN
ejpam-5514	118	2	pure	pure	PROPN
ejpam-5514	118	3	appl	appl	PROPN
ejpam-5514	118	4	.	.	PROPN
ejpam-5514	118	5	math	math	PROPN
ejpam-5514	118	6	,	,	PUNCT
ejpam-5514	118	7	18	18	NUM
ejpam-5514	118	8	(	(	PUNCT
ejpam-5514	118	9	1	1	NUM
ejpam-5514	118	10	)	)	PUNCT
ejpam-5514	118	11	(	(	PUNCT
ejpam-5514	118	12	2025	2025	NUM
ejpam-5514	118	13	)	)	PUNCT
ejpam-5514	118	14	,	,	PUNCT
ejpam-5514	118	15	5514	5514	NUM
ejpam-5514	118	16	6	6	NUM
ejpam-5514	118	17	of	of	ADP
ejpam-5514	118	18	9	9	NUM
ejpam-5514	118	19	example	example	NOUN
ejpam-5514	118	20	3	3	NUM
ejpam-5514	118	21	.	.	X
ejpam-5514	118	22	consider	consider	VERB
ejpam-5514	118	23	r	r	NOUN
ejpam-5514	118	24	=	=	SYM
ejpam-5514	118	25	z12	z12	NUM
ejpam-5514	118	26	as	as	ADP
ejpam-5514	118	27	the	the	DET
ejpam-5514	118	28	ring	ring	NOUN
ejpam-5514	118	29	,	,	PUNCT
ejpam-5514	118	30	j	j	PROPN
ejpam-5514	118	31	=	=	PUNCT
ejpam-5514	118	32	{	{	PUNCT
ejpam-5514	118	33	0	0	NUM
ejpam-5514	118	34	,	,	PUNCT
ejpam-5514	118	35	6	6	NUM
ejpam-5514	118	36	}	}	PUNCT
ejpam-5514	118	37	and	and	CCONJ
ejpam-5514	118	38	k	k	NOUN
ejpam-5514	118	39	=	=	X
ejpam-5514	118	40	{	{	PUNCT
ejpam-5514	118	41	0	0	NUM
ejpam-5514	118	42	,	,	PUNCT
ejpam-5514	118	43	2	2	NUM
ejpam-5514	118	44	,	,	PUNCT
ejpam-5514	118	45	4	4	NUM
ejpam-5514	118	46	,	,	PUNCT
ejpam-5514	118	47	6	6	NUM
ejpam-5514	118	48	,	,	PUNCT
ejpam-5514	118	49	8	8	NUM
ejpam-5514	118	50	,	,	PUNCT
ejpam-5514	118	51	10	10	NUM
ejpam-5514	118	52	}	}	PUNCT
ejpam-5514	118	53	as	as	ADP
ejpam-5514	118	54	congruence	congruence	NOUN
ejpam-5514	118	55	relations	relation	NOUN
ejpam-5514	118	56	on	on	ADP
ejpam-5514	118	57	z12	z12	PROPN
ejpam-5514	118	58	.	.	PUNCT
ejpam-5514	119	1	it	it	PRON
ejpam-5514	119	2	’s	’	VERB
ejpam-5514	119	3	clear	clear	ADJ
ejpam-5514	119	4	that	that	SCONJ
ejpam-5514	119	5	j	j	PROPN
ejpam-5514	119	6	and	and	CCONJ
ejpam-5514	119	7	k	k	PROPN
ejpam-5514	119	8	are	be	AUX
ejpam-5514	119	9	ideals	ideal	NOUN
ejpam-5514	119	10	on	on	ADP
ejpam-5514	119	11	r.	r.	PROPN
ejpam-5514	119	12	in	in	ADP
ejpam-5514	119	13	this	this	DET
ejpam-5514	119	14	scenario	scenario	NOUN
ejpam-5514	119	15	,	,	PUNCT
ejpam-5514	119	16	j	j	PROPN
ejpam-5514	119	17	is	be	AUX
ejpam-5514	119	18	a	a	DET
ejpam-5514	119	19	subset	subset	NOUN
ejpam-5514	119	20	of	of	ADP
ejpam-5514	119	21	k.	k.	PROPN
ejpam-5514	119	22	let	let	VERB
ejpam-5514	119	23	b	b	NOUN
ejpam-5514	119	24	=	=	PRON
ejpam-5514	119	25	{	{	PUNCT
ejpam-5514	119	26	0	0	NUM
ejpam-5514	119	27	,	,	PUNCT
ejpam-5514	119	28	1	1	NUM
ejpam-5514	119	29	,	,	PUNCT
ejpam-5514	119	30	2	2	NUM
ejpam-5514	119	31	,	,	PUNCT
ejpam-5514	119	32	5	5	NUM
ejpam-5514	119	33	}	}	PUNCT
ejpam-5514	119	34	as	as	ADP
ejpam-5514	119	35	a	a	DET
ejpam-5514	119	36	subset	subset	NOUN
ejpam-5514	119	37	of	of	ADP
ejpam-5514	119	38	z12	z12	PROPN
ejpam-5514	119	39	.	.	PUNCT
ejpam-5514	120	1	by	by	ADP
ejpam-5514	120	2	calculating	calculate	VERB
ejpam-5514	120	3	the	the	DET
ejpam-5514	120	4	respective	respective	ADJ
ejpam-5514	120	5	classes	class	NOUN
ejpam-5514	120	6	,	,	PUNCT
ejpam-5514	120	7	we	we	PRON
ejpam-5514	120	8	obtain	obtain	VERB
ejpam-5514	120	9	the	the	DET
ejpam-5514	120	10	lower	low	ADJ
ejpam-5514	120	11	approximations	approximation	NOUN
ejpam-5514	120	12	:	:	PUNCT
ejpam-5514	120	13	j−(b	j−(b	PROPN
ejpam-5514	120	14	)	)	PUNCT
ejpam-5514	121	1	=	=	PUNCT
ejpam-5514	121	2	{	{	PUNCT
ejpam-5514	121	3	0	0	NUM
ejpam-5514	121	4	,	,	PUNCT
ejpam-5514	121	5	1	1	NUM
ejpam-5514	121	6	,	,	PUNCT
ejpam-5514	121	7	2	2	NUM
ejpam-5514	121	8	,	,	PUNCT
ejpam-5514	121	9	5	5	NUM
ejpam-5514	121	10	,	,	PUNCT
ejpam-5514	121	11	6	6	NUM
ejpam-5514	121	12	,	,	PUNCT
ejpam-5514	121	13	7	7	NUM
ejpam-5514	121	14	,	,	PUNCT
ejpam-5514	121	15	8	8	NUM
ejpam-5514	121	16	,	,	PUNCT
ejpam-5514	121	17	11	11	NUM
ejpam-5514	121	18	}	}	PUNCT
ejpam-5514	121	19	k−(b	k−(b	PROPN
ejpam-5514	121	20	)	)	PUNCT
ejpam-5514	121	21	=	=	PUNCT
ejpam-5514	121	22	{	{	PUNCT
ejpam-5514	121	23	0	0	NUM
ejpam-5514	121	24	,	,	PUNCT
ejpam-5514	121	25	1	1	NUM
ejpam-5514	121	26	,	,	PUNCT
ejpam-5514	121	27	2	2	NUM
ejpam-5514	121	28	,	,	PUNCT
ejpam-5514	121	29	3	3	NUM
ejpam-5514	121	30	,	,	PUNCT
ejpam-5514	121	31	4	4	NUM
ejpam-5514	121	32	,	,	PUNCT
ejpam-5514	121	33	5	5	NUM
ejpam-5514	121	34	,	,	PUNCT
ejpam-5514	121	35	6	6	NUM
ejpam-5514	121	36	,	,	PUNCT
ejpam-5514	121	37	7	7	NUM
ejpam-5514	121	38	,	,	PUNCT
ejpam-5514	121	39	8	8	NUM
ejpam-5514	121	40	,	,	PUNCT
ejpam-5514	121	41	9	9	NUM
ejpam-5514	121	42	,	,	PUNCT
ejpam-5514	121	43	10	10	NUM
ejpam-5514	121	44	,	,	PUNCT
ejpam-5514	121	45	11	11	NUM
ejpam-5514	121	46	}	}	PUNCT
ejpam-5514	121	47	(	(	PUNCT
ejpam-5514	121	48	j	j	PROPN
ejpam-5514	121	49	∩k)−(b	∩k)−(b	PROPN
ejpam-5514	121	50	)	)	PUNCT
ejpam-5514	122	1	=	=	PRON
ejpam-5514	122	2	{	{	PUNCT
ejpam-5514	122	3	0	0	NUM
ejpam-5514	122	4	,	,	PUNCT
ejpam-5514	122	5	1	1	NUM
ejpam-5514	122	6	,	,	PUNCT
ejpam-5514	122	7	2	2	NUM
ejpam-5514	122	8	,	,	PUNCT
ejpam-5514	122	9	5	5	NUM
ejpam-5514	122	10	,	,	PUNCT
ejpam-5514	122	11	6	6	NUM
ejpam-5514	122	12	,	,	PUNCT
ejpam-5514	122	13	7	7	NUM
ejpam-5514	122	14	,	,	PUNCT
ejpam-5514	122	15	8	8	NUM
ejpam-5514	122	16	,	,	PUNCT
ejpam-5514	122	17	11	11	NUM
ejpam-5514	122	18	}	}	PUNCT
ejpam-5514	122	19	the	the	DET
ejpam-5514	122	20	intersection	intersection	NOUN
ejpam-5514	122	21	of	of	ADP
ejpam-5514	122	22	j−(b	j−(b	PROPN
ejpam-5514	122	23	)	)	PUNCT
ejpam-5514	122	24	and	and	CCONJ
ejpam-5514	122	25	k−(b	k−(b	PROPN
ejpam-5514	122	26	)	)	PUNCT
ejpam-5514	122	27	equals	equal	VERB
ejpam-5514	122	28	(	(	PUNCT
ejpam-5514	122	29	j	j	PROPN
ejpam-5514	122	30	∩	∩	NOUN
ejpam-5514	122	31	k)−(b	k)−(b	X
ejpam-5514	122	32	)	)	PUNCT
ejpam-5514	122	33	,	,	PUNCT
ejpam-5514	122	34	showing	show	VERB
ejpam-5514	122	35	that	that	SCONJ
ejpam-5514	122	36	the	the	DET
ejpam-5514	122	37	lower	low	ADJ
ejpam-5514	122	38	approximation	approximation	NOUN
ejpam-5514	122	39	of	of	ADP
ejpam-5514	122	40	b	b	NOUN
ejpam-5514	122	41	under	under	ADP
ejpam-5514	122	42	congruence	congruence	NOUN
ejpam-5514	122	43	relation	relation	NOUN
ejpam-5514	122	44	(	(	PUNCT
ejpam-5514	122	45	j	j	PROPN
ejpam-5514	122	46	∩k	∩k	PROPN
ejpam-5514	122	47	)	)	PUNCT
ejpam-5514	122	48	is	be	AUX
ejpam-5514	122	49	(	(	PUNCT
ejpam-5514	122	50	j	j	PROPN
ejpam-5514	122	51	∩k)−(b	∩k)−(b	PROPN
ejpam-5514	122	52	)	)	PUNCT
ejpam-5514	122	53	.	.	PUNCT
ejpam-5514	123	1	theorem	theorem	VERB
ejpam-5514	123	2	6	6	NUM
ejpam-5514	123	3	.	.	PUNCT
ejpam-5514	124	1	if	if	SCONJ
ejpam-5514	124	2	ρ	ρ	PROPN
ejpam-5514	124	3	and	and	CCONJ
ejpam-5514	124	4	λ	λ	PROPN
ejpam-5514	124	5	are	be	AUX
ejpam-5514	124	6	congruence	congruence	NOUN
ejpam-5514	124	7	relations	relation	NOUN
ejpam-5514	124	8	on	on	ADP
ejpam-5514	124	9	a	a	DET
ejpam-5514	124	10	ring	ring	NOUN
ejpam-5514	124	11	r	r	NOUN
ejpam-5514	124	12	,	,	PUNCT
ejpam-5514	124	13	and	and	CCONJ
ejpam-5514	124	14	a	a	PRON
ejpam-5514	124	15	is	be	AUX
ejpam-5514	124	16	a	a	DET
ejpam-5514	124	17	non	non	ADJ
ejpam-5514	124	18	-	-	ADJ
ejpam-5514	124	19	empty	empty	ADJ
ejpam-5514	124	20	subset	subset	NOUN
ejpam-5514	124	21	of	of	ADP
ejpam-5514	124	22	r	r	NOUN
ejpam-5514	124	23	,	,	PUNCT
ejpam-5514	124	24	then	then	ADV
ejpam-5514	124	25	(	(	PUNCT
ejpam-5514	124	26	ρ	ρ	PROPN
ejpam-5514	124	27	∩	∩	X
ejpam-5514	124	28	λ)−(a	λ)−(a	NUM
ejpam-5514	124	29	)	)	PUNCT
ejpam-5514	124	30	=	=	SYM
ejpam-5514	124	31	ρ−(a	ρ−(a	ADV
ejpam-5514	124	32	)	)	PUNCT
ejpam-5514	124	33	∩	∩	ADJ
ejpam-5514	124	34	λ−(a	λ−(a	NOUN
ejpam-5514	124	35	)	)	PUNCT
ejpam-5514	124	36	.	.	PUNCT
ejpam-5514	125	1	proof	proof	NOUN
ejpam-5514	125	2	.	.	PUNCT
ejpam-5514	126	1	the	the	DET
ejpam-5514	126	2	proof	proof	NOUN
ejpam-5514	126	3	for	for	ADP
ejpam-5514	126	4	this	this	DET
ejpam-5514	126	5	theorem	theorem	NOUN
ejpam-5514	126	6	follows	follow	VERB
ejpam-5514	126	7	a	a	DET
ejpam-5514	126	8	similar	similar	ADJ
ejpam-5514	126	9	logic	logic	NOUN
ejpam-5514	126	10	to	to	ADP
ejpam-5514	126	11	the	the	DET
ejpam-5514	126	12	previous	previous	ADJ
ejpam-5514	126	13	theorem	theorem	NOUN
ejpam-5514	126	14	,	,	PUNCT
ejpam-5514	126	15	with	with	ADP
ejpam-5514	126	16	the	the	DET
ejpam-5514	126	17	focus	focus	NOUN
ejpam-5514	126	18	on	on	ADP
ejpam-5514	126	19	the	the	DET
ejpam-5514	126	20	lower	low	ADJ
ejpam-5514	126	21	approximations	approximation	NOUN
ejpam-5514	126	22	.	.	PUNCT
ejpam-5514	127	1	example	example	NOUN
ejpam-5514	127	2	4	4	NUM
ejpam-5514	127	3	.	.	PUNCT
ejpam-5514	128	1	let	let	VERB
ejpam-5514	128	2	’s	’s	NOUN
ejpam-5514	128	3	consider	consider	VERB
ejpam-5514	128	4	r	r	NOUN
ejpam-5514	128	5	=	=	SYM
ejpam-5514	128	6	z12	z12	NUM
ejpam-5514	128	7	as	as	ADP
ejpam-5514	128	8	the	the	DET
ejpam-5514	128	9	ring	ring	NOUN
ejpam-5514	128	10	,	,	PUNCT
ejpam-5514	128	11	i	i	PRON
ejpam-5514	128	12	=	=	PUNCT
ejpam-5514	128	13	{	{	PUNCT
ejpam-5514	128	14	0	0	NUM
ejpam-5514	128	15	,	,	PUNCT
ejpam-5514	128	16	4	4	NUM
ejpam-5514	128	17	,	,	PUNCT
ejpam-5514	128	18	8	8	NUM
ejpam-5514	128	19	}	}	PUNCT
ejpam-5514	128	20	,	,	PUNCT
ejpam-5514	128	21	and	and	CCONJ
ejpam-5514	128	22	k	k	PROPN
ejpam-5514	128	23	=	=	X
ejpam-5514	128	24	{	{	PUNCT
ejpam-5514	128	25	0	0	NUM
ejpam-5514	128	26	,	,	PUNCT
ejpam-5514	128	27	2	2	NUM
ejpam-5514	128	28	,	,	PUNCT
ejpam-5514	128	29	4	4	NUM
ejpam-5514	128	30	,	,	PUNCT
ejpam-5514	128	31	6	6	NUM
ejpam-5514	128	32	,	,	PUNCT
ejpam-5514	128	33	8	8	NUM
ejpam-5514	128	34	,	,	PUNCT
ejpam-5514	128	35	10	10	NUM
ejpam-5514	128	36	}	}	PUNCT
ejpam-5514	128	37	as	as	ADP
ejpam-5514	128	38	congruence	congruence	NOUN
ejpam-5514	128	39	relations	relation	NOUN
ejpam-5514	128	40	on	on	ADP
ejpam-5514	128	41	z12	z12	PROPN
ejpam-5514	128	42	.	.	PUNCT
ejpam-5514	129	1	we	we	PRON
ejpam-5514	129	2	note	note	VERB
ejpam-5514	129	3	that	that	SCONJ
ejpam-5514	129	4	i	i	PRON
ejpam-5514	129	5	and	and	CCONJ
ejpam-5514	129	6	k	k	PROPN
ejpam-5514	129	7	are	be	AUX
ejpam-5514	129	8	ideals	ideal	NOUN
ejpam-5514	129	9	on	on	ADP
ejpam-5514	129	10	z12	z12	PROPN
ejpam-5514	129	11	.	.	PUNCT
ejpam-5514	130	1	in	in	ADP
ejpam-5514	130	2	this	this	DET
ejpam-5514	130	3	case	case	NOUN
ejpam-5514	130	4	,	,	PUNCT
ejpam-5514	130	5	i	i	PRON
ejpam-5514	130	6	is	be	AUX
ejpam-5514	130	7	a	a	DET
ejpam-5514	130	8	subset	subset	NOUN
ejpam-5514	130	9	of	of	ADP
ejpam-5514	130	10	k.	k.	PROPN
ejpam-5514	130	11	let	let	VERB
ejpam-5514	130	12	a	a	PRON
ejpam-5514	130	13	=	=	PUNCT
ejpam-5514	130	14	{	{	PUNCT
ejpam-5514	130	15	1	1	NUM
ejpam-5514	130	16	,	,	PUNCT
ejpam-5514	130	17	3	3	NUM
ejpam-5514	130	18	,	,	PUNCT
ejpam-5514	130	19	5	5	NUM
ejpam-5514	130	20	,	,	PUNCT
ejpam-5514	130	21	7	7	NUM
ejpam-5514	130	22	,	,	PUNCT
ejpam-5514	130	23	9	9	NUM
ejpam-5514	130	24	,	,	PUNCT
ejpam-5514	130	25	11	11	NUM
ejpam-5514	130	26	}	}	PUNCT
ejpam-5514	130	27	as	as	ADP
ejpam-5514	130	28	a	a	DET
ejpam-5514	130	29	subset	subset	NOUN
ejpam-5514	130	30	of	of	ADP
ejpam-5514	130	31	z12	z12	PROPN
ejpam-5514	130	32	.	.	PUNCT
ejpam-5514	131	1	by	by	ADP
ejpam-5514	131	2	calculating	calculate	VERB
ejpam-5514	131	3	the	the	DET
ejpam-5514	131	4	classes	class	NOUN
ejpam-5514	131	5	for	for	ADP
ejpam-5514	131	6	i	i	PRON
ejpam-5514	131	7	and	and	CCONJ
ejpam-5514	131	8	k	k	NOUN
ejpam-5514	131	9	,	,	PUNCT
ejpam-5514	131	10	we	we	PRON
ejpam-5514	131	11	obtain	obtain	VERB
ejpam-5514	131	12	the	the	DET
ejpam-5514	131	13	lower	low	ADJ
ejpam-5514	131	14	approximations	approximation	NOUN
ejpam-5514	131	15	:	:	PUNCT
ejpam-5514	131	16	i−(a	i−(a	ADV
ejpam-5514	131	17	)	)	PUNCT
ejpam-5514	131	18	=	=	SYM
ejpam-5514	131	19	{	{	PUNCT
ejpam-5514	131	20	1	1	NUM
ejpam-5514	131	21	,	,	PUNCT
ejpam-5514	131	22	3	3	NUM
ejpam-5514	131	23	,	,	PUNCT
ejpam-5514	131	24	5	5	NUM
ejpam-5514	131	25	,	,	PUNCT
ejpam-5514	131	26	7	7	NUM
ejpam-5514	131	27	,	,	PUNCT
ejpam-5514	131	28	9	9	NUM
ejpam-5514	131	29	,	,	PUNCT
ejpam-5514	131	30	11	11	NUM
ejpam-5514	131	31	}	}	PUNCT
ejpam-5514	131	32	k−(a	k−(a	NOUN
ejpam-5514	131	33	)	)	PUNCT
ejpam-5514	131	34	=	=	PUNCT
ejpam-5514	131	35	{	{	PUNCT
ejpam-5514	131	36	1	1	NUM
ejpam-5514	131	37	,	,	PUNCT
ejpam-5514	131	38	3	3	NUM
ejpam-5514	131	39	,	,	PUNCT
ejpam-5514	131	40	5	5	NUM
ejpam-5514	131	41	,	,	PUNCT
ejpam-5514	131	42	7	7	NUM
ejpam-5514	131	43	,	,	PUNCT
ejpam-5514	131	44	9	9	NUM
ejpam-5514	131	45	,	,	PUNCT
ejpam-5514	131	46	11	11	NUM
ejpam-5514	131	47	}	}	PUNCT
ejpam-5514	131	48	(	(	PUNCT
ejpam-5514	131	49	i	i	PRON
ejpam-5514	131	50	∩k)−(a	∩k)−(a	ADV
ejpam-5514	131	51	)	)	PUNCT
ejpam-5514	131	52	=	=	PRON
ejpam-5514	131	53	{	{	PUNCT
ejpam-5514	131	54	1	1	NUM
ejpam-5514	131	55	,	,	PUNCT
ejpam-5514	131	56	3	3	NUM
ejpam-5514	131	57	,	,	PUNCT
ejpam-5514	131	58	5	5	NUM
ejpam-5514	131	59	,	,	PUNCT
ejpam-5514	131	60	7	7	NUM
ejpam-5514	131	61	,	,	PUNCT
ejpam-5514	131	62	9	9	NUM
ejpam-5514	131	63	,	,	PUNCT
ejpam-5514	131	64	11	11	NUM
ejpam-5514	131	65	}	}	PUNCT
ejpam-5514	131	66	in	in	ADP
ejpam-5514	131	67	this	this	DET
ejpam-5514	131	68	case	case	NOUN
ejpam-5514	131	69	,	,	PUNCT
ejpam-5514	131	70	the	the	DET
ejpam-5514	131	71	intersection	intersection	NOUN
ejpam-5514	131	72	of	of	ADP
ejpam-5514	131	73	i−(a	i−(a	ADV
ejpam-5514	131	74	)	)	PUNCT
ejpam-5514	131	75	and	and	CCONJ
ejpam-5514	131	76	k−(a	k−(a	NOUN
ejpam-5514	131	77	)	)	PUNCT
ejpam-5514	131	78	equals	equal	VERB
ejpam-5514	131	79	(	(	PUNCT
ejpam-5514	131	80	i	i	NOUN
ejpam-5514	131	81	∩k)−(a	∩k)−(a	NOUN
ejpam-5514	131	82	)	)	PUNCT
ejpam-5514	131	83	,	,	PUNCT
ejpam-5514	131	84	showing	show	VERB
ejpam-5514	131	85	that	that	SCONJ
ejpam-5514	131	86	the	the	DET
ejpam-5514	131	87	lower	low	ADJ
ejpam-5514	131	88	approximation	approximation	NOUN
ejpam-5514	131	89	of	of	ADP
ejpam-5514	131	90	a	a	PRON
ejpam-5514	131	91	under	under	ADP
ejpam-5514	131	92	congruence	congruence	NOUN
ejpam-5514	131	93	relation	relation	NOUN
ejpam-5514	131	94	(	(	PUNCT
ejpam-5514	131	95	i	i	PRON
ejpam-5514	131	96	∩k	∩k	ADJ
ejpam-5514	131	97	)	)	PUNCT
ejpam-5514	131	98	is	be	AUX
ejpam-5514	131	99	(	(	PUNCT
ejpam-5514	131	100	i	i	NOUN
ejpam-5514	131	101	∩k)−(a	∩k)−(a	ADV
ejpam-5514	131	102	)	)	PUNCT
ejpam-5514	131	103	.	.	PUNCT
ejpam-5514	132	1	definition	definition	NOUN
ejpam-5514	132	2	2	2	NUM
ejpam-5514	132	3	.	.	PUNCT
ejpam-5514	133	1	let	let	VERB
ejpam-5514	133	2	r	r	PRON
ejpam-5514	133	3	be	be	AUX
ejpam-5514	133	4	a	a	DET
ejpam-5514	133	5	ring	ring	NOUN
ejpam-5514	133	6	in	in	ADP
ejpam-5514	133	7	the	the	DET
ejpam-5514	133	8	usual	usual	ADJ
ejpam-5514	133	9	sense	sense	NOUN
ejpam-5514	133	10	.	.	PUNCT
ejpam-5514	134	1	an	an	DET
ejpam-5514	134	2	ideal	ideal	NOUN
ejpam-5514	134	3	a	a	PRON
ejpam-5514	134	4	of	of	ADP
ejpam-5514	134	5	a	a	DET
ejpam-5514	134	6	ring	ring	NOUN
ejpam-5514	134	7	r	r	NOUN
ejpam-5514	134	8	is	be	AUX
ejpam-5514	134	9	termed	term	VERB
ejpam-5514	134	10	a	a	DET
ejpam-5514	134	11	prime	prime	ADJ
ejpam-5514	134	12	ideal	ideal	NOUN
ejpam-5514	134	13	of	of	ADP
ejpam-5514	134	14	the	the	DET
ejpam-5514	134	15	ring	ring	NOUN
ejpam-5514	134	16	r	r	NOUN
ejpam-5514	134	17	if	if	SCONJ
ejpam-5514	134	18	,	,	PUNCT
ejpam-5514	134	19	for	for	ADP
ejpam-5514	134	20	any	any	DET
ejpam-5514	134	21	x	x	NOUN
ejpam-5514	134	22	,	,	PUNCT
ejpam-5514	134	23	y	y	PROPN
ejpam-5514	134	24	∈	∈	PROPN
ejpam-5514	134	25	r	r	NOUN
ejpam-5514	134	26	,	,	PUNCT
ejpam-5514	134	27	xy	xy	PROPN
ejpam-5514	134	28	∈	∈	PROPN
ejpam-5514	134	29	a	a	DET
ejpam-5514	134	30	implies	imply	VERB
ejpam-5514	134	31	that	that	SCONJ
ejpam-5514	134	32	either	either	CCONJ
ejpam-5514	134	33	x	x	SYM
ejpam-5514	134	34	∈	∈	PROPN
ejpam-5514	134	35	a	a	DET
ejpam-5514	134	36	or	or	CCONJ
ejpam-5514	134	37	y	y	PROPN
ejpam-5514	134	38	∈	∈	PROPN
ejpam-5514	134	39	a.	a.	NOUN
ejpam-5514	134	40	now	now	ADV
ejpam-5514	134	41	,	,	PUNCT
ejpam-5514	134	42	let	let	VERB
ejpam-5514	134	43	ρ	ρ	NOUN
ejpam-5514	134	44	be	be	AUX
ejpam-5514	134	45	a	a	DET
ejpam-5514	134	46	congruence	congruence	NOUN
ejpam-5514	134	47	relation	relation	NOUN
ejpam-5514	134	48	on	on	ADP
ejpam-5514	134	49	a	a	DET
ejpam-5514	134	50	ring	ring	NOUN
ejpam-5514	134	51	r.	r.	NOUN
ejpam-5514	134	52	we	we	PRON
ejpam-5514	134	53	define	define	VERB
ejpam-5514	134	54	a	a	DET
ejpam-5514	134	55	subset	subset	NOUN
ejpam-5514	134	56	i	i	PRON
ejpam-5514	134	57	of	of	ADP
ejpam-5514	134	58	ring	ring	NOUN
ejpam-5514	134	59	r	r	NOUN
ejpam-5514	134	60	as	as	ADP
ejpam-5514	134	61	a	a	DET
ejpam-5514	134	62	lower	low	ADJ
ejpam-5514	134	63	rough	rough	ADJ
ejpam-5514	134	64	prime	prime	ADJ
ejpam-5514	134	65	ideal	ideal	NOUN
ejpam-5514	134	66	of	of	ADP
ejpam-5514	134	67	ring	ring	NOUN
ejpam-5514	134	68	r	r	NOUN
ejpam-5514	134	69	if	if	SCONJ
ejpam-5514	134	70	ρ−(i	ρ−(i	VERB
ejpam-5514	134	71	)	)	PUNCT
ejpam-5514	134	72	is	be	AUX
ejpam-5514	134	73	a	a	DET
ejpam-5514	134	74	prime	prime	ADJ
ejpam-5514	134	75	ideal	ideal	NOUN
ejpam-5514	134	76	of	of	ADP
ejpam-5514	134	77	ring	ring	NOUN
ejpam-5514	134	78	r	r	NOUN
ejpam-5514	134	79	,	,	PUNCT
ejpam-5514	134	80	and	and	CCONJ
ejpam-5514	134	81	an	an	DET
ejpam-5514	134	82	upper	upper	ADJ
ejpam-5514	134	83	rough	rough	ADJ
ejpam-5514	134	84	prime	prime	ADJ
ejpam-5514	134	85	ideal	ideal	NOUN
ejpam-5514	134	86	of	of	ADP
ejpam-5514	134	87	ring	ring	NOUN
ejpam-5514	134	88	r	r	NOUN
ejpam-5514	134	89	if	if	SCONJ
ejpam-5514	134	90	ρ−(i	ρ−(i	VERB
ejpam-5514	134	91	)	)	PUNCT
ejpam-5514	134	92	is	be	AUX
ejpam-5514	134	93	a	a	DET
ejpam-5514	134	94	prime	prime	ADJ
ejpam-5514	134	95	ideal	ideal	NOUN
ejpam-5514	134	96	of	of	ADP
ejpam-5514	134	97	ring	ring	PROPN
ejpam-5514	134	98	r.	r.	PROPN
ejpam-5514	134	99	we	we	PRON
ejpam-5514	134	100	then	then	ADV
ejpam-5514	134	101	denote	denote	VERB
ejpam-5514	134	102	ρ(i	ρ(i	PROPN
ejpam-5514	134	103	)	)	PUNCT
ejpam-5514	134	104	=	=	PUNCT
ejpam-5514	134	105	(	(	PUNCT
ejpam-5514	134	106	ρ−(i	ρ−(i	PROPN
ejpam-5514	134	107	)	)	PUNCT
ejpam-5514	134	108	,	,	PUNCT
ejpam-5514	134	109	ρ−(i	ρ−(i	PROPN
ejpam-5514	134	110	)	)	PUNCT
ejpam-5514	134	111	)	)	PUNCT
ejpam-5514	134	112	as	as	ADP
ejpam-5514	134	113	a	a	DET
ejpam-5514	134	114	rough	rough	ADJ
ejpam-5514	134	115	prime	prime	ADJ
ejpam-5514	134	116	ideal	ideal	NOUN
ejpam-5514	134	117	of	of	ADP
ejpam-5514	134	118	r.	r.	PROPN
ejpam-5514	134	119	example	example	NOUN
ejpam-5514	135	1	5	5	NUM
ejpam-5514	135	2	.	.	X
ejpam-5514	135	3	consider	consider	VERB
ejpam-5514	135	4	r	r	NOUN
ejpam-5514	135	5	=	=	SYM
ejpam-5514	135	6	z12	z12	NUM
ejpam-5514	135	7	as	as	ADP
ejpam-5514	135	8	a	a	DET
ejpam-5514	135	9	ring	ring	NOUN
ejpam-5514	135	10	,	,	PUNCT
ejpam-5514	135	11	with	with	ADP
ejpam-5514	135	12	i	i	PRON
ejpam-5514	135	13	=	=	PUNCT
ejpam-5514	135	14	{	{	PUNCT
ejpam-5514	135	15	0	0	NUM
ejpam-5514	135	16	,	,	PUNCT
ejpam-5514	135	17	6	6	NUM
ejpam-5514	135	18	}	}	PUNCT
ejpam-5514	135	19	and	and	CCONJ
ejpam-5514	135	20	subsets	subset	VERB
ejpam-5514	135	21	a	a	PRON
ejpam-5514	135	22	=	=	PUNCT
ejpam-5514	135	23	{	{	PUNCT
ejpam-5514	135	24	0	0	NUM
ejpam-5514	135	25	,	,	PUNCT
ejpam-5514	135	26	1	1	NUM
ejpam-5514	135	27	,	,	PUNCT
ejpam-5514	135	28	2	2	NUM
ejpam-5514	135	29	,	,	PUNCT
ejpam-5514	135	30	5	5	NUM
ejpam-5514	135	31	,	,	PUNCT
ejpam-5514	135	32	6	6	NUM
ejpam-5514	135	33	,	,	PUNCT
ejpam-5514	135	34	8	8	NUM
ejpam-5514	135	35	}	}	PUNCT
ejpam-5514	135	36	and	and	CCONJ
ejpam-5514	135	37	b	b	X
ejpam-5514	135	38	=	=	SYM
ejpam-5514	135	39	{	{	PUNCT
ejpam-5514	135	40	0	0	NUM
ejpam-5514	135	41	,	,	PUNCT
ejpam-5514	135	42	3	3	NUM
ejpam-5514	135	43	,	,	PUNCT
ejpam-5514	135	44	4	4	NUM
ejpam-5514	135	45	,	,	PUNCT
ejpam-5514	135	46	6	6	NUM
ejpam-5514	135	47	,	,	PUNCT
ejpam-5514	135	48	9	9	NUM
ejpam-5514	135	49	}	}	PUNCT
ejpam-5514	135	50	.	.	PUNCT
ejpam-5514	136	1	in	in	ADP
ejpam-5514	136	2	this	this	DET
ejpam-5514	136	3	context	context	NOUN
ejpam-5514	136	4	,	,	PUNCT
ejpam-5514	136	5	i	i	PRON
ejpam-5514	136	6	is	be	AUX
ejpam-5514	136	7	an	an	DET
ejpam-5514	136	8	ideal	ideal	NOUN
ejpam-5514	136	9	of	of	ADP
ejpam-5514	136	10	ring	ring	NOUN
ejpam-5514	136	11	z12	z12	PROPN
ejpam-5514	136	12	.	.	PUNCT
ejpam-5514	137	1	a	a	PRON
ejpam-5514	137	2	is	be	AUX
ejpam-5514	137	3	a	a	DET
ejpam-5514	137	4	subset	subset	NOUN
ejpam-5514	137	5	of	of	ADP
ejpam-5514	137	6	z12	z12	PROPN
ejpam-5514	137	7	,	,	PUNCT
ejpam-5514	137	8	and	and	CCONJ
ejpam-5514	137	9	it	it	PRON
ejpam-5514	137	10	’s	’	VERB
ejpam-5514	137	11	evident	evident	ADJ
ejpam-5514	137	12	that	that	SCONJ
ejpam-5514	137	13	i	i	PRON
ejpam-5514	137	14	is	be	AUX
ejpam-5514	137	15	a	a	DET
ejpam-5514	137	16	congruence	congruence	NOUN
ejpam-5514	137	17	relation	relation	NOUN
ejpam-5514	137	18	in	in	ADP
ejpam-5514	137	19	z12	z12	PROPN
ejpam-5514	137	20	.	.	PUNCT
ejpam-5514	138	1	now	now	ADV
ejpam-5514	138	2	,	,	PUNCT
ejpam-5514	138	3	we	we	PRON
ejpam-5514	138	4	define	define	VERB
ejpam-5514	138	5	the	the	DET
ejpam-5514	138	6	lower	low	ADJ
ejpam-5514	138	7	approximation	approximation	NOUN
ejpam-5514	138	8	of	of	ADP
ejpam-5514	138	9	a	a	DET
ejpam-5514	138	10	with	with	ADP
ejpam-5514	138	11	respect	respect	NOUN
ejpam-5514	138	12	to	to	ADP
ejpam-5514	138	13	the	the	DET
ejpam-5514	138	14	congruence	congruence	NOUN
ejpam-5514	138	15	relation	relation	NOUN
ejpam-5514	138	16	i	i	PRON
ejpam-5514	138	17	as	as	ADP
ejpam-5514	138	18	i−(a	i−(a	ADV
ejpam-5514	138	19	)	)	PUNCT
ejpam-5514	138	20	=	=	PRON
ejpam-5514	139	1	{	{	PUNCT
ejpam-5514	139	2	x	x	PUNCT
ejpam-5514	139	3	∈	∈	PROPN
ejpam-5514	139	4	z12|(x	z12|(x	PROPN
ejpam-5514	139	5	+	+	NUM
ejpam-5514	139	6	i	i	PROPN
ejpam-5514	139	7	)	)	PUNCT
ejpam-5514	140	1	⊆	⊆	X
ejpam-5514	140	2	a	a	PRON
ejpam-5514	140	3	}	}	PUNCT
ejpam-5514	140	4	and	and	CCONJ
ejpam-5514	140	5	the	the	DET
ejpam-5514	140	6	upper	upper	ADJ
ejpam-5514	140	7	approximation	approximation	NOUN
ejpam-5514	140	8	as	as	ADP
ejpam-5514	140	9	i−(a	i−(a	ADV
ejpam-5514	140	10	)	)	PUNCT
ejpam-5514	140	11	=	=	PRON
ejpam-5514	140	12	{	{	PUNCT
ejpam-5514	140	13	x	x	PUNCT
ejpam-5514	140	14	∈	∈	PROPN
ejpam-5514	140	15	z12|(x	z12|(x	PROPN
ejpam-5514	140	16	+	+	NUM
ejpam-5514	140	17	i	i	NOUN
ejpam-5514	140	18	)	)	PUNCT
ejpam-5514	140	19	∩a	∩a	PROPN
ejpam-5514	140	20	̸=	̸=	PROPN
ejpam-5514	140	21	∅	∅	NOUN
ejpam-5514	140	22	}	}	PUNCT
ejpam-5514	140	23	.	.	PUNCT
ejpam-5514	141	1	for	for	ADP
ejpam-5514	141	2	this	this	DET
ejpam-5514	141	3	example	example	NOUN
ejpam-5514	141	4	,	,	PUNCT
ejpam-5514	141	5	the	the	DET
ejpam-5514	141	6	classes	class	NOUN
ejpam-5514	141	7	of	of	ADP
ejpam-5514	141	8	i	i	PRON
ejpam-5514	141	9	are	be	AUX
ejpam-5514	141	10	{	{	PUNCT
ejpam-5514	141	11	2	2	NUM
ejpam-5514	141	12	,	,	PUNCT
ejpam-5514	141	13	8	8	NUM
ejpam-5514	141	14	}	}	PUNCT
ejpam-5514	141	15	,	,	PUNCT
ejpam-5514	141	16	{	{	PUNCT
ejpam-5514	141	17	4	4	NUM
ejpam-5514	141	18	,	,	PUNCT
ejpam-5514	141	19	10	10	NUM
ejpam-5514	141	20	}	}	PUNCT
ejpam-5514	141	21	,	,	PUNCT
ejpam-5514	141	22	{	{	PUNCT
ejpam-5514	141	23	3	3	NUM
ejpam-5514	141	24	,	,	PUNCT
ejpam-5514	141	25	9	9	NUM
ejpam-5514	141	26	}	}	PUNCT
ejpam-5514	141	27	,	,	PUNCT
ejpam-5514	141	28	{	{	PUNCT
ejpam-5514	141	29	1	1	NUM
ejpam-5514	141	30	,	,	PUNCT
ejpam-5514	141	31	7	7	NUM
ejpam-5514	141	32	}	}	PUNCT
ejpam-5514	141	33	,	,	PUNCT
ejpam-5514	141	34	{	{	PUNCT
ejpam-5514	141	35	6	6	NUM
ejpam-5514	141	36	,	,	PUNCT
ejpam-5514	141	37	0	0	NUM
ejpam-5514	141	38	}	}	PUNCT
ejpam-5514	141	39	,	,	PUNCT
ejpam-5514	141	40	{	{	PUNCT
ejpam-5514	141	41	5	5	NUM
ejpam-5514	141	42	,	,	PUNCT
ejpam-5514	141	43	11	11	NUM
ejpam-5514	141	44	}	}	PUNCT
ejpam-5514	141	45	in	in	ADP
ejpam-5514	141	46	z12	z12	PROPN
ejpam-5514	141	47	.	.	PUNCT
ejpam-5514	142	1	consequently	consequently	ADV
ejpam-5514	142	2	,	,	PUNCT
ejpam-5514	142	3	i−(a	i−(a	ADV
ejpam-5514	142	4	)	)	PUNCT
ejpam-5514	142	5	=	=	SYM
ejpam-5514	142	6	{	{	PUNCT
ejpam-5514	142	7	0	0	NUM
ejpam-5514	142	8	,	,	PUNCT
ejpam-5514	142	9	2	2	NUM
ejpam-5514	142	10	,	,	PUNCT
ejpam-5514	142	11	6	6	NUM
ejpam-5514	142	12	,	,	PUNCT
ejpam-5514	142	13	8	8	NUM
ejpam-5514	142	14	}	}	PUNCT
ejpam-5514	142	15	and	and	CCONJ
ejpam-5514	142	16	i−(a	i−(a	ADV
ejpam-5514	142	17	)	)	PUNCT
ejpam-5514	142	18	=	=	PRON
ejpam-5514	142	19	{	{	PUNCT
ejpam-5514	142	20	0	0	NUM
ejpam-5514	142	21	,	,	PUNCT
ejpam-5514	142	22	1	1	NUM
ejpam-5514	142	23	,	,	PUNCT
ejpam-5514	142	24	2	2	NUM
ejpam-5514	142	25	,	,	PUNCT
ejpam-5514	142	26	5	5	NUM
ejpam-5514	142	27	,	,	PUNCT
ejpam-5514	142	28	6	6	NUM
ejpam-5514	142	29	,	,	PUNCT
ejpam-5514	142	30	7	7	NUM
ejpam-5514	142	31	,	,	PUNCT
ejpam-5514	142	32	8	8	NUM
ejpam-5514	142	33	,	,	PUNCT
ejpam-5514	142	34	11	11	NUM
ejpam-5514	142	35	}	}	PUNCT
ejpam-5514	142	36	.	.	PUNCT
ejpam-5514	143	1	this	this	PRON
ejpam-5514	143	2	implies	imply	VERB
ejpam-5514	143	3	that	that	SCONJ
ejpam-5514	143	4	i−(a	i−(a	ADV
ejpam-5514	143	5	)	)	PUNCT
ejpam-5514	143	6	and	and	CCONJ
ejpam-5514	143	7	i−(a	i−(a	ADV
ejpam-5514	143	8	)	)	PUNCT
ejpam-5514	143	9	are	be	AUX
ejpam-5514	143	10	prime	prime	ADJ
ejpam-5514	143	11	ideals	ideal	NOUN
ejpam-5514	143	12	of	of	ADP
ejpam-5514	143	13	z12	z12	PROPN
ejpam-5514	143	14	,	,	PUNCT
ejpam-5514	143	15	demonstrating	demonstrate	VERB
ejpam-5514	143	16	that	that	SCONJ
ejpam-5514	143	17	apr(a	apr(a	PROPN
ejpam-5514	143	18	)	)	PUNCT
ejpam-5514	143	19	=	=	SYM
ejpam-5514	143	20	(	(	PUNCT
ejpam-5514	143	21	i−(a	i−(a	ADV
ejpam-5514	143	22	)	)	PUNCT
ejpam-5514	143	23	,	,	PUNCT
ejpam-5514	143	24	i−(a	i−(a	ADV
ejpam-5514	143	25	)	)	PUNCT
ejpam-5514	143	26	)	)	PUNCT
ejpam-5514	143	27	is	be	AUX
ejpam-5514	143	28	a	a	DET
ejpam-5514	143	29	rough	rough	ADJ
ejpam-5514	143	30	prime	prime	ADJ
ejpam-5514	143	31	ideal	ideal	NOUN
ejpam-5514	143	32	.	.	PUNCT
ejpam-5514	144	1	the	the	DET
ejpam-5514	144	2	roughness	roughness	NOUN
ejpam-5514	144	3	of	of	ADP
ejpam-5514	144	4	set	set	NOUN
ejpam-5514	144	5	a	a	PRON
ejpam-5514	144	6	is	be	AUX
ejpam-5514	144	7	calculated	calculate	VERB
ejpam-5514	144	8	as	as	ADP
ejpam-5514	144	9	ρ(a	ρ(a	PROPN
ejpam-5514	144	10	)	)	PUNCT
ejpam-5514	144	11	=	=	SYM
ejpam-5514	144	12	0.5	0.5	NUM
ejpam-5514	144	13	.	.	PUNCT
ejpam-5514	144	14	a.	a.	PROPN
ejpam-5514	144	15	prakash	prakash	PROPN
ejpam-5514	144	16	,	,	PUNCT
ejpam-5514	144	17	r.	r.	PROPN
ejpam-5514	144	18	shukla	shukla	PROPN
ejpam-5514	144	19	/	/	SYM
ejpam-5514	144	20	eur	eur	PROPN
ejpam-5514	144	21	.	.	PUNCT
ejpam-5514	145	1	j.	j.	PROPN
ejpam-5514	145	2	pure	pure	PROPN
ejpam-5514	145	3	appl	appl	PROPN
ejpam-5514	145	4	.	.	PROPN
ejpam-5514	145	5	math	math	PROPN
ejpam-5514	145	6	,	,	PUNCT
ejpam-5514	145	7	18	18	NUM
ejpam-5514	145	8	(	(	PUNCT
ejpam-5514	145	9	1	1	NUM
ejpam-5514	145	10	)	)	PUNCT
ejpam-5514	145	11	(	(	PUNCT
ejpam-5514	145	12	2025	2025	NUM
ejpam-5514	145	13	)	)	PUNCT
ejpam-5514	145	14	,	,	PUNCT
ejpam-5514	145	15	5514	5514	NUM
ejpam-5514	145	16	7	7	NUM
ejpam-5514	145	17	of	of	ADP
ejpam-5514	145	18	9	9	NUM
ejpam-5514	145	19	example	example	NOUN
ejpam-5514	145	20	6	6	NUM
ejpam-5514	145	21	.	.	PUNCT
ejpam-5514	146	1	let	let	VERB
ejpam-5514	146	2	’s	’s	NOUN
ejpam-5514	146	3	consider	consider	VERB
ejpam-5514	146	4	the	the	DET
ejpam-5514	146	5	same	same	ADJ
ejpam-5514	146	6	ring	ring	NOUN
ejpam-5514	146	7	r	r	NOUN
ejpam-5514	146	8	=	=	SYM
ejpam-5514	146	9	z12	z12	NUM
ejpam-5514	146	10	,	,	PUNCT
ejpam-5514	146	11	but	but	CCONJ
ejpam-5514	146	12	this	this	DET
ejpam-5514	146	13	time	time	NOUN
ejpam-5514	146	14	with	with	ADP
ejpam-5514	146	15	the	the	DET
ejpam-5514	146	16	congruence	congruence	NOUN
ejpam-5514	146	17	relation	relation	NOUN
ejpam-5514	147	1	i	i	PRON
ejpam-5514	147	2	=	=	PUNCT
ejpam-5514	147	3	{	{	PUNCT
ejpam-5514	147	4	0	0	NUM
ejpam-5514	147	5	,	,	PUNCT
ejpam-5514	147	6	4	4	NUM
ejpam-5514	147	7	,	,	PUNCT
ejpam-5514	147	8	8	8	NUM
ejpam-5514	147	9	}	}	PUNCT
ejpam-5514	147	10	and	and	CCONJ
ejpam-5514	147	11	the	the	DET
ejpam-5514	147	12	subset	subset	NOUN
ejpam-5514	147	13	b	b	NOUN
ejpam-5514	147	14	=	=	PUNCT
ejpam-5514	147	15	{	{	PUNCT
ejpam-5514	147	16	0	0	NUM
ejpam-5514	147	17	,	,	PUNCT
ejpam-5514	147	18	1	1	NUM
ejpam-5514	147	19	,	,	PUNCT
ejpam-5514	147	20	3	3	NUM
ejpam-5514	147	21	,	,	PUNCT
ejpam-5514	147	22	4	4	NUM
ejpam-5514	147	23	,	,	PUNCT
ejpam-5514	147	24	6	6	NUM
ejpam-5514	147	25	,	,	PUNCT
ejpam-5514	147	26	8	8	NUM
ejpam-5514	147	27	,	,	PUNCT
ejpam-5514	147	28	10	10	NUM
ejpam-5514	147	29	}	}	PUNCT
ejpam-5514	147	30	.	.	PUNCT
ejpam-5514	148	1	here	here	ADV
ejpam-5514	148	2	,	,	PUNCT
ejpam-5514	148	3	i	i	PRON
ejpam-5514	148	4	is	be	AUX
ejpam-5514	148	5	an	an	DET
ejpam-5514	148	6	ideal	ideal	NOUN
ejpam-5514	148	7	over	over	ADP
ejpam-5514	148	8	ring	ring	NOUN
ejpam-5514	148	9	r	r	NOUN
ejpam-5514	148	10	,	,	PUNCT
ejpam-5514	148	11	and	and	CCONJ
ejpam-5514	148	12	b	b	NOUN
ejpam-5514	148	13	is	be	AUX
ejpam-5514	148	14	any	any	DET
ejpam-5514	148	15	subset	subset	NOUN
ejpam-5514	148	16	of	of	ADP
ejpam-5514	148	17	r.	r.	NOUN
ejpam-5514	148	18	by	by	ADP
ejpam-5514	148	19	defining	define	VERB
ejpam-5514	148	20	the	the	DET
ejpam-5514	148	21	lower	low	ADJ
ejpam-5514	148	22	approximation	approximation	NOUN
ejpam-5514	148	23	of	of	ADP
ejpam-5514	148	24	b	b	NOUN
ejpam-5514	148	25	with	with	ADP
ejpam-5514	148	26	respect	respect	NOUN
ejpam-5514	148	27	to	to	ADP
ejpam-5514	148	28	congruence	congruence	NOUN
ejpam-5514	148	29	relation	relation	NOUN
ejpam-5514	148	30	i	i	PRON
ejpam-5514	148	31	,	,	PUNCT
ejpam-5514	148	32	we	we	PRON
ejpam-5514	148	33	get	get	VERB
ejpam-5514	148	34	i−(b	i−(b	NOUN
ejpam-5514	148	35	)	)	PUNCT
ejpam-5514	148	36	=	=	PRON
ejpam-5514	149	1	{	{	PUNCT
ejpam-5514	149	2	x	x	SYM
ejpam-5514	149	3	∈	∈	NOUN
ejpam-5514	149	4	r|x	r|x	VERB
ejpam-5514	150	1	+	+	CCONJ
ejpam-5514	150	2	i	i	PRON
ejpam-5514	150	3	⊆	⊆	NUM
ejpam-5514	150	4	b	b	NOUN
ejpam-5514	150	5	}	}	PUNCT
ejpam-5514	150	6	and	and	CCONJ
ejpam-5514	150	7	the	the	DET
ejpam-5514	150	8	upper	upper	ADJ
ejpam-5514	150	9	approximation	approximation	NOUN
ejpam-5514	150	10	is	be	AUX
ejpam-5514	150	11	i−(b	i−(b	NOUN
ejpam-5514	150	12	)	)	PUNCT
ejpam-5514	150	13	=	=	PRON
ejpam-5514	151	1	{	{	PUNCT
ejpam-5514	151	2	x	x	PUNCT
ejpam-5514	151	3	∈	∈	PROPN
ejpam-5514	151	4	r|(x	r|(x	NOUN
ejpam-5514	151	5	+	+	CCONJ
ejpam-5514	151	6	i	i	NOUN
ejpam-5514	151	7	)	)	PUNCT
ejpam-5514	151	8	∩b	∩b	NOUN
ejpam-5514	151	9	̸=	̸=	PROPN
ejpam-5514	151	10	∅	∅	NOUN
ejpam-5514	151	11	}	}	PUNCT
ejpam-5514	151	12	.	.	PUNCT
ejpam-5514	152	1	in	in	ADP
ejpam-5514	152	2	this	this	DET
ejpam-5514	152	3	scenario	scenario	NOUN
ejpam-5514	152	4	,	,	PUNCT
ejpam-5514	152	5	the	the	DET
ejpam-5514	152	6	classes	class	NOUN
ejpam-5514	152	7	of	of	ADP
ejpam-5514	152	8	ρ	ρ	PROPN
ejpam-5514	152	9	are	be	AUX
ejpam-5514	152	10	{	{	PUNCT
ejpam-5514	152	11	0	0	NUM
ejpam-5514	152	12	,	,	PUNCT
ejpam-5514	152	13	4	4	NUM
ejpam-5514	152	14	,	,	PUNCT
ejpam-5514	152	15	8	8	NUM
ejpam-5514	152	16	}	}	PUNCT
ejpam-5514	152	17	,	,	PUNCT
ejpam-5514	152	18	{	{	PUNCT
ejpam-5514	152	19	1	1	NUM
ejpam-5514	152	20	,	,	PUNCT
ejpam-5514	152	21	5	5	NUM
ejpam-5514	152	22	,	,	PUNCT
ejpam-5514	152	23	9	9	NUM
ejpam-5514	152	24	}	}	PUNCT
ejpam-5514	152	25	,	,	PUNCT
ejpam-5514	152	26	{	{	PUNCT
ejpam-5514	152	27	2	2	NUM
ejpam-5514	152	28	,	,	PUNCT
ejpam-5514	152	29	6	6	NUM
ejpam-5514	152	30	,	,	PUNCT
ejpam-5514	152	31	10	10	NUM
ejpam-5514	152	32	}	}	PUNCT
ejpam-5514	152	33	,	,	PUNCT
ejpam-5514	152	34	{	{	PUNCT
ejpam-5514	152	35	3	3	NUM
ejpam-5514	152	36	,	,	PUNCT
ejpam-5514	152	37	7	7	NUM
ejpam-5514	152	38	,	,	PUNCT
ejpam-5514	152	39	11	11	NUM
ejpam-5514	152	40	}	}	PUNCT
ejpam-5514	152	41	within	within	ADP
ejpam-5514	152	42	z12	z12	PROPN
ejpam-5514	152	43	.	.	PUNCT
ejpam-5514	153	1	thus	thus	ADV
ejpam-5514	153	2	ρ−(b	ρ−(b	NOUN
ejpam-5514	153	3	)	)	PUNCT
ejpam-5514	153	4	=	=	PUNCT
ejpam-5514	153	5	{	{	PUNCT
ejpam-5514	153	6	0	0	NUM
ejpam-5514	153	7	,	,	PUNCT
ejpam-5514	153	8	2	2	NUM
ejpam-5514	153	9	,	,	PUNCT
ejpam-5514	153	10	4	4	NUM
ejpam-5514	153	11	,	,	PUNCT
ejpam-5514	153	12	6	6	NUM
ejpam-5514	153	13	,	,	PUNCT
ejpam-5514	153	14	8	8	NUM
ejpam-5514	153	15	,	,	PUNCT
ejpam-5514	153	16	10	10	NUM
ejpam-5514	153	17	}	}	PUNCT
ejpam-5514	153	18	and	and	CCONJ
ejpam-5514	153	19	ρ−(b	ρ−(b	NOUN
ejpam-5514	153	20	)	)	PUNCT
ejpam-5514	153	21	=	=	PUNCT
ejpam-5514	153	22	{	{	PUNCT
ejpam-5514	153	23	0	0	NUM
ejpam-5514	153	24	,	,	PUNCT
ejpam-5514	153	25	2	2	NUM
ejpam-5514	153	26	,	,	PUNCT
ejpam-5514	153	27	3	3	NUM
ejpam-5514	153	28	,	,	PUNCT
ejpam-5514	153	29	4	4	NUM
ejpam-5514	153	30	,	,	PUNCT
ejpam-5514	153	31	6	6	NUM
ejpam-5514	153	32	,	,	PUNCT
ejpam-5514	153	33	7	7	NUM
ejpam-5514	153	34	,	,	PUNCT
ejpam-5514	153	35	8	8	NUM
ejpam-5514	153	36	,	,	PUNCT
ejpam-5514	153	37	11	11	NUM
ejpam-5514	153	38	}	}	PUNCT
ejpam-5514	153	39	.	.	PUNCT
ejpam-5514	154	1	this	this	PRON
ejpam-5514	154	2	indicates	indicate	VERB
ejpam-5514	154	3	that	that	SCONJ
ejpam-5514	154	4	ρ−(b	ρ−(b	NOUN
ejpam-5514	154	5	)	)	PUNCT
ejpam-5514	154	6	and	and	CCONJ
ejpam-5514	154	7	ρ−(b	ρ−(b	NOUN
ejpam-5514	154	8	)	)	PUNCT
ejpam-5514	154	9	are	be	AUX
ejpam-5514	154	10	prime	prime	ADJ
ejpam-5514	154	11	ideals	ideal	NOUN
ejpam-5514	154	12	of	of	ADP
ejpam-5514	154	13	r	r	NOUN
ejpam-5514	154	14	=	=	SYM
ejpam-5514	154	15	z12	z12	PROPN
ejpam-5514	154	16	,	,	PUNCT
ejpam-5514	154	17	suggesting	suggest	VERB
ejpam-5514	154	18	that	that	SCONJ
ejpam-5514	154	19	ρ(b	ρ(b	NOUN
ejpam-5514	154	20	)	)	PUNCT
ejpam-5514	154	21	=	=	SYM
ejpam-5514	154	22	(	(	PUNCT
ejpam-5514	154	23	ρ−(b	ρ−(b	PROPN
ejpam-5514	154	24	)	)	PUNCT
ejpam-5514	154	25	,	,	PUNCT
ejpam-5514	154	26	ρ−(b	ρ−(b	PROPN
ejpam-5514	154	27	)	)	PUNCT
ejpam-5514	154	28	)	)	PUNCT
ejpam-5514	154	29	is	be	AUX
ejpam-5514	154	30	a	a	DET
ejpam-5514	154	31	prime	prime	ADJ
ejpam-5514	154	32	ideal	ideal	NOUN
ejpam-5514	154	33	of	of	ADP
ejpam-5514	154	34	r	r	NOUN
ejpam-5514	154	35	=	=	SYM
ejpam-5514	154	36	z12	z12	NUM
ejpam-5514	154	37	.	.	PROPN
ejpam-5514	154	38	4	4	NUM
ejpam-5514	154	39	.	.	X
ejpam-5514	154	40	conclusion	conclusion	NOUN
ejpam-5514	154	41	in	in	ADP
ejpam-5514	154	42	conclusion	conclusion	NOUN
ejpam-5514	154	43	,	,	PUNCT
ejpam-5514	154	44	traditional	traditional	ADJ
ejpam-5514	154	45	algebraic	algebraic	ADJ
ejpam-5514	154	46	interpretations	interpretation	NOUN
ejpam-5514	154	47	of	of	ADP
ejpam-5514	154	48	ideals	ideal	NOUN
ejpam-5514	154	49	and	and	CCONJ
ejpam-5514	154	50	prime	prime	ADJ
ejpam-5514	154	51	ideals	ideal	NOUN
ejpam-5514	154	52	are	be	AUX
ejpam-5514	154	53	grounded	ground	VERB
ejpam-5514	154	54	in	in	ADP
ejpam-5514	154	55	precise	precise	ADJ
ejpam-5514	154	56	mathematical	mathematical	ADJ
ejpam-5514	154	57	definitions	definition	NOUN
ejpam-5514	154	58	,	,	PUNCT
ejpam-5514	154	59	often	often	ADV
ejpam-5514	154	60	struggling	struggle	VERB
ejpam-5514	154	61	to	to	PART
ejpam-5514	154	62	accommodate	accommodate	VERB
ejpam-5514	154	63	the	the	DET
ejpam-5514	154	64	inherent	inherent	ADJ
ejpam-5514	154	65	imprecision	imprecision	NOUN
ejpam-5514	154	66	and	and	CCONJ
ejpam-5514	154	67	uncertainty	uncertainty	NOUN
ejpam-5514	154	68	encountered	encounter	VERB
ejpam-5514	154	69	in	in	ADP
ejpam-5514	154	70	real	real	ADJ
ejpam-5514	154	71	-	-	PUNCT
ejpam-5514	154	72	world	world	NOUN
ejpam-5514	154	73	data	datum	NOUN
ejpam-5514	154	74	.	.	PUNCT
ejpam-5514	155	1	in	in	ADP
ejpam-5514	155	2	contrast	contrast	NOUN
ejpam-5514	155	3	,	,	PUNCT
ejpam-5514	155	4	rough	rough	ADJ
ejpam-5514	155	5	set	set	NOUN
ejpam-5514	155	6	theory	theory	NOUN
ejpam-5514	155	7	offers	offer	VERB
ejpam-5514	155	8	a	a	DET
ejpam-5514	155	9	refreshing	refreshing	ADJ
ejpam-5514	155	10	perspective	perspective	NOUN
ejpam-5514	155	11	,	,	PUNCT
ejpam-5514	155	12	introducing	introduce	VERB
ejpam-5514	155	13	a	a	DET
ejpam-5514	155	14	high	high	ADJ
ejpam-5514	155	15	degree	degree	NOUN
ejpam-5514	155	16	of	of	ADP
ejpam-5514	155	17	flexibility	flexibility	NOUN
ejpam-5514	155	18	by	by	ADP
ejpam-5514	155	19	considering	consider	VERB
ejpam-5514	155	20	lower	low	ADJ
ejpam-5514	155	21	and	and	CCONJ
ejpam-5514	155	22	upper	upper	ADJ
ejpam-5514	155	23	approximations	approximation	NOUN
ejpam-5514	155	24	.	.	PUNCT
ejpam-5514	156	1	these	these	DET
ejpam-5514	156	2	rough	rough	ADJ
ejpam-5514	156	3	interpretations	interpretation	NOUN
ejpam-5514	156	4	bridge	bridge	VERB
ejpam-5514	156	5	the	the	DET
ejpam-5514	156	6	gap	gap	NOUN
ejpam-5514	156	7	between	between	ADP
ejpam-5514	156	8	classical	classical	ADJ
ejpam-5514	156	9	algebraic	algebraic	ADJ
ejpam-5514	156	10	structures	structure	NOUN
ejpam-5514	156	11	and	and	CCONJ
ejpam-5514	156	12	practical	practical	ADJ
ejpam-5514	156	13	applications	application	NOUN
ejpam-5514	156	14	,	,	PUNCT
ejpam-5514	156	15	providing	provide	VERB
ejpam-5514	156	16	a	a	DET
ejpam-5514	156	17	versatile	versatile	ADJ
ejpam-5514	156	18	mathematical	mathematical	ADJ
ejpam-5514	156	19	framework	framework	NOUN
ejpam-5514	156	20	that	that	PRON
ejpam-5514	156	21	excels	excel	VERB
ejpam-5514	156	22	in	in	ADP
ejpam-5514	156	23	scenarios	scenario	NOUN
ejpam-5514	156	24	where	where	SCONJ
ejpam-5514	156	25	precision	precision	NOUN
ejpam-5514	156	26	remains	remain	VERB
ejpam-5514	156	27	elusive	elusive	ADJ
ejpam-5514	156	28	.	.	PUNCT
ejpam-5514	157	1	the	the	DET
ejpam-5514	157	2	fusion	fusion	NOUN
ejpam-5514	157	3	of	of	ADP
ejpam-5514	157	4	abstract	abstract	ADJ
ejpam-5514	157	5	algebraic	algebraic	ADJ
ejpam-5514	157	6	concepts	concept	NOUN
ejpam-5514	157	7	with	with	ADP
ejpam-5514	157	8	the	the	DET
ejpam-5514	157	9	adeptness	adeptness	NOUN
ejpam-5514	157	10	of	of	ADP
ejpam-5514	157	11	rough	rough	ADJ
ejpam-5514	157	12	set	set	NOUN
ejpam-5514	157	13	theory	theory	NOUN
ejpam-5514	157	14	in	in	ADP
ejpam-5514	157	15	handling	handle	VERB
ejpam-5514	157	16	imprecision	imprecision	NOUN
ejpam-5514	157	17	and	and	CCONJ
ejpam-5514	157	18	uncertainty	uncertainty	NOUN
ejpam-5514	157	19	opens	open	VERB
ejpam-5514	157	20	up	up	ADP
ejpam-5514	157	21	a	a	DET
ejpam-5514	157	22	promising	promising	ADJ
ejpam-5514	157	23	avenue	avenue	NOUN
ejpam-5514	157	24	for	for	ADP
ejpam-5514	157	25	further	further	ADJ
ejpam-5514	157	26	research	research	NOUN
ejpam-5514	157	27	and	and	CCONJ
ejpam-5514	157	28	practical	practical	ADJ
ejpam-5514	157	29	applications	application	NOUN
ejpam-5514	157	30	.	.	PUNCT
ejpam-5514	158	1	the	the	DET
ejpam-5514	158	2	exploration	exploration	NOUN
ejpam-5514	158	3	of	of	ADP
ejpam-5514	158	4	algebraic	algebraic	ADJ
ejpam-5514	158	5	concepts	concept	NOUN
ejpam-5514	158	6	within	within	ADP
ejpam-5514	158	7	rough	rough	ADJ
ejpam-5514	158	8	sets	set	NOUN
ejpam-5514	158	9	has	have	AUX
ejpam-5514	158	10	proven	prove	VERB
ejpam-5514	158	11	to	to	PART
ejpam-5514	158	12	be	be	AUX
ejpam-5514	158	13	captivating	captivate	VERB
ejpam-5514	158	14	.	.	PUNCT
ejpam-5514	159	1	we	we	PRON
ejpam-5514	159	2	explained	explain	VERB
ejpam-5514	159	3	the	the	DET
ejpam-5514	159	4	concepts	concept	NOUN
ejpam-5514	159	5	of	of	ADP
ejpam-5514	159	6	upper	upper	ADJ
ejpam-5514	159	7	and	and	CCONJ
ejpam-5514	159	8	lower	low	ADJ
ejpam-5514	159	9	approximations	approximation	NOUN
ejpam-5514	159	10	of	of	ADP
ejpam-5514	159	11	ideals	ideal	NOUN
ejpam-5514	159	12	within	within	ADP
ejpam-5514	159	13	the	the	DET
ejpam-5514	159	14	context	context	NOUN
ejpam-5514	159	15	of	of	ADP
ejpam-5514	159	16	congruence	congruence	PROPN
ejpam-5514	159	17	relations	relation	NOUN
ejpam-5514	159	18	,	,	PUNCT
ejpam-5514	159	19	shedding	shed	VERB
ejpam-5514	159	20	light	light	NOUN
ejpam-5514	159	21	on	on	ADP
ejpam-5514	159	22	the	the	DET
ejpam-5514	159	23	notion	notion	NOUN
ejpam-5514	159	24	of	of	ADP
ejpam-5514	159	25	rough	rough	ADJ
ejpam-5514	159	26	ideals	ideal	NOUN
ejpam-5514	159	27	with	with	ADP
ejpam-5514	159	28	examples	example	NOUN
ejpam-5514	159	29	.	.	PUNCT
ejpam-5514	160	1	we	we	PRON
ejpam-5514	160	2	also	also	ADV
ejpam-5514	160	3	investigated	investigate	VERB
ejpam-5514	160	4	the	the	DET
ejpam-5514	160	5	study	study	NOUN
ejpam-5514	160	6	of	of	ADP
ejpam-5514	160	7	upper	upper	ADJ
ejpam-5514	160	8	and	and	CCONJ
ejpam-5514	160	9	lower	low	ADJ
ejpam-5514	160	10	prime	prime	ADJ
ejpam-5514	160	11	ideals	ideal	NOUN
ejpam-5514	160	12	,	,	PUNCT
ejpam-5514	160	13	examining	examine	VERB
ejpam-5514	160	14	their	their	PRON
ejpam-5514	160	15	associated	associated	ADJ
ejpam-5514	160	16	roughness	roughness	NOUN
ejpam-5514	160	17	and	and	CCONJ
ejpam-5514	160	18	bridging	bridge	VERB
ejpam-5514	160	19	the	the	DET
ejpam-5514	160	20	theory	theory	NOUN
ejpam-5514	160	21	and	and	CCONJ
ejpam-5514	160	22	applications	application	NOUN
ejpam-5514	160	23	.	.	PUNCT
ejpam-5514	161	1	these	these	DET
ejpam-5514	161	2	findings	finding	NOUN
ejpam-5514	161	3	are	be	AUX
ejpam-5514	161	4	poised	poise	VERB
ejpam-5514	161	5	to	to	PART
ejpam-5514	161	6	make	make	VERB
ejpam-5514	161	7	significant	significant	ADJ
ejpam-5514	161	8	contributions	contribution	NOUN
ejpam-5514	161	9	to	to	ADP
ejpam-5514	161	10	the	the	DET
ejpam-5514	161	11	mathematical	mathematical	ADJ
ejpam-5514	161	12	foundation	foundation	NOUN
ejpam-5514	161	13	of	of	ADP
ejpam-5514	161	14	rough	rough	ADJ
ejpam-5514	161	15	set	set	NOUN
ejpam-5514	161	16	theory	theory	NOUN
ejpam-5514	161	17	.	.	PUNCT
ejpam-5514	162	1	they	they	PRON
ejpam-5514	162	2	not	not	PART
ejpam-5514	162	3	only	only	ADV
ejpam-5514	162	4	extend	extend	VERB
ejpam-5514	162	5	our	our	PRON
ejpam-5514	162	6	understanding	understanding	NOUN
ejpam-5514	162	7	of	of	ADP
ejpam-5514	162	8	algebraic	algebraic	ADJ
ejpam-5514	162	9	structures	structure	NOUN
ejpam-5514	162	10	within	within	ADP
ejpam-5514	162	11	rough	rough	ADJ
ejpam-5514	162	12	set	set	NOUN
ejpam-5514	162	13	theory	theory	NOUN
ejpam-5514	162	14	but	but	CCONJ
ejpam-5514	162	15	also	also	ADV
ejpam-5514	162	16	pave	pave	VERB
ejpam-5514	162	17	the	the	DET
ejpam-5514	162	18	way	way	NOUN
ejpam-5514	162	19	for	for	ADP
ejpam-5514	162	20	further	further	ADJ
ejpam-5514	162	21	explorations	exploration	NOUN
ejpam-5514	162	22	and	and	CCONJ
ejpam-5514	162	23	applications	application	NOUN
ejpam-5514	162	24	within	within	ADP
ejpam-5514	162	25	this	this	DET
ejpam-5514	162	26	intriguing	intriguing	ADJ
ejpam-5514	162	27	field	field	NOUN
ejpam-5514	162	28	.	.	PUNCT
ejpam-5514	163	1	the	the	DET
ejpam-5514	163	2	mathematical	mathematical	ADJ
ejpam-5514	163	3	interpretations	interpretation	NOUN
ejpam-5514	163	4	of	of	ADP
ejpam-5514	163	5	rough	rough	ADJ
ejpam-5514	163	6	ideals	ideal	NOUN
ejpam-5514	163	7	and	and	CCONJ
ejpam-5514	163	8	rough	rough	ADJ
ejpam-5514	163	9	prime	prime	ADJ
ejpam-5514	163	10	ideals	ideal	NOUN
ejpam-5514	163	11	empower	empower	VERB
ejpam-5514	163	12	us	we	PRON
ejpam-5514	163	13	to	to	PART
ejpam-5514	163	14	adapt	adapt	VERB
ejpam-5514	163	15	and	and	CCONJ
ejpam-5514	163	16	extend	extend	VERB
ejpam-5514	163	17	traditional	traditional	ADJ
ejpam-5514	163	18	algebraic	algebraic	ADJ
ejpam-5514	163	19	concepts	concept	NOUN
ejpam-5514	163	20	effectively	effectively	ADV
ejpam-5514	163	21	,	,	PUNCT
ejpam-5514	163	22	providing	provide	VERB
ejpam-5514	163	23	the	the	DET
ejpam-5514	163	24	tools	tool	NOUN
ejpam-5514	163	25	needed	need	VERB
ejpam-5514	163	26	to	to	PART
ejpam-5514	163	27	tackle	tackle	VERB
ejpam-5514	163	28	imprecise	imprecise	ADJ
ejpam-5514	163	29	information	information	NOUN
ejpam-5514	163	30	efficiently	efficiently	ADV
ejpam-5514	163	31	.	.	PUNCT
ejpam-5514	164	1	this	this	PRON
ejpam-5514	164	2	,	,	PUNCT
ejpam-5514	164	3	enables	enable	VERB
ejpam-5514	164	4	us	we	PRON
ejpam-5514	164	5	to	to	PART
ejpam-5514	164	6	gain	gain	VERB
ejpam-5514	164	7	deeper	deep	ADJ
ejpam-5514	164	8	insights	insight	NOUN
ejpam-5514	164	9	and	and	CCONJ
ejpam-5514	164	10	develop	develop	VERB
ejpam-5514	164	11	solutions	solution	NOUN
ejpam-5514	164	12	applicable	applicable	ADJ
ejpam-5514	164	13	in	in	ADP
ejpam-5514	164	14	various	various	ADJ
ejpam-5514	164	15	domains	domain	NOUN
ejpam-5514	164	16	where	where	SCONJ
ejpam-5514	164	17	uncertainty	uncertainty	NOUN
ejpam-5514	164	18	and	and	CCONJ
ejpam-5514	164	19	data	datum	NOUN
ejpam-5514	164	20	imperfections	imperfection	NOUN
ejpam-5514	164	21	prevail	prevail	VERB
ejpam-5514	164	22	.	.	PUNCT
ejpam-5514	165	1	as	as	SCONJ
ejpam-5514	165	2	mathematical	mathematical	ADJ
ejpam-5514	165	3	tools	tool	NOUN
ejpam-5514	165	4	continue	continue	VERB
ejpam-5514	165	5	to	to	PART
ejpam-5514	165	6	evolve	evolve	VERB
ejpam-5514	165	7	to	to	PART
ejpam-5514	165	8	meet	meet	VERB
ejpam-5514	165	9	the	the	DET
ejpam-5514	165	10	demands	demand	NOUN
ejpam-5514	165	11	of	of	ADP
ejpam-5514	165	12	the	the	DET
ejpam-5514	165	13	modern	modern	ADJ
ejpam-5514	165	14	world	world	NOUN
ejpam-5514	165	15	,	,	PUNCT
ejpam-5514	165	16	rough	rough	ADJ
ejpam-5514	165	17	set	set	NOUN
ejpam-5514	165	18	theory	theory	NOUN
ejpam-5514	165	19	stands	stand	VERB
ejpam-5514	165	20	as	as	ADP
ejpam-5514	165	21	a	a	DET
ejpam-5514	165	22	testament	testament	NOUN
ejpam-5514	165	23	to	to	ADP
ejpam-5514	165	24	the	the	DET
ejpam-5514	165	25	adaptability	adaptability	NOUN
ejpam-5514	165	26	and	and	CCONJ
ejpam-5514	165	27	versatility	versatility	NOUN
ejpam-5514	165	28	of	of	ADP
ejpam-5514	165	29	mathematics	mathematic	NOUN
ejpam-5514	165	30	in	in	ADP
ejpam-5514	165	31	addressing	address	VERB
ejpam-5514	165	32	real	real	ADJ
ejpam-5514	165	33	-	-	PUNCT
ejpam-5514	165	34	world	world	NOUN
ejpam-5514	165	35	challenges	challenge	NOUN
ejpam-5514	165	36	.	.	PUNCT
ejpam-5514	166	1	acknowledgements	acknowledgement	NOUN
ejpam-5514	166	2	the	the	DET
ejpam-5514	166	3	authors	author	NOUN
ejpam-5514	166	4	sincerely	sincerely	ADV
ejpam-5514	166	5	thank	thank	VERB
ejpam-5514	166	6	the	the	DET
ejpam-5514	166	7	reviewers	reviewer	NOUN
ejpam-5514	166	8	for	for	ADP
ejpam-5514	166	9	their	their	PRON
ejpam-5514	166	10	careful	careful	ADJ
ejpam-5514	166	11	reading	reading	NOUN
ejpam-5514	166	12	,	,	PUNCT
ejpam-5514	166	13	constructive	constructive	ADJ
ejpam-5514	166	14	comments	comment	NOUN
ejpam-5514	166	15	,	,	PUNCT
ejpam-5514	166	16	and	and	CCONJ
ejpam-5514	166	17	fruitful	fruitful	ADJ
ejpam-5514	166	18	suggestions	suggestion	NOUN
ejpam-5514	166	19	,	,	PUNCT
ejpam-5514	166	20	which	which	PRON
ejpam-5514	166	21	have	have	AUX
ejpam-5514	166	22	been	be	AUX
ejpam-5514	166	23	incorporated	incorporate	VERB
ejpam-5514	166	24	to	to	PART
ejpam-5514	166	25	improve	improve	VERB
ejpam-5514	166	26	the	the	DET
ejpam-5514	166	27	manuscript	manuscript	NOUN
ejpam-5514	166	28	.	.	PUNCT
ejpam-5514	167	1	a.	a.	PROPN
ejpam-5514	167	2	prakash	prakash	PROPN
ejpam-5514	167	3	,	,	PUNCT
ejpam-5514	167	4	r.	r.	PROPN
ejpam-5514	167	5	shukla	shukla	PROPN
ejpam-5514	167	6	/	/	SYM
ejpam-5514	167	7	eur	eur	PROPN
ejpam-5514	167	8	.	.	PUNCT
ejpam-5514	168	1	j.	j.	PROPN
ejpam-5514	168	2	pure	pure	PROPN
ejpam-5514	168	3	appl	appl	PROPN
ejpam-5514	168	4	.	.	PROPN
ejpam-5514	168	5	math	math	PROPN
ejpam-5514	168	6	,	,	PUNCT
ejpam-5514	168	7	18	18	NUM
ejpam-5514	168	8	(	(	PUNCT
ejpam-5514	168	9	1	1	NUM
ejpam-5514	168	10	)	)	PUNCT
ejpam-5514	168	11	(	(	PUNCT
ejpam-5514	168	12	2025	2025	NUM
ejpam-5514	168	13	)	)	PUNCT
ejpam-5514	168	14	,	,	PUNCT
ejpam-5514	168	15	5514	5514	NUM
ejpam-5514	168	16	8	8	NUM
ejpam-5514	168	17	of	of	ADP
ejpam-5514	168	18	9	9	NUM
ejpam-5514	168	19	funding	funding	NOUN
ejpam-5514	168	20	information	information	NOUN
ejpam-5514	168	21	this	this	DET
ejpam-5514	168	22	work	work	NOUN
ejpam-5514	168	23	was	be	AUX
ejpam-5514	168	24	supported	support	VERB
ejpam-5514	168	25	by	by	ADP
ejpam-5514	168	26	directorate	directorate	NOUN
ejpam-5514	168	27	of	of	ADP
ejpam-5514	168	28	research	research	NOUN
ejpam-5514	168	29	and	and	CCONJ
ejpam-5514	168	30	innovation	innovation	NOUN
ejpam-5514	168	31	,	,	PUNCT
ejpam-5514	168	32	walter	walter	PROPN
ejpam-5514	168	33	sisulu	sisulu	PROPN
ejpam-5514	168	34	university	university	PROPN
ejpam-5514	168	35	,	,	PUNCT
ejpam-5514	168	36	south	south	PROPN
ejpam-5514	168	37	africa	africa	PROPN
ejpam-5514	168	38	.	.	PUNCT
ejpam-5514	169	1	author	author	NOUN
ejpam-5514	169	2	contributions	contribution	NOUN
ejpam-5514	169	3	all	all	DET
ejpam-5514	169	4	authors	author	NOUN
ejpam-5514	169	5	contributed	contribute	VERB
ejpam-5514	169	6	equally	equally	ADV
ejpam-5514	169	7	to	to	ADP
ejpam-5514	169	8	the	the	DET
ejpam-5514	169	9	writing	writing	NOUN
ejpam-5514	169	10	of	of	ADP
ejpam-5514	169	11	this	this	DET
ejpam-5514	169	12	article	article	NOUN
ejpam-5514	169	13	.	.	PUNCT
ejpam-5514	170	1	all	all	DET
ejpam-5514	170	2	authors	author	NOUN
ejpam-5514	170	3	read	read	VERB
ejpam-5514	170	4	and	and	CCONJ
ejpam-5514	170	5	approved	approve	VERB
ejpam-5514	170	6	the	the	DET
ejpam-5514	170	7	final	final	ADJ
ejpam-5514	170	8	manuscript	manuscript	NOUN
ejpam-5514	170	9	.	.	PUNCT
ejpam-5514	171	1	conflict	conflict	NOUN
ejpam-5514	171	2	of	of	ADP
ejpam-5514	171	3	interest	interest	NOUN
ejpam-5514	171	4	the	the	DET
ejpam-5514	171	5	authors	author	NOUN
ejpam-5514	171	6	state	state	VERB
ejpam-5514	171	7	no	no	DET
ejpam-5514	171	8	conflict	conflict	NOUN
ejpam-5514	171	9	of	of	ADP
ejpam-5514	171	10	interest	interest	NOUN
ejpam-5514	171	11	.	.	PUNCT
ejpam-5514	172	1	references	reference	NOUN
ejpam-5514	172	2	[	[	X
ejpam-5514	172	3	1	1	NUM
ejpam-5514	172	4	]	]	PUNCT
ejpam-5514	172	5	tareq	tareq	PROPN
ejpam-5514	172	6	m.	m.	PROPN
ejpam-5514	172	7	al	al	PROPN
ejpam-5514	172	8	-	-	PUNCT
ejpam-5514	172	9	shami	shami	PROPN
ejpam-5514	172	10	.	.	PUNCT
ejpam-5514	173	1	an	an	DET
ejpam-5514	173	2	improvement	improvement	NOUN
ejpam-5514	173	3	of	of	ADP
ejpam-5514	173	4	rough	rough	ADJ
ejpam-5514	173	5	sets	set	NOUN
ejpam-5514	173	6	’	'	PUNCT
ejpam-5514	173	7	accuracy	accuracy	NOUN
ejpam-5514	173	8	measure	measure	NOUN
ejpam-5514	173	9	using	use	VERB
ejpam-5514	173	10	containment	containment	NOUN
ejpam-5514	173	11	neighborhoods	neighborhood	NOUN
ejpam-5514	173	12	with	with	ADP
ejpam-5514	173	13	a	a	DET
ejpam-5514	173	14	medical	medical	ADJ
ejpam-5514	173	15	application	application	NOUN
ejpam-5514	173	16	.	.	PUNCT
ejpam-5514	174	1	information	information	NOUN
ejpam-5514	174	2	sciences	sciences	PROPN
ejpam-5514	174	3	,	,	PUNCT
ejpam-5514	174	4	569:110–124	569:110–124	NUM
ejpam-5514	174	5	,	,	PUNCT
ejpam-5514	174	6	august	august	PROPN
ejpam-5514	174	7	2021	2021	NUM
ejpam-5514	174	8	.	.	PUNCT
ejpam-5514	175	1	[	[	X
ejpam-5514	175	2	2	2	NUM
ejpam-5514	175	3	]	]	PUNCT
ejpam-5514	175	4	tareq	tareq	PROPN
ejpam-5514	175	5	m.	m.	PROPN
ejpam-5514	175	6	al	al	PROPN
ejpam-5514	175	7	-	-	PUNCT
ejpam-5514	175	8	shami	shami	PROPN
ejpam-5514	175	9	.	.	PUNCT
ejpam-5514	176	1	improvement	improvement	NOUN
ejpam-5514	176	2	of	of	ADP
ejpam-5514	176	3	the	the	DET
ejpam-5514	176	4	approximations	approximation	NOUN
ejpam-5514	176	5	and	and	CCONJ
ejpam-5514	176	6	accuracy	accuracy	NOUN
ejpam-5514	176	7	measure	measure	NOUN
ejpam-5514	176	8	of	of	ADP
ejpam-5514	176	9	a	a	DET
ejpam-5514	176	10	rough	rough	ADJ
ejpam-5514	176	11	set	set	NOUN
ejpam-5514	176	12	using	use	VERB
ejpam-5514	176	13	somewhere	somewhere	ADV
ejpam-5514	176	14	dense	dense	ADJ
ejpam-5514	176	15	sets	set	NOUN
ejpam-5514	176	16	.	.	PUNCT
ejpam-5514	177	1	soft	soft	ADJ
ejpam-5514	177	2	computing	computing	NOUN
ejpam-5514	177	3	,	,	PUNCT
ejpam-5514	177	4	25(23):14449–14460	25(23):14449–14460	NUM
ejpam-5514	177	5	,	,	PUNCT
ejpam-5514	177	6	october	october	PROPN
ejpam-5514	177	7	2021	2021	NUM
ejpam-5514	177	8	.	.	PUNCT
ejpam-5514	178	1	[	[	X
ejpam-5514	178	2	3	3	X
ejpam-5514	178	3	]	]	X
ejpam-5514	178	4	tareq	tareq	PROPN
ejpam-5514	178	5	m.	m.	PROPN
ejpam-5514	178	6	al	al	PROPN
ejpam-5514	178	7	-	-	PUNCT
ejpam-5514	178	8	shami	shami	PROPN
ejpam-5514	178	9	.	.	PUNCT
ejpam-5514	179	1	topological	topological	ADJ
ejpam-5514	179	2	approach	approach	NOUN
ejpam-5514	179	3	to	to	PART
ejpam-5514	179	4	generate	generate	VERB
ejpam-5514	179	5	new	new	ADJ
ejpam-5514	179	6	rough	rough	ADJ
ejpam-5514	179	7	set	set	NOUN
ejpam-5514	179	8	models	model	NOUN
ejpam-5514	179	9	.	.	PUNCT
ejpam-5514	180	1	complex	complex	ADJ
ejpam-5514	180	2	amp	amp	NOUN
ejpam-5514	180	3	;	;	PUNCT
ejpam-5514	180	4	intelligent	intelligent	ADJ
ejpam-5514	180	5	systems	system	NOUN
ejpam-5514	180	6	,	,	PUNCT
ejpam-5514	180	7	8(5):4101–4113	8(5):4101–4113	PROPN
ejpam-5514	180	8	,	,	PUNCT
ejpam-5514	180	9	march	march	PROPN
ejpam-5514	180	10	2022	2022	NUM
ejpam-5514	180	11	.	.	PUNCT
ejpam-5514	181	1	[	[	X
ejpam-5514	181	2	4	4	X
ejpam-5514	181	3	]	]	PUNCT
ejpam-5514	181	4	tareq	tareq	PROPN
ejpam-5514	181	5	m.	m.	PROPN
ejpam-5514	181	6	al	al	PROPN
ejpam-5514	181	7	-	-	PUNCT
ejpam-5514	181	8	shami	shami	PROPN
ejpam-5514	181	9	and	and	CCONJ
ejpam-5514	181	10	ibtesam	ibtesam	PROPN
ejpam-5514	181	11	alshammari	alshammari	PROPN
ejpam-5514	181	12	.	.	PUNCT
ejpam-5514	182	1	rough	rough	ADJ
ejpam-5514	182	2	sets	set	NOUN
ejpam-5514	182	3	models	model	NOUN
ejpam-5514	182	4	inspired	inspire	VERB
ejpam-5514	182	5	by	by	ADP
ejpam-5514	182	6	supratopology	supratopology	NOUN
ejpam-5514	182	7	structures	structure	NOUN
ejpam-5514	182	8	.	.	PUNCT
ejpam-5514	183	1	artificial	artificial	ADJ
ejpam-5514	183	2	intelligence	intelligence	NOUN
ejpam-5514	183	3	review	review	NOUN
ejpam-5514	183	4	,	,	PUNCT
ejpam-5514	183	5	56(7):6855–6883	56(7):6855–6883	NUM
ejpam-5514	183	6	,	,	PUNCT
ejpam-5514	183	7	december	december	PROPN
ejpam-5514	183	8	2022	2022	NUM
ejpam-5514	183	9	.	.	PUNCT
ejpam-5514	184	1	[	[	X
ejpam-5514	184	2	5	5	X
ejpam-5514	184	3	]	]	PUNCT
ejpam-5514	184	4	tareq	tareq	PROPN
ejpam-5514	184	5	m.	m.	PROPN
ejpam-5514	184	6	al	al	PROPN
ejpam-5514	184	7	-	-	PUNCT
ejpam-5514	184	8	shami	shami	PROPN
ejpam-5514	184	9	and	and	CCONJ
ejpam-5514	184	10	m.	m.	PROPN
ejpam-5514	184	11	hosny	hosny	PROPN
ejpam-5514	184	12	.	.	PUNCT
ejpam-5514	185	1	generalized	generalized	ADJ
ejpam-5514	185	2	approximation	approximation	NOUN
ejpam-5514	185	3	spaces	space	NOUN
ejpam-5514	185	4	generation	generation	NOUN
ejpam-5514	185	5	from	from	ADP
ejpam-5514	185	6	ij	ij	NOUN
ejpam-5514	185	7	-	-	PUNCT
ejpam-5514	185	8	neighborhoods	neighborhood	NOUN
ejpam-5514	185	9	and	and	CCONJ
ejpam-5514	185	10	ideals	ideal	NOUN
ejpam-5514	185	11	with	with	ADP
ejpam-5514	185	12	application	application	NOUN
ejpam-5514	185	13	to	to	ADP
ejpam-5514	185	14	chikungunya	chikungunya	NOUN
ejpam-5514	185	15	disease	disease	NOUN
ejpam-5514	185	16	.	.	PUNCT
ejpam-5514	186	1	aims	aim	VERB
ejpam-5514	186	2	mathematics	mathematic	NOUN
ejpam-5514	186	3	,	,	PUNCT
ejpam-5514	186	4	9(4):10050–10077	9(4):10050–10077	NUM
ejpam-5514	186	5	,	,	PUNCT
ejpam-5514	186	6	2024	2024	NUM
ejpam-5514	186	7	.	.	PUNCT
ejpam-5514	187	1	[	[	X
ejpam-5514	187	2	6	6	NUM
ejpam-5514	187	3	]	]	PUNCT
ejpam-5514	187	4	ranjit	ranjit	PROPN
ejpam-5514	187	5	biswas	biswas	PROPN
ejpam-5514	187	6	and	and	CCONJ
ejpam-5514	187	7	sudarshan	sudarshan	PROPN
ejpam-5514	187	8	nanda	nanda	PROPN
ejpam-5514	187	9	.	.	PUNCT
ejpam-5514	188	1	rough	rough	ADJ
ejpam-5514	188	2	groups	group	NOUN
ejpam-5514	188	3	and	and	CCONJ
ejpam-5514	188	4	rough	rough	ADJ
ejpam-5514	188	5	subgroups	subgroup	NOUN
ejpam-5514	188	6	.	.	PUNCT
ejpam-5514	189	1	bull	bull	NOUN
ejpam-5514	189	2	.	.	PUNCT
ejpam-5514	190	1	pol	pol	PROPN
ejpam-5514	190	2	.	.	PUNCT
ejpam-5514	191	1	acad	acad	PROPN
ejpam-5514	191	2	.	.	PUNCT
ejpam-5514	192	1	sci	sci	PROPN
ejpam-5514	192	2	.	.	PROPN
ejpam-5514	192	3	,	,	PUNCT
ejpam-5514	192	4	math	math	NOUN
ejpam-5514	192	5	.	.	PUNCT
ejpam-5514	192	6	,	,	PUNCT
ejpam-5514	192	7	42(3):251–254	42(3):251–254	PROPN
ejpam-5514	192	8	,	,	PUNCT
ejpam-5514	192	9	1994	1994	NUM
ejpam-5514	192	10	.	.	PUNCT
ejpam-5514	193	1	[	[	X
ejpam-5514	193	2	7	7	X
ejpam-5514	193	3	]	]	X
ejpam-5514	193	4	b.	b.	PROPN
ejpam-5514	193	5	davvaz	davvaz	PROPN
ejpam-5514	193	6	.	.	PUNCT
ejpam-5514	194	1	roughness	roughness	NOUN
ejpam-5514	194	2	in	in	ADP
ejpam-5514	194	3	rings	ring	NOUN
ejpam-5514	194	4	.	.	PUNCT
ejpam-5514	195	1	information	information	NOUN
ejpam-5514	195	2	sciences	sciences	PROPN
ejpam-5514	195	3	,	,	PUNCT
ejpam-5514	195	4	164(1):147–163	164(1):147–163	NUM
ejpam-5514	195	5	,	,	PUNCT
ejpam-5514	195	6	2004	2004	NUM
ejpam-5514	195	7	.	.	PUNCT
ejpam-5514	196	1	[	[	X
ejpam-5514	196	2	8	8	NUM
ejpam-5514	196	3	]	]	X
ejpam-5514	196	4	b.	b.	PROPN
ejpam-5514	196	5	davvaz	davvaz	PROPN
ejpam-5514	196	6	.	.	PUNCT
ejpam-5514	197	1	roughness	roughness	NOUN
ejpam-5514	197	2	based	base	VERB
ejpam-5514	197	3	on	on	ADP
ejpam-5514	197	4	fuzzy	fuzzy	ADJ
ejpam-5514	197	5	ideals	ideal	NOUN
ejpam-5514	197	6	.	.	PUNCT
ejpam-5514	198	1	information	information	NOUN
ejpam-5514	198	2	sciences	sciences	PROPN
ejpam-5514	198	3	,	,	PUNCT
ejpam-5514	198	4	176(16):2417	176(16):2417	NUM
ejpam-5514	198	5	–	–	PUNCT
ejpam-5514	198	6	2437	2437	NUM
ejpam-5514	198	7	,	,	PUNCT
ejpam-5514	198	8	2006	2006	NUM
ejpam-5514	198	9	.	.	PUNCT
ejpam-5514	199	1	[	[	X
ejpam-5514	199	2	9	9	NUM
ejpam-5514	199	3	]	]	X
ejpam-5514	199	4	mona	mona	PROPN
ejpam-5514	199	5	hosny	hosny	PROPN
ejpam-5514	199	6	.	.	PUNCT
ejpam-5514	199	7	generalization	generalization	NOUN
ejpam-5514	199	8	of	of	ADP
ejpam-5514	199	9	rough	rough	ADJ
ejpam-5514	199	10	sets	set	NOUN
ejpam-5514	199	11	using	use	VERB
ejpam-5514	199	12	maximal	maximal	ADJ
ejpam-5514	199	13	right	right	ADJ
ejpam-5514	199	14	neighborhood	neighborhood	NOUN
ejpam-5514	199	15	systems	system	NOUN
ejpam-5514	199	16	and	and	CCONJ
ejpam-5514	199	17	ideals	ideal	NOUN
ejpam-5514	199	18	with	with	ADP
ejpam-5514	199	19	medical	medical	ADJ
ejpam-5514	199	20	applications	application	NOUN
ejpam-5514	199	21	.	.	PUNCT
ejpam-5514	200	1	aims	aim	VERB
ejpam-5514	200	2	mathematics	mathematic	NOUN
ejpam-5514	200	3	,	,	PUNCT
ejpam-5514	200	4	7(7):13104–13138	7(7):13104–13138	NUM
ejpam-5514	200	5	,	,	PUNCT
ejpam-5514	200	6	2022	2022	NUM
ejpam-5514	200	7	.	.	PUNCT
ejpam-5514	201	1	[	[	X
ejpam-5514	201	2	10	10	NUM
ejpam-5514	201	3	]	]	X
ejpam-5514	201	4	moslem	moslem	PROPN
ejpam-5514	201	5	imani	imani	PROPN
ejpam-5514	201	6	,	,	PUNCT
ejpam-5514	201	7	hoda	hoda	PROPN
ejpam-5514	201	8	fakour	fakour	PROPN
ejpam-5514	201	9	,	,	PUNCT
ejpam-5514	201	10	wen	wen	PROPN
ejpam-5514	201	11	-	-	PUNCT
ejpam-5514	201	12	hau	hau	PROPN
ejpam-5514	201	13	lan	lan	PROPN
ejpam-5514	201	14	,	,	PUNCT
ejpam-5514	201	15	huan	huan	PROPN
ejpam-5514	201	16	-	-	PUNCT
ejpam-5514	201	17	chin	chin	PROPN
ejpam-5514	201	18	kao	kao	PROPN
ejpam-5514	201	19	,	,	PUNCT
ejpam-5514	201	20	chi	chi	PROPN
ejpam-5514	201	21	ming	ming	PROPN
ejpam-5514	201	22	lee	lee	PROPN
ejpam-5514	201	23	,	,	PUNCT
ejpam-5514	201	24	yu	yu	PROPN
ejpam-5514	201	25	-	-	PROPN
ejpam-5514	201	26	shen	shen	PROPN
ejpam-5514	201	27	hsiao	hsiao	PROPN
ejpam-5514	201	28	,	,	PUNCT
ejpam-5514	201	29	and	and	CCONJ
ejpam-5514	201	30	chung	chung	ADJ
ejpam-5514	201	31	-	-	PUNCT
ejpam-5514	201	32	yen	yen	NOUN
ejpam-5514	201	33	kuo	kuo	PROPN
ejpam-5514	201	34	.	.	PUNCT
ejpam-5514	201	35	application	application	NOUN
ejpam-5514	201	36	of	of	ADP
ejpam-5514	201	37	rough	rough	ADJ
ejpam-5514	201	38	and	and	CCONJ
ejpam-5514	201	39	fuzzy	fuzzy	ADJ
ejpam-5514	201	40	set	set	NOUN
ejpam-5514	201	41	theory	theory	NOUN
ejpam-5514	201	42	for	for	ADP
ejpam-5514	201	43	prediction	prediction	NOUN
ejpam-5514	201	44	of	of	ADP
ejpam-5514	201	45	stochastic	stochastic	ADJ
ejpam-5514	201	46	wind	wind	NOUN
ejpam-5514	201	47	speed	speed	NOUN
ejpam-5514	201	48	data	datum	NOUN
ejpam-5514	201	49	using	use	VERB
ejpam-5514	201	50	long	long	ADJ
ejpam-5514	201	51	short	short	ADJ
ejpam-5514	201	52	-	-	PUNCT
ejpam-5514	201	53	term	term	NOUN
ejpam-5514	201	54	memory	memory	NOUN
ejpam-5514	201	55	.	.	PUNCT
ejpam-5514	202	1	atmosphere	atmosphere	NOUN
ejpam-5514	202	2	,	,	PUNCT
ejpam-5514	202	3	12(7	12(7	NUM
ejpam-5514	202	4	)	)	PUNCT
ejpam-5514	202	5	,	,	PUNCT
ejpam-5514	202	6	2021	2021	NUM
ejpam-5514	202	7	.	.	PUNCT
ejpam-5514	203	1	[	[	X
ejpam-5514	203	2	11	11	NUM
ejpam-5514	203	3	]	]	X
ejpam-5514	203	4	nobuaki	nobuaki	ADJ
ejpam-5514	203	5	kuroki	kuroki	PROPN
ejpam-5514	203	6	.	.	PUNCT
ejpam-5514	204	1	rough	rough	ADJ
ejpam-5514	204	2	ideals	ideal	NOUN
ejpam-5514	204	3	in	in	ADP
ejpam-5514	204	4	semigroups	semigroup	NOUN
ejpam-5514	204	5	.	.	PUNCT
ejpam-5514	205	1	information	information	NOUN
ejpam-5514	205	2	sciences	science	NOUN
ejpam-5514	205	3	,	,	PUNCT
ejpam-5514	205	4	100(1–4):139–163	100(1–4):139–163	NUM
ejpam-5514	205	5	,	,	PUNCT
ejpam-5514	205	6	august	august	PROPN
ejpam-5514	205	7	1997	1997	NUM
ejpam-5514	205	8	.	.	PUNCT
ejpam-5514	206	1	a.	a.	PROPN
ejpam-5514	206	2	prakash	prakash	PROPN
ejpam-5514	206	3	,	,	PUNCT
ejpam-5514	206	4	r.	r.	PROPN
ejpam-5514	206	5	shukla	shukla	PROPN
ejpam-5514	206	6	/	/	SYM
ejpam-5514	206	7	eur	eur	PROPN
ejpam-5514	206	8	.	.	PUNCT
ejpam-5514	207	1	j.	j.	PROPN
ejpam-5514	207	2	pure	pure	PROPN
ejpam-5514	207	3	appl	appl	PROPN
ejpam-5514	207	4	.	.	PROPN
ejpam-5514	207	5	math	math	PROPN
ejpam-5514	207	6	,	,	PUNCT
ejpam-5514	207	7	18	18	NUM
ejpam-5514	207	8	(	(	PUNCT
ejpam-5514	207	9	1	1	NUM
ejpam-5514	207	10	)	)	PUNCT
ejpam-5514	207	11	(	(	PUNCT
ejpam-5514	207	12	2025	2025	NUM
ejpam-5514	207	13	)	)	PUNCT
ejpam-5514	207	14	,	,	PUNCT
ejpam-5514	207	15	5514	5514	NUM
ejpam-5514	207	16	9	9	NUM
ejpam-5514	207	17	of	of	ADP
ejpam-5514	207	18	9	9	NUM
ejpam-5514	207	19	[	[	X
ejpam-5514	207	20	12	12	NUM
ejpam-5514	207	21	]	]	PUNCT
ejpam-5514	207	22	duoqian	duoqian	ADJ
ejpam-5514	207	23	miao	miao	PROPN
ejpam-5514	207	24	,	,	PUNCT
ejpam-5514	207	25	suqing	suqe	VERB
ejpam-5514	207	26	han	han	PROPN
ejpam-5514	207	27	,	,	PUNCT
ejpam-5514	207	28	daoguo	daoguo	PROPN
ejpam-5514	207	29	li	li	PROPN
ejpam-5514	207	30	,	,	PUNCT
ejpam-5514	207	31	and	and	CCONJ
ejpam-5514	207	32	lijun	lijun	PROPN
ejpam-5514	207	33	sun	sun	PROPN
ejpam-5514	207	34	.	.	PUNCT
ejpam-5514	208	1	rough	rough	ADJ
ejpam-5514	208	2	group	group	NOUN
ejpam-5514	208	3	,	,	PUNCT
ejpam-5514	208	4	rough	rough	ADJ
ejpam-5514	208	5	subgroup	subgroup	NOUN
ejpam-5514	208	6	and	and	CCONJ
ejpam-5514	208	7	their	their	PRON
ejpam-5514	208	8	properties	property	NOUN
ejpam-5514	208	9	.	.	PUNCT
ejpam-5514	209	1	in	in	ADP
ejpam-5514	209	2	dominik	dominik	PROPN
ejpam-5514	209	3	ślezak	ślezak	PROPN
ejpam-5514	209	4	,	,	PUNCT
ejpam-5514	209	5	guoyin	guoyin	PROPN
ejpam-5514	209	6	wang	wang	PROPN
ejpam-5514	209	7	,	,	PUNCT
ejpam-5514	209	8	marcin	marcin	PROPN
ejpam-5514	209	9	szczuka	szczuka	PROPN
ejpam-5514	209	10	,	,	PUNCT
ejpam-5514	209	11	ivo	ivo	PROPN
ejpam-5514	209	12	düntsch	düntsch	PROPN
ejpam-5514	209	13	,	,	PUNCT
ejpam-5514	209	14	and	and	CCONJ
ejpam-5514	209	15	yiyu	yiyu	NOUN
ejpam-5514	209	16	yao	yao	PROPN
ejpam-5514	209	17	,	,	PUNCT
ejpam-5514	209	18	editors	editor	NOUN
ejpam-5514	209	19	,	,	PUNCT
ejpam-5514	209	20	rough	rough	ADJ
ejpam-5514	209	21	sets	set	NOUN
ejpam-5514	209	22	,	,	PUNCT
ejpam-5514	209	23	fuzzy	fuzzy	ADJ
ejpam-5514	209	24	sets	set	NOUN
ejpam-5514	209	25	,	,	PUNCT
ejpam-5514	209	26	data	datum	NOUN
ejpam-5514	209	27	mining	mining	NOUN
ejpam-5514	209	28	,	,	PUNCT
ejpam-5514	209	29	and	and	CCONJ
ejpam-5514	209	30	granular	granular	ADJ
ejpam-5514	209	31	computing	computing	NOUN
ejpam-5514	209	32	,	,	PUNCT
ejpam-5514	209	33	pages	page	NOUN
ejpam-5514	209	34	104–113	104–113	NUM
ejpam-5514	209	35	,	,	PUNCT
ejpam-5514	209	36	berlin	berlin	PROPN
ejpam-5514	209	37	,	,	PUNCT
ejpam-5514	209	38	heidelberg	heidelberg	PROPN
ejpam-5514	209	39	,	,	PUNCT
ejpam-5514	209	40	2005	2005	NUM
ejpam-5514	209	41	.	.	PUNCT
ejpam-5514	210	1	springer	springer	PROPN
ejpam-5514	210	2	berlin	berlin	PROPN
ejpam-5514	210	3	heidelberg	heidelberg	PROPN
ejpam-5514	210	4	.	.	PUNCT
ejpam-5514	211	1	[	[	X
ejpam-5514	211	2	13	13	NUM
ejpam-5514	211	3	]	]	X
ejpam-5514	211	4	t.k	t.k	PROPN
ejpam-5514	211	5	.	.	PROPN
ejpam-5514	211	6	mukherjee	mukherjee	PROPN
ejpam-5514	211	7	and	and	CCONJ
ejpam-5514	211	8	m.k	m.k	PROPN
ejpam-5514	211	9	.	.	PUNCT
ejpam-5514	211	10	sen	sen	PROPN
ejpam-5514	211	11	.	.	PROPN
ejpam-5514	211	12	on	on	ADP
ejpam-5514	211	13	fuzzy	fuzzy	ADJ
ejpam-5514	211	14	ideals	ideal	NOUN
ejpam-5514	211	15	of	of	ADP
ejpam-5514	211	16	a	a	DET
ejpam-5514	211	17	ring	ring	NOUN
ejpam-5514	211	18	i.	i.	NOUN
ejpam-5514	211	19	fuzzy	fuzzy	ADJ
ejpam-5514	211	20	sets	set	NOUN
ejpam-5514	211	21	and	and	CCONJ
ejpam-5514	211	22	systems	system	NOUN
ejpam-5514	211	23	,	,	PUNCT
ejpam-5514	211	24	21(1):99–104	21(1):99–104	NUM
ejpam-5514	211	25	,	,	PUNCT
ejpam-5514	211	26	1987	1987	NUM
ejpam-5514	211	27	.	.	PUNCT
ejpam-5514	212	1	[	[	X
ejpam-5514	212	2	14	14	NUM
ejpam-5514	212	3	]	]	PUNCT
ejpam-5514	212	4	z.	z.	PROPN
ejpam-5514	212	5	pawlak	pawlak	PROPN
ejpam-5514	212	6	.	.	PUNCT
ejpam-5514	213	1	rough	rough	ADJ
ejpam-5514	213	2	sets	set	NOUN
ejpam-5514	213	3	.	.	PUNCT
ejpam-5514	214	1	int	int	NOUN
ejpam-5514	214	2	.	.	PUNCT
ejpam-5514	215	1	j.	j.	PROPN
ejpam-5514	215	2	inf	inf	PROPN
ejpam-5514	215	3	.	.	PUNCT
ejpam-5514	215	4	comput	comput	PROPN
ejpam-5514	215	5	.	.	PUNCT
ejpam-5514	216	1	sci	sci	PROPN
ejpam-5514	216	2	.	.	PROPN
ejpam-5514	216	3	,	,	PUNCT
ejpam-5514	216	4	11:341	11:341	NUM
ejpam-5514	216	5	–	–	PUNCT
ejpam-5514	216	6	356	356	NUM
ejpam-5514	216	7	,	,	PUNCT
ejpam-5514	216	8	1982	1982	NUM
ejpam-5514	216	9	.	.	PUNCT
ejpam-5514	217	1	[	[	X
ejpam-5514	217	2	15	15	NUM
ejpam-5514	217	3	]	]	PUNCT
ejpam-5514	217	4	z.	z.	PROPN
ejpam-5514	217	5	pawlak	pawlak	PROPN
ejpam-5514	217	6	.	.	PUNCT
ejpam-5514	218	1	why	why	SCONJ
ejpam-5514	218	2	rough	rough	ADJ
ejpam-5514	218	3	sets	set	NOUN
ejpam-5514	218	4	?	?	PUNCT
ejpam-5514	219	1	in	in	ADP
ejpam-5514	219	2	proceedings	proceeding	NOUN
ejpam-5514	219	3	of	of	ADP
ejpam-5514	219	4	ieee	ieee	NOUN
ejpam-5514	219	5	5th	5th	ADJ
ejpam-5514	219	6	international	international	ADJ
ejpam-5514	219	7	fuzzy	fuzzy	ADJ
ejpam-5514	219	8	systems	system	NOUN
ejpam-5514	219	9	,	,	PUNCT
ejpam-5514	219	10	volume	volume	NOUN
ejpam-5514	219	11	2	2	NUM
ejpam-5514	219	12	,	,	PUNCT
ejpam-5514	219	13	pages	page	VERB
ejpam-5514	219	14	738–743	738–743	NUM
ejpam-5514	219	15	vol.2	vol.2	PROPN
ejpam-5514	219	16	,	,	PUNCT
ejpam-5514	219	17	1996	1996	NUM
ejpam-5514	219	18	.	.	PUNCT
ejpam-5514	220	1	[	[	X
ejpam-5514	220	2	16	16	NUM
ejpam-5514	220	3	]	]	PUNCT
ejpam-5514	220	4	zdzislaw	zdzislaw	NOUN
ejpam-5514	220	5	pawlak	pawlak	ADJ
ejpam-5514	220	6	.	.	PUNCT
ejpam-5514	221	1	rough	rough	ADJ
ejpam-5514	221	2	sets	set	NOUN
ejpam-5514	221	3	theoretical	theoretical	ADJ
ejpam-5514	221	4	aspects	aspect	NOUN
ejpam-5514	221	5	of	of	ADP
ejpam-5514	221	6	reasoning	reasoning	NOUN
ejpam-5514	221	7	about	about	ADP
ejpam-5514	221	8	data	datum	NOUN
ejpam-5514	221	9	,	,	PUNCT
ejpam-5514	221	10	volume	volume	NOUN
ejpam-5514	221	11	9	9	NUM
ejpam-5514	221	12	of	of	ADP
ejpam-5514	221	13	theory	theory	NOUN
ejpam-5514	221	14	and	and	CCONJ
ejpam-5514	221	15	decision	decision	NOUN
ejpam-5514	221	16	library	library	NOUN
ejpam-5514	221	17	:	:	PUNCT
ejpam-5514	221	18	series	series	PROPN
ejpam-5514	221	19	d.	d.	PROPN
ejpam-5514	221	20	kluwer	kluwer	PROPN
ejpam-5514	221	21	,	,	PUNCT
ejpam-5514	221	22	1991	1991	NUM
ejpam-5514	221	23	.	.	PUNCT
ejpam-5514	222	1	[	[	X
ejpam-5514	222	2	17	17	NUM
ejpam-5514	222	3	]	]	X
ejpam-5514	222	4	jihong	jihong	PROPN
ejpam-5514	222	5	qu	qu	PROPN
ejpam-5514	222	6	,	,	PUNCT
ejpam-5514	222	7	xiao	xiao	PROPN
ejpam-5514	222	8	bai	bai	PROPN
ejpam-5514	222	9	,	,	PUNCT
ejpam-5514	222	10	jiajun	jiajun	PROPN
ejpam-5514	222	11	gu	gu	PROPN
ejpam-5514	222	12	,	,	PUNCT
ejpam-5514	222	13	farhad	farhad	ADJ
ejpam-5514	222	14	taghizadeh	taghizadeh	PROPN
ejpam-5514	222	15	-	-	PUNCT
ejpam-5514	222	16	hesary	hesary	ADJ
ejpam-5514	222	17	,	,	PUNCT
ejpam-5514	222	18	and	and	CCONJ
ejpam-5514	222	19	ji	ji	PROPN
ejpam-5514	222	20	lin	lin	PROPN
ejpam-5514	222	21	.	.	PUNCT
ejpam-5514	223	1	assessment	assessment	NOUN
ejpam-5514	223	2	of	of	ADP
ejpam-5514	223	3	rough	rough	ADJ
ejpam-5514	223	4	set	set	NOUN
ejpam-5514	223	5	theory	theory	NOUN
ejpam-5514	223	6	in	in	ADP
ejpam-5514	223	7	relation	relation	NOUN
ejpam-5514	223	8	to	to	ADP
ejpam-5514	223	9	risks	risk	NOUN
ejpam-5514	223	10	regarding	regard	VERB
ejpam-5514	223	11	hydraulic	hydraulic	ADJ
ejpam-5514	223	12	engineering	engineering	NOUN
ejpam-5514	223	13	investment	investment	NOUN
ejpam-5514	223	14	decisions	decision	NOUN
ejpam-5514	223	15	.	.	PUNCT
ejpam-5514	224	1	mathematics	mathematic	NOUN
ejpam-5514	224	2	,	,	PUNCT
ejpam-5514	224	3	8(8	8(8	NUM
ejpam-5514	224	4	)	)	PUNCT
ejpam-5514	224	5	,	,	PUNCT
ejpam-5514	224	6	2020	2020	NUM
ejpam-5514	224	7	.	.	PUNCT
ejpam-5514	225	1	[	[	X
ejpam-5514	225	2	18	18	NUM
ejpam-5514	225	3	]	]	PUNCT
ejpam-5514	225	4	a.	a.	PROPN
ejpam-5514	225	5	k.	k.	PROPN
ejpam-5514	225	6	sinha	sinha	PROPN
ejpam-5514	225	7	and	and	CCONJ
ejpam-5514	225	8	anand	anand	PROPN
ejpam-5514	225	9	prakash	prakash	PROPN
ejpam-5514	225	10	.	.	PUNCT
ejpam-5514	226	1	rough	rough	ADJ
ejpam-5514	226	2	exact	exact	ADJ
ejpam-5514	226	3	sequences	sequence	NOUN
ejpam-5514	226	4	of	of	ADP
ejpam-5514	226	5	modules	module	NOUN
ejpam-5514	226	6	.	.	PUNCT
ejpam-5514	227	1	international	international	ADJ
ejpam-5514	227	2	journal	journal	NOUN
ejpam-5514	227	3	of	of	ADP
ejpam-5514	227	4	applied	apply	VERB
ejpam-5514	227	5	engineering	engineering	NOUN
ejpam-5514	227	6	research	research	NOUN
ejpam-5514	227	7	,	,	PUNCT
ejpam-5514	227	8	11:2513–2517	11:2513–2517	PROPN
ejpam-5514	227	9	,	,	PUNCT
ejpam-5514	227	10	2016	2016	NUM
ejpam-5514	227	11	.	.	PUNCT
ejpam-5514	228	1	[	[	X
ejpam-5514	228	2	19	19	NUM
ejpam-5514	228	3	]	]	PUNCT
ejpam-5514	228	4	arvind	arvind	PROPN
ejpam-5514	228	5	kumar	kumar	PROPN
ejpam-5514	228	6	sinha	sinha	PROPN
ejpam-5514	228	7	and	and	CCONJ
ejpam-5514	228	8	anand	anand	PROPN
ejpam-5514	228	9	prakash	prakash	PROPN
ejpam-5514	228	10	.	.	PUNCT
ejpam-5514	228	11	injective	injective	ADJ
ejpam-5514	228	12	module	module	NOUN
ejpam-5514	228	13	based	base	VERB
ejpam-5514	228	14	on	on	ADP
ejpam-5514	228	15	rough	rough	ADJ
ejpam-5514	228	16	set	set	NOUN
ejpam-5514	228	17	theory	theory	NOUN
ejpam-5514	228	18	.	.	PUNCT
ejpam-5514	229	1	cogent	cogent	NOUN
ejpam-5514	229	2	mathematics	mathematic	NOUN
ejpam-5514	229	3	,	,	PUNCT
ejpam-5514	229	4	2(1):1069481	2(1):1069481	NUM
ejpam-5514	229	5	,	,	PUNCT
ejpam-5514	229	6	2015	2015	NUM
ejpam-5514	229	7	.	.	PUNCT
ejpam-5514	230	1	[	[	X
ejpam-5514	230	2	20	20	NUM
ejpam-5514	230	3	]	]	PUNCT
ejpam-5514	230	4	qi	qi	PROPN
ejpam-5514	230	5	-	-	PUNCT
ejpam-5514	230	6	mei	mei	PROPN
ejpam-5514	230	7	xiao	xiao	PROPN
ejpam-5514	230	8	and	and	CCONJ
ejpam-5514	230	9	zhen	zhen	PROPN
ejpam-5514	230	10	-	-	PUNCT
ejpam-5514	230	11	liang	liang	PROPN
ejpam-5514	230	12	zhang	zhang	PROPN
ejpam-5514	230	13	.	.	PUNCT
ejpam-5514	231	1	rough	rough	ADJ
ejpam-5514	231	2	prime	prime	ADJ
ejpam-5514	231	3	ideals	ideal	NOUN
ejpam-5514	231	4	and	and	CCONJ
ejpam-5514	231	5	rough	rough	ADJ
ejpam-5514	231	6	fuzzy	fuzzy	ADJ
ejpam-5514	231	7	prime	prime	ADJ
ejpam-5514	231	8	ideals	ideal	NOUN
ejpam-5514	231	9	in	in	ADP
ejpam-5514	231	10	semigroups	semigroup	NOUN
ejpam-5514	231	11	.	.	PUNCT
ejpam-5514	232	1	information	information	NOUN
ejpam-5514	232	2	sciences	sciences	PROPN
ejpam-5514	232	3	,	,	PUNCT
ejpam-5514	232	4	176(6):725–733	176(6):725–733	NUM
ejpam-5514	232	5	,	,	PUNCT
ejpam-5514	232	6	2006	2006	NUM
ejpam-5514	232	7	.	.	PUNCT
ejpam-5514	233	1	[	[	X
ejpam-5514	233	2	21	21	NUM
ejpam-5514	233	3	]	]	SYM
ejpam-5514	233	4	yongwei	yongwei	PROPN
ejpam-5514	233	5	yang	yang	PROPN
ejpam-5514	233	6	,	,	PUNCT
ejpam-5514	233	7	kuanyun	kuanyun	PROPN
ejpam-5514	233	8	zhu	zhu	PROPN
ejpam-5514	233	9	,	,	PUNCT
ejpam-5514	233	10	and	and	CCONJ
ejpam-5514	233	11	xiaolong	xiaolong	PROPN
ejpam-5514	233	12	xin	xin	PROPN
ejpam-5514	233	13	.	.	PUNCT
ejpam-5514	234	1	rough	rough	ADJ
ejpam-5514	234	2	sets	set	NOUN
ejpam-5514	234	3	based	base	VERB
ejpam-5514	234	4	on	on	ADP
ejpam-5514	234	5	fuzzy	fuzzy	ADJ
ejpam-5514	234	6	ideals	ideal	NOUN
ejpam-5514	234	7	in	in	ADP
ejpam-5514	234	8	distributive	distributive	ADJ
ejpam-5514	234	9	lattices	lattice	NOUN
ejpam-5514	234	10	.	.	PUNCT
ejpam-5514	235	1	open	open	ADJ
ejpam-5514	235	2	mathematics	mathematic	NOUN
ejpam-5514	235	3	,	,	PUNCT
ejpam-5514	235	4	18(1):122–137	18(1):122–137	NUM
ejpam-5514	235	5	,	,	PUNCT
ejpam-5514	235	6	2020	2020	NUM
ejpam-5514	235	7	.	.	PUNCT