id	sid	tid	token	lemma	pos
cana-2486	1	1	communications	communication	NOUN
cana-2486	1	2	on	on	ADP
cana-2486	1	3	applied	apply	VERB
cana-2486	1	4	nonlinear	nonlinear	ADJ
cana-2486	1	5	analysis	analysis	NOUN
cana-2486	1	6	issn	issn	NOUN
cana-2486	1	7	:	:	PUNCT
cana-2486	1	8	1074	1074	NUM
cana-2486	1	9	-	-	PUNCT
cana-2486	1	10	133x	133x	NUM
cana-2486	1	11	vol	vol	NOUN
cana-2486	1	12	.	.	PROPN
cana-2486	2	1	32	32	NUM
cana-2486	2	2	no	no	INTJ
cana-2486	2	3	.	.	PUNCT
cana-2486	3	1	2s	2s	NUM
cana-2486	3	2	(	(	PUNCT
cana-2486	3	3	2024	2024	NUM
cana-2486	3	4	)	)	PUNCT
cana-2486	3	5	538	538	NUM
cana-2486	3	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2486	3	7	pure	pure	ADJ
cana-2486	3	8	and	and	CCONJ
cana-2486	3	9	weakly	weakly	ADJ
cana-2486	3	10	pure	pure	ADJ
cana-2486	3	11	elements	element	NOUN
cana-2486	3	12	in	in	ADP
cana-2486	3	13	lattice	lattice	NOUN
cana-2486	3	14	modules	module	NOUN
cana-2486	3	15	santosh	santosh	NOUN
cana-2486	3	16	mitkari1	mitkari1	PROPN
cana-2486	3	17	,	,	PUNCT
cana-2486	3	18	renu	renu	PROPN
cana-2486	3	19	pathak2	pathak2	PROPN
cana-2486	3	20	,	,	PUNCT
cana-2486	3	21	smita	smita	PROPN
cana-2486	3	22	nigam3	nigam3	PROPN
cana-2486	3	23	,	,	PUNCT
cana-2486	3	24	pradip	pradip	NOUN
cana-2486	3	25	girase4	girase4	PROPN
cana-2486	3	26	,	,	PUNCT
cana-2486	3	27	lakpa	lakpa	ADP
cana-2486	3	28	sherpa5	sherpa5	PROPN
cana-2486	3	29	,	,	PUNCT
cana-2486	3	30	narayan	narayan	ADJ
cana-2486	3	31	phadatare6	phadatare6	NOUN
cana-2486	3	32	1	1	NUM
cana-2486	3	33	,	,	PUNCT
cana-2486	3	34	2department	2department	NUM
cana-2486	3	35	of	of	ADP
cana-2486	3	36	mathematics	mathematic	NOUN
cana-2486	3	37	,	,	PUNCT
cana-2486	3	38	school	school	NOUN
cana-2486	3	39	of	of	ADP
cana-2486	3	40	science	science	NOUN
cana-2486	3	41	,	,	PUNCT
cana-2486	3	42	sandip	sandip	PROPN
cana-2486	3	43	university	university	NOUN
cana-2486	3	44	,	,	PUNCT
cana-2486	3	45	nashik	nashik	PROPN
cana-2486	3	46	(	(	PUNCT
cana-2486	3	47	india	india	PROPN
cana-2486	3	48	)	)	PUNCT
cana-2486	3	49	3genba	3genba	NUM
cana-2486	3	50	sopanrao	sopanrao	NOUN
cana-2486	3	51	moze	moze	NOUN
cana-2486	3	52	college	college	NOUN
cana-2486	3	53	of	of	ADP
cana-2486	3	54	engineering	engineering	PROPN
cana-2486	3	55	,	,	PUNCT
cana-2486	3	56	pune	pune	PROPN
cana-2486	3	57	(	(	PUNCT
cana-2486	3	58	india	india	PROPN
cana-2486	3	59	)	)	PUNCT
cana-2486	3	60	4department	4department	NUM
cana-2486	3	61	of	of	ADP
cana-2486	3	62	mathematics	mathematic	NOUN
cana-2486	3	63	,	,	PUNCT
cana-2486	3	64	k.	k.	PROPN
cana-2486	3	65	k.	k.	PROPN
cana-2486	3	66	m.	m.	PROPN
cana-2486	3	67	college	college	PROPN
cana-2486	3	68	manwath	manwath	PROPN
cana-2486	3	69	,	,	PUNCT
cana-2486	3	70	parbhani	parbhani	NOUN
cana-2486	3	71	(	(	PUNCT
cana-2486	3	72	india	india	PROPN
cana-2486	3	73	)	)	PUNCT
cana-2486	4	1	5department	5department	NUM
cana-2486	4	2	of	of	ADP
cana-2486	4	3	mathematics	mathematic	NOUN
cana-2486	4	4	,	,	PUNCT
cana-2486	4	5	savitribai	savitribai	VERB
cana-2486	4	6	phule	phule	PROPN
cana-2486	4	7	pune	pune	PROPN
cana-2486	4	8	university	university	NOUN
cana-2486	4	9	,	,	PUNCT
cana-2486	4	10	pune-411	pune-411	NOUN
cana-2486	4	11	007	007	NUM
cana-2486	4	12	(	(	PUNCT
cana-2486	4	13	india	india	PROPN
cana-2486	4	14	)	)	PUNCT
cana-2486	4	15	6bharati	6bharati	PROPN
cana-2486	4	16	vidyapeeth	vidyapeeth	PROPN
cana-2486	4	17	deemed	deem	VERB
cana-2486	4	18	to	to	PART
cana-2486	4	19	be	be	AUX
cana-2486	4	20	university	university	NOUN
cana-2486	4	21	college	college	NOUN
cana-2486	4	22	of	of	ADP
cana-2486	4	23	engineering	engineering	NOUN
cana-2486	4	24	,	,	PUNCT
cana-2486	4	25	pune-411	pune-411	NOUN
cana-2486	4	26	043	043	NUM
cana-2486	4	27	(	(	PUNCT
cana-2486	4	28	india	india	PROPN
cana-2486	4	29	)	)	PUNCT
cana-2486	4	30	santosh.mitkari@bharatividyapeeth.edu	santosh.mitkari@bharatividyapeeth.edu	PROPN
cana-2486	4	31	,	,	PUNCT
cana-2486	4	32	renu.pathak@sandipuniversity.edu.in	renu.pathak@sandipuniversity.edu.in	NOUN
cana-2486	4	33	,	,	PUNCT
cana-2486	4	34	smita.nigam03@gmail.com	smita.nigam03@gmail.com	PROPN
cana-2486	4	35	,	,	PUNCT
cana-2486	4	36	gpradipmaths22@gmail.com	gpradipmaths22@gmail.com	PROPN
cana-2486	4	37	,	,	PUNCT
cana-2486	4	38	csherpaap@gmail.com	csherpaap@gmail.com	NOUN
cana-2486	4	39	,	,	PUNCT
cana-2486	4	40	nmphadatare@bvucoep.edu.in	nmphadatare@bvucoep.edu.in	NOUN
cana-2486	4	41	article	article	NOUN
cana-2486	4	42	history	history	NOUN
cana-2486	4	43	:	:	PUNCT
cana-2486	4	44	received	receive	VERB
cana-2486	4	45	:	:	PUNCT
cana-2486	4	46	25	25	NUM
cana-2486	4	47	-	-	PUNCT
cana-2486	4	48	09	09	NUM
cana-2486	4	49	-	-	PUNCT
cana-2486	4	50	2024	2024	NUM
cana-2486	4	51	revised	revise	VERB
cana-2486	4	52	:	:	PUNCT
cana-2486	4	53	02	02	NUM
cana-2486	4	54	-	-	SYM
cana-2486	4	55	11	11	NUM
cana-2486	4	56	-	-	PUNCT
cana-2486	4	57	2024	2024	NUM
cana-2486	4	58	accepted	accept	VERB
cana-2486	4	59	:	:	PUNCT
cana-2486	4	60	13	13	NUM
cana-2486	4	61	-	-	SYM
cana-2486	4	62	11	11	NUM
cana-2486	4	63	-	-	PUNCT
cana-2486	4	64	2024	2024	NUM
cana-2486	4	65	abstract	abstract	NOUN
cana-2486	4	66	:	:	PUNCT
cana-2486	4	67	this	this	DET
cana-2486	4	68	study	study	NOUN
cana-2486	4	69	concerns	concern	NOUN
cana-2486	4	70	with	with	ADP
cana-2486	4	71	investigation	investigation	NOUN
cana-2486	4	72	of	of	ADP
cana-2486	4	73	pure	pure	ADJ
cana-2486	4	74	and	and	CCONJ
cana-2486	4	75	weakly	weakly	ADJ
cana-2486	4	76	pure	pure	ADJ
cana-2486	4	77	elements	element	NOUN
cana-2486	4	78	of	of	ADP
cana-2486	4	79	lattice	lattice	NOUN
cana-2486	4	80	modules	module	NOUN
cana-2486	4	81	.	.	PUNCT
cana-2486	5	1	an	an	DET
cana-2486	5	2	element	element	NOUN
cana-2486	5	3	n	n	PROPN
cana-2486	5	4	of	of	ADP
cana-2486	5	5	m	m	PROPN
cana-2486	5	6	is	be	AUX
cana-2486	5	7	called	call	VERB
cana-2486	5	8	pure	pure	ADJ
cana-2486	5	9	,	,	PUNCT
cana-2486	5	10	if	if	SCONJ
cana-2486	5	11	an	an	DET
cana-2486	5	12	=	=	NOUN
cana-2486	5	13	n	n	X
cana-2486	5	14	∧	∧	PROPN
cana-2486	5	15	a1	a1	PROPN
cana-2486	5	16	m	m	PROPN
cana-2486	5	17	,	,	PUNCT
cana-2486	5	18	for	for	ADP
cana-2486	5	19	each	each	DET
cana-2486	5	20	a	a	PRON
cana-2486	5	21	of	of	ADP
cana-2486	5	22	l.	l.	PROPN
cana-2486	5	23	an	an	DET
cana-2486	5	24	element	element	NOUN
cana-2486	5	25	k	k	PROPN
cana-2486	5	26	of	of	ADP
cana-2486	5	27	m	m	PROPN
cana-2486	5	28	is	be	AUX
cana-2486	5	29	called	call	VERB
cana-2486	5	30	weakly	weakly	ADV
cana-2486	5	31	pure	pure	ADJ
cana-2486	5	32	,	,	PUNCT
cana-2486	5	33	if	if	SCONJ
cana-2486	5	34	an	an	DET
cana-2486	5	35	=	=	NOUN
cana-2486	5	36	n	n	X
cana-2486	5	37	∧	∧	PROPN
cana-2486	5	38	a1	a1	PROPN
cana-2486	5	39	m	m	PROPN
cana-2486	5	40	,	,	PUNCT
cana-2486	5	41	for	for	ADP
cana-2486	5	42	each	each	DET
cana-2486	5	43	idempotent	idempotent	NOUN
cana-2486	5	44	element	element	NOUN
cana-2486	5	45	a	a	PRON
cana-2486	5	46	of	of	ADP
cana-2486	5	47	l.	l.	NOUN
cana-2486	5	48	also	also	ADV
cana-2486	5	49	,	,	PUNCT
cana-2486	5	50	this	this	DET
cana-2486	5	51	study	study	NOUN
cana-2486	5	52	obtains	obtain	VERB
cana-2486	5	53	the	the	DET
cana-2486	5	54	relation	relation	NOUN
cana-2486	5	55	between	between	ADP
cana-2486	5	56	pure	pure	ADJ
cana-2486	5	57	,	,	PUNCT
cana-2486	5	58	idempotent	idempotent	ADJ
cana-2486	5	59	and	and	CCONJ
cana-2486	5	60	multiplication	multiplication	NOUN
cana-2486	5	61	elements	element	NOUN
cana-2486	5	62	of	of	ADP
cana-2486	5	63	lattice	lattice	NOUN
cana-2486	5	64	modules	module	NOUN
cana-2486	5	65	.	.	PUNCT
cana-2486	6	1	keywords	keyword	NOUN
cana-2486	6	2	:	:	PUNCT
cana-2486	6	3	pure	pure	ADJ
cana-2486	6	4	element	element	NOUN
cana-2486	6	5	,	,	PUNCT
cana-2486	6	6	weakly	weakly	ADJ
cana-2486	6	7	pure	pure	ADJ
cana-2486	6	8	element	element	NOUN
cana-2486	6	9	,	,	PUNCT
cana-2486	6	10	idempotent	idempotent	ADJ
cana-2486	6	11	element	element	NOUN
cana-2486	6	12	,	,	PUNCT
cana-2486	6	13	multiplication	multiplication	NOUN
cana-2486	6	14	element	element	NOUN
cana-2486	6	15	.	.	PUNCT
cana-2486	7	1	1	1	X
cana-2486	7	2	.	.	X
cana-2486	7	3	introduction	introduction	NOUN
cana-2486	7	4	a	a	DET
cana-2486	7	5	lattice	lattice	NOUN
cana-2486	7	6	l	l	NOUN
cana-2486	7	7	is	be	AUX
cana-2486	7	8	called	call	VERB
cana-2486	7	9	as	as	ADP
cana-2486	7	10	a	a	DET
cana-2486	7	11	multiplicative	multiplicative	ADJ
cana-2486	7	12	lattice	lattice	NOUN
cana-2486	7	13	,	,	PUNCT
cana-2486	7	14	if	if	SCONJ
cana-2486	7	15	l	l	NOUN
cana-2486	7	16	is	be	AUX
cana-2486	7	17	complete	complete	ADJ
cana-2486	7	18	with	with	ADP
cana-2486	7	19	commutative	commutative	ADJ
cana-2486	7	20	,	,	PUNCT
cana-2486	7	21	associative	associative	ADJ
cana-2486	7	22	and	and	CCONJ
cana-2486	7	23	join	join	VERB
cana-2486	7	24	distributive	distributive	ADJ
cana-2486	7	25	binary	binary	ADJ
cana-2486	7	26	operation	operation	NOUN
cana-2486	7	27	called	call	VERB
cana-2486	7	28	as	as	ADP
cana-2486	7	29	multiplication	multiplication	NOUN
cana-2486	7	30	.	.	PUNCT
cana-2486	8	1	an	an	DET
cana-2486	8	2	element	element	NOUN
cana-2486	8	3	1l	1l	NUM
cana-2486	8	4	of	of	ADP
cana-2486	8	5	l	l	PROPN
cana-2486	8	6	act	act	NOUN
cana-2486	8	7	as	as	ADP
cana-2486	8	8	a	a	DET
cana-2486	8	9	identity	identity	NOUN
cana-2486	8	10	with	with	ADP
cana-2486	8	11	respect	respect	NOUN
cana-2486	8	12	to	to	ADP
cana-2486	8	13	multiplication	multiplication	NOUN
cana-2486	8	14	.	.	PUNCT
cana-2486	9	1	for	for	ADP
cana-2486	9	2	a1	a1	PROPN
cana-2486	9	3	,	,	PUNCT
cana-2486	9	4	a2	a2	PROPN
cana-2486	9	5	∈	∈	PROPN
cana-2486	9	6	l	l	PROPN
cana-2486	9	7	,	,	PUNCT
cana-2486	9	8	(	(	PUNCT
cana-2486	9	9	a1	a1	NOUN
cana-2486	9	10	:	:	PUNCT
cana-2486	9	11	a2	a2	NOUN
cana-2486	9	12	)	)	PUNCT
cana-2486	9	13	=	=	SYM
cana-2486	9	14	∨{x	∨{x	NOUN
cana-2486	9	15	∈	∈	PROPN
cana-2486	9	16	l|a2x	l|a2x	NOUN
cana-2486	9	17	≤	≤	NUM
cana-2486	9	18	a1	a1	NOUN
cana-2486	9	19	}	}	PUNCT
cana-2486	9	20	.	.	PUNCT
cana-2486	10	1	element	element	NOUN
cana-2486	10	2	p	p	PROPN
cana-2486	10	3	∈	∈	PROPN
cana-2486	10	4	l	l	NOUN
cana-2486	10	5	such	such	ADJ
cana-2486	10	6	that	that	SCONJ
cana-2486	10	7	p	p	NOUN
cana-2486	10	8	≠1l	≠1l	X
cana-2486	10	9	is	be	AUX
cana-2486	10	10	prime	prime	ADJ
cana-2486	10	11	,	,	PUNCT
cana-2486	10	12	if	if	SCONJ
cana-2486	10	13	p1.p2	p1.p2	NOUN
cana-2486	10	14	≤	≤	NOUN
cana-2486	10	15	p	p	NOUN
cana-2486	10	16	implies	imply	VERB
cana-2486	10	17	p1	p1	NOUN
cana-2486	10	18	≤	≤	ADJ
cana-2486	10	19	p	p	NOUN
cana-2486	10	20	or	or	CCONJ
cana-2486	10	21	p2	p2	PROPN
cana-2486	10	22	≤	≤	NUM
cana-2486	11	1	p.	p.	NOUN
cana-2486	11	2	the	the	DET
cana-2486	11	3	radical	radical	NOUN
cana-2486	11	4	of	of	ADP
cana-2486	11	5	a	a	DET
cana-2486	11	6	∈	∈	NOUN
cana-2486	11	7	lis	li	NOUN
cana-2486	11	8	denoted	denote	VERB
cana-2486	11	9	by	by	ADP
cana-2486	11	10	√𝑎	√𝑎	PRON
cana-2486	11	11	and	and	CCONJ
cana-2486	11	12	is	be	AUX
cana-2486	11	13	defined	define	VERB
cana-2486	11	14	as	as	ADP
cana-2486	11	15	∨{x	∨{x	PROPN
cana-2486	11	16	∈	∈	PROPN
cana-2486	11	17	l|xk	l|xk	X
cana-2486	11	18	≤	≤	NOUN
cana-2486	11	19	a	a	X
cana-2486	11	20	,	,	PUNCT
cana-2486	11	21	for	for	ADP
cana-2486	11	22	some	some	DET
cana-2486	11	23	k	k	PROPN
cana-2486	11	24	∈	∈	PROPN
cana-2486	11	25	z+	z+	PRON
cana-2486	11	26	}	}	PUNCT
cana-2486	11	27	=	=	PUNCT
cana-2486	11	28	∧{p	∧{p	PROPN
cana-2486	11	29	∈	∈	PROPN
cana-2486	11	30	l|a	l|a	NOUN
cana-2486	12	1	≤	≤	PROPN
cana-2486	12	2	p	p	NOUN
cana-2486	13	1	and	and	CCONJ
cana-2486	13	2	p	p	NOUN
cana-2486	13	3	is	be	AUX
cana-2486	13	4	a	a	DET
cana-2486	13	5	prime	prime	ADJ
cana-2486	13	6	element	element	NOUN
cana-2486	13	7	}	}	PUNCT
cana-2486	13	8	.	.	PUNCT
cana-2486	14	1	an	an	DET
cana-2486	14	2	element	element	NOUN
cana-2486	14	3	c	c	PROPN
cana-2486	14	4	∈	∈	PROPN
cana-2486	14	5	l	l	NOUN
cana-2486	14	6	is	be	AUX
cana-2486	14	7	called	call	VERB
cana-2486	14	8	compact	compact	ADJ
cana-2486	14	9	,	,	PUNCT
cana-2486	14	10	if	if	SCONJ
cana-2486	14	11	for	for	ADP
cana-2486	14	12	t	t	PROPN
cana-2486	14	13	∈	∈	PROPN
cana-2486	14	14	i(i	i(i	PROPN
cana-2486	14	15	is	be	AUX
cana-2486	14	16	an	an	DET
cana-2486	14	17	index	index	NOUN
cana-2486	14	18	set	set	NOUN
cana-2486	14	19	)	)	PUNCT
cana-2486	14	20	,	,	PUNCT
cana-2486	14	21	c	c	NOUN
cana-2486	14	22	≤	≤	X
cana-2486	14	23	∨tat	∨tat	ADJ
cana-2486	14	24	⇒	⇒	NOUN
cana-2486	14	25	c	c	PROPN
cana-2486	14	26	≤	≤	PROPN
cana-2486	14	27	⋁𝑖=0	⋁𝑖=0	PROPN
cana-2486	14	28	𝑛	𝑛	DET
cana-2486	14	29	𝑎𝑡𝑖	𝑎𝑡𝑖	NOUN
cana-2486	14	30	,	,	PUNCT
cana-2486	14	31	for	for	ADP
cana-2486	14	32	some	some	DET
cana-2486	14	33	n	n	PRON
cana-2486	14	34	∈	∈	NOUN
cana-2486	14	35	z+	z+	NOUN
cana-2486	14	36	.	.	PUNCT
cana-2486	15	1	if	if	SCONJ
cana-2486	15	2	each	each	DET
cana-2486	15	3	element	element	NOUN
cana-2486	15	4	of	of	ADP
cana-2486	15	5	l	l	NOUN
cana-2486	15	6	is	be	AUX
cana-2486	15	7	a	a	DET
cana-2486	15	8	join	join	NOUN
cana-2486	15	9	of	of	ADP
cana-2486	15	10	compact	compact	ADJ
cana-2486	15	11	elements	element	NOUN
cana-2486	15	12	of	of	ADP
cana-2486	15	13	l	l	NOUN
cana-2486	15	14	,	,	PUNCT
cana-2486	15	15	then	then	ADV
cana-2486	15	16	l	l	NOUN
cana-2486	15	17	is	be	AUX
cana-2486	15	18	called	call	VERB
cana-2486	15	19	a	a	DET
cana-2486	15	20	cg	cg	NOUN
cana-2486	15	21	-	-	PUNCT
cana-2486	15	22	lattice	lattice	NOUN
cana-2486	15	23	.	.	PUNCT
cana-2486	16	1	an	an	DET
cana-2486	16	2	element	element	NOUN
cana-2486	16	3	p	p	PROPN
cana-2486	16	4	∈	∈	PROPN
cana-2486	16	5	l	l	NOUN
cana-2486	16	6	is	be	AUX
cana-2486	16	7	called	call	VERB
cana-2486	16	8	meet	meet	NOUN
cana-2486	16	9	[	[	X
cana-2486	16	10	join	join	VERB
cana-2486	16	11	]	]	X
cana-2486	16	12	principal	principal	NOUN
cana-2486	16	13	,	,	PUNCT
cana-2486	16	14	if	if	SCONJ
cana-2486	16	15	a1	a1	PROPN
cana-2486	16	16	∧	∧	PROPN
cana-2486	16	17	a2p=((a1	a2p=((a1	PUNCT
cana-2486	16	18	:	:	PUNCT
cana-2486	16	19	p	p	X
cana-2486	16	20	)	)	PUNCT
cana-2486	16	21	∧	∧	NOUN
cana-2486	16	22	a2)p	a2)p	NOUN
cana-2486	17	1	[	[	X
cana-2486	17	2	(	(	PUNCT
cana-2486	17	3	(	(	PUNCT
cana-2486	17	4	a1p	a1p	PROPN
cana-2486	17	5	∨	∨	NUM
cana-2486	17	6	a2	a2	PROPN
cana-2486	17	7	)	)	PUNCT
cana-2486	17	8	:	:	PUNCT
cana-2486	18	1	p	p	X
cana-2486	18	2	)	)	PUNCT
cana-2486	18	3	=	=	SYM
cana-2486	18	4	a1	a1	NOUN
cana-2486	18	5	∨	∨	X
cana-2486	18	6	(	(	PUNCT
cana-2486	18	7	a2	a2	PROPN
cana-2486	18	8	:	:	PUNCT
cana-2486	18	9	p	p	X
cana-2486	18	10	)	)	PUNCT
cana-2486	18	11	]	]	PUNCT
cana-2486	18	12	,	,	PUNCT
cana-2486	18	13	∀	∀	X
cana-2486	18	14	a1	a1	NOUN
cana-2486	18	15	,	,	PUNCT
cana-2486	18	16	a2	a2	PROPN
cana-2486	18	17	∈	∈	PROPN
cana-2486	18	18	l.	l.	NOUN
cana-2486	18	19	if	if	SCONJ
cana-2486	18	20	p	p	PROPN
cana-2486	18	21	∈	∈	PROPN
cana-2486	18	22	l	l	NOUN
cana-2486	18	23	is	be	AUX
cana-2486	18	24	both	both	PRON
cana-2486	18	25	meet	meet	VERB
cana-2486	18	26	and	and	CCONJ
cana-2486	18	27	join	join	VERB
cana-2486	18	28	principal	principal	NOUN
cana-2486	18	29	,	,	PUNCT
cana-2486	18	30	then	then	ADV
cana-2486	18	31	p	p	PROPN
cana-2486	18	32	is	be	AUX
cana-2486	18	33	called	call	VERB
cana-2486	18	34	principal	principal	ADJ
cana-2486	18	35	element	element	NOUN
cana-2486	18	36	.	.	PUNCT
cana-2486	19	1	if	if	SCONJ
cana-2486	19	2	every	every	DET
cana-2486	19	3	element	element	NOUN
cana-2486	19	4	of	of	ADP
cana-2486	19	5	l	l	NOUN
cana-2486	19	6	is	be	AUX
cana-2486	19	7	a	a	DET
cana-2486	19	8	join	join	NOUN
cana-2486	19	9	of	of	ADP
cana-2486	19	10	principal	principal	ADJ
cana-2486	19	11	elements	element	NOUN
cana-2486	19	12	of	of	ADP
cana-2486	19	13	l	l	NOUN
cana-2486	19	14	,	,	PUNCT
cana-2486	19	15	then	then	ADV
cana-2486	19	16	l	l	NOUN
cana-2486	19	17	is	be	AUX
cana-2486	19	18	called	call	VERB
cana-2486	19	19	a	a	DET
cana-2486	19	20	pg	pg	NOUN
cana-2486	19	21	-	-	PUNCT
cana-2486	19	22	lattice	lattice	NOUN
cana-2486	19	23	.	.	PUNCT
cana-2486	20	1	an	an	DET
cana-2486	20	2	element	element	NOUN
cana-2486	20	3	p	p	PROPN
cana-2486	20	4	∈	∈	PROPN
cana-2486	20	5	l	l	NOUN
cana-2486	20	6	is	be	AUX
cana-2486	20	7	said	say	VERB
cana-2486	20	8	to	to	PART
cana-2486	20	9	be	be	AUX
cana-2486	20	10	weak	weak	ADJ
cana-2486	20	11	meet	meet	NOUN
cana-2486	20	12	[	[	X
cana-2486	20	13	join	join	NOUN
cana-2486	20	14	]	]	X
cana-2486	20	15	principal	principal	NOUN
cana-2486	20	16	,	,	PUNCT
cana-2486	21	1	if	if	SCONJ
cana-2486	21	2	a	a	DET
cana-2486	21	3	∧	∧	PROPN
cana-2486	21	4	p	p	NOUN
cana-2486	21	5	=	=	NOUN
cana-2486	21	6	p(a	p(a	NOUN
cana-2486	21	7	:	:	PUNCT
cana-2486	21	8	p	p	X
cana-2486	21	9	)	)	PUNCT
cana-2486	22	1	[	[	X
cana-2486	22	2	a	a	DET
cana-2486	22	3	∨	∨	NOUN
cana-2486	22	4	(	(	PUNCT
cana-2486	22	5	0l	0l	X
cana-2486	22	6	:	:	PUNCT
cana-2486	22	7	p	p	X
cana-2486	22	8	)	)	PUNCT
cana-2486	22	9	=	=	SYM
cana-2486	22	10	(	(	PUNCT
cana-2486	22	11	pa	pa	NOUN
cana-2486	22	12	:	:	PUNCT
cana-2486	22	13	p	p	X
cana-2486	22	14	)	)	PUNCT
cana-2486	22	15	]	]	PUNCT
cana-2486	22	16	,	,	PUNCT
cana-2486	22	17	∀	∀	X
cana-2486	22	18	a	a	DET
cana-2486	22	19	∈	∈	PROPN
cana-2486	22	20	l.	l.	NOUN
cana-2486	22	21	an	an	DET
cana-2486	22	22	element	element	NOUN
cana-2486	22	23	a	a	DET
cana-2486	22	24	∈	∈	PROPN
cana-2486	22	25	l	l	NOUN
cana-2486	22	26	is	be	AUX
cana-2486	22	27	called	call	VERB
cana-2486	22	28	semiprime	semiprime	NOUN
cana-2486	22	29	or	or	CCONJ
cana-2486	22	30	radical	radical	ADJ
cana-2486	22	31	,	,	PUNCT
cana-2486	22	32	if	if	SCONJ
cana-2486	22	33	√𝑎	√𝑎	ADJ
cana-2486	22	34	=	=	PUNCT
cana-2486	22	35	a.	a.	NOUN
cana-2486	22	36	if	if	SCONJ
cana-2486	22	37	a	a	DET
cana-2486	22	38	∈	∈	NOUN
cana-2486	22	39	l	l	NOUN
cana-2486	22	40	such	such	ADJ
cana-2486	22	41	that	that	DET
cana-2486	22	42	a2	a2	PROPN
cana-2486	22	43	=	=	SYM
cana-2486	22	44	a	a	PROPN
cana-2486	22	45	,	,	PUNCT
cana-2486	22	46	then	then	ADV
cana-2486	22	47	a	a	PRON
cana-2486	22	48	is	be	AUX
cana-2486	22	49	called	call	VERB
cana-2486	22	50	an	an	DET
cana-2486	22	51	idempotent	idempotent	NOUN
cana-2486	22	52	.	.	PUNCT
cana-2486	23	1	let	let	VERB
cana-2486	23	2	c	c	PROPN
cana-2486	23	3	∈	∈	PROPN
cana-2486	23	4	l.	l.	NOUN
cana-2486	23	5	if	if	SCONJ
cana-2486	23	6	for	for	ADP
cana-2486	23	7	each	each	PRON
cana-2486	23	8	a	a	DET
cana-2486	23	9	∈	∈	NOUN
cana-2486	23	10	l	l	NOUN
cana-2486	23	11	such	such	ADJ
cana-2486	23	12	that	that	SCONJ
cana-2486	23	13	a	a	DET
cana-2486	23	14	≤	≤	ADJ
cana-2486	23	15	c	c	NOUN
cana-2486	23	16	there	there	PRON
cana-2486	23	17	exists	exist	VERB
cana-2486	23	18	an	an	DET
cana-2486	23	19	element	element	NOUN
cana-2486	23	20	d	d	PROPN
cana-2486	23	21	∈	∈	PROPN
cana-2486	23	22	l	l	NOUN
cana-2486	23	23	such	such	ADJ
cana-2486	23	24	that	that	SCONJ
cana-2486	23	25	a	a	DET
cana-2486	23	26	=	=	X
cana-2486	23	27	cd	cd	PROPN
cana-2486	23	28	,	,	PUNCT
cana-2486	23	29	then	then	ADV
cana-2486	23	30	c	c	PROPN
cana-2486	23	31	is	be	AUX
cana-2486	23	32	called	call	VERB
cana-2486	23	33	multiplication	multiplication	NOUN
cana-2486	23	34	element	element	NOUN
cana-2486	23	35	.	.	PUNCT
cana-2486	24	1	note	note	VERB
cana-2486	24	2	that	that	SCONJ
cana-2486	24	3	,	,	PUNCT
cana-2486	24	4	a	a	DET
cana-2486	24	5	∈	∈	PROPN
cana-2486	24	6	l	l	NOUN
cana-2486	24	7	is	be	AUX
cana-2486	24	8	a	a	DET
cana-2486	24	9	multiplication	multiplication	NOUN
cana-2486	24	10	element	element	NOUN
cana-2486	24	11	if	if	SCONJ
cana-2486	24	12	and	and	CCONJ
cana-2486	24	13	only	only	ADV
cana-2486	24	14	if	if	SCONJ
cana-2486	24	15	it	it	PRON
cana-2486	24	16	is	be	AUX
cana-2486	24	17	weak	weak	ADJ
cana-2486	24	18	meet	meet	ADJ
cana-2486	24	19	principal	principal	ADJ
cana-2486	24	20	element	element	NOUN
cana-2486	24	21	in	in	ADP
cana-2486	24	22	l.	l.	PROPN
cana-2486	24	23	a	a	DET
cana-2486	24	24	complete	complete	ADJ
cana-2486	24	25	lattice	lattice	NOUN
cana-2486	24	26	m	m	VERB
cana-2486	24	27	is	be	AUX
cana-2486	24	28	called	call	VERB
cana-2486	24	29	a	a	DET
cana-2486	24	30	lattice	lattice	NOUN
cana-2486	24	31	module	module	NOUN
cana-2486	24	32	(	(	PUNCT
cana-2486	24	33	l	l	NOUN
cana-2486	24	34	-	-	NOUN
cana-2486	24	35	module	module	NOUN
cana-2486	24	36	)	)	PUNCT
cana-2486	24	37	,	,	PUNCT
cana-2486	24	38	where	where	SCONJ
cana-2486	24	39	l	l	NOUN
cana-2486	24	40	is	be	AUX
cana-2486	24	41	a	a	DET
cana-2486	24	42	multiplicative	multiplicative	ADJ
cana-2486	24	43	lattice	lattice	NOUN
cana-2486	24	44	,	,	PUNCT
cana-2486	24	45	if	if	SCONJ
cana-2486	24	46	the	the	DET
cana-2486	24	47	multiplication	multiplication	NOUN
cana-2486	24	48	an	an	DET
cana-2486	24	49	∈	∈	NOUN
cana-2486	24	50	m	m	VERB
cana-2486	24	51	,	,	PUNCT
cana-2486	24	52	for	for	ADP
cana-2486	24	53	a	a	DET
cana-2486	24	54	∈	∈	PROPN
cana-2486	24	55	l	l	NOUN
cana-2486	24	56	and	and	CCONJ
cana-2486	24	57	n	n	PRON
cana-2486	24	58	∈	∈	PROPN
cana-2486	24	59	m	m	VERB
cana-2486	24	60	satisfies,(ab)n	satisfies,(ab)n	NOUN
cana-2486	24	61	=	=	PUNCT
cana-2486	24	62	a(bn	a(bn	PROPN
cana-2486	24	63	)	)	PUNCT
cana-2486	24	64	;	;	PUNCT
cana-2486	24	65	for	for	ADP
cana-2486	24	66	all	all	DET
cana-2486	24	67	a	a	PRON
cana-2486	24	68	,	,	PUNCT
cana-2486	24	69	communications	communication	NOUN
cana-2486	24	70	on	on	ADP
cana-2486	24	71	applied	apply	VERB
cana-2486	24	72	nonlinear	nonlinear	ADJ
cana-2486	24	73	analysis	analysis	NOUN
cana-2486	24	74	issn	issn	NOUN
cana-2486	24	75	:	:	PUNCT
cana-2486	24	76	1074	1074	NUM
cana-2486	24	77	-	-	PUNCT
cana-2486	24	78	133x	133x	NUM
cana-2486	24	79	vol	vol	NOUN
cana-2486	24	80	.	.	PROPN
cana-2486	25	1	32	32	NUM
cana-2486	25	2	no	no	INTJ
cana-2486	25	3	.	.	PUNCT
cana-2486	26	1	2s	2s	NUM
cana-2486	26	2	(	(	PUNCT
cana-2486	26	3	2024	2024	NUM
cana-2486	26	4	)	)	PUNCT
cana-2486	26	5	539	539	NUM
cana-2486	26	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2486	26	7	b	b	PROPN
cana-2486	26	8	in	in	ADP
cana-2486	26	9	l	l	PROPN
cana-2486	26	10	and	and	CCONJ
cana-2486	26	11	for	for	ADP
cana-2486	26	12	all	all	DET
cana-2486	26	13	n	n	NOUN
cana-2486	26	14	in	in	ADP
cana-2486	26	15	m	m	PROPN
cana-2486	26	16	.	.	PUNCT
cana-2486	27	1	1	1	X
cana-2486	27	2	.	.	X
cana-2486	27	3	(	(	PUNCT
cana-2486	27	4	∨α	∨α	NOUN
cana-2486	27	5	lα	lα	NOUN
cana-2486	27	6	)	)	PUNCT
cana-2486	27	7	(	(	PUNCT
cana-2486	27	8	∨β	∨β	NOUN
cana-2486	27	9	nβ	nβ	PROPN
cana-2486	27	10	)	)	PUNCT
cana-2486	27	11	=	=	PUNCT
cana-2486	27	12	(	(	PUNCT
cana-2486	27	13	∨αβ	∨αβ	PROPN
cana-2486	27	14	lαnβ	lαnβ	NOUN
cana-2486	27	15	)	)	PUNCT
cana-2486	27	16	;	;	PUNCT
cana-2486	27	17	for	for	ADP
cana-2486	27	18	all	all	DET
cana-2486	27	19	lα	lα	NOUN
cana-2486	27	20	in	in	ADP
cana-2486	27	21	l	l	NOUN
cana-2486	27	22	and	and	CCONJ
cana-2486	27	23	for	for	ADP
cana-2486	27	24	all	all	DET
cana-2486	27	25	nβ	nβ	NOUN
cana-2486	27	26	in	in	ADP
cana-2486	27	27	m	m	PROPN
cana-2486	27	28	.	.	PUNCT
cana-2486	28	1	2	2	X
cana-2486	28	2	.	.	X
cana-2486	28	3	1ln	1ln	NOUN
cana-2486	28	4	=	=	SYM
cana-2486	28	5	n	n	CCONJ
cana-2486	28	6	;	;	PUNCT
cana-2486	28	7	for	for	ADP
cana-2486	28	8	1l	1l	NUM
cana-2486	28	9	∈	∈	PROPN
cana-2486	28	10	l	l	NOUN
cana-2486	28	11	and	and	CCONJ
cana-2486	28	12	n	n	PRON
cana-2486	28	13	∈	∈	NOUN
cana-2486	28	14	m	m	NOUN
cana-2486	28	15	.	.	PUNCT
cana-2486	29	1	3	3	X
cana-2486	29	2	.	.	X
cana-2486	29	3	0ln	0ln	NOUN
cana-2486	30	1	=	=	PUNCT
cana-2486	30	2	0	0	NUM
cana-2486	30	3	m	m	VERB
cana-2486	30	4	;	;	PUNCT
cana-2486	30	5	for	for	ADP
cana-2486	30	6	0l	0l	NUM
cana-2486	30	7	∈	∈	PROPN
cana-2486	30	8	l	l	NOUN
cana-2486	30	9	and	and	CCONJ
cana-2486	30	10	n	n	PRON
cana-2486	30	11	∈	∈	NOUN
cana-2486	30	12	m	m	VERB
cana-2486	30	13	.	.	PUNCT
cana-2486	31	1	note	note	VERB
cana-2486	31	2	that	that	SCONJ
cana-2486	31	3	0	0	NUM
cana-2486	31	4	m	m	NOUN
cana-2486	31	5	is	be	AUX
cana-2486	31	6	a	a	DET
cana-2486	31	7	least	least	ADJ
cana-2486	31	8	and	and	CCONJ
cana-2486	31	9	1	1	NUM
cana-2486	31	10	m	m	NOUN
cana-2486	31	11	is	be	AUX
cana-2486	31	12	a	a	DET
cana-2486	31	13	greatest	great	ADJ
cana-2486	31	14	element	element	NOUN
cana-2486	31	15	of	of	ADP
cana-2486	31	16	m.	m.	NOUN
cana-2486	31	17	for	for	ADP
cana-2486	31	18	n1,n2	n1,n2	PROPN
cana-2486	31	19	∈	∈	PROPN
cana-2486	31	20	m	m	PRON
cana-2486	31	21	,	,	PUNCT
cana-2486	31	22	(	(	PUNCT
cana-2486	31	23	n1	n1	NOUN
cana-2486	31	24	:	:	PUNCT
cana-2486	31	25	n2	n2	ADJ
cana-2486	31	26	)	)	PUNCT
cana-2486	32	1	=	=	SYM
cana-2486	32	2	∨{x	∨{x	NOUN
cana-2486	32	3	∈	∈	PROPN
cana-2486	32	4	l|xn2	l|xn2	PRON
cana-2486	32	5	≤	≤	NUM
cana-2486	32	6	n1	n1	NOUN
cana-2486	32	7	}	}	PUNCT
cana-2486	32	8	.	.	PUNCT
cana-2486	33	1	for	for	ADP
cana-2486	33	2	n	n	DET
cana-2486	33	3	∈	∈	PROPN
cana-2486	33	4	m	m	NOUN
cana-2486	33	5	and	and	CCONJ
cana-2486	33	6	a	a	DET
cana-2486	33	7	∈	∈	PROPN
cana-2486	33	8	l	l	NOUN
cana-2486	33	9	,	,	PUNCT
cana-2486	33	10	(	(	PUNCT
cana-2486	33	11	n	n	CCONJ
cana-2486	33	12	:	:	PUNCT
cana-2486	33	13	a	a	X
cana-2486	33	14	)	)	PUNCT
cana-2486	33	15	=	=	PUNCT
cana-2486	33	16	∨{k	∨{k	NOUN
cana-2486	33	17	∈	∈	PROPN
cana-2486	33	18	m	m	NOUN
cana-2486	33	19	|ak	|ak	NOUN
cana-2486	33	20	≤	≤	NUM
cana-2486	33	21	n	n	CCONJ
cana-2486	33	22	}	}	PUNCT
cana-2486	33	23	.	.	PUNCT
cana-2486	34	1	an	an	DET
cana-2486	34	2	element	element	NOUN
cana-2486	34	3	n∈	n∈	NOUN
cana-2486	34	4	m	m	VERB
cana-2486	34	5	is	be	AUX
cana-2486	34	6	called	call	VERB
cana-2486	34	7	compact	compact	ADJ
cana-2486	34	8	,	,	PUNCT
cana-2486	34	9	if	if	SCONJ
cana-2486	34	10	for	for	ADP
cana-2486	34	11	t	t	PROPN
cana-2486	34	12	∈	∈	PROPN
cana-2486	34	13	i(i	i(i	PROPN
cana-2486	34	14	is	be	AUX
cana-2486	34	15	an	an	DET
cana-2486	34	16	index	index	NOUN
cana-2486	34	17	set	set	NOUN
cana-2486	34	18	)	)	PUNCT
cana-2486	34	19	,	,	PUNCT
cana-2486	35	1	n	n	CCONJ
cana-2486	35	2	≤	≤	X
cana-2486	35	3	∨tbt	∨tbt	PROPN
cana-2486	35	4	⇒	⇒	VERB
cana-2486	35	5	n≤	n≤	PRON
cana-2486	35	6	∨𝑖=0	∨𝑖=0	PROPN
cana-2486	35	7	𝑛	𝑛	DET
cana-2486	35	8	𝐵𝑡𝑖	𝐵𝑡𝑖	PROPN
cana-2486	35	9	,	,	PUNCT
cana-2486	35	10	for	for	ADP
cana-2486	35	11	some	some	DET
cana-2486	35	12	n	n	PRON
cana-2486	35	13	∈	∈	NOUN
cana-2486	35	14	z+	z+	NOUN
cana-2486	35	15	.	.	PUNCT
cana-2486	36	1	if	if	SCONJ
cana-2486	36	2	each	each	DET
cana-2486	36	3	element	element	NOUN
cana-2486	36	4	of	of	ADP
cana-2486	36	5	m	m	PROPN
cana-2486	36	6	is	be	AUX
cana-2486	36	7	a	a	DET
cana-2486	36	8	join	join	NOUN
cana-2486	36	9	of	of	ADP
cana-2486	36	10	compact	compact	ADJ
cana-2486	36	11	elements	element	NOUN
cana-2486	36	12	of	of	ADP
cana-2486	36	13	m	m	PROPN
cana-2486	36	14	,	,	PUNCT
cana-2486	36	15	then	then	ADV
cana-2486	36	16	m	m	VERB
cana-2486	36	17	is	be	AUX
cana-2486	36	18	called	call	VERB
cana-2486	36	19	a	a	DET
cana-2486	36	20	cglattice	cglattice	NOUN
cana-2486	36	21	module	module	NOUN
cana-2486	36	22	.	.	PUNCT
cana-2486	37	1	an	an	DET
cana-2486	37	2	element	element	NOUN
cana-2486	37	3	n	n	CCONJ
cana-2486	37	4	∈	∈	NOUN
cana-2486	37	5	m	m	VERB
cana-2486	37	6	is	be	AUX
cana-2486	37	7	called	call	VERB
cana-2486	37	8	meet	meet	NOUN
cana-2486	37	9	[	[	X
cana-2486	37	10	join	join	VERB
cana-2486	37	11	]	]	PUNCT
cana-2486	37	12	principal	principal	NOUN
cana-2486	37	13	,	,	PUNCT
cana-2486	37	14	if	if	SCONJ
cana-2486	37	15	(	(	PUNCT
cana-2486	37	16	a	a	DET
cana-2486	37	17	∧	∧	PROPN
cana-2486	37	18	(	(	PUNCT
cana-2486	37	19	b	b	NOUN
cana-2486	37	20	:	:	PUNCT
cana-2486	37	21	n	n	NUM
cana-2486	37	22	)	)	PUNCT
cana-2486	37	23	)	)	PUNCT
cana-2486	38	1	n	n	X
cana-2486	38	2	=	=	NOUN
cana-2486	38	3	an	an	DET
cana-2486	38	4	∧	∧	PROPN
cana-2486	38	5	b	b	PROPN
cana-2486	38	6	[	[	X
cana-2486	38	7	(	(	PUNCT
cana-2486	38	8	a	a	DET
cana-2486	38	9	∨	∨	NOUN
cana-2486	38	10	(	(	PUNCT
cana-2486	38	11	b	b	NOUN
cana-2486	38	12	:	:	PUNCT
cana-2486	38	13	n	n	CCONJ
cana-2486	38	14	)	)	PUNCT
cana-2486	38	15	=(	=(	NOUN
cana-2486	38	16	(	(	PUNCT
cana-2486	38	17	an	an	DET
cana-2486	38	18	∨	∨	NUM
cana-2486	38	19	b	b	NOUN
cana-2486	38	20	)	)	PUNCT
cana-2486	38	21	:	:	PUNCT
cana-2486	38	22	n	n	X
cana-2486	38	23	)	)	PUNCT
cana-2486	38	24	]	]	PUNCT
cana-2486	38	25	,	,	PUNCT
cana-2486	38	26	∀	∀	X
cana-2486	38	27	a	a	DET
cana-2486	38	28	∈	∈	PROPN
cana-2486	38	29	l	l	NOUN
cana-2486	38	30	and	and	CCONJ
cana-2486	38	31	b	b	PROPN
cana-2486	38	32	∈	∈	PROPN
cana-2486	38	33	m.	m.	NOUN
cana-2486	38	34	if	if	SCONJ
cana-2486	38	35	b	b	X
cana-2486	38	36	∈	∈	PROPN
cana-2486	38	37	m	m	VERB
cana-2486	38	38	is	be	AUX
cana-2486	38	39	both	both	PRON
cana-2486	38	40	meet	meet	VERB
cana-2486	38	41	and	and	CCONJ
cana-2486	38	42	join	join	VERB
cana-2486	38	43	principal	principal	NOUN
cana-2486	38	44	,	,	PUNCT
cana-2486	38	45	then	then	ADV
cana-2486	38	46	b	b	PROPN
cana-2486	38	47	is	be	AUX
cana-2486	38	48	called	call	VERB
cana-2486	38	49	principal	principal	ADJ
cana-2486	38	50	element	element	NOUN
cana-2486	38	51	.	.	PUNCT
cana-2486	39	1	if	if	SCONJ
cana-2486	39	2	each	each	DET
cana-2486	39	3	element	element	NOUN
cana-2486	39	4	of	of	ADP
cana-2486	39	5	m	m	PROPN
cana-2486	39	6	is	be	AUX
cana-2486	39	7	a	a	DET
cana-2486	39	8	join	join	NOUN
cana-2486	39	9	of	of	ADP
cana-2486	39	10	principal	principal	ADJ
cana-2486	39	11	elements	element	NOUN
cana-2486	39	12	of	of	ADP
cana-2486	39	13	m	m	PROPN
cana-2486	39	14	,	,	PUNCT
cana-2486	39	15	then	then	ADV
cana-2486	39	16	m	m	VERB
cana-2486	39	17	is	be	AUX
cana-2486	39	18	called	call	VERB
cana-2486	39	19	a	a	DET
cana-2486	39	20	pg	pg	ADJ
cana-2486	39	21	-	-	PUNCT
cana-2486	39	22	lattice	lattice	NOUN
cana-2486	39	23	module	module	NOUN
cana-2486	39	24	.	.	PUNCT
cana-2486	40	1	an	an	DET
cana-2486	40	2	element	element	NOUN
cana-2486	40	3	n	n	CCONJ
cana-2486	40	4	∈	∈	NOUN
cana-2486	40	5	m	m	VERB
cana-2486	40	6	is	be	AUX
cana-2486	40	7	said	say	VERB
cana-2486	40	8	to	to	PART
cana-2486	40	9	be	be	AUX
cana-2486	40	10	weak	weak	ADJ
cana-2486	40	11	meet	meet	NOUN
cana-2486	40	12	[	[	X
cana-2486	40	13	join	join	NOUN
cana-2486	40	14	]	]	PUNCT
cana-2486	40	15	principal	principal	NOUN
cana-2486	40	16	,	,	PUNCT
cana-2486	40	17	if	if	SCONJ
cana-2486	40	18	(	(	PUNCT
cana-2486	40	19	b	b	NOUN
cana-2486	40	20	:	:	PUNCT
cana-2486	40	21	n	n	X
cana-2486	40	22	)	)	PUNCT
cana-2486	40	23	n	n	NOUN
cana-2486	40	24	=	=	SYM
cana-2486	40	25	b	b	PROPN
cana-2486	40	26	∧	∧	PROPN
cana-2486	40	27	n	n	CCONJ
cana-2486	40	28	[	[	X
cana-2486	40	29	(	(	PUNCT
cana-2486	40	30	an	an	DET
cana-2486	40	31	:	:	PUNCT
cana-2486	40	32	n	n	NOUN
cana-2486	40	33	)	)	PUNCT
cana-2486	40	34	=	=	PUNCT
cana-2486	40	35	a	a	DET
cana-2486	40	36	∨	∨	NUM
cana-2486	40	37	(	(	PUNCT
cana-2486	40	38	0	0	NUM
cana-2486	40	39	m	m	NOUN
cana-2486	40	40	:	:	PUNCT
cana-2486	40	41	n	n	X
cana-2486	40	42	)	)	PUNCT
cana-2486	40	43	]	]	PUNCT
cana-2486	40	44	,	,	PUNCT
cana-2486	40	45	∀	∀	X
cana-2486	40	46	a	a	DET
cana-2486	40	47	∈	∈	PROPN
cana-2486	40	48	l	l	NOUN
cana-2486	40	49	and	and	CCONJ
cana-2486	40	50	b	b	X
cana-2486	40	51	∈	∈	ADV
cana-2486	40	52	m	m	VERB
cana-2486	40	53	.	.	PUNCT
cana-2486	41	1	an	an	DET
cana-2486	41	2	element	element	NOUN
cana-2486	41	3	n	n	CCONJ
cana-2486	41	4	∈	∈	NOUN
cana-2486	41	5	m	m	VERB
cana-2486	41	6	is	be	AUX
cana-2486	41	7	said	say	VERB
cana-2486	41	8	to	to	PART
cana-2486	41	9	be	be	AUX
cana-2486	41	10	proper	proper	ADJ
cana-2486	41	11	,	,	PUNCT
cana-2486	41	12	if	if	SCONJ
cana-2486	41	13	n	n	CCONJ
cana-2486	41	14	<	<	X
cana-2486	41	15	1	1	NUM
cana-2486	41	16	m.	m.	NOUN
cana-2486	41	17	if	if	SCONJ
cana-2486	41	18	n	n	NOUN
cana-2486	41	19	∈	∈	NOUN
cana-2486	41	20	m	m	VERB
cana-2486	41	21	such	such	ADJ
cana-2486	41	22	that	that	SCONJ
cana-2486	41	23	n	n	NOUN
cana-2486	41	24	=	=	SYM
cana-2486	41	25	(	(	PUNCT
cana-2486	41	26	n	n	NOUN
cana-2486	41	27	:	:	PUNCT
cana-2486	41	28	1	1	NUM
cana-2486	41	29	m	m	NOUN
cana-2486	41	30	)	)	PUNCT
cana-2486	41	31	n	n	CCONJ
cana-2486	41	32	,	,	PUNCT
cana-2486	41	33	then	then	ADV
cana-2486	41	34	n	n	PRON
cana-2486	41	35	is	be	AUX
cana-2486	41	36	an	an	DET
cana-2486	41	37	idempotent	idempotent	ADJ
cana-2486	41	38	element	element	NOUN
cana-2486	41	39	of	of	ADP
cana-2486	41	40	m	m	PROPN
cana-2486	41	41	.	.	PUNCT
cana-2486	42	1	element	element	NOUN
cana-2486	42	2	n	n	PROPN
cana-2486	42	3	∈	∈	NOUN
cana-2486	42	4	m	m	VERB
cana-2486	42	5	is	be	AUX
cana-2486	42	6	said	say	VERB
cana-2486	42	7	to	to	PART
cana-2486	42	8	be	be	AUX
cana-2486	42	9	multiplication	multiplication	NOUN
cana-2486	42	10	,	,	PUNCT
cana-2486	42	11	if	if	SCONJ
cana-2486	42	12	for	for	ADP
cana-2486	42	13	every	every	DET
cana-2486	42	14	k	k	PROPN
cana-2486	42	15	∈	∈	PROPN
cana-2486	42	16	m	m	VERB
cana-2486	42	17	with	with	ADP
cana-2486	42	18	k	k	PROPN
cana-2486	42	19	≤	≤	PROPN
cana-2486	42	20	n	n	CCONJ
cana-2486	42	21	there	there	ADV
cana-2486	42	22	exists	exist	VERB
cana-2486	42	23	an	an	DET
cana-2486	42	24	element	element	NOUN
cana-2486	42	25	a	a	DET
cana-2486	42	26	∈	∈	NOUN
cana-2486	42	27	l	l	NOUN
cana-2486	42	28	such	such	ADJ
cana-2486	42	29	that	that	SCONJ
cana-2486	42	30	k	k	PROPN
cana-2486	42	31	=	=	PUNCT
cana-2486	42	32	an	an	PROPN
cana-2486	42	33	.	.	PUNCT
cana-2486	43	1	it	it	PRON
cana-2486	43	2	is	be	AUX
cana-2486	43	3	also	also	ADV
cana-2486	43	4	noted	note	VERB
cana-2486	43	5	that	that	SCONJ
cana-2486	43	6	,	,	PUNCT
cana-2486	43	7	n	n	PRON
cana-2486	43	8	∈	∈	NOUN
cana-2486	43	9	m	m	VERB
cana-2486	43	10	is	be	AUX
cana-2486	43	11	a	a	DET
cana-2486	43	12	multiplication	multiplication	NOUN
cana-2486	43	13	element	element	NOUN
cana-2486	43	14	if	if	SCONJ
cana-2486	43	15	and	and	CCONJ
cana-2486	43	16	only	only	ADV
cana-2486	43	17	if	if	SCONJ
cana-2486	43	18	n	n	PRON
cana-2486	43	19	is	be	AUX
cana-2486	43	20	weak	weak	ADJ
cana-2486	43	21	meet	meet	ADJ
cana-2486	43	22	principal	principal	NOUN
cana-2486	43	23	in	in	ADP
cana-2486	43	24	m	m	PROPN
cana-2486	43	25	.	.	PUNCT
cana-2486	44	1	a	a	DET
cana-2486	44	2	l−lattice	l−lattice	PROPN
cana-2486	44	3	module	module	NOUN
cana-2486	44	4	m	m	NOUN
cana-2486	44	5	is	be	AUX
cana-2486	44	6	called	call	VERB
cana-2486	44	7	second	second	ADJ
cana-2486	44	8	,	,	PUNCT
cana-2486	44	9	if	if	SCONJ
cana-2486	44	10	for	for	ADP
cana-2486	44	11	each	each	DET
cana-2486	44	12	a	a	DET
cana-2486	44	13	∈	∈	PROPN
cana-2486	44	14	l	l	NOUN
cana-2486	44	15	,	,	PUNCT
cana-2486	44	16	a1	a1	PROPN
cana-2486	44	17	m	m	NOUN
cana-2486	44	18	=	=	NOUN
cana-2486	44	19	1	1	NUM
cana-2486	44	20	m	m	NOUN
cana-2486	44	21	or	or	CCONJ
cana-2486	44	22	a1	a1	NOUN
cana-2486	44	23	m	m	NOUN
cana-2486	44	24	=	=	SYM
cana-2486	44	25	0	0	NUM
cana-2486	44	26	m.	m.	NOUN
cana-2486	44	27	a	a	DET
cana-2486	44	28	l−lattice	l−lattice	NOUN
cana-2486	44	29	module	module	NOUN
cana-2486	44	30	m	m	NOUN
cana-2486	44	31	is	be	AUX
cana-2486	44	32	called	call	VERB
cana-2486	44	33	secondary	secondary	ADJ
cana-2486	44	34	,	,	PUNCT
cana-2486	44	35	if	if	SCONJ
cana-2486	44	36	for	for	ADP
cana-2486	44	37	each	each	DET
cana-2486	44	38	a	a	DET
cana-2486	44	39	∈	∈	PROPN
cana-2486	44	40	l	l	NOUN
cana-2486	44	41	,	,	PUNCT
cana-2486	44	42	a1	a1	PROPN
cana-2486	44	43	m	m	NOUN
cana-2486	44	44	=	=	NOUN
cana-2486	44	45	1	1	NUM
cana-2486	44	46	m	m	NOUN
cana-2486	44	47	or	or	CCONJ
cana-2486	44	48	an1	an1	NOUN
cana-2486	44	49	m	m	PROPN
cana-2486	44	50	=	=	SYM
cana-2486	44	51	0	0	NUM
cana-2486	44	52	m	m	VERB
cana-2486	44	53	for	for	ADP
cana-2486	44	54	some	some	DET
cana-2486	44	55	n>0	n>0	NOUN
cana-2486	44	56	.	.	PUNCT
cana-2486	45	1	if	if	SCONJ
cana-2486	45	2	annm	annm	NOUN
cana-2486	45	3	=	=	SYM
cana-2486	45	4	(	(	PUNCT
cana-2486	45	5	0	0	NUM
cana-2486	45	6	m	m	VERB
cana-2486	45	7	:	:	PUNCT
cana-2486	45	8	1	1	NUM
cana-2486	45	9	m	m	NOUN
cana-2486	45	10	)	)	PUNCT
cana-2486	46	1	=	=	SYM
cana-2486	46	2	0l	0l	NOUN
cana-2486	46	3	,	,	PUNCT
cana-2486	46	4	then	then	ADV
cana-2486	46	5	m	m	VERB
cana-2486	46	6	is	be	AUX
cana-2486	46	7	called	call	VERB
cana-2486	46	8	faithful	faithful	ADJ
cana-2486	46	9	l−module	l−module	NOUN
cana-2486	46	10	.	.	PUNCT
cana-2486	47	1	a	a	DET
cana-2486	47	2	l−module	l−module	NOUN
cana-2486	47	3	m	m	VERB
cana-2486	47	4	is	be	AUX
cana-2486	47	5	called	call	VERB
cana-2486	47	6	torsion	torsion	NOUN
cana-2486	47	7	-	-	PUNCT
cana-2486	47	8	free	free	ADJ
cana-2486	47	9	,	,	PUNCT
cana-2486	47	10	whenever	whenever	SCONJ
cana-2486	47	11	ak	ak	PROPN
cana-2486	47	12	=	=	PROPN
cana-2486	47	13	0	0	PROPN
cana-2486	47	14	m	m	NOUN
cana-2486	47	15	implies	imply	VERB
cana-2486	47	16	k	k	PROPN
cana-2486	47	17	=	=	PUNCT
cana-2486	47	18	0	0	NUM
cana-2486	47	19	m	m	VERB
cana-2486	47	20	or	or	CCONJ
cana-2486	47	21	a	a	DET
cana-2486	47	22	=	=	PUNCT
cana-2486	47	23	0l	0l	NOUN
cana-2486	47	24	,	,	PUNCT
cana-2486	47	25	for	for	ADP
cana-2486	47	26	any	any	DET
cana-2486	47	27	a	a	DET
cana-2486	47	28	∈	∈	ADJ
cana-2486	47	29	l	l	NOUN
cana-2486	47	30	and	and	CCONJ
cana-2486	47	31	k	k	PROPN
cana-2486	47	32	∈	∈	PROPN
cana-2486	47	33	m.	m.	NOUN
cana-2486	47	34	a	a	DET
cana-2486	47	35	l−module	l−module	NOUN
cana-2486	47	36	m	m	VERB
cana-2486	47	37	is	be	AUX
cana-2486	47	38	multiplication	multiplication	NOUN
cana-2486	47	39	,	,	PUNCT
cana-2486	47	40	if	if	SCONJ
cana-2486	47	41	for	for	ADP
cana-2486	47	42	each	each	DET
cana-2486	47	43	element	element	NOUN
cana-2486	47	44	n	n	PRON
cana-2486	47	45	∈	∈	NOUN
cana-2486	47	46	m	m	VERB
cana-2486	47	47	there	there	PRON
cana-2486	47	48	exists	exist	VERB
cana-2486	47	49	a	a	DET
cana-2486	47	50	∈	∈	NOUN
cana-2486	47	51	l	l	NOUN
cana-2486	47	52	such	such	ADJ
cana-2486	47	53	that	that	SCONJ
cana-2486	47	54	n	n	NOUN
cana-2486	47	55	=	=	PUNCT
cana-2486	47	56	a1	a1	PROPN
cana-2486	47	57	m.	m.	NOUN
cana-2486	47	58	note	note	NOUN
cana-2486	47	59	that	that	SCONJ
cana-2486	47	60	,	,	PUNCT
cana-2486	47	61	l−module	l−module	NOUN
cana-2486	47	62	m	m	VERB
cana-2486	47	63	is	be	AUX
cana-2486	47	64	a	a	DET
cana-2486	47	65	multiplication	multiplication	NOUN
cana-2486	47	66	if	if	SCONJ
cana-2486	47	67	and	and	CCONJ
cana-2486	47	68	only	only	ADV
cana-2486	47	69	if	if	SCONJ
cana-2486	47	70	n	n	ADV
cana-2486	47	71	=	=	SYM
cana-2486	47	72	(	(	PUNCT
cana-2486	47	73	n	n	NOUN
cana-2486	47	74	:	:	PUNCT
cana-2486	47	75	1	1	NUM
cana-2486	47	76	m	m	NOUN
cana-2486	47	77	)	)	PUNCT
cana-2486	47	78	1	1	NUM
cana-2486	47	79	m	m	NOUN
cana-2486	47	80	for	for	ADP
cana-2486	47	81	all	all	PRON
cana-2486	47	82	n	n	DET
cana-2486	47	83	∈	∈	NOUN
cana-2486	47	84	m	m	AUX
cana-2486	47	85	(	(	PUNCT
cana-2486	47	86	see	see	VERB
cana-2486	47	87	[	[	X
cana-2486	47	88	4	4	NUM
cana-2486	47	89	]	]	NUM
cana-2486	47	90	)	)	PUNCT
cana-2486	47	91	.	.	PUNCT
cana-2486	48	1	for	for	ADP
cana-2486	48	2	n	n	PRON
cana-2486	48	3	∈	∈	PROPN
cana-2486	48	4	m	m	NOUN
cana-2486	48	5	,	,	PUNCT
cana-2486	48	6	[	[	X
cana-2486	48	7	n	n	CCONJ
cana-2486	48	8	,	,	PUNCT
cana-2486	48	9	1	1	NUM
cana-2486	48	10	m	m	NOUN
cana-2486	48	11	]	]	PUNCT
cana-2486	48	12	is	be	AUX
cana-2486	48	13	a	a	DET
cana-2486	48	14	set	set	NOUN
cana-2486	48	15	of	of	ADP
cana-2486	48	16	all	all	PRON
cana-2486	48	17	k	k	PROPN
cana-2486	48	18	∈	∈	PROPN
cana-2486	48	19	m	m	VERB
cana-2486	48	20	such	such	ADJ
cana-2486	48	21	that	that	SCONJ
cana-2486	48	22	n	n	NOUN
cana-2486	48	23	≤	≤	ADV
cana-2486	48	24	k	k	X
cana-2486	48	25	≤	≤	ADJ
cana-2486	48	26	1	1	NUM
cana-2486	48	27	m.	m.	NOUN
cana-2486	48	28	note	note	NOUN
cana-2486	48	29	that	that	SCONJ
cana-2486	48	30	,	,	PUNCT
cana-2486	48	31	[	[	X
cana-2486	48	32	n	n	CCONJ
cana-2486	48	33	,	,	PUNCT
cana-2486	48	34	1	1	NUM
cana-2486	48	35	m	m	NOUN
cana-2486	48	36	]	]	PUNCT
cana-2486	48	37	is	be	AUX
cana-2486	48	38	a	a	DET
cana-2486	48	39	l	l	ADJ
cana-2486	48	40	-	-	PUNCT
cana-2486	48	41	lattice	lattice	NOUN
cana-2486	48	42	module	module	NOUN
cana-2486	48	43	with	with	ADP
cana-2486	48	44	multiplication	multiplication	NOUN
cana-2486	48	45	a	a	DET
cana-2486	48	46	◦	◦	NOUN
cana-2486	48	47	k	k	X
cana-2486	48	48	=	=	PROPN
cana-2486	48	49	ak	ak	PROPN
cana-2486	48	50	∨	∨	PROPN
cana-2486	48	51	n	n	CCONJ
cana-2486	48	52	,	,	PUNCT
cana-2486	48	53	where	where	SCONJ
cana-2486	48	54	a	a	DET
cana-2486	48	55	∈	∈	PROPN
cana-2486	48	56	l	l	NOUN
cana-2486	48	57	and	and	CCONJ
cana-2486	48	58	k	k	PROPN
cana-2486	48	59	∈	∈	PROPN
cana-2486	48	60	m	m	VERB
cana-2486	48	61	such	such	ADJ
cana-2486	48	62	that	that	SCONJ
cana-2486	48	63	n	n	ADV
cana-2486	48	64	≤	≤	NOUN
cana-2486	48	65	k.	k.	NOUN
cana-2486	49	1	this	this	DET
cana-2486	49	2	study	study	NOUN
cana-2486	49	3	aims	aim	VERB
cana-2486	49	4	the	the	DET
cana-2486	49	5	generalization	generalization	NOUN
cana-2486	49	6	of	of	ADP
cana-2486	49	7	some	some	DET
cana-2486	49	8	important	important	ADJ
cana-2486	49	9	results	result	NOUN
cana-2486	49	10	studied	study	VERB
cana-2486	49	11	in	in	ADP
cana-2486	49	12	[	[	X
cana-2486	49	13	1	1	NUM
cana-2486	49	14	]	]	PUNCT
cana-2486	49	15	,	,	PUNCT
cana-2486	49	16	[	[	X
cana-2486	49	17	2	2	NUM
cana-2486	49	18	]	]	PUNCT
cana-2486	49	19	for	for	ADP
cana-2486	49	20	submodules	submodule	NOUN
cana-2486	49	21	of	of	ADP
cana-2486	49	22	module	module	NOUN
cana-2486	49	23	over	over	ADP
cana-2486	49	24	commutative	commutative	ADJ
cana-2486	49	25	ring	ring	NOUN
cana-2486	49	26	to	to	ADP
cana-2486	49	27	the	the	DET
cana-2486	49	28	lattice	lattice	NOUN
cana-2486	49	29	modules	module	NOUN
cana-2486	49	30	over	over	ADP
cana-2486	49	31	multiplicative	multiplicative	ADJ
cana-2486	49	32	lattices	lattice	NOUN
cana-2486	49	33	and	and	CCONJ
cana-2486	49	34	examine	examine	VERB
cana-2486	49	35	the	the	DET
cana-2486	49	36	concepts	concept	NOUN
cana-2486	49	37	in	in	ADP
cana-2486	49	38	multiplicative	multiplicative	ADJ
cana-2486	49	39	lattices	lattice	NOUN
cana-2486	49	40	and	and	CCONJ
cana-2486	49	41	multiplication	multiplication	NOUN
cana-2486	49	42	lattice	lattice	NOUN
cana-2486	49	43	modules	module	NOUN
cana-2486	49	44	.	.	PUNCT
cana-2486	50	1	remark	remark	VERB
cana-2486	50	2	1.1	1.1	NUM
cana-2486	50	3	.	.	PUNCT
cana-2486	51	1	let	let	VERB
cana-2486	51	2	m	m	PRON
cana-2486	51	3	be	be	AUX
cana-2486	51	4	a	a	DET
cana-2486	51	5	multiplication	multiplication	NOUN
cana-2486	51	6	lattice	lattice	NOUN
cana-2486	51	7	module	module	NOUN
cana-2486	51	8	and	and	CCONJ
cana-2486	51	9	n	n	DET
cana-2486	51	10	a	a	DET
cana-2486	51	11	element	element	NOUN
cana-2486	51	12	of	of	ADP
cana-2486	51	13	m.	m.	NOUN
cana-2486	51	14	if	if	SCONJ
cana-2486	51	15	(	(	PUNCT
cana-2486	51	16	n	n	X
cana-2486	51	17	:	:	PUNCT
cana-2486	51	18	1	1	NUM
cana-2486	51	19	m	m	NOUN
cana-2486	51	20	)	)	PUNCT
cana-2486	51	21	is	be	AUX
cana-2486	51	22	an	an	DET
cana-2486	51	23	idempotent	idempotent	NOUN
cana-2486	51	24	,	,	PUNCT
cana-2486	51	25	then	then	ADV
cana-2486	51	26	n	n	NOUN
cana-2486	51	27	=	=	SYM
cana-2486	51	28	(	(	PUNCT
cana-2486	51	29	n	n	X
cana-2486	51	30	:	:	PUNCT
cana-2486	51	31	1	1	NUM
cana-2486	51	32	m	m	NOUN
cana-2486	51	33	)	)	PUNCT
cana-2486	51	34	1	1	NUM
cana-2486	51	35	m	m	NOUN
cana-2486	51	36	=	=	PUNCT
cana-2486	51	37	(	(	PUNCT
cana-2486	51	38	n	n	NOUN
cana-2486	51	39	:	:	PUNCT
cana-2486	51	40	1	1	NUM
cana-2486	51	41	m	m	NOUN
cana-2486	51	42	)	)	PUNCT
cana-2486	51	43	21	21	NUM
cana-2486	51	44	m	m	NOUN
cana-2486	51	45	=	=	PUNCT
cana-2486	51	46	(	(	PUNCT
cana-2486	51	47	n	n	NOUN
cana-2486	51	48	:	:	PUNCT
cana-2486	51	49	1	1	NUM
cana-2486	51	50	m	m	NOUN
cana-2486	51	51	)	)	PUNCT
cana-2486	51	52	n	n	NOUN
cana-2486	51	53	,	,	PUNCT
cana-2486	51	54	and	and	CCONJ
cana-2486	51	55	n	n	PRON
cana-2486	51	56	is	be	AUX
cana-2486	51	57	idempotent	idempotent	ADJ
cana-2486	51	58	in	in	ADP
cana-2486	51	59	m	m	PROPN
cana-2486	51	60	.	.	PUNCT
cana-2486	52	1	conversely	conversely	ADV
cana-2486	52	2	,	,	PUNCT
cana-2486	52	3	if	if	SCONJ
cana-2486	52	4	m	m	NOUN
cana-2486	52	5	is	be	AUX
cana-2486	52	6	a	a	DET
cana-2486	52	7	cg	cg	NOUN
cana-2486	52	8	and	and	CCONJ
cana-2486	52	9	faithful	faithful	ADJ
cana-2486	52	10	multiplication	multiplication	NOUN
cana-2486	52	11	l	l	NOUN
cana-2486	52	12	-	-	NOUN
cana-2486	52	13	module	module	NOUN
cana-2486	52	14	with	with	ADP
cana-2486	52	15	n	n	PRON
cana-2486	52	16	is	be	AUX
cana-2486	52	17	idempotent	idempotent	ADJ
cana-2486	52	18	in	in	ADP
cana-2486	52	19	m	m	PROPN
cana-2486	52	20	,	,	PUNCT
cana-2486	52	21	then	then	ADV
cana-2486	52	22	n	n	PROPN
cana-2486	52	23	=	=	SYM
cana-2486	52	24	(	(	PUNCT
cana-2486	52	25	n	n	X
cana-2486	52	26	:	:	PUNCT
cana-2486	52	27	1	1	NUM
cana-2486	52	28	m	m	NOUN
cana-2486	52	29	)	)	PUNCT
cana-2486	52	30	1	1	NUM
cana-2486	52	31	m	m	NOUN
cana-2486	52	32	=	=	PUNCT
cana-2486	52	33	(	(	PUNCT
cana-2486	52	34	n	n	NOUN
cana-2486	52	35	:	:	PUNCT
cana-2486	52	36	1	1	NUM
cana-2486	52	37	m	m	NOUN
cana-2486	52	38	)	)	PUNCT
cana-2486	52	39	n	n	CCONJ
cana-2486	52	40	,	,	PUNCT
cana-2486	52	41	and	and	CCONJ
cana-2486	52	42	hence	hence	ADV
cana-2486	52	43	n	n	NOUN
cana-2486	52	44	=	=	SYM
cana-2486	52	45	(	(	PUNCT
cana-2486	52	46	n	n	X
cana-2486	52	47	:	:	PUNCT
cana-2486	52	48	1	1	NUM
cana-2486	52	49	m	m	NOUN
cana-2486	52	50	)	)	PUNCT
cana-2486	52	51	21	21	NUM
cana-2486	52	52	m	m	NOUN
cana-2486	52	53	=	=	PUNCT
cana-2486	52	54	(	(	PUNCT
cana-2486	52	55	n	n	NOUN
cana-2486	52	56	:	:	PUNCT
cana-2486	52	57	1	1	NUM
cana-2486	52	58	m	m	NOUN
cana-2486	52	59	)	)	PUNCT
cana-2486	52	60	1	1	NUM
cana-2486	52	61	m	m	PROPN
cana-2486	52	62	,	,	PUNCT
cana-2486	52	63	which	which	PRON
cana-2486	52	64	shows	show	VERB
cana-2486	52	65	that	that	SCONJ
cana-2486	52	66	(	(	PUNCT
cana-2486	52	67	n	n	X
cana-2486	52	68	:	:	PUNCT
cana-2486	52	69	1	1	NUM
cana-2486	52	70	m	m	NOUN
cana-2486	52	71	)	)	PUNCT
cana-2486	52	72	2	2	NUM
cana-2486	52	73	=	=	SYM
cana-2486	52	74	(	(	PUNCT
cana-2486	52	75	n	n	NOUN
cana-2486	52	76	:	:	PUNCT
cana-2486	52	77	1	1	NUM
cana-2486	52	78	m	m	NOUN
cana-2486	52	79	)	)	PUNCT
cana-2486	52	80	is	be	AUX
cana-2486	52	81	an	an	DET
cana-2486	52	82	idempotent	idempotent	NOUN
cana-2486	52	83	.	.	PUNCT
cana-2486	53	1	further	far	ADV
cana-2486	53	2	,	,	PUNCT
cana-2486	53	3	for	for	ADP
cana-2486	53	4	more	more	ADJ
cana-2486	53	5	information	information	NOUN
cana-2486	53	6	on	on	ADP
cana-2486	53	7	modules	module	NOUN
cana-2486	53	8	,	,	PUNCT
cana-2486	53	9	multiplicatice	multiplicatice	NOUN
cana-2486	53	10	lattices	lattice	NOUN
cana-2486	53	11	,	,	PUNCT
cana-2486	53	12	lattice	lattice	NOUN
cana-2486	53	13	modules	module	NOUN
cana-2486	53	14	,	,	PUNCT
cana-2486	53	15	the	the	DET
cana-2486	53	16	reader	reader	NOUN
cana-2486	53	17	may	may	AUX
cana-2486	53	18	refer	refer	VERB
cana-2486	53	19	to	to	ADP
cana-2486	53	20	[	[	X
cana-2486	53	21	3	3	NUM
cana-2486	53	22	]	]	PUNCT
cana-2486	53	23	,	,	PUNCT
cana-2486	53	24	[	[	X
cana-2486	53	25	7	7	NUM
cana-2486	53	26	]	]	PUNCT
cana-2486	53	27	,	,	PUNCT
cana-2486	53	28	[	[	X
cana-2486	53	29	8	8	NUM
cana-2486	53	30	]	]	PUNCT
cana-2486	53	31	,	,	PUNCT
cana-2486	53	32	[	[	X
cana-2486	53	33	9	9	NUM
cana-2486	53	34	]	]	PUNCT
cana-2486	53	35	,	,	PUNCT
cana-2486	53	36	[	[	X
cana-2486	53	37	10	10	NUM
cana-2486	53	38	]	]	PUNCT
cana-2486	53	39	.	.	PUNCT
cana-2486	54	1	communications	communication	NOUN
cana-2486	54	2	on	on	ADP
cana-2486	54	3	applied	apply	VERB
cana-2486	54	4	nonlinear	nonlinear	ADJ
cana-2486	54	5	analysis	analysis	NOUN
cana-2486	54	6	issn	issn	NOUN
cana-2486	54	7	:	:	PUNCT
cana-2486	54	8	1074	1074	NUM
cana-2486	54	9	-	-	PUNCT
cana-2486	54	10	133x	133x	NUM
cana-2486	54	11	vol	vol	NOUN
cana-2486	54	12	.	.	PROPN
cana-2486	55	1	32	32	NUM
cana-2486	55	2	no	no	INTJ
cana-2486	55	3	.	.	PUNCT
cana-2486	56	1	2s	2s	NUM
cana-2486	56	2	(	(	PUNCT
cana-2486	56	3	2024	2024	NUM
cana-2486	56	4	)	)	PUNCT
cana-2486	56	5	540	540	NUM
cana-2486	56	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2486	56	7	2	2	NUM
cana-2486	56	8	.	.	PUNCT
cana-2486	56	9	pure	pure	ADJ
cana-2486	56	10	element	element	NOUN
cana-2486	56	11	we	we	PRON
cana-2486	56	12	begin	begin	VERB
cana-2486	56	13	this	this	DET
cana-2486	56	14	section	section	NOUN
cana-2486	56	15	with	with	ADP
cana-2486	56	16	the	the	DET
cana-2486	56	17	following	follow	VERB
cana-2486	56	18	definitions	definition	NOUN
cana-2486	56	19	:	:	PUNCT
cana-2486	56	20	definition	definition	NOUN
cana-2486	56	21	2.1	2.1	NUM
cana-2486	56	22	.	.	PUNCT
cana-2486	57	1	let	let	VERB
cana-2486	57	2	l	l	NOUN
cana-2486	57	3	be	be	AUX
cana-2486	57	4	a	a	DET
cana-2486	57	5	multiplicative	multiplicative	ADJ
cana-2486	57	6	lattice	lattice	NOUN
cana-2486	57	7	and	and	CCONJ
cana-2486	57	8	c	c	NOUN
cana-2486	57	9	∈	∈	PROPN
cana-2486	57	10	l.	l.	PROPN
cana-2486	57	11	c	c	PROPN
cana-2486	57	12	is	be	AUX
cana-2486	57	13	said	say	VERB
cana-2486	57	14	to	to	PART
cana-2486	57	15	be	be	AUX
cana-2486	57	16	a	a	DET
cana-2486	57	17	multiplication	multiplication	NOUN
cana-2486	57	18	element	element	NOUN
cana-2486	57	19	,	,	PUNCT
cana-2486	57	20	if	if	SCONJ
cana-2486	57	21	for	for	ADP
cana-2486	57	22	every	every	DET
cana-2486	57	23	element	element	NOUN
cana-2486	57	24	a	a	PRON
cana-2486	57	25	of	of	ADP
cana-2486	57	26	l	l	NOUN
cana-2486	57	27	such	such	ADJ
cana-2486	57	28	that	that	SCONJ
cana-2486	57	29	a	a	DET
cana-2486	57	30	≤	≤	ADJ
cana-2486	57	31	c	c	NOUN
cana-2486	57	32	there	there	PRON
cana-2486	57	33	exists	exist	VERB
cana-2486	57	34	an	an	DET
cana-2486	57	35	element	element	NOUN
cana-2486	57	36	d	d	PROPN
cana-2486	57	37	∈	∈	PROPN
cana-2486	57	38	l	l	NOUN
cana-2486	57	39	such	such	ADJ
cana-2486	57	40	that	that	SCONJ
cana-2486	57	41	a	a	DET
cana-2486	57	42	=	=	X
cana-2486	57	43	cd	cd	NOUN
cana-2486	57	44	.	.	PUNCT
cana-2486	58	1	[	[	X
cana-2486	58	2	5	5	NUM
cana-2486	58	3	]	]	PUNCT
cana-2486	58	4	.	.	PUNCT
cana-2486	59	1	definition	definition	NOUN
cana-2486	59	2	2.2	2.2	NUM
cana-2486	59	3	.	.	PUNCT
cana-2486	60	1	[	[	X
cana-2486	60	2	5	5	X
cana-2486	60	3	]	]	PUNCT
cana-2486	60	4	let	let	AUX
cana-2486	60	5	l	l	NOUN
cana-2486	60	6	be	be	AUX
cana-2486	60	7	a	a	DET
cana-2486	60	8	multiplicative	multiplicative	ADJ
cana-2486	60	9	lattice	lattice	NOUN
cana-2486	60	10	and	and	CCONJ
cana-2486	60	11	m	m	VERB
cana-2486	60	12	a	a	DET
cana-2486	60	13	lattice	lattice	ADJ
cana-2486	60	14	l	l	NOUN
cana-2486	60	15	-	-	NOUN
cana-2486	60	16	module	module	NOUN
cana-2486	60	17	.	.	PUNCT
cana-2486	61	1	n	n	PRON
cana-2486	61	2	∈	∈	NOUN
cana-2486	61	3	m	m	VERB
cana-2486	61	4	is	be	AUX
cana-2486	61	5	said	say	VERB
cana-2486	61	6	to	to	PART
cana-2486	61	7	be	be	AUX
cana-2486	61	8	a	a	DET
cana-2486	61	9	multiplication	multiplication	NOUN
cana-2486	61	10	element	element	NOUN
cana-2486	61	11	,	,	PUNCT
cana-2486	61	12	if	if	SCONJ
cana-2486	61	13	for	for	ADP
cana-2486	61	14	every	every	DET
cana-2486	61	15	element	element	NOUN
cana-2486	61	16	k	k	PROPN
cana-2486	61	17	of	of	ADP
cana-2486	61	18	m	m	PRON
cana-2486	61	19	such	such	ADJ
cana-2486	61	20	that	that	SCONJ
cana-2486	61	21	k	k	PROPN
cana-2486	61	22	≤	≤	PROPN
cana-2486	61	23	n	n	CCONJ
cana-2486	61	24	there	there	ADV
cana-2486	61	25	exists	exist	VERB
cana-2486	61	26	an	an	DET
cana-2486	61	27	element	element	NOUN
cana-2486	61	28	a	a	DET
cana-2486	61	29	∈	∈	NOUN
cana-2486	61	30	l	l	NOUN
cana-2486	61	31	such	such	ADJ
cana-2486	61	32	that	that	SCONJ
cana-2486	61	33	k	k	PROPN
cana-2486	61	34	=	=	PUNCT
cana-2486	61	35	an	an	PROPN
cana-2486	61	36	.	.	PUNCT
cana-2486	61	37	definition	definition	NOUN
cana-2486	61	38	2.3	2.3	NUM
cana-2486	61	39	.	.	PUNCT
cana-2486	62	1	let	let	VERB
cana-2486	62	2	l	l	NOUN
cana-2486	62	3	be	be	AUX
cana-2486	62	4	a	a	DET
cana-2486	62	5	multiplicative	multiplicative	ADJ
cana-2486	62	6	lattice	lattice	NOUN
cana-2486	62	7	and	and	CCONJ
cana-2486	62	8	m	m	VERB
cana-2486	62	9	a	a	DET
cana-2486	62	10	lattice	lattice	ADJ
cana-2486	62	11	l	l	NOUN
cana-2486	62	12	-	-	NOUN
cana-2486	62	13	module	module	NOUN
cana-2486	62	14	.	.	PUNCT
cana-2486	63	1	n	n	PRON
cana-2486	63	2	∈	∈	NOUN
cana-2486	63	3	m	m	VERB
cana-2486	63	4	is	be	AUX
cana-2486	63	5	said	say	VERB
cana-2486	63	6	to	to	PART
cana-2486	63	7	be	be	AUX
cana-2486	63	8	a	a	DET
cana-2486	63	9	idempotent	idempotent	ADJ
cana-2486	63	10	element	element	NOUN
cana-2486	63	11	in	in	ADP
cana-2486	63	12	m	m	PROPN
cana-2486	63	13	,	,	PUNCT
cana-2486	63	14	if	if	SCONJ
cana-2486	63	15	n	n	ADV
cana-2486	63	16	=	=	SYM
cana-2486	63	17	(	(	PUNCT
cana-2486	63	18	n	n	NOUN
cana-2486	63	19	:	:	PUNCT
cana-2486	63	20	1	1	NUM
cana-2486	63	21	m	m	NOUN
cana-2486	63	22	)	)	PUNCT
cana-2486	63	23	n.	n.	NOUN
cana-2486	63	24	proposition	proposition	NOUN
cana-2486	63	25	2.4	2.4	NUM
cana-2486	63	26	.	.	PUNCT
cana-2486	64	1	let	let	VERB
cana-2486	64	2	l	l	NOUN
cana-2486	64	3	be	be	AUX
cana-2486	64	4	a	a	DET
cana-2486	64	5	cg	cg	NOUN
cana-2486	64	6	-	-	PUNCT
cana-2486	64	7	lattice	lattice	NOUN
cana-2486	64	8	,	,	PUNCT
cana-2486	64	9	m	m	AUX
cana-2486	64	10	be	be	VERB
cana-2486	64	11	a	a	DET
cana-2486	64	12	nonzero	nonzero	ADJ
cana-2486	64	13	l	l	ADJ
cana-2486	64	14	-	-	PUNCT
cana-2486	64	15	lattice	lattice	NOUN
cana-2486	64	16	module	module	NOUN
cana-2486	64	17	and	and	CCONJ
cana-2486	64	18	0	0	NUM
cana-2486	64	19	m	m	NOUN
cana-2486	64	20	≠	≠	NOUN
cana-2486	64	21	n	n	NUM
cana-2486	64	22	is	be	AUX
cana-2486	64	23	pure	pure	ADJ
cana-2486	64	24	element	element	NOUN
cana-2486	64	25	of	of	ADP
cana-2486	64	26	m.	m.	NOUN
cana-2486	64	27	if	if	SCONJ
cana-2486	64	28	m	m	PROPN
cana-2486	64	29	is	be	AUX
cana-2486	64	30	p	p	ADJ
cana-2486	64	31	-	-	PUNCT
cana-2486	64	32	secondary	secondary	ADJ
cana-2486	64	33	lattice	lattice	NOUN
cana-2486	64	34	module	module	NOUN
cana-2486	64	35	,	,	PUNCT
cana-2486	64	36	then	then	ADV
cana-2486	64	37	[	[	X
cana-2486	64	38	n	n	CCONJ
cana-2486	64	39	,	,	PUNCT
cana-2486	64	40	1	1	NUM
cana-2486	64	41	m	m	NOUN
cana-2486	64	42	]	]	PUNCT
cana-2486	64	43	and	and	CCONJ
cana-2486	64	44	[	[	X
cana-2486	64	45	0	0	NUM
cana-2486	64	46	m	m	NOUN
cana-2486	64	47	,	,	PUNCT
cana-2486	64	48	n	n	CCONJ
cana-2486	64	49	]	]	X
cana-2486	64	50	are	be	AUX
cana-2486	64	51	both	both	DET
cana-2486	64	52	p	p	ADJ
cana-2486	64	53	-	-	PUNCT
cana-2486	64	54	secondary	secondary	ADJ
cana-2486	64	55	lattice	lattice	NOUN
cana-2486	64	56	modules	module	NOUN
cana-2486	64	57	.	.	PUNCT
cana-2486	65	1	proof	proof	NOUN
cana-2486	65	2	.	.	PUNCT
cana-2486	66	1	see	see	VERB
cana-2486	66	2	[	[	X
cana-2486	66	3	6	6	NUM
cana-2486	66	4	]	]	PUNCT
cana-2486	66	5	,	,	PUNCT
cana-2486	66	6	proposition	proposition	NOUN
cana-2486	66	7	13	13	NUM
cana-2486	66	8	.	.	PUNCT
cana-2486	67	1	proposition	proposition	NOUN
cana-2486	67	2	2.5	2.5	NUM
cana-2486	67	3	.	.	PUNCT
cana-2486	68	1	let	let	VERB
cana-2486	68	2	l	l	NOUN
cana-2486	68	3	be	be	AUX
cana-2486	68	4	a	a	DET
cana-2486	68	5	domain	domain	NOUN
cana-2486	68	6	.	.	PUNCT
cana-2486	69	1	if	if	SCONJ
cana-2486	69	2	m	m	NOUN
cana-2486	69	3	is	be	AUX
cana-2486	69	4	a	a	DET
cana-2486	69	5	multiplication	multiplication	NOUN
cana-2486	69	6	second	second	ADJ
cana-2486	69	7	l	l	NOUN
cana-2486	69	8	-	-	NOUN
cana-2486	69	9	module	module	NOUN
cana-2486	69	10	,	,	PUNCT
cana-2486	69	11	then	then	ADV
cana-2486	69	12	every	every	DET
cana-2486	69	13	element	element	NOUN
cana-2486	69	14	in	in	ADP
cana-2486	69	15	m	m	PROPN
cana-2486	69	16	is	be	AUX
cana-2486	69	17	pure	pure	ADJ
cana-2486	69	18	.	.	PUNCT
cana-2486	70	1	proof	proof	NOUN
cana-2486	70	2	.	.	PUNCT
cana-2486	71	1	let	let	VERB
cana-2486	71	2	n	n	PRON
cana-2486	71	3	be	be	AUX
cana-2486	71	4	any	any	DET
cana-2486	71	5	element	element	NOUN
cana-2486	71	6	of	of	ADP
cana-2486	71	7	m.	m.	NOUN
cana-2486	71	8	since	since	SCONJ
cana-2486	71	9	m	m	PROPN
cana-2486	71	10	is	be	AUX
cana-2486	71	11	a	a	DET
cana-2486	71	12	multiplication	multiplication	NOUN
cana-2486	71	13	second	second	ADJ
cana-2486	71	14	l	l	NOUN
cana-2486	71	15	-	-	NOUN
cana-2486	71	16	module	module	NOUN
cana-2486	71	17	,	,	PUNCT
cana-2486	71	18	so	so	SCONJ
cana-2486	71	19	m	m	NOUN
cana-2486	71	20	is	be	AUX
cana-2486	71	21	either	either	CCONJ
cana-2486	71	22	divisible	divisible	ADJ
cana-2486	71	23	or	or	CCONJ
cana-2486	71	24	torsion	torsion	NOUN
cana-2486	72	1	[	[	X
cana-2486	72	2	6	6	NUM
cana-2486	72	3	]	]	PUNCT
cana-2486	72	4	.	.	PUNCT
cana-2486	73	1	if	if	SCONJ
cana-2486	73	2	m	m	NOUN
cana-2486	73	3	is	be	AUX
cana-2486	73	4	divisible	divisible	ADJ
cana-2486	73	5	,	,	PUNCT
cana-2486	73	6	then	then	ADV
cana-2486	73	7	a1	a1	PROPN
cana-2486	73	8	m	m	NOUN
cana-2486	73	9	=	=	NOUN
cana-2486	73	10	1	1	NUM
cana-2486	73	11	m	m	VERB
cana-2486	73	12	,	,	PUNCT
cana-2486	73	13	for	for	ADP
cana-2486	73	14	every	every	DET
cana-2486	73	15	0l	0l	NUM
cana-2486	73	16	≠	≠	PROPN
cana-2486	73	17	a	a	DET
cana-2486	73	18	∈	∈	PROPN
cana-2486	73	19	l.	l.	NOUN
cana-2486	74	1	so	so	SCONJ
cana-2486	74	2	an	an	DET
cana-2486	74	3	=	=	NOUN
cana-2486	74	4	n	n	NOUN
cana-2486	74	5	=	=	SYM
cana-2486	74	6	n	n	CCONJ
cana-2486	74	7	∧	∧	PROPN
cana-2486	74	8	a1	a1	PROPN
cana-2486	74	9	m	m	PROPN
cana-2486	74	10	,	,	PUNCT
cana-2486	74	11	since	since	SCONJ
cana-2486	74	12	m	m	PROPN
cana-2486	74	13	is	be	AUX
cana-2486	74	14	multiplication	multiplication	NOUN
cana-2486	74	15	l	l	NOUN
cana-2486	74	16	-	-	NOUN
cana-2486	74	17	module	module	NOUN
cana-2486	74	18	.	.	PUNCT
cana-2486	75	1	if	if	SCONJ
cana-2486	75	2	m	m	NOUN
cana-2486	75	3	is	be	AUX
cana-2486	75	4	torsion	torsion	NOUN
cana-2486	75	5	,	,	PUNCT
cana-2486	75	6	then	then	ADV
cana-2486	75	7	a1	a1	PROPN
cana-2486	75	8	m	m	NOUN
cana-2486	75	9	=	=	NOUN
cana-2486	75	10	0	0	NUM
cana-2486	75	11	m	m	PROPN
cana-2486	75	12	,	,	PUNCT
cana-2486	75	13	for	for	ADP
cana-2486	75	14	every	every	DET
cana-2486	75	15	0l	0l	NUM
cana-2486	75	16	≠	≠	PROPN
cana-2486	75	17	a	a	DET
cana-2486	75	18	∈	∈	PROPN
cana-2486	75	19	l.	l.	NOUN
cana-2486	76	1	so	so	ADV
cana-2486	76	2	,	,	PUNCT
cana-2486	76	3	an	an	DET
cana-2486	76	4	=	=	NOUN
cana-2486	76	5	0	0	NUM
cana-2486	76	6	m	m	NOUN
cana-2486	76	7	=	=	SYM
cana-2486	76	8	n	n	CCONJ
cana-2486	76	9	∧	∧	PROPN
cana-2486	76	10	a1	a1	PROPN
cana-2486	76	11	m.	m.	NOUN
cana-2486	76	12	lemma	lemma	PROPN
cana-2486	76	13	2.6	2.6	NUM
cana-2486	76	14	.	.	PUNCT
cana-2486	77	1	let	let	VERB
cana-2486	77	2	m	m	PRON
cana-2486	77	3	be	be	AUX
cana-2486	77	4	a	a	DET
cana-2486	77	5	multiplication	multiplication	NOUN
cana-2486	77	6	l−module	l−module	NOUN
cana-2486	77	7	,	,	PUNCT
cana-2486	77	8	and	and	CCONJ
cana-2486	77	9	0	0	NUM
cana-2486	77	10	m	m	NOUN
cana-2486	77	11	≠	≠	NOUN
cana-2486	77	12	n	n	VERB
cana-2486	77	13	be	be	VERB
cana-2486	77	14	a	a	DET
cana-2486	77	15	pure	pure	ADJ
cana-2486	77	16	element	element	NOUN
cana-2486	77	17	of	of	ADP
cana-2486	77	18	m.	m.	NOUN
cana-2486	77	19	then	then	ADV
cana-2486	77	20	m	m	VERB
cana-2486	77	21	is	be	AUX
cana-2486	77	22	a	a	DET
cana-2486	77	23	p	p	ADJ
cana-2486	77	24	-	-	PUNCT
cana-2486	77	25	second	second	ADJ
cana-2486	77	26	lattice	lattice	NOUN
cana-2486	77	27	module	module	NOUN
cana-2486	77	28	if	if	SCONJ
cana-2486	77	29	and	and	CCONJ
cana-2486	77	30	only	only	ADV
cana-2486	77	31	if	if	SCONJ
cana-2486	77	32	[	[	X
cana-2486	77	33	0	0	NUM
cana-2486	77	34	m	m	NOUN
cana-2486	77	35	,	,	PUNCT
cana-2486	77	36	n	n	CCONJ
cana-2486	77	37	]	]	PUNCT
cana-2486	77	38	and	and	CCONJ
cana-2486	77	39	[	[	X
cana-2486	77	40	n	n	CCONJ
cana-2486	77	41	,	,	PUNCT
cana-2486	77	42	1	1	NUM
cana-2486	77	43	m	m	NOUN
cana-2486	77	44	]	]	PUNCT
cana-2486	77	45	are	be	AUX
cana-2486	77	46	both	both	PRON
cana-2486	77	47	p	p	ADJ
cana-2486	77	48	-	-	PUNCT
cana-2486	77	49	second	second	ADJ
cana-2486	77	50	lattice	lattice	NOUN
cana-2486	77	51	modules	module	NOUN
cana-2486	77	52	.	.	PUNCT
cana-2486	78	1	proof	proof	NOUN
cana-2486	78	2	.	.	PUNCT
cana-2486	79	1	see	see	VERB
cana-2486	79	2	[	[	X
cana-2486	79	3	6	6	NUM
cana-2486	79	4	]	]	PUNCT
cana-2486	79	5	,	,	PUNCT
cana-2486	79	6	proposition	proposition	NOUN
cana-2486	79	7	14	14	NUM
cana-2486	79	8	.	.	PUNCT
cana-2486	80	1	lemma	lemma	PROPN
cana-2486	80	2	2.7	2.7	NUM
cana-2486	80	3	.	.	PUNCT
cana-2486	81	1	let	let	VERB
cana-2486	81	2	m	m	PRON
cana-2486	81	3	be	be	AUX
cana-2486	81	4	a	a	DET
cana-2486	81	5	faithful	faithful	ADJ
cana-2486	81	6	multiplication	multiplication	NOUN
cana-2486	81	7	l−module	l−module	NOUN
cana-2486	81	8	.	.	PUNCT
cana-2486	82	1	if	if	SCONJ
cana-2486	82	2	n	n	PRON
cana-2486	82	3	is	be	AUX
cana-2486	82	4	a	a	DET
cana-2486	82	5	pure	pure	ADJ
cana-2486	82	6	element	element	NOUN
cana-2486	82	7	of	of	ADP
cana-2486	82	8	m	m	PROPN
cana-2486	82	9	,	,	PUNCT
cana-2486	82	10	then	then	ADV
cana-2486	82	11	n	n	PROPN
cana-2486	82	12	is	be	AUX
cana-2486	82	13	multiplication	multiplication	NOUN
cana-2486	82	14	and	and	CCONJ
cana-2486	82	15	is	be	AUX
cana-2486	82	16	idempotent	idempotent	ADJ
cana-2486	82	17	in	in	ADP
cana-2486	82	18	m.	m.	NOUN
cana-2486	82	19	proof	proof	NOUN
cana-2486	82	20	.	.	PUNCT
cana-2486	83	1	let	let	VERB
cana-2486	83	2	k	k	PRON
cana-2486	83	3	be	be	AUX
cana-2486	83	4	a	a	DET
cana-2486	83	5	element	element	NOUN
cana-2486	83	6	of	of	ADP
cana-2486	83	7	m.	m.	NOUN
cana-2486	83	8	then	then	ADV
cana-2486	83	9	k	k	PROPN
cana-2486	83	10	=	=	PUNCT
cana-2486	83	11	(	(	PUNCT
cana-2486	83	12	k	k	X
cana-2486	83	13	:	:	PUNCT
cana-2486	83	14	1	1	NUM
cana-2486	83	15	m	m	NOUN
cana-2486	83	16	)	)	PUNCT
cana-2486	83	17	1	1	NUM
cana-2486	83	18	m	m	NOUN
cana-2486	83	19	.	.	PUNCT
cana-2486	84	1	since	since	SCONJ
cana-2486	84	2	n	n	NUM
cana-2486	84	3	is	be	AUX
cana-2486	84	4	a	a	DET
cana-2486	84	5	pure	pure	ADJ
cana-2486	84	6	element	element	NOUN
cana-2486	84	7	of	of	ADP
cana-2486	84	8	m	m	PROPN
cana-2486	84	9	,	,	PUNCT
cana-2486	84	10	we	we	PRON
cana-2486	84	11	have	have	VERB
cana-2486	84	12	,	,	PUNCT
cana-2486	84	13	(	(	PUNCT
cana-2486	84	14	k	k	NOUN
cana-2486	84	15	:	:	PUNCT
cana-2486	84	16	n	n	X
cana-2486	84	17	)	)	PUNCT
cana-2486	84	18	n	n	NOUN
cana-2486	84	19	=	=	SYM
cana-2486	84	20	n	n	PRON
cana-2486	84	21	∧	∧	PROPN
cana-2486	84	22	(	(	PUNCT
cana-2486	84	23	k	k	NOUN
cana-2486	84	24	:	:	PUNCT
cana-2486	84	25	n	n	X
cana-2486	84	26	)	)	PUNCT
cana-2486	84	27	1	1	NUM
cana-2486	84	28	m	m	NOUN
cana-2486	84	29	≥	≥	NOUN
cana-2486	84	30	n	n	PRON
cana-2486	84	31	∧	∧	PROPN
cana-2486	84	32	(	(	PUNCT
cana-2486	84	33	k	k	NOUN
cana-2486	84	34	:	:	PUNCT
cana-2486	84	35	1	1	NUM
cana-2486	84	36	m	m	NOUN
cana-2486	84	37	)	)	PUNCT
cana-2486	84	38	1	1	NUM
cana-2486	84	39	m	m	NOUN
cana-2486	84	40	=	=	SYM
cana-2486	84	41	n	n	PROPN
cana-2486	84	42	∧	∧	PROPN
cana-2486	84	43	k	k	PROPN
cana-2486	84	44	≥	≥	X
cana-2486	84	45	(	(	PUNCT
cana-2486	84	46	k	k	NOUN
cana-2486	84	47	:	:	PUNCT
cana-2486	84	48	n	n	X
cana-2486	84	49	)	)	PUNCT
cana-2486	84	50	n	n	CCONJ
cana-2486	84	51	,	,	PUNCT
cana-2486	84	52	so	so	SCONJ
cana-2486	84	53	that	that	SCONJ
cana-2486	84	54	(	(	PUNCT
cana-2486	84	55	k	k	NOUN
cana-2486	84	56	:	:	PUNCT
cana-2486	84	57	n	n	X
cana-2486	84	58	)	)	PUNCT
cana-2486	84	59	n	n	NOUN
cana-2486	84	60	=	=	SYM
cana-2486	85	1	k	k	PROPN
cana-2486	85	2	∧	∧	PROPN
cana-2486	85	3	n	n	PRON
cana-2486	85	4	⇒	⇒	NOUN
cana-2486	85	5	n	n	CCONJ
cana-2486	85	6	a	a	DET
cana-2486	85	7	weak	weak	ADJ
cana-2486	85	8	meet	meet	ADJ
cana-2486	85	9	principal	principal	ADJ
cana-2486	85	10	element	element	NOUN
cana-2486	85	11	in	in	ADP
cana-2486	85	12	m	m	PROPN
cana-2486	85	13	,	,	PUNCT
cana-2486	85	14	and	and	CCONJ
cana-2486	85	15	n	n	PRON
cana-2486	85	16	is	be	AUX
cana-2486	85	17	multiplication	multiplication	NOUN
cana-2486	85	18	.	.	PUNCT
cana-2486	86	1	since	since	SCONJ
cana-2486	86	2	n	n	PRON
cana-2486	86	3	is	be	AUX
cana-2486	86	4	pure	pure	ADJ
cana-2486	86	5	in	in	ADP
cana-2486	86	6	m	m	PROPN
cana-2486	86	7	,	,	PUNCT
cana-2486	86	8	we	we	PRON
cana-2486	86	9	have	have	VERB
cana-2486	86	10	that	that	PRON
cana-2486	86	11	(	(	PUNCT
cana-2486	86	12	n	n	X
cana-2486	86	13	:	:	PUNCT
cana-2486	86	14	1	1	NUM
cana-2486	86	15	m	m	NOUN
cana-2486	86	16	)	)	PUNCT
cana-2486	87	1	n	n	NOUN
cana-2486	87	2	=	=	SYM
cana-2486	87	3	n	n	PRON
cana-2486	87	4	∧	∧	PROPN
cana-2486	87	5	(	(	PUNCT
cana-2486	87	6	n	n	NOUN
cana-2486	87	7	:	:	PUNCT
cana-2486	87	8	1	1	NUM
cana-2486	87	9	m	m	NOUN
cana-2486	87	10	)	)	PUNCT
cana-2486	88	1	1	1	NUM
cana-2486	88	2	m	m	NOUN
cana-2486	88	3	=	=	SYM
cana-2486	88	4	n	n	CCONJ
cana-2486	88	5	,	,	PUNCT
cana-2486	88	6	and	and	CCONJ
cana-2486	88	7	hence	hence	ADV
cana-2486	88	8	n	n	PRON
cana-2486	88	9	is	be	AUX
cana-2486	88	10	idempotent	idempotent	ADJ
cana-2486	88	11	in	in	ADP
cana-2486	88	12	m	m	PROPN
cana-2486	88	13	.	.	PUNCT
cana-2486	89	1	lemma	lemma	PROPN
cana-2486	89	2	2.8	2.8	NUM
cana-2486	89	3	.	.	PUNCT
cana-2486	90	1	let	let	VERB
cana-2486	90	2	m	m	PRON
cana-2486	90	3	be	be	AUX
cana-2486	90	4	a	a	DET
cana-2486	90	5	multiplication	multiplication	NOUN
cana-2486	90	6	l−module	l−module	NOUN
cana-2486	90	7	.	.	PUNCT
cana-2486	91	1	if	if	SCONJ
cana-2486	91	2	n	n	PRON
cana-2486	91	3	is	be	AUX
cana-2486	91	4	a	a	DET
cana-2486	91	5	pure	pure	ADJ
cana-2486	91	6	element	element	NOUN
cana-2486	91	7	of	of	ADP
cana-2486	91	8	m	m	PROPN
cana-2486	91	9	,	,	PUNCT
cana-2486	91	10	then	then	ADV
cana-2486	91	11	k	k	PROPN
cana-2486	91	12	=	=	PUNCT
cana-2486	91	13	(	(	PUNCT
cana-2486	91	14	n	n	NOUN
cana-2486	91	15	:	:	PUNCT
cana-2486	91	16	1	1	NUM
cana-2486	91	17	m	m	NOUN
cana-2486	91	18	)	)	PUNCT
cana-2486	92	1	k	k	NOUN
cana-2486	92	2	and	and	CCONJ
cana-2486	92	3	(	(	PUNCT
cana-2486	92	4	k	k	NOUN
cana-2486	92	5	:	:	PUNCT
cana-2486	92	6	n	n	X
cana-2486	92	7	)	)	PUNCT
cana-2486	92	8	n	n	NOUN
cana-2486	92	9	=	=	SYM
cana-2486	92	10	(	(	PUNCT
cana-2486	92	11	k	k	NOUN
cana-2486	92	12	:	:	PUNCT
cana-2486	92	13	1	1	NUM
cana-2486	92	14	m	m	NOUN
cana-2486	92	15	)	)	PUNCT
cana-2486	92	16	n	n	CCONJ
cana-2486	92	17	,	,	PUNCT
cana-2486	92	18	for	for	ADP
cana-2486	92	19	each	each	DET
cana-2486	92	20	k	k	NOUN
cana-2486	92	21	of	of	ADP
cana-2486	92	22	m.	m.	NOUN
cana-2486	92	23	proof	proof	NOUN
cana-2486	92	24	.	.	PUNCT
cana-2486	93	1	by	by	ADP
cana-2486	93	2	lemma	lemma	PROPN
cana-2486	93	3	2.7	2.7	NUM
cana-2486	93	4	,	,	PUNCT
cana-2486	93	5	n	n	X
cana-2486	93	6	is	be	AUX
cana-2486	93	7	multiplication	multiplication	NOUN
cana-2486	93	8	and	and	CCONJ
cana-2486	93	9	is	be	AUX
cana-2486	93	10	idempotent	idempotent	ADJ
cana-2486	93	11	in	in	ADP
cana-2486	93	12	m.	m.	NOUN
cana-2486	93	13	let	let	VERB
cana-2486	93	14	k	k	PROPN
cana-2486	93	15	≤	≤	PROPN
cana-2486	93	16	n	n	CCONJ
cana-2486	93	17	,	,	PUNCT
cana-2486	93	18	then	then	ADV
cana-2486	93	19	k	k	PROPN
cana-2486	93	20	=	=	PUNCT
cana-2486	93	21	(	(	PUNCT
cana-2486	93	22	k	k	NOUN
cana-2486	93	23	:	:	PUNCT
cana-2486	93	24	n	n	X
cana-2486	93	25	)	)	PUNCT
cana-2486	93	26	n	n	NOUN
cana-2486	94	1	=	=	SYM
cana-2486	94	2	(	(	PUNCT
cana-2486	94	3	k	k	NOUN
cana-2486	94	4	:	:	PUNCT
cana-2486	94	5	n	n	X
cana-2486	94	6	)	)	PUNCT
cana-2486	94	7	(	(	PUNCT
cana-2486	94	8	n	n	X
cana-2486	94	9	:	:	PUNCT
cana-2486	94	10	1	1	NUM
cana-2486	94	11	m	m	NOUN
cana-2486	94	12	)	)	PUNCT
cana-2486	94	13	n	n	NOUN
cana-2486	94	14	=	=	SYM
cana-2486	94	15	(	(	PUNCT
cana-2486	94	16	n	n	NOUN
cana-2486	94	17	:	:	PUNCT
cana-2486	94	18	1	1	NUM
cana-2486	94	19	m	m	NOUN
cana-2486	94	20	)	)	PUNCT
cana-2486	94	21	k.	k.	PROPN
cana-2486	95	1	also	also	ADV
cana-2486	95	2	,	,	PUNCT
cana-2486	95	3	for	for	ADP
cana-2486	95	4	k	k	PROPN
cana-2486	95	5	≤	≤	PROPN
cana-2486	95	6	n	n	PRON
cana-2486	95	7	,	,	PUNCT
cana-2486	95	8	(	(	PUNCT
cana-2486	95	9	k	k	NOUN
cana-2486	95	10	:	:	PUNCT
cana-2486	95	11	n	n	X
cana-2486	95	12	)	)	PUNCT
cana-2486	95	13	n	n	NOUN
cana-2486	95	14	=	=	SYM
cana-2486	95	15	(	(	PUNCT
cana-2486	95	16	k	k	NOUN
cana-2486	95	17	:	:	PUNCT
cana-2486	95	18	n	n	X
cana-2486	95	19	)	)	PUNCT
cana-2486	95	20	(	(	PUNCT
cana-2486	95	21	n	n	X
cana-2486	95	22	:	:	PUNCT
cana-2486	95	23	1	1	NUM
cana-2486	95	24	m	m	NOUN
cana-2486	95	25	)	)	PUNCT
cana-2486	95	26	n	n	NOUN
cana-2486	95	27	≤	≤	NOUN
cana-2486	95	28	(	(	PUNCT
cana-2486	95	29	k	k	NOUN
cana-2486	95	30	:	:	PUNCT
cana-2486	95	31	1	1	NUM
cana-2486	95	32	m	m	NOUN
cana-2486	95	33	)	)	PUNCT
cana-2486	95	34	n	n	NOUN
cana-2486	95	35	≤	≤	NOUN
cana-2486	95	36	(	(	PUNCT
cana-2486	95	37	k	k	NOUN
cana-2486	95	38	:	:	PUNCT
cana-2486	95	39	n	n	X
cana-2486	95	40	)	)	PUNCT
cana-2486	95	41	n	n	CCONJ
cana-2486	95	42	,	,	PUNCT
cana-2486	95	43	so	so	SCONJ
cana-2486	95	44	that	that	SCONJ
cana-2486	95	45	(	(	PUNCT
cana-2486	95	46	k	k	NOUN
cana-2486	95	47	:	:	PUNCT
cana-2486	95	48	n	n	X
cana-2486	95	49	)	)	PUNCT
cana-2486	95	50	n	n	NOUN
cana-2486	95	51	=	=	SYM
cana-2486	95	52	(	(	PUNCT
cana-2486	95	53	k	k	NOUN
cana-2486	95	54	:	:	PUNCT
cana-2486	95	55	1	1	NUM
cana-2486	95	56	m	m	NOUN
cana-2486	95	57	)	)	PUNCT
cana-2486	95	58	n	n	NOUN
cana-2486	95	59	.	.	PUNCT
cana-2486	96	1	lemma	lemma	PROPN
cana-2486	96	2	2.9	2.9	NUM
cana-2486	96	3	.	.	PUNCT
cana-2486	97	1	let	let	VERB
cana-2486	97	2	m	m	PRON
cana-2486	97	3	be	be	AUX
cana-2486	97	4	a	a	DET
cana-2486	97	5	faithful	faithful	ADJ
cana-2486	97	6	multiplication	multiplication	NOUN
cana-2486	97	7	l−module	l−module	NOUN
cana-2486	97	8	.	.	PUNCT
cana-2486	98	1	if	if	SCONJ
cana-2486	98	2	n	n	PRON
cana-2486	98	3	is	be	AUX
cana-2486	98	4	a	a	DET
cana-2486	98	5	pure	pure	ADJ
cana-2486	98	6	element	element	NOUN
cana-2486	98	7	of	of	ADP
cana-2486	98	8	m	m	PRON
cana-2486	98	9	,	,	PUNCT
cana-2486	98	10	then	then	ADV
cana-2486	98	11	a(n	a(n	ADV
cana-2486	98	12	:	:	PUNCT
cana-2486	98	13	1	1	NUM
cana-2486	98	14	m	m	NOUN
cana-2486	98	15	)	)	PUNCT
cana-2486	98	16	=	=	PUNCT
cana-2486	98	17	a	a	DET
cana-2486	98	18	∧	∧	PROPN
cana-2486	98	19	(	(	PUNCT
cana-2486	98	20	n	n	NOUN
cana-2486	98	21	:	:	PUNCT
cana-2486	98	22	1	1	NUM
cana-2486	98	23	m	m	NOUN
cana-2486	98	24	)	)	PUNCT
cana-2486	98	25	,	,	PUNCT
cana-2486	98	26	for	for	SCONJ
cana-2486	98	27	every	every	DET
cana-2486	98	28	a	a	PRON
cana-2486	98	29	in	in	ADP
cana-2486	98	30	l.	l.	PROPN
cana-2486	98	31	communications	communication	NOUN
cana-2486	98	32	on	on	ADP
cana-2486	98	33	applied	apply	VERB
cana-2486	98	34	nonlinear	nonlinear	ADJ
cana-2486	98	35	analysis	analysis	NOUN
cana-2486	98	36	issn	issn	NOUN
cana-2486	98	37	:	:	PUNCT
cana-2486	98	38	1074	1074	NUM
cana-2486	98	39	-	-	PUNCT
cana-2486	98	40	133x	133x	NUM
cana-2486	98	41	vol	vol	NOUN
cana-2486	98	42	.	.	PROPN
cana-2486	98	43	32	32	NUM
cana-2486	99	1	no	no	INTJ
cana-2486	99	2	.	.	PUNCT
cana-2486	100	1	2s	2s	NUM
cana-2486	100	2	(	(	PUNCT
cana-2486	100	3	2024	2024	NUM
cana-2486	100	4	)	)	PUNCT
cana-2486	100	5	541	541	NUM
cana-2486	100	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2486	100	7	proof	proof	NOUN
cana-2486	100	8	.	.	PUNCT
cana-2486	101	1	since	since	SCONJ
cana-2486	101	2	n	n	NUM
cana-2486	101	3	is	be	AUX
cana-2486	101	4	a	a	DET
cana-2486	101	5	pure	pure	ADJ
cana-2486	101	6	element	element	NOUN
cana-2486	101	7	of	of	ADP
cana-2486	101	8	m	m	PROPN
cana-2486	101	9	,	,	PUNCT
cana-2486	101	10	so	so	ADV
cana-2486	101	11	an	an	DET
cana-2486	101	12	=	=	PUNCT
cana-2486	101	13	n	n	CCONJ
cana-2486	101	14	∧	∧	PROPN
cana-2486	101	15	a1	a1	NOUN
cana-2486	101	16	m.	m.	NOUN
cana-2486	101	17	hence	hence	ADV
cana-2486	101	18	(	(	PUNCT
cana-2486	101	19	an	an	DET
cana-2486	101	20	:	:	SYM
cana-2486	101	21	1	1	NUM
cana-2486	101	22	m	m	NOUN
cana-2486	101	23	)	)	PUNCT
cana-2486	102	1	=	=	SYM
cana-2486	102	2	(	(	PUNCT
cana-2486	102	3	(	(	PUNCT
cana-2486	102	4	n	n	CCONJ
cana-2486	102	5	∧	∧	PROPN
cana-2486	102	6	a1	a1	PROPN
cana-2486	102	7	m	m	NOUN
cana-2486	102	8	)	)	PUNCT
cana-2486	102	9	:	:	PUNCT
cana-2486	103	1	1	1	NUM
cana-2486	103	2	m	m	NOUN
cana-2486	103	3	)	)	PUNCT
cana-2486	104	1	=	=	SYM
cana-2486	104	2	(	(	PUNCT
cana-2486	104	3	n	n	X
cana-2486	104	4	:	:	PUNCT
cana-2486	104	5	1	1	NUM
cana-2486	104	6	m	m	NOUN
cana-2486	104	7	)	)	PUNCT
cana-2486	104	8	∧	∧	PROPN
cana-2486	104	9	(	(	PUNCT
cana-2486	104	10	a1	a1	PROPN
cana-2486	104	11	m	m	NOUN
cana-2486	104	12	:	:	PUNCT
cana-2486	104	13	1	1	NUM
cana-2486	104	14	m	m	NOUN
cana-2486	104	15	)	)	PUNCT
cana-2486	104	16	=	=	SYM
cana-2486	104	17	(	(	PUNCT
cana-2486	104	18	n	n	X
cana-2486	104	19	:	:	PUNCT
cana-2486	104	20	1	1	NUM
cana-2486	104	21	m	m	NOUN
cana-2486	104	22	)	)	PUNCT
cana-2486	105	1	∧	∧	NOUN
cana-2486	105	2	a.	a.	NOUN
cana-2486	105	3	we	we	PRON
cana-2486	105	4	need	need	VERB
cana-2486	105	5	to	to	PART
cana-2486	105	6	show	show	VERB
cana-2486	105	7	that	that	SCONJ
cana-2486	105	8	(	(	PUNCT
cana-2486	105	9	an	an	DET
cana-2486	105	10	:	:	SYM
cana-2486	105	11	1	1	NUM
cana-2486	105	12	m	m	NOUN
cana-2486	105	13	)	)	PUNCT
cana-2486	106	1	=	=	PUNCT
cana-2486	106	2	a(n	a(n	ADV
cana-2486	106	3	:	:	PUNCT
cana-2486	106	4	1	1	NUM
cana-2486	106	5	m	m	NOUN
cana-2486	106	6	)	)	PUNCT
cana-2486	106	7	.	.	PUNCT
cana-2486	107	1	obviously	obviously	ADV
cana-2486	107	2	,	,	PUNCT
cana-2486	107	3	a(n	a(n	ADV
cana-2486	107	4	:	:	PUNCT
cana-2486	107	5	1	1	NUM
cana-2486	107	6	m	m	NOUN
cana-2486	107	7	)	)	PUNCT
cana-2486	107	8	≤	≤	NOUN
cana-2486	107	9	(	(	PUNCT
cana-2486	107	10	an	an	DET
cana-2486	107	11	:	:	SYM
cana-2486	107	12	1	1	NUM
cana-2486	107	13	m	m	NOUN
cana-2486	107	14	)	)	PUNCT
cana-2486	107	15	.	.	PUNCT
cana-2486	108	1	conversely	conversely	ADV
cana-2486	108	2	,	,	PUNCT
cana-2486	108	3	let	let	VERB
cana-2486	108	4	x	x	SYM
cana-2486	108	5	≤	≤	X
cana-2486	108	6	(	(	PUNCT
cana-2486	108	7	an	an	DET
cana-2486	108	8	:	:	SYM
cana-2486	108	9	1	1	NUM
cana-2486	108	10	m	m	NOUN
cana-2486	108	11	)	)	PUNCT
cana-2486	108	12	.	.	PUNCT
cana-2486	109	1	then	then	ADV
cana-2486	109	2	x1	x1	NUM
cana-2486	109	3	m	m	VERB
cana-2486	109	4	≤	≤	NOUN
cana-2486	109	5	an	an	DET
cana-2486	109	6	=	=	NOUN
cana-2486	109	7	a(n	a(n	NOUN
cana-2486	109	8	:	:	PUNCT
cana-2486	109	9	1	1	NUM
cana-2486	109	10	m	m	NOUN
cana-2486	109	11	)	)	PUNCT
cana-2486	109	12	1	1	NUM
cana-2486	109	13	m.	m.	NOUN
cana-2486	109	14	thus	thus	ADV
cana-2486	109	15	x	x	SYM
cana-2486	109	16	≤	≤	PUNCT
cana-2486	109	17	a(n	a(n	ADV
cana-2486	109	18	:	:	PUNCT
cana-2486	109	19	1	1	NUM
cana-2486	109	20	m	m	NOUN
cana-2486	109	21	)	)	PUNCT
cana-2486	109	22	,	,	PUNCT
cana-2486	109	23	and	and	CCONJ
cana-2486	109	24	hence	hence	ADV
cana-2486	109	25	(	(	PUNCT
cana-2486	109	26	an	an	DET
cana-2486	109	27	:	:	SYM
cana-2486	109	28	1	1	NUM
cana-2486	109	29	m	m	NOUN
cana-2486	109	30	)	)	PUNCT
cana-2486	109	31	≤	≤	PUNCT
cana-2486	110	1	a(n	a(n	ADV
cana-2486	110	2	:	:	PUNCT
cana-2486	110	3	1	1	NUM
cana-2486	110	4	m	m	NOUN
cana-2486	110	5	)	)	PUNCT
cana-2486	110	6	.	.	PUNCT
cana-2486	111	1	lemma	lemma	PROPN
cana-2486	111	2	2.10	2.10	NUM
cana-2486	111	3	.	.	PUNCT
cana-2486	112	1	let	let	VERB
cana-2486	112	2	m	m	PRON
cana-2486	112	3	be	be	AUX
cana-2486	112	4	a	a	DET
cana-2486	112	5	faithful	faithful	ADJ
cana-2486	112	6	multiplication	multiplication	NOUN
cana-2486	112	7	l−module	l−module	NOUN
cana-2486	112	8	.	.	PUNCT
cana-2486	113	1	if	if	SCONJ
cana-2486	113	2	a(n	a(n	ADV
cana-2486	113	3	:	:	PUNCT
cana-2486	113	4	1	1	NUM
cana-2486	113	5	m	m	NOUN
cana-2486	113	6	)	)	PUNCT
cana-2486	114	1	=	=	PUNCT
cana-2486	114	2	a	a	DET
cana-2486	114	3	∧	∧	PROPN
cana-2486	114	4	(	(	PUNCT
cana-2486	114	5	n	n	NOUN
cana-2486	114	6	:	:	PUNCT
cana-2486	114	7	1	1	NUM
cana-2486	114	8	m	m	NOUN
cana-2486	114	9	)	)	PUNCT
cana-2486	114	10	,	,	PUNCT
cana-2486	114	11	for	for	ADP
cana-2486	114	12	every	every	DET
cana-2486	114	13	a	a	PRON
cana-2486	114	14	of	of	ADP
cana-2486	114	15	l	l	NOUN
cana-2486	114	16	,	,	PUNCT
cana-2486	114	17	then	then	ADV
cana-2486	114	18	n	n	PROPN
cana-2486	114	19	is	be	AUX
cana-2486	114	20	multiplication	multiplication	NOUN
cana-2486	114	21	and	and	CCONJ
cana-2486	114	22	is	be	AUX
cana-2486	114	23	idempotent	idempotent	ADJ
cana-2486	114	24	in	in	ADP
cana-2486	114	25	m.	m.	NOUN
cana-2486	114	26	proof	proof	NOUN
cana-2486	114	27	.	.	PUNCT
cana-2486	115	1	assume	assume	VERB
cana-2486	115	2	a(n	a(n	ADV
cana-2486	115	3	:	:	PUNCT
cana-2486	115	4	1	1	NUM
cana-2486	115	5	m	m	NOUN
cana-2486	115	6	)	)	PUNCT
cana-2486	116	1	=	=	PUNCT
cana-2486	116	2	a	a	DET
cana-2486	116	3	∧	∧	PROPN
cana-2486	116	4	(	(	PUNCT
cana-2486	116	5	n	n	NOUN
cana-2486	116	6	:	:	PUNCT
cana-2486	116	7	1	1	NUM
cana-2486	116	8	m	m	NOUN
cana-2486	116	9	)	)	PUNCT
cana-2486	116	10	,	,	PUNCT
cana-2486	116	11	for	for	ADP
cana-2486	116	12	all	all	DET
cana-2486	116	13	a	a	PRON
cana-2486	116	14	of	of	ADP
cana-2486	116	15	l.	l.	NOUN
cana-2486	116	16	take	take	VERB
cana-2486	116	17	a	a	DET
cana-2486	116	18	=	=	PUNCT
cana-2486	116	19	(	(	PUNCT
cana-2486	116	20	n	n	NOUN
cana-2486	116	21	:	:	PUNCT
cana-2486	116	22	1	1	NUM
cana-2486	116	23	m	m	NOUN
cana-2486	116	24	)	)	PUNCT
cana-2486	116	25	.	.	PUNCT
cana-2486	117	1	then	then	ADV
cana-2486	117	2	(	(	PUNCT
cana-2486	117	3	n	n	X
cana-2486	117	4	:	:	PUNCT
cana-2486	117	5	1	1	NUM
cana-2486	117	6	m	m	NOUN
cana-2486	117	7	)	)	PUNCT
cana-2486	117	8	2	2	NUM
cana-2486	117	9	=	=	SYM
cana-2486	117	10	(	(	PUNCT
cana-2486	117	11	n	n	NOUN
cana-2486	117	12	:	:	PUNCT
cana-2486	117	13	1	1	NUM
cana-2486	117	14	m	m	NOUN
cana-2486	117	15	)	)	PUNCT
cana-2486	117	16	and	and	CCONJ
cana-2486	117	17	hence	hence	ADV
cana-2486	117	18	(	(	PUNCT
cana-2486	117	19	n	n	X
cana-2486	117	20	:	:	PUNCT
cana-2486	117	21	1	1	NUM
cana-2486	117	22	m	m	NOUN
cana-2486	117	23	)	)	PUNCT
cana-2486	117	24	is	be	AUX
cana-2486	117	25	an	an	DET
cana-2486	117	26	idempotent	idempotent	ADJ
cana-2486	117	27	element	element	NOUN
cana-2486	117	28	of	of	ADP
cana-2486	117	29	l.	l.	PROPN
cana-2486	117	30	hence	hence	ADV
cana-2486	117	31	n	n	PROPN
cana-2486	117	32	=	=	SYM
cana-2486	117	33	(	(	PUNCT
cana-2486	117	34	n	n	X
cana-2486	117	35	:	:	PUNCT
cana-2486	117	36	1	1	NUM
cana-2486	117	37	m	m	NOUN
cana-2486	117	38	)	)	PUNCT
cana-2486	117	39	1	1	NUM
cana-2486	117	40	m	m	NOUN
cana-2486	117	41	=	=	PUNCT
cana-2486	117	42	(	(	PUNCT
cana-2486	117	43	n	n	NOUN
cana-2486	117	44	:	:	PUNCT
cana-2486	117	45	1	1	NUM
cana-2486	117	46	m	m	NOUN
cana-2486	117	47	)	)	PUNCT
cana-2486	117	48	21	21	NUM
cana-2486	117	49	m	m	NOUN
cana-2486	117	50	=	=	PUNCT
cana-2486	117	51	(	(	PUNCT
cana-2486	117	52	n	n	NOUN
cana-2486	117	53	:	:	PUNCT
cana-2486	117	54	1	1	NUM
cana-2486	117	55	m	m	NOUN
cana-2486	117	56	)	)	PUNCT
cana-2486	117	57	(	(	PUNCT
cana-2486	117	58	n	n	X
cana-2486	117	59	:	:	PUNCT
cana-2486	117	60	1	1	NUM
cana-2486	117	61	m	m	NOUN
cana-2486	117	62	)	)	PUNCT
cana-2486	117	63	1	1	NUM
cana-2486	117	64	m	m	NOUN
cana-2486	117	65	=	=	PUNCT
cana-2486	117	66	(	(	PUNCT
cana-2486	117	67	n	n	NUM
cana-2486	117	68	:	:	PUNCT
cana-2486	117	69	1m)n	1m)n	NUM
cana-2486	117	70	,	,	PUNCT
cana-2486	117	71	and	and	CCONJ
cana-2486	117	72	hence	hence	ADV
cana-2486	117	73	n	n	PRON
cana-2486	117	74	is	be	AUX
cana-2486	117	75	idempotent	idempotent	ADJ
cana-2486	117	76	in	in	ADP
cana-2486	117	77	m.	m.	NOUN
cana-2486	117	78	to	to	PART
cana-2486	117	79	prove	prove	VERB
cana-2486	117	80	that	that	SCONJ
cana-2486	117	81	n	n	NOUN
cana-2486	117	82	is	be	AUX
cana-2486	117	83	multiplication	multiplication	NOUN
cana-2486	117	84	,	,	PUNCT
cana-2486	117	85	let	let	VERB
cana-2486	117	86	k	k	PRON
cana-2486	117	87	be	be	AUX
cana-2486	117	88	any	any	DET
cana-2486	117	89	element	element	NOUN
cana-2486	117	90	of	of	ADP
cana-2486	117	91	m.	m.	NOUN
cana-2486	117	92	let	let	VERB
cana-2486	117	93	a	a	DET
cana-2486	117	94	=	=	X
cana-2486	117	95	(	(	PUNCT
cana-2486	117	96	k	k	NOUN
cana-2486	117	97	:	:	PUNCT
cana-2486	117	98	1	1	NUM
cana-2486	117	99	m	m	NOUN
cana-2486	117	100	)	)	PUNCT
cana-2486	117	101	.	.	PUNCT
cana-2486	118	1	then	then	ADV
cana-2486	118	2	(	(	PUNCT
cana-2486	118	3	(	(	PUNCT
cana-2486	118	4	k	k	X
cana-2486	118	5	∧	∧	PROPN
cana-2486	118	6	n	n	PROPN
cana-2486	118	7	)	)	PUNCT
cana-2486	118	8	:	:	PUNCT
cana-2486	118	9	1	1	NUM
cana-2486	118	10	m	m	NOUN
cana-2486	118	11	)	)	PUNCT
cana-2486	118	12	=	=	SYM
cana-2486	119	1	(	(	PUNCT
cana-2486	119	2	k	k	X
cana-2486	119	3	:	:	PUNCT
cana-2486	119	4	1	1	NUM
cana-2486	119	5	m	m	NOUN
cana-2486	119	6	)	)	PUNCT
cana-2486	119	7	∧	∧	PROPN
cana-2486	119	8	(	(	PUNCT
cana-2486	119	9	n	n	NOUN
cana-2486	119	10	:	:	PUNCT
cana-2486	119	11	1	1	NUM
cana-2486	119	12	m	m	NOUN
cana-2486	119	13	)	)	PUNCT
cana-2486	120	1	=	=	SYM
cana-2486	120	2	(	(	PUNCT
cana-2486	120	3	k	k	X
cana-2486	120	4	:	:	PUNCT
cana-2486	120	5	1	1	NUM
cana-2486	120	6	m	m	NOUN
cana-2486	120	7	)	)	PUNCT
cana-2486	120	8	(	(	PUNCT
cana-2486	120	9	n	n	X
cana-2486	120	10	:	:	PUNCT
cana-2486	120	11	1	1	NUM
cana-2486	120	12	m	m	NOUN
cana-2486	120	13	)	)	PUNCT
cana-2486	120	14	≤	≤	NOUN
cana-2486	120	15	(	(	PUNCT
cana-2486	120	16	k	k	NOUN
cana-2486	120	17	:	:	PUNCT
cana-2486	120	18	n)(n	n)(n	NOUN
cana-2486	120	19	:	:	PUNCT
cana-2486	120	20	1	1	NUM
cana-2486	120	21	m	m	NOUN
cana-2486	120	22	)	)	PUNCT
cana-2486	120	23	,	,	PUNCT
cana-2486	120	24	and	and	CCONJ
cana-2486	120	25	hence	hence	ADV
cana-2486	120	26	k	k	PROPN
cana-2486	120	27	∧	∧	PROPN
cana-2486	120	28	n	n	PROPN
cana-2486	120	29	=	=	SYM
cana-2486	120	30	(	(	PUNCT
cana-2486	120	31	(	(	PUNCT
cana-2486	120	32	k	k	PROPN
cana-2486	120	33	∧	∧	PROPN
cana-2486	120	34	n	n	CCONJ
cana-2486	120	35	)	)	PUNCT
cana-2486	120	36	:	:	PUNCT
cana-2486	121	1	1	1	NUM
cana-2486	121	2	m	m	NOUN
cana-2486	121	3	)	)	PUNCT
cana-2486	121	4	1	1	NUM
cana-2486	121	5	m	m	NOUN
cana-2486	121	6	≤	≤	NOUN
cana-2486	121	7	(	(	PUNCT
cana-2486	121	8	k	k	NOUN
cana-2486	121	9	:	:	PUNCT
cana-2486	121	10	n)(n	n)(n	NOUN
cana-2486	121	11	:	:	PUNCT
cana-2486	121	12	1	1	NUM
cana-2486	121	13	m	m	NOUN
cana-2486	121	14	)	)	PUNCT
cana-2486	121	15	1	1	NUM
cana-2486	121	16	m	m	NOUN
cana-2486	121	17	≤	≤	NOUN
cana-2486	121	18	(	(	PUNCT
cana-2486	121	19	k	k	NOUN
cana-2486	121	20	:	:	PUNCT
cana-2486	121	21	n	n	X
cana-2486	121	22	)	)	PUNCT
cana-2486	122	1	n	n	PRON
cana-2486	122	2	≤	≤	NOUN
cana-2486	122	3	k	k	PROPN
cana-2486	122	4	∧	∧	PROPN
cana-2486	122	5	n	n	CCONJ
cana-2486	122	6	,	,	PUNCT
cana-2486	122	7	so	so	SCONJ
cana-2486	122	8	that	that	SCONJ
cana-2486	122	9	k	k	PROPN
cana-2486	122	10	∧	∧	PROPN
cana-2486	122	11	n	n	PROPN
cana-2486	122	12	=	=	SYM
cana-2486	122	13	(	(	PUNCT
cana-2486	122	14	k	k	NOUN
cana-2486	122	15	:	:	PUNCT
cana-2486	122	16	n	n	X
cana-2486	122	17	)	)	PUNCT
cana-2486	122	18	n	n	NOUN
cana-2486	122	19	and	and	CCONJ
cana-2486	122	20	n	n	PROPN
cana-2486	122	21	is	be	AUX
cana-2486	122	22	multiplication	multiplication	NOUN
cana-2486	122	23	.	.	PUNCT
cana-2486	123	1	this	this	PRON
cana-2486	123	2	completes	complete	VERB
cana-2486	123	3	the	the	DET
cana-2486	123	4	proof	proof	NOUN
cana-2486	123	5	of	of	ADP
cana-2486	123	6	the	the	DET
cana-2486	123	7	theorem	theorem	PROPN
cana-2486	123	8	.	.	PUNCT
cana-2486	123	9	theorem	theorem	PROPN
cana-2486	123	10	2.11	2.11	NUM
cana-2486	123	11	.	.	PUNCT
cana-2486	124	1	let	let	VERB
cana-2486	124	2	l	l	NOUN
cana-2486	124	3	be	be	AUX
cana-2486	124	4	a	a	DET
cana-2486	124	5	cg	cg	NOUN
cana-2486	124	6	-	-	PUNCT
cana-2486	124	7	multiplicative	multiplicative	ADJ
cana-2486	124	8	lattice	lattice	NOUN
cana-2486	124	9	and	and	CCONJ
cana-2486	124	10	m	m	AUX
cana-2486	124	11	be	be	AUX
cana-2486	124	12	a	a	DET
cana-2486	124	13	multiplication	multiplication	NOUN
cana-2486	124	14	l	l	NOUN
cana-2486	124	15	-	-	NOUN
cana-2486	124	16	module	module	NOUN
cana-2486	124	17	.	.	PUNCT
cana-2486	125	1	for	for	ADP
cana-2486	125	2	n	n	PRON
cana-2486	125	3	,	,	PUNCT
cana-2486	125	4	k	k	PROPN
cana-2486	125	5	in	in	ADP
cana-2486	125	6	m	m	PROPN
cana-2486	125	7	and	and	CCONJ
cana-2486	125	8	a	a	PRON
cana-2486	125	9	in	in	ADP
cana-2486	125	10	l.	l.	PROPN
cana-2486	125	11	4	4	NUM
cana-2486	125	12	.	.	PUNCT
cana-2486	126	1	if	if	SCONJ
cana-2486	126	2	a	a	PRON
cana-2486	126	3	is	be	AUX
cana-2486	126	4	pure	pure	ADJ
cana-2486	126	5	in	in	ADP
cana-2486	126	6	l	l	NOUN
cana-2486	126	7	and	and	CCONJ
cana-2486	126	8	n	n	CCONJ
cana-2486	126	9	pure	pure	ADJ
cana-2486	126	10	in	in	ADP
cana-2486	126	11	m	m	PROPN
cana-2486	126	12	,	,	PUNCT
cana-2486	126	13	then	then	ADV
cana-2486	126	14	an	an	PRON
cana-2486	126	15	is	be	AUX
cana-2486	126	16	pure	pure	ADJ
cana-2486	126	17	in	in	ADP
cana-2486	126	18	m.	m.	NOUN
cana-2486	126	19	in	in	ADP
cana-2486	126	20	particular	particular	ADJ
cana-2486	126	21	,	,	PUNCT
cana-2486	126	22	if	if	SCONJ
cana-2486	126	23	a	a	PRON
cana-2486	126	24	is	be	AUX
cana-2486	126	25	pure	pure	ADJ
cana-2486	126	26	in	in	ADP
cana-2486	126	27	l	l	NOUN
cana-2486	126	28	,	,	PUNCT
cana-2486	126	29	then	then	ADV
cana-2486	126	30	a1	a1	PROPN
cana-2486	126	31	m	m	PROPN
cana-2486	126	32	is	be	AUX
cana-2486	126	33	a	a	DET
cana-2486	126	34	pure	pure	ADJ
cana-2486	126	35	element	element	NOUN
cana-2486	126	36	of	of	ADP
cana-2486	126	37	m.	m.	NOUN
cana-2486	126	38	5	5	NUM
cana-2486	126	39	.	.	PUNCT
cana-2486	127	1	if	if	SCONJ
cana-2486	127	2	k	k	PROPN
cana-2486	127	3	is	be	AUX
cana-2486	127	4	pure	pure	ADJ
cana-2486	127	5	in	in	ADP
cana-2486	127	6	n	n	PROPN
cana-2486	127	7	and	and	CCONJ
cana-2486	127	8	n	n	ADV
cana-2486	127	9	pure	pure	ADJ
cana-2486	127	10	in	in	ADP
cana-2486	127	11	m	m	PROPN
cana-2486	127	12	,	,	PUNCT
cana-2486	127	13	then	then	ADV
cana-2486	127	14	k	k	PROPN
cana-2486	127	15	is	be	AUX
cana-2486	127	16	pure	pure	ADJ
cana-2486	127	17	in	in	ADP
cana-2486	127	18	m.	m.	NOUN
cana-2486	127	19	6	6	NUM
cana-2486	127	20	.	.	PUNCT
cana-2486	128	1	let	let	VERB
cana-2486	128	2	k	k	PROPN
cana-2486	128	3	∨	∨	PROPN
cana-2486	128	4	n	n	CCONJ
cana-2486	128	5	be	be	AUX
cana-2486	128	6	a	a	DET
cana-2486	128	7	multiplication	multiplication	NOUN
cana-2486	128	8	element	element	NOUN
cana-2486	128	9	.	.	PUNCT
cana-2486	129	1	if	if	SCONJ
cana-2486	129	2	each	each	PRON
cana-2486	129	3	of	of	ADP
cana-2486	129	4	k	k	PROPN
cana-2486	129	5	and	and	CCONJ
cana-2486	129	6	n	n	PROPN
cana-2486	129	7	is	be	AUX
cana-2486	129	8	pure	pure	ADJ
cana-2486	129	9	in	in	ADP
cana-2486	129	10	m	m	PROPN
cana-2486	129	11	,	,	PUNCT
cana-2486	129	12	then	then	ADV
cana-2486	129	13	k	k	PROPN
cana-2486	129	14	∨	∨	PROPN
cana-2486	129	15	n	n	PROPN
cana-2486	129	16	and	and	CCONJ
cana-2486	129	17	k	k	PROPN
cana-2486	129	18	∧	∧	PROPN
cana-2486	129	19	n	n	NOUN
cana-2486	129	20	are	be	AUX
cana-2486	129	21	pure	pure	ADJ
cana-2486	129	22	in	in	ADP
cana-2486	129	23	m.	m.	NOUN
cana-2486	129	24	proof	proof	NOUN
cana-2486	129	25	.	.	PUNCT
cana-2486	130	1	1	1	NUM
cana-2486	130	2	:	:	PUNCT
cana-2486	130	3	⇒	⇒	NOUN
cana-2486	130	4	let	let	VERB
cana-2486	130	5	b	b	X
cana-2486	130	6	∈	∈	PROPN
cana-2486	130	7	l.	l.	NOUN
cana-2486	130	8	we	we	PRON
cana-2486	130	9	show	show	VERB
cana-2486	130	10	that	that	SCONJ
cana-2486	130	11	,	,	PUNCT
cana-2486	130	12	b(an	b(an	PROPN
cana-2486	130	13	)	)	PUNCT
cana-2486	130	14	=	=	PUNCT
cana-2486	130	15	an	an	DET
cana-2486	130	16	∧	∧	PROPN
cana-2486	130	17	b1	b1	NOUN
cana-2486	130	18	m.	m.	NOUN
cana-2486	130	19	assume	assume	VERB
cana-2486	130	20	that	that	SCONJ
cana-2486	130	21	,	,	PUNCT
cana-2486	130	22	l	l	NOUN
cana-2486	130	23	is	be	AUX
cana-2486	130	24	local	local	ADJ
cana-2486	130	25	multiplicative	multiplicative	ADJ
cana-2486	130	26	lattice	lattice	NOUN
cana-2486	130	27	.	.	PUNCT
cana-2486	131	1	since	since	SCONJ
cana-2486	131	2	a	a	PRON
cana-2486	131	3	is	be	AUX
cana-2486	131	4	a	a	DET
cana-2486	131	5	pure	pure	ADJ
cana-2486	131	6	in	in	ADP
cana-2486	131	7	l	l	NOUN
cana-2486	131	8	,	,	PUNCT
cana-2486	131	9	then	then	ADV
cana-2486	131	10	a	a	DET
cana-2486	131	11	=	=	X
cana-2486	131	12	0l	0l	X
cana-2486	131	13	or	or	CCONJ
cana-2486	131	14	a	a	DET
cana-2486	131	15	=	=	NOUN
cana-2486	131	16	1l	1l	NUM
cana-2486	131	17	.	.	PUNCT
cana-2486	132	1	if	if	SCONJ
cana-2486	132	2	a	a	DET
cana-2486	132	3	=	=	SYM
cana-2486	132	4	0l	0l	NUM
cana-2486	132	5	,	,	PUNCT
cana-2486	132	6	then	then	ADV
cana-2486	132	7	we	we	PRON
cana-2486	132	8	are	be	AUX
cana-2486	132	9	through	through	ADP
cana-2486	132	10	.	.	PUNCT
cana-2486	133	1	if	if	SCONJ
cana-2486	133	2	a	a	DET
cana-2486	133	3	=	=	NOUN
cana-2486	133	4	1l	1l	NUM
cana-2486	133	5	,	,	PUNCT
cana-2486	133	6	then	then	ADV
cana-2486	133	7	the	the	DET
cana-2486	133	8	purity	purity	NOUN
cana-2486	133	9	of	of	ADP
cana-2486	133	10	n	n	PROPN
cana-2486	133	11	implies	imply	VERB
cana-2486	133	12	that	that	SCONJ
cana-2486	133	13	b(an	b(an	NOUN
cana-2486	133	14	)	)	PUNCT
cana-2486	133	15	=	=	PUNCT
cana-2486	134	1	bn	bn	NOUN
cana-2486	134	2	=	=	SYM
cana-2486	134	3	n	n	CCONJ
cana-2486	134	4	∧	∧	PROPN
cana-2486	134	5	b1	b1	NOUN
cana-2486	134	6	m	m	NOUN
cana-2486	134	7	=	=	SYM
cana-2486	134	8	an	an	DET
cana-2486	134	9	∧	∧	PROPN
cana-2486	134	10	b1	b1	PROPN
cana-2486	134	11	m	m	NOUN
cana-2486	134	12	.	.	PUNCT
cana-2486	135	1	2:⇒	2:⇒	NUM
cana-2486	135	2	let	let	VERB
cana-2486	135	3	b	b	PROPN
cana-2486	135	4	∈	∈	PROPN
cana-2486	135	5	l.	l.	NOUN
cana-2486	135	6	then	then	ADV
cana-2486	135	7	bk	bk	ADP
cana-2486	135	8	=	=	PUNCT
cana-2486	135	9	k	k	PROPN
cana-2486	135	10	∧	∧	PROPN
cana-2486	135	11	bn	bn	NOUN
cana-2486	135	12	and	and	CCONJ
cana-2486	135	13	bn	bn	NOUN
cana-2486	135	14	=	=	SYM
cana-2486	135	15	n	n	CCONJ
cana-2486	135	16	∧	∧	PROPN
cana-2486	135	17	b1	b1	NOUN
cana-2486	135	18	m	m	PROPN
cana-2486	135	19	and	and	CCONJ
cana-2486	135	20	hence	hence	ADV
cana-2486	135	21	,	,	PUNCT
cana-2486	135	22	bk	bk	ADP
cana-2486	135	23	=	=	SYM
cana-2486	135	24	(	(	PUNCT
cana-2486	135	25	k	k	X
cana-2486	135	26	∧	∧	PROPN
cana-2486	135	27	n	n	NOUN
cana-2486	135	28	)	)	PUNCT
cana-2486	135	29	∧	∧	PROPN
cana-2486	135	30	b1	b1	NOUN
cana-2486	135	31	m	m	NOUN
cana-2486	135	32	=	=	SYM
cana-2486	135	33	k	k	PROPN
cana-2486	135	34	∧	∧	PROPN
cana-2486	135	35	b1	b1	PROPN
cana-2486	135	36	m	m	PROPN
cana-2486	135	37	,	,	PUNCT
cana-2486	135	38	since	since	SCONJ
cana-2486	135	39	k	k	PROPN
cana-2486	135	40	≤	≤	PUNCT
cana-2486	135	41	n	n	ADV
cana-2486	135	42	.	.	PUNCT
cana-2486	136	1	so	so	ADV
cana-2486	136	2	k	k	PROPN
cana-2486	136	3	is	be	AUX
cana-2486	136	4	pure	pure	ADJ
cana-2486	136	5	in	in	ADP
cana-2486	136	6	m	m	PROPN
cana-2486	136	7	.	.	PUNCT
cana-2486	137	1	3	3	NUM
cana-2486	137	2	:	:	PUNCT
cana-2486	137	3	⇒	⇒	NOUN
cana-2486	137	4	given	give	VERB
cana-2486	137	5	k	k	PROPN
cana-2486	137	6	and	and	CCONJ
cana-2486	137	7	n	n	PROPN
cana-2486	137	8	are	be	AUX
cana-2486	137	9	pure	pure	ADJ
cana-2486	137	10	in	in	ADP
cana-2486	137	11	m	m	PROPN
cana-2486	137	12	,	,	PUNCT
cana-2486	137	13	ak	ak	PROPN
cana-2486	137	14	=	=	SYM
cana-2486	137	15	k	k	PROPN
cana-2486	137	16	∧	∧	PROPN
cana-2486	137	17	a1	a1	PROPN
cana-2486	137	18	m	m	PROPN
cana-2486	137	19	and	and	CCONJ
cana-2486	137	20	an	an	DET
cana-2486	137	21	=	=	NOUN
cana-2486	137	22	n	n	CCONJ
cana-2486	137	23	∧	∧	PROPN
cana-2486	137	24	a1	a1	PROPN
cana-2486	137	25	m	m	NOUN
cana-2486	137	26	.	.	PUNCT
cana-2486	138	1	so	so	ADV
cana-2486	138	2	ak	ak	PROPN
cana-2486	138	3	∧	∧	PROPN
cana-2486	138	4	an	an	PRON
cana-2486	139	1	=	=	X
cana-2486	139	2	(	(	PUNCT
cana-2486	139	3	k	k	PROPN
cana-2486	139	4	∧	∧	PROPN
cana-2486	139	5	n	n	NOUN
cana-2486	139	6	)	)	PUNCT
cana-2486	139	7	∧	∧	PROPN
cana-2486	139	8	a1	a1	PROPN
cana-2486	139	9	m	m	PROPN
cana-2486	139	10	and	and	CCONJ
cana-2486	139	11	a(k	a(k	PROPN
cana-2486	139	12	∨	∨	NUM
cana-2486	139	13	n	n	CCONJ
cana-2486	139	14	)	)	PUNCT
cana-2486	139	15	=	=	SYM
cana-2486	140	1	(	(	PUNCT
cana-2486	140	2	k	k	X
cana-2486	140	3	∧	∧	PROPN
cana-2486	140	4	a1	a1	PROPN
cana-2486	140	5	m	m	PROPN
cana-2486	140	6	)	)	PUNCT
cana-2486	140	7	∨	∨	NOUN
cana-2486	140	8	(	(	PUNCT
cana-2486	140	9	n	n	CCONJ
cana-2486	140	10	∧	∧	PROPN
cana-2486	140	11	a1	a1	PROPN
cana-2486	140	12	m	m	PROPN
cana-2486	140	13	)	)	PUNCT
cana-2486	140	14	.	.	PUNCT
cana-2486	141	1	since	since	SCONJ
cana-2486	141	2	k	k	PROPN
cana-2486	141	3	∨	∨	PROPN
cana-2486	141	4	n	n	PRON
cana-2486	141	5	is	be	AUX
cana-2486	141	6	multiplication	multiplication	NOUN
cana-2486	141	7	,	,	PUNCT
cana-2486	141	8	so	so	ADV
cana-2486	141	9	a(k	a(k	PROPN
cana-2486	141	10	∧	∧	PROPN
cana-2486	141	11	n	n	NOUN
cana-2486	141	12	)	)	PUNCT
cana-2486	142	1	=	=	PUNCT
cana-2486	142	2	ak	ak	PROPN
cana-2486	142	3	∧	∧	PROPN
cana-2486	142	4	an	an	PROPN
cana-2486	142	5	and	and	CCONJ
cana-2486	142	6	(	(	PUNCT
cana-2486	142	7	k	k	PROPN
cana-2486	142	8	∨	∨	PROPN
cana-2486	142	9	n	n	CCONJ
cana-2486	142	10	)	)	PUNCT
cana-2486	142	11	∧	∧	PROPN
cana-2486	142	12	a1	a1	PROPN
cana-2486	142	13	m	m	NOUN
cana-2486	142	14	=	=	PUNCT
cana-2486	142	15	(	(	PUNCT
cana-2486	142	16	k	k	X
cana-2486	142	17	∧	∧	PROPN
cana-2486	142	18	a1	a1	PROPN
cana-2486	142	19	m	m	PROPN
cana-2486	142	20	)	)	PUNCT
cana-2486	142	21	∨	∨	NOUN
cana-2486	142	22	(	(	PUNCT
cana-2486	142	23	n	n	CCONJ
cana-2486	142	24	∧	∧	PROPN
cana-2486	142	25	a1	a1	PROPN
cana-2486	142	26	m	m	NOUN
cana-2486	142	27	)	)	PUNCT
cana-2486	142	28	and	and	CCONJ
cana-2486	142	29	this	this	PRON
cana-2486	142	30	shows	show	VERB
cana-2486	142	31	that	that	SCONJ
cana-2486	142	32	k	k	PROPN
cana-2486	142	33	∧	∧	PROPN
cana-2486	142	34	n	n	PROPN
cana-2486	142	35	and	and	CCONJ
cana-2486	142	36	k	k	PROPN
cana-2486	142	37	∨	∨	PROPN
cana-2486	142	38	n	n	CCONJ
cana-2486	142	39	are	be	AUX
cana-2486	142	40	pure	pure	ADJ
cana-2486	142	41	elements	element	NOUN
cana-2486	142	42	of	of	ADP
cana-2486	142	43	m	m	PROPN
cana-2486	142	44	.	.	PUNCT
cana-2486	143	1	in	in	ADP
cana-2486	143	2	the	the	DET
cana-2486	143	3	following	follow	VERB
cana-2486	143	4	theorem	theorem	NOUN
cana-2486	143	5	we	we	PRON
cana-2486	143	6	give	give	VERB
cana-2486	143	7	a	a	DET
cana-2486	143	8	relation	relation	NOUN
cana-2486	143	9	between	between	ADP
cana-2486	143	10	pure	pure	ADJ
cana-2486	143	11	elements	element	NOUN
cana-2486	143	12	,	,	PUNCT
cana-2486	143	13	multiplication	multiplication	NOUN
cana-2486	143	14	elements	element	NOUN
cana-2486	143	15	and	and	CCONJ
cana-2486	143	16	idempotent	idempotent	ADJ
cana-2486	143	17	elements	element	NOUN
cana-2486	143	18	.	.	PUNCT
cana-2486	144	1	theorem	theorem	VERB
cana-2486	144	2	2.12	2.12	NUM
cana-2486	144	3	.	.	PUNCT
cana-2486	145	1	let	let	VERB
cana-2486	145	2	l	l	NOUN
cana-2486	145	3	be	be	AUX
cana-2486	145	4	a	a	DET
cana-2486	145	5	cg	cg	NOUN
cana-2486	145	6	-	-	PUNCT
cana-2486	145	7	multiplicative	multiplicative	ADJ
cana-2486	145	8	lattice	lattice	NOUN
cana-2486	145	9	and	and	CCONJ
cana-2486	145	10	m	m	AUX
cana-2486	145	11	be	be	AUX
cana-2486	145	12	a	a	DET
cana-2486	145	13	faithful	faithful	ADJ
cana-2486	145	14	multiplication	multiplication	NOUN
cana-2486	145	15	lmodule	lmodule	VERB
cana-2486	145	16	such	such	ADJ
cana-2486	145	17	that	that	SCONJ
cana-2486	145	18	1	1	NUM
cana-2486	145	19	m	m	NOUN
cana-2486	145	20	compact	compact	ADJ
cana-2486	145	21	.	.	PUNCT
cana-2486	146	1	for	for	ADP
cana-2486	146	2	n	n	PROPN
cana-2486	146	3	in	in	ADP
cana-2486	146	4	m	m	PROPN
cana-2486	146	5	,	,	PUNCT
cana-2486	146	6	the	the	DET
cana-2486	146	7	following	follow	VERB
cana-2486	146	8	are	be	AUX
cana-2486	146	9	equivalent	equivalent	ADJ
cana-2486	146	10	:	:	PUNCT
cana-2486	146	11	1	1	X
cana-2486	146	12	.	.	X
cana-2486	146	13	n	n	PRON
cana-2486	146	14	is	be	AUX
cana-2486	146	15	a	a	DET
cana-2486	146	16	pure	pure	ADJ
cana-2486	146	17	element	element	NOUN
cana-2486	146	18	of	of	ADP
cana-2486	146	19	m.	m.	NOUN
cana-2486	146	20	2	2	NUM
cana-2486	146	21	.	.	PUNCT
cana-2486	147	1	n	n	PRON
cana-2486	147	2	is	be	AUX
cana-2486	147	3	multiplication	multiplication	NOUN
cana-2486	147	4	and	and	CCONJ
cana-2486	147	5	is	be	AUX
cana-2486	147	6	idempotent	idempotent	ADJ
cana-2486	147	7	in	in	ADP
cana-2486	147	8	m.	m.	NOUN
cana-2486	147	9	3	3	NUM
cana-2486	147	10	.	.	PUNCT
cana-2486	148	1	(	(	PUNCT
cana-2486	148	2	n	n	X
cana-2486	148	3	:	:	PUNCT
cana-2486	148	4	1	1	NUM
cana-2486	148	5	m	m	NOUN
cana-2486	148	6	)	)	PUNCT
cana-2486	149	1	=	=	PUNCT
cana-2486	149	2	a	a	DET
cana-2486	149	3	∧	∧	PROPN
cana-2486	149	4	(	(	PUNCT
cana-2486	149	5	n	n	NOUN
cana-2486	149	6	:	:	PUNCT
cana-2486	149	7	1	1	NUM
cana-2486	149	8	m	m	NOUN
cana-2486	149	9	)	)	PUNCT
cana-2486	149	10	,	,	PUNCT
cana-2486	149	11	for	for	ADP
cana-2486	149	12	every	every	DET
cana-2486	149	13	a	a	DET
cana-2486	149	14	∈	∈	PROPN
cana-2486	149	15	l.	l.	NOUN
cana-2486	149	16	proof	proof	NOUN
cana-2486	149	17	.	.	PUNCT
cana-2486	150	1	1	1	NUM
cana-2486	150	2	⇒	⇒	NOUN
cana-2486	150	3	2	2	NUM
cana-2486	150	4	:	:	PUNCT
cana-2486	150	5	assume	assume	VERB
cana-2486	150	6	that	that	SCONJ
cana-2486	150	7	n	n	PRON
cana-2486	150	8	is	be	AUX
cana-2486	150	9	a	a	DET
cana-2486	150	10	pure	pure	ADJ
cana-2486	150	11	element	element	NOUN
cana-2486	150	12	of	of	ADP
cana-2486	150	13	m	m	PROPN
cana-2486	150	14	.	.	PUNCT
cana-2486	151	1	let	let	VERB
cana-2486	151	2	k	k	X
cana-2486	151	3	be	be	AUX
cana-2486	151	4	a	a	DET
cana-2486	151	5	element	element	NOUN
cana-2486	151	6	of	of	ADP
cana-2486	151	7	m.	m.	NOUN
cana-2486	151	8	we	we	PRON
cana-2486	151	9	will	will	AUX
cana-2486	151	10	communications	communication	NOUN
cana-2486	151	11	on	on	ADP
cana-2486	151	12	applied	apply	VERB
cana-2486	151	13	nonlinear	nonlinear	ADJ
cana-2486	151	14	analysis	analysis	NOUN
cana-2486	151	15	issn	issn	NOUN
cana-2486	151	16	:	:	PUNCT
cana-2486	151	17	1074	1074	NUM
cana-2486	151	18	-	-	PUNCT
cana-2486	151	19	133x	133x	NUM
cana-2486	151	20	vol	vol	NOUN
cana-2486	151	21	.	.	PROPN
cana-2486	152	1	32	32	NUM
cana-2486	152	2	no	no	INTJ
cana-2486	152	3	.	.	PUNCT
cana-2486	153	1	2s	2s	NUM
cana-2486	153	2	(	(	PUNCT
cana-2486	153	3	2024	2024	NUM
cana-2486	153	4	)	)	PUNCT
cana-2486	153	5	542	542	NUM
cana-2486	153	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2486	153	7	show	show	VERB
cana-2486	153	8	that	that	SCONJ
cana-2486	153	9	,	,	PUNCT
cana-2486	153	10	n	n	PROPN
cana-2486	153	11	∧	∧	PROPN
cana-2486	153	12	k	k	NOUN
cana-2486	153	13	=	=	PRON
cana-2486	153	14	(	(	PUNCT
cana-2486	153	15	k	k	NOUN
cana-2486	153	16	:	:	PUNCT
cana-2486	153	17	n	n	X
cana-2486	153	18	)	)	PUNCT
cana-2486	153	19	n	n	CCONJ
cana-2486	153	20	.	.	PUNCT
cana-2486	154	1	since	since	SCONJ
cana-2486	154	2	m	m	PROPN
cana-2486	154	3	is	be	AUX
cana-2486	154	4	multiplication	multiplication	NOUN
cana-2486	154	5	,	,	PUNCT
cana-2486	154	6	k	k	X
cana-2486	154	7	=	=	PRON
cana-2486	154	8	(	(	PUNCT
cana-2486	154	9	k	k	X
cana-2486	154	10	:	:	PUNCT
cana-2486	154	11	1	1	NUM
cana-2486	154	12	m	m	NOUN
cana-2486	154	13	)	)	PUNCT
cana-2486	154	14	1	1	NUM
cana-2486	154	15	m	m	NOUN
cana-2486	154	16	.	.	PUNCT
cana-2486	155	1	since	since	SCONJ
cana-2486	155	2	n	n	NUM
cana-2486	155	3	is	be	AUX
cana-2486	155	4	a	a	DET
cana-2486	155	5	pure	pure	ADJ
cana-2486	155	6	element	element	NOUN
cana-2486	155	7	of	of	ADP
cana-2486	155	8	m	m	PROPN
cana-2486	155	9	,	,	PUNCT
cana-2486	155	10	so	so	SCONJ
cana-2486	155	11	we	we	PRON
cana-2486	155	12	have	have	VERB
cana-2486	155	13	,	,	PUNCT
cana-2486	155	14	(	(	PUNCT
cana-2486	155	15	k	k	NOUN
cana-2486	155	16	:	:	PUNCT
cana-2486	155	17	n	n	X
cana-2486	155	18	)	)	PUNCT
cana-2486	155	19	n	n	NOUN
cana-2486	155	20	=	=	SYM
cana-2486	156	1	n	n	PRON
cana-2486	156	2	∧	∧	PROPN
cana-2486	156	3	(	(	PUNCT
cana-2486	156	4	k	k	NOUN
cana-2486	156	5	:	:	PUNCT
cana-2486	156	6	n	n	X
cana-2486	156	7	)	)	PUNCT
cana-2486	156	8	1	1	NUM
cana-2486	156	9	m	m	NOUN
cana-2486	156	10	.	.	PUNCT
cana-2486	157	1	now	now	ADV
cana-2486	157	2	(	(	PUNCT
cana-2486	157	3	k	k	NOUN
cana-2486	157	4	:	:	PUNCT
cana-2486	157	5	n	n	X
cana-2486	157	6	)	)	PUNCT
cana-2486	157	7	n	n	NOUN
cana-2486	157	8	=	=	SYM
cana-2486	157	9	n	n	PRON
cana-2486	157	10	∧	∧	PROPN
cana-2486	157	11	(	(	PUNCT
cana-2486	157	12	k	k	NOUN
cana-2486	157	13	:	:	PUNCT
cana-2486	157	14	n	n	X
cana-2486	157	15	)	)	PUNCT
cana-2486	157	16	1	1	NUM
cana-2486	157	17	m	m	NOUN
cana-2486	157	18	≥	≥	NOUN
cana-2486	157	19	n	n	PRON
cana-2486	157	20	∧	∧	PROPN
cana-2486	157	21	(	(	PUNCT
cana-2486	157	22	k	k	NOUN
cana-2486	157	23	:	:	PUNCT
cana-2486	157	24	1	1	NUM
cana-2486	157	25	m	m	NOUN
cana-2486	157	26	)	)	PUNCT
cana-2486	157	27	1	1	NUM
cana-2486	157	28	m	m	NOUN
cana-2486	157	29	=	=	SYM
cana-2486	157	30	n	n	PROPN
cana-2486	157	31	∧	∧	PROPN
cana-2486	157	32	k	k	PROPN
cana-2486	157	33	≥	≥	X
cana-2486	157	34	(	(	PUNCT
cana-2486	157	35	k	k	NOUN
cana-2486	157	36	:	:	PUNCT
cana-2486	157	37	n	n	X
cana-2486	157	38	)	)	PUNCT
cana-2486	157	39	n.	n.	NOUN
cana-2486	157	40	hence	hence	ADV
cana-2486	157	41	,	,	PUNCT
cana-2486	157	42	we	we	PRON
cana-2486	157	43	get	get	VERB
cana-2486	157	44	(	(	PUNCT
cana-2486	157	45	k	k	NOUN
cana-2486	157	46	:	:	PUNCT
cana-2486	157	47	n	n	X
cana-2486	157	48	)	)	PUNCT
cana-2486	157	49	n	n	NOUN
cana-2486	157	50	=	=	SYM
cana-2486	157	51	k	k	PROPN
cana-2486	157	52	∧	∧	PROPN
cana-2486	157	53	n.	n.	PROPN
cana-2486	157	54	this	this	PRON
cana-2486	157	55	implies	imply	VERB
cana-2486	157	56	that	that	SCONJ
cana-2486	157	57	n	n	X
cana-2486	157	58	is	be	AUX
cana-2486	157	59	a	a	DET
cana-2486	157	60	multiplication	multiplication	NOUN
cana-2486	157	61	in	in	ADP
cana-2486	157	62	m.	m.	NOUN
cana-2486	157	63	since	since	SCONJ
cana-2486	157	64	n	n	NUM
cana-2486	157	65	is	be	AUX
cana-2486	157	66	pure	pure	ADJ
cana-2486	157	67	element	element	NOUN
cana-2486	157	68	,	,	PUNCT
cana-2486	157	69	so	so	SCONJ
cana-2486	157	70	we	we	PRON
cana-2486	157	71	have	have	VERB
cana-2486	157	72	(	(	PUNCT
cana-2486	157	73	n	n	X
cana-2486	157	74	:	:	PUNCT
cana-2486	157	75	1	1	NUM
cana-2486	157	76	m	m	NOUN
cana-2486	157	77	)	)	PUNCT
cana-2486	157	78	n	n	NOUN
cana-2486	157	79	=	=	SYM
cana-2486	157	80	n	n	PRON
cana-2486	157	81	∧	∧	PROPN
cana-2486	157	82	(	(	PUNCT
cana-2486	157	83	n	n	NOUN
cana-2486	157	84	:	:	PUNCT
cana-2486	157	85	1	1	NUM
cana-2486	157	86	m	m	NOUN
cana-2486	157	87	)	)	PUNCT
cana-2486	157	88	1	1	NUM
cana-2486	157	89	m	m	NOUN
cana-2486	157	90	=	=	NOUN
cana-2486	157	91	n.	n.	NOUN
cana-2486	157	92	n	n	NOUN
cana-2486	157	93	=	=	SYM
cana-2486	157	94	(	(	PUNCT
cana-2486	157	95	n	n	NOUN
cana-2486	157	96	:	:	PUNCT
cana-2486	157	97	1	1	NUM
cana-2486	157	98	m	m	NOUN
cana-2486	157	99	)	)	PUNCT
cana-2486	157	100	n	n	NOUN
cana-2486	157	101	=	=	SYM
cana-2486	157	102	(	(	PUNCT
cana-2486	157	103	n	n	NOUN
cana-2486	157	104	:	:	PUNCT
cana-2486	157	105	1	1	NUM
cana-2486	157	106	m	m	NOUN
cana-2486	157	107	)	)	PUNCT
cana-2486	157	108	(	(	PUNCT
cana-2486	157	109	n	n	X
cana-2486	157	110	:	:	PUNCT
cana-2486	157	111	1	1	NUM
cana-2486	157	112	m	m	NOUN
cana-2486	157	113	)	)	PUNCT
cana-2486	157	114	1	1	NUM
cana-2486	157	115	m	m	NOUN
cana-2486	157	116	=	=	PUNCT
cana-2486	157	117	(	(	PUNCT
cana-2486	157	118	n	n	NOUN
cana-2486	157	119	:	:	PUNCT
cana-2486	157	120	1	1	NUM
cana-2486	157	121	m	m	NOUN
cana-2486	157	122	)	)	PUNCT
cana-2486	157	123	21	21	NUM
cana-2486	157	124	m.	m.	NOUN
cana-2486	157	125	hence	hence	ADV
cana-2486	157	126	,	,	PUNCT
cana-2486	157	127	we	we	PRON
cana-2486	157	128	get	get	VERB
cana-2486	157	129	(	(	PUNCT
cana-2486	157	130	n	n	NOUN
cana-2486	157	131	:	:	PUNCT
cana-2486	157	132	1	1	NUM
cana-2486	157	133	m	m	NOUN
cana-2486	157	134	)	)	PUNCT
cana-2486	157	135	21	21	NUM
cana-2486	157	136	m	m	NOUN
cana-2486	157	137	=	=	PUNCT
cana-2486	157	138	(	(	PUNCT
cana-2486	157	139	n	n	NOUN
cana-2486	157	140	:	:	PUNCT
cana-2486	157	141	1	1	NUM
cana-2486	157	142	m	m	NOUN
cana-2486	157	143	)	)	PUNCT
cana-2486	157	144	1	1	NUM
cana-2486	157	145	m.	m.	NOUN
cana-2486	158	1	so	so	SCONJ
cana-2486	158	2	we	we	PRON
cana-2486	158	3	have	have	AUX
cana-2486	158	4	(	(	PUNCT
cana-2486	158	5	n	n	X
cana-2486	158	6	:	:	PUNCT
cana-2486	158	7	1	1	NUM
cana-2486	158	8	m	m	NOUN
cana-2486	158	9	)	)	PUNCT
cana-2486	158	10	is	be	AUX
cana-2486	158	11	an	an	DET
cana-2486	158	12	idempotent	idempotent	ADJ
cana-2486	158	13	element	element	NOUN
cana-2486	158	14	of	of	ADP
cana-2486	158	15	l.	l.	PROPN
cana-2486	158	16	and	and	CCONJ
cana-2486	158	17	hence	hence	ADV
cana-2486	158	18	n	n	PRON
cana-2486	158	19	is	be	AUX
cana-2486	158	20	idempotent	idempotent	ADJ
cana-2486	158	21	in	in	ADP
cana-2486	158	22	m.	m.	NOUN
cana-2486	158	23	2	2	NUM
cana-2486	158	24	⇒	⇒	NOUN
cana-2486	158	25	3	3	NUM
cana-2486	158	26	:	:	PUNCT
cana-2486	158	27	assume	assume	VERB
cana-2486	158	28	that	that	SCONJ
cana-2486	158	29	n	n	PRON
cana-2486	158	30	is	be	AUX
cana-2486	158	31	multiplication	multiplication	NOUN
cana-2486	158	32	and	and	CCONJ
cana-2486	158	33	idempotent	idempotent	NOUN
cana-2486	158	34	in	in	ADP
cana-2486	158	35	m	m	PROPN
cana-2486	158	36	.	.	PUNCT
cana-2486	159	1	so	so	ADV
cana-2486	159	2	(	(	PUNCT
cana-2486	159	3	n	n	X
cana-2486	159	4	:	:	PUNCT
cana-2486	159	5	1	1	NUM
cana-2486	159	6	m	m	NOUN
cana-2486	159	7	)	)	PUNCT
cana-2486	159	8	is	be	AUX
cana-2486	159	9	an	an	DET
cana-2486	159	10	idempotent	idempotent	ADJ
cana-2486	159	11	element	element	NOUN
cana-2486	159	12	,	,	PUNCT
cana-2486	159	13	then	then	ADV
cana-2486	159	14	we	we	PRON
cana-2486	159	15	have	have	VERB
cana-2486	159	16	n	n	NOUN
cana-2486	159	17	=	=	SYM
cana-2486	159	18	(	(	PUNCT
cana-2486	159	19	n	n	NOUN
cana-2486	159	20	:	:	PUNCT
cana-2486	159	21	1	1	NUM
cana-2486	159	22	m	m	NOUN
cana-2486	159	23	)	)	PUNCT
cana-2486	159	24	1	1	NUM
cana-2486	159	25	m	m	NOUN
cana-2486	159	26	=	=	PUNCT
cana-2486	159	27	(	(	PUNCT
cana-2486	159	28	n	n	NOUN
cana-2486	159	29	:	:	PUNCT
cana-2486	159	30	1	1	NUM
cana-2486	159	31	m	m	NOUN
cana-2486	159	32	)	)	PUNCT
cana-2486	159	33	21	21	NUM
cana-2486	159	34	m	m	NOUN
cana-2486	159	35	=	=	PUNCT
cana-2486	159	36	(	(	PUNCT
cana-2486	159	37	n	n	NOUN
cana-2486	159	38	:	:	PUNCT
cana-2486	159	39	1	1	NUM
cana-2486	159	40	m	m	NOUN
cana-2486	159	41	)	)	PUNCT
cana-2486	159	42	(	(	PUNCT
cana-2486	159	43	n	n	X
cana-2486	159	44	:	:	PUNCT
cana-2486	159	45	1	1	NUM
cana-2486	159	46	m	m	NOUN
cana-2486	159	47	)	)	PUNCT
cana-2486	159	48	1	1	NUM
cana-2486	159	49	m	m	NOUN
cana-2486	159	50	=	=	PUNCT
cana-2486	159	51	(	(	PUNCT
cana-2486	159	52	n	n	NOUN
cana-2486	159	53	:	:	PUNCT
cana-2486	159	54	1	1	NUM
cana-2486	159	55	m	m	NOUN
cana-2486	159	56	)	)	PUNCT
cana-2486	159	57	n.	n.	NOUN
cana-2486	160	1	so	so	ADV
cana-2486	160	2	for	for	ADP
cana-2486	160	3	any	any	DET
cana-2486	160	4	element	element	NOUN
cana-2486	160	5	k	k	PROPN
cana-2486	160	6	of	of	ADP
cana-2486	160	7	m	m	PROPN
cana-2486	160	8	,	,	PUNCT
cana-2486	160	9	we	we	PRON
cana-2486	160	10	have	have	VERB
cana-2486	160	11	,	,	PUNCT
cana-2486	160	12	(	(	PUNCT
cana-2486	160	13	k	k	NOUN
cana-2486	160	14	:	:	PUNCT
cana-2486	160	15	n	n	X
cana-2486	160	16	)	)	PUNCT
cana-2486	160	17	n	n	NOUN
cana-2486	160	18	=	=	SYM
cana-2486	160	19	(	(	PUNCT
cana-2486	160	20	k	k	NOUN
cana-2486	160	21	:	:	PUNCT
cana-2486	160	22	n	n	X
cana-2486	160	23	)	)	PUNCT
cana-2486	160	24	(	(	PUNCT
cana-2486	160	25	n	n	X
cana-2486	160	26	:	:	PUNCT
cana-2486	160	27	1	1	NUM
cana-2486	160	28	m	m	NOUN
cana-2486	160	29	)	)	PUNCT
cana-2486	161	1	n	n	NOUN
cana-2486	161	2	≤	≤	NOUN
cana-2486	161	3	(	(	PUNCT
cana-2486	161	4	k	k	NOUN
cana-2486	161	5	:	:	PUNCT
cana-2486	161	6	1	1	NUM
cana-2486	161	7	m	m	NOUN
cana-2486	161	8	)	)	PUNCT
cana-2486	161	9	n	n	NOUN
cana-2486	161	10	≤	≤	NOUN
cana-2486	161	11	(	(	PUNCT
cana-2486	161	12	k	k	NOUN
cana-2486	161	13	:	:	PUNCT
cana-2486	161	14	n	n	X
cana-2486	161	15	)	)	PUNCT
cana-2486	161	16	n	n	CCONJ
cana-2486	161	17	,	,	PUNCT
cana-2486	161	18	that	that	PRON
cana-2486	161	19	implies	imply	VERB
cana-2486	161	20	(	(	PUNCT
cana-2486	161	21	k	k	NOUN
cana-2486	161	22	:	:	PUNCT
cana-2486	161	23	n	n	X
cana-2486	161	24	)	)	PUNCT
cana-2486	161	25	n	n	NOUN
cana-2486	161	26	=	=	SYM
cana-2486	161	27	(	(	PUNCT
cana-2486	161	28	k	k	NOUN
cana-2486	161	29	:	:	PUNCT
cana-2486	161	30	1	1	NUM
cana-2486	161	31	m	m	NOUN
cana-2486	161	32	)	)	PUNCT
cana-2486	161	33	n	n	NOUN
cana-2486	161	34	.	.	PUNCT
cana-2486	162	1	since	since	SCONJ
cana-2486	162	2	n	n	ADV
cana-2486	162	3	is	be	AUX
cana-2486	162	4	multiplication	multiplication	NOUN
cana-2486	162	5	element	element	NOUN
cana-2486	162	6	of	of	ADP
cana-2486	162	7	m	m	PROPN
cana-2486	162	8	,	,	PUNCT
cana-2486	162	9	so	so	CCONJ
cana-2486	162	10	for	for	ADP
cana-2486	162	11	every	every	DET
cana-2486	162	12	a	a	PRON
cana-2486	162	13	of	of	ADP
cana-2486	162	14	l	l	NOUN
cana-2486	162	15	,	,	PUNCT
cana-2486	162	16	a1	a1	PROPN
cana-2486	162	17	m	m	PROPN
cana-2486	162	18	∧	∧	PROPN
cana-2486	162	19	n	n	NOUN
cana-2486	162	20	=	=	SYM
cana-2486	162	21	(	(	PUNCT
cana-2486	162	22	a1	a1	PROPN
cana-2486	162	23	m	m	NOUN
cana-2486	162	24	:	:	PUNCT
cana-2486	162	25	n	n	X
cana-2486	162	26	)	)	PUNCT
cana-2486	162	27	n	n	NOUN
cana-2486	162	28	=	=	SYM
cana-2486	162	29	(	(	PUNCT
cana-2486	162	30	a1	a1	PROPN
cana-2486	162	31	m	m	NOUN
cana-2486	162	32	:	:	PUNCT
cana-2486	162	33	1	1	NUM
cana-2486	162	34	m	m	NOUN
cana-2486	162	35	)	)	PUNCT
cana-2486	163	1	n	n	NOUN
cana-2486	163	2	=	=	SYM
cana-2486	163	3	an	an	DET
cana-2486	163	4	=	=	PUNCT
cana-2486	163	5	a1	a1	PROPN
cana-2486	163	6	m	m	PROPN
cana-2486	163	7	∧	∧	NOUN
cana-2486	163	8	(	(	PUNCT
cana-2486	163	9	n	n	NOUN
cana-2486	163	10	:	:	PUNCT
cana-2486	163	11	1	1	NUM
cana-2486	163	12	m	m	NOUN
cana-2486	163	13	)	)	PUNCT
cana-2486	163	14	1	1	NUM
cana-2486	163	15	m	m	NOUN
cana-2486	163	16	.	.	PUNCT
cana-2486	164	1	also	also	ADV
cana-2486	164	2	an	an	DET
cana-2486	164	3	=	=	X
cana-2486	164	4	a(n	a(n	NOUN
cana-2486	164	5	:	:	PUNCT
cana-2486	164	6	1	1	NUM
cana-2486	164	7	m	m	NOUN
cana-2486	164	8	)	)	PUNCT
cana-2486	164	9	n	n	NOUN
cana-2486	164	10	=	=	PUNCT
cana-2486	165	1	a(n	a(n	NOUN
cana-2486	165	2	:	:	PUNCT
cana-2486	165	3	1	1	NUM
cana-2486	165	4	m	m	NOUN
cana-2486	165	5	)	)	PUNCT
cana-2486	165	6	1	1	NUM
cana-2486	165	7	m	m	NOUN
cana-2486	165	8	,	,	PUNCT
cana-2486	165	9	so	so	ADV
cana-2486	165	10	a1	a1	PROPN
cana-2486	165	11	m	m	PROPN
cana-2486	165	12	∧	∧	NOUN
cana-2486	165	13	(	(	PUNCT
cana-2486	165	14	n	n	NOUN
cana-2486	165	15	:	:	PUNCT
cana-2486	165	16	1	1	NUM
cana-2486	165	17	m	m	NOUN
cana-2486	165	18	)	)	PUNCT
cana-2486	165	19	1	1	NUM
cana-2486	165	20	m	m	NOUN
cana-2486	165	21	=	=	PUNCT
cana-2486	165	22	a(n	a(n	NOUN
cana-2486	165	23	:	:	PUNCT
cana-2486	165	24	1	1	NUM
cana-2486	165	25	m	m	NOUN
cana-2486	165	26	)	)	PUNCT
cana-2486	165	27	1	1	NUM
cana-2486	165	28	m	m	NOUN
cana-2486	165	29	for	for	ADP
cana-2486	165	30	any	any	DET
cana-2486	165	31	a	a	DET
cana-2486	165	32	∈	∈	PROPN
cana-2486	165	33	l	l	NOUN
cana-2486	165	34	,	,	PUNCT
cana-2486	165	35	hence	hence	ADV
cana-2486	165	36	a1	a1	PROPN
cana-2486	165	37	m	m	PROPN
cana-2486	165	38	∧	∧	NOUN
cana-2486	165	39	(	(	PUNCT
cana-2486	165	40	n	n	NOUN
cana-2486	165	41	:	:	PUNCT
cana-2486	165	42	1	1	NUM
cana-2486	165	43	m	m	NOUN
cana-2486	165	44	)	)	PUNCT
cana-2486	165	45	1	1	NUM
cana-2486	165	46	m	m	NOUN
cana-2486	165	47	=	=	PUNCT
cana-2486	165	48	(	(	PUNCT
cana-2486	165	49	a	a	DET
cana-2486	165	50	∧	∧	PROPN
cana-2486	165	51	(	(	PUNCT
cana-2486	165	52	n	n	NOUN
cana-2486	165	53	:	:	PUNCT
cana-2486	165	54	1	1	NUM
cana-2486	165	55	m	m	NOUN
cana-2486	165	56	)	)	PUNCT
cana-2486	165	57	)	)	PUNCT
cana-2486	166	1	1	1	NUM
cana-2486	166	2	m	m	NOUN
cana-2486	166	3	.	.	PUNCT
cana-2486	167	1	so	so	ADV
cana-2486	167	2	we	we	PRON
cana-2486	167	3	have	have	VERB
cana-2486	167	4	,	,	PUNCT
cana-2486	167	5	a(n	a(n	ADV
cana-2486	167	6	:	:	PUNCT
cana-2486	167	7	1	1	NUM
cana-2486	167	8	m	m	NOUN
cana-2486	167	9	)	)	PUNCT
cana-2486	168	1	=	=	PUNCT
cana-2486	168	2	a	a	DET
cana-2486	168	3	∧	∧	PROPN
cana-2486	168	4	(	(	PUNCT
cana-2486	168	5	n	n	NOUN
cana-2486	168	6	:	:	PUNCT
cana-2486	168	7	1	1	NUM
cana-2486	168	8	m	m	NOUN
cana-2486	168	9	)	)	PUNCT
cana-2486	168	10	.	.	PUNCT
cana-2486	169	1	3	3	NUM
cana-2486	169	2	⇒	⇒	NOUN
cana-2486	169	3	1	1	NUM
cana-2486	169	4	:	:	PUNCT
cana-2486	169	5	let	let	VERB
cana-2486	169	6	a	a	DET
cana-2486	169	7	∈	∈	PROPN
cana-2486	169	8	l.	l.	NOUN
cana-2486	169	9	we	we	PRON
cana-2486	169	10	have	have	VERB
cana-2486	169	11	(	(	PUNCT
cana-2486	169	12	n	n	X
cana-2486	169	13	:	:	PUNCT
cana-2486	169	14	1	1	NUM
cana-2486	169	15	m	m	NOUN
cana-2486	169	16	)	)	PUNCT
cana-2486	169	17	1	1	NUM
cana-2486	169	18	m	m	NOUN
cana-2486	169	19	∧	∧	NOUN
cana-2486	169	20	a1	a1	PROPN
cana-2486	169	21	m	m	NOUN
cana-2486	169	22	=	=	PUNCT
cana-2486	169	23	(	(	PUNCT
cana-2486	169	24	(	(	PUNCT
cana-2486	169	25	n	n	X
cana-2486	169	26	:	:	PUNCT
cana-2486	169	27	1	1	NUM
cana-2486	169	28	m	m	NOUN
cana-2486	169	29	)	)	PUNCT
cana-2486	169	30	∧	∧	PROPN
cana-2486	169	31	a)1	a)1	PROPN
cana-2486	169	32	m	m	PROPN
cana-2486	169	33	.	.	PUNCT
cana-2486	170	1	since	since	SCONJ
cana-2486	170	2	(	(	PUNCT
cana-2486	170	3	n	n	X
cana-2486	170	4	:	:	PUNCT
cana-2486	170	5	1	1	NUM
cana-2486	170	6	m	m	NOUN
cana-2486	170	7	)	)	PUNCT
cana-2486	170	8	∧	∧	NOUN
cana-2486	170	9	a	a	DET
cana-2486	170	10	=	=	X
cana-2486	170	11	a(n	a(n	NOUN
cana-2486	170	12	:	:	PUNCT
cana-2486	170	13	1	1	NUM
cana-2486	170	14	m	m	NOUN
cana-2486	170	15	)	)	PUNCT
cana-2486	170	16	,	,	PUNCT
cana-2486	170	17	implies	imply	VERB
cana-2486	170	18	that	that	SCONJ
cana-2486	170	19	n	n	NUM
cana-2486	170	20	∧	∧	PROPN
cana-2486	170	21	a1	a1	PROPN
cana-2486	170	22	m	m	NOUN
cana-2486	170	23	=	=	PUNCT
cana-2486	170	24	(	(	PUNCT
cana-2486	170	25	n	n	NOUN
cana-2486	170	26	:	:	PUNCT
cana-2486	170	27	1	1	NUM
cana-2486	170	28	m	m	NOUN
cana-2486	170	29	)	)	PUNCT
cana-2486	170	30	1	1	NUM
cana-2486	170	31	m	m	NOUN
cana-2486	170	32	∧	∧	NOUN
cana-2486	170	33	a1	a1	PROPN
cana-2486	170	34	m	m	NOUN
cana-2486	170	35	=	=	PUNCT
cana-2486	170	36	(	(	PUNCT
cana-2486	170	37	(	(	PUNCT
cana-2486	170	38	n	n	X
cana-2486	170	39	:	:	PUNCT
cana-2486	170	40	1	1	NUM
cana-2486	170	41	m	m	NOUN
cana-2486	170	42	)	)	PUNCT
cana-2486	170	43	∧	∧	PROPN
cana-2486	170	44	a)1	a)1	NOUN
cana-2486	170	45	m	m	NOUN
cana-2486	170	46	=	=	PUNCT
cana-2486	170	47	a(n	a(n	NOUN
cana-2486	170	48	:	:	PUNCT
cana-2486	170	49	1	1	NUM
cana-2486	170	50	m	m	NOUN
cana-2486	170	51	)	)	PUNCT
cana-2486	170	52	1	1	NUM
cana-2486	170	53	m	m	NOUN
cana-2486	170	54	=	=	SYM
cana-2486	170	55	an	an	PRON
cana-2486	170	56	.	.	PUNCT
cana-2486	171	1	hence	hence	ADV
cana-2486	171	2	n	n	ADV
cana-2486	171	3	is	be	AUX
cana-2486	171	4	a	a	DET
cana-2486	171	5	pure	pure	ADJ
cana-2486	171	6	element	element	NOUN
cana-2486	171	7	in	in	ADP
cana-2486	171	8	m.	m.	NOUN
cana-2486	171	9	theorem	theorem	VERB
cana-2486	171	10	2.13	2.13	NUM
cana-2486	171	11	.	.	PUNCT
cana-2486	172	1	let	let	VERB
cana-2486	172	2	l	l	NOUN
cana-2486	172	3	be	be	AUX
cana-2486	172	4	a	a	DET
cana-2486	172	5	cg	cg	NOUN
cana-2486	172	6	-	-	PUNCT
cana-2486	172	7	multiplicative	multiplicative	ADJ
cana-2486	172	8	lattice	lattice	NOUN
cana-2486	172	9	and	and	CCONJ
cana-2486	172	10	m	m	VERB
cana-2486	172	11	a	a	DET
cana-2486	172	12	faithful	faithful	ADJ
cana-2486	172	13	multiplication	multiplication	NOUN
cana-2486	172	14	l	l	NOUN
cana-2486	172	15	-	-	NOUN
cana-2486	172	16	module	module	NOUN
cana-2486	172	17	.	.	PUNCT
cana-2486	173	1	if	if	SCONJ
cana-2486	173	2	n	n	PRON
cana-2486	173	3	is	be	AUX
cana-2486	173	4	pure	pure	ADJ
cana-2486	173	5	in	in	ADP
cana-2486	173	6	m	m	PROPN
cana-2486	173	7	,	,	PUNCT
cana-2486	173	8	then	then	ADV
cana-2486	173	9	(	(	PUNCT
cana-2486	173	10	n	n	X
cana-2486	173	11	:	:	PUNCT
cana-2486	173	12	1	1	NUM
cana-2486	173	13	m	m	NOUN
cana-2486	173	14	)	)	PUNCT
cana-2486	173	15	is	be	AUX
cana-2486	173	16	the	the	DET
cana-2486	173	17	smallest	small	ADJ
cana-2486	173	18	element	element	NOUN
cana-2486	173	19	a	a	DET
cana-2486	173	20	∈	∈	PROPN
cana-2486	173	21	l	l	NOUN
cana-2486	173	22	,	,	PUNCT
cana-2486	173	23	such	such	ADJ
cana-2486	173	24	that	that	SCONJ
cana-2486	173	25	n	n	NOUN
cana-2486	173	26	=	=	SYM
cana-2486	173	27	an	an	PROPN
cana-2486	173	28	.	.	PUNCT
cana-2486	173	29	proof	proof	NOUN
cana-2486	173	30	.	.	PUNCT
cana-2486	174	1	let	let	VERB
cana-2486	174	2	λ	λ	NOUN
cana-2486	174	3	be	be	AUX
cana-2486	174	4	the	the	DET
cana-2486	174	5	collection	collection	NOUN
cana-2486	174	6	of	of	ADP
cana-2486	174	7	all	all	DET
cana-2486	174	8	elements	element	NOUN
cana-2486	174	9	a	a	PRON
cana-2486	174	10	of	of	ADP
cana-2486	174	11	l	l	NOUN
cana-2486	174	12	with	with	ADP
cana-2486	174	13	the	the	DET
cana-2486	174	14	property	property	NOUN
cana-2486	174	15	that	that	PRON
cana-2486	174	16	n	n	X
cana-2486	174	17	=	=	SYM
cana-2486	174	18	an	an	PROPN
cana-2486	174	19	.	.	PUNCT
cana-2486	175	1	then	then	ADV
cana-2486	175	2	n	n	PROPN
cana-2486	175	3	=	=	SYM
cana-2486	175	4	∧a∈λ	∧a∈λ	NUM
cana-2486	175	5	an	an	X
cana-2486	175	6	=	=	X
cana-2486	175	7	(	(	PUNCT
cana-2486	175	8	∧	∧	PROPN
cana-2486	175	9	a∈λa)n	a∈λa)n	ADV
cana-2486	175	10	.	.	PUNCT
cana-2486	176	1	it	it	PRON
cana-2486	176	2	follows	follow	VERB
cana-2486	176	3	that	that	PRON
cana-2486	176	4	(	(	PUNCT
cana-2486	176	5	n	n	X
cana-2486	176	6	:	:	PUNCT
cana-2486	176	7	1	1	NUM
cana-2486	176	8	m	m	NOUN
cana-2486	176	9	)	)	PUNCT
cana-2486	177	1	=	=	SYM
cana-2486	177	2	(	(	PUNCT
cana-2486	177	3	(	(	PUNCT
cana-2486	177	4	∧	∧	PROPN
cana-2486	177	5	a∈λ	a∈λ	NOUN
cana-2486	177	6	a)n	a)n	NOUN
cana-2486	177	7	:	:	PUNCT
cana-2486	177	8	1	1	NUM
cana-2486	177	9	m	m	NOUN
cana-2486	177	10	)	)	PUNCT
cana-2486	177	11	=	=	SYM
cana-2486	178	1	(	(	PUNCT
cana-2486	178	2	∧	∧	PROPN
cana-2486	178	3	a∈λ	a∈λ	NOUN
cana-2486	178	4	a)(n	a)(n	PROPN
cana-2486	178	5	:	:	PUNCT
cana-2486	178	6	1	1	NUM
cana-2486	178	7	m	m	NOUN
cana-2486	178	8	)	)	PUNCT
cana-2486	178	9	,	,	PUNCT
cana-2486	178	10	and	and	CCONJ
cana-2486	178	11	hence	hence	ADV
cana-2486	178	12	(	(	PUNCT
cana-2486	178	13	n	n	X
cana-2486	178	14	:	:	PUNCT
cana-2486	178	15	1	1	NUM
cana-2486	178	16	m	m	NOUN
cana-2486	178	17	)	)	PUNCT
cana-2486	178	18	≤(∧	≤(∧	PROPN
cana-2486	178	19	a∈λ	a∈λ	VERB
cana-2486	178	20	a	a	PRON
cana-2486	178	21	)	)	PUNCT
cana-2486	178	22	.	.	PUNCT
cana-2486	179	1	but	but	CCONJ
cana-2486	179	2	n	n	PRON
cana-2486	179	3	is	be	AUX
cana-2486	179	4	pure	pure	ADJ
cana-2486	179	5	,	,	PUNCT
cana-2486	179	6	and	and	CCONJ
cana-2486	179	7	hence	hence	ADV
cana-2486	179	8	an	an	DET
cana-2486	179	9	idempotent	idempotent	NOUN
cana-2486	179	10	.	.	PUNCT
cana-2486	180	1	thus	thus	ADV
cana-2486	180	2	n=	n=	X
cana-2486	180	3	(	(	PUNCT
cana-2486	180	4	n	n	X
cana-2486	180	5	:	:	PUNCT
cana-2486	180	6	1	1	NUM
cana-2486	180	7	m	m	NOUN
cana-2486	180	8	)	)	PUNCT
cana-2486	180	9	n	n	CCONJ
cana-2486	180	10	,	,	PUNCT
cana-2486	180	11	and	and	CCONJ
cana-2486	180	12	this	this	PRON
cana-2486	180	13	means	mean	VERB
cana-2486	180	14	that	that	SCONJ
cana-2486	180	15	(	(	PUNCT
cana-2486	180	16	n	n	X
cana-2486	180	17	:	:	PUNCT
cana-2486	180	18	1	1	NUM
cana-2486	180	19	m	m	NOUN
cana-2486	180	20	)	)	PUNCT
cana-2486	180	21	∈	∈	PROPN
cana-2486	180	22	λ	λ	PROPN
cana-2486	180	23	.	.	PUNCT
cana-2486	181	1	so	so	ADV
cana-2486	181	2	(	(	PUNCT
cana-2486	181	3	n	n	X
cana-2486	181	4	:	:	PUNCT
cana-2486	181	5	1	1	NUM
cana-2486	181	6	m	m	NOUN
cana-2486	181	7	)	)	PUNCT
cana-2486	181	8	is	be	AUX
cana-2486	181	9	the	the	DET
cana-2486	181	10	smallest	small	ADJ
cana-2486	181	11	element	element	NOUN
cana-2486	181	12	of	of	ADP
cana-2486	181	13	λ	λ	PROPN
cana-2486	181	14	.	.	PUNCT
cana-2486	182	1	let	let	VERB
cana-2486	182	2	m	m	PRON
cana-2486	182	3	be	be	AUX
cana-2486	182	4	a	a	DET
cana-2486	182	5	l	l	NOUN
cana-2486	182	6	-	-	NOUN
cana-2486	182	7	module	module	NOUN
cana-2486	182	8	.	.	PUNCT
cana-2486	183	1	a	a	DET
cana-2486	183	2	proper	proper	ADJ
cana-2486	183	3	element	element	NOUN
cana-2486	183	4	p	p	NOUN
cana-2486	183	5	of	of	ADP
cana-2486	183	6	m	m	PROPN
cana-2486	183	7	is	be	AUX
cana-2486	183	8	called	call	VERB
cana-2486	183	9	a	a	DET
cana-2486	183	10	prime	prime	ADJ
cana-2486	183	11	element	element	NOUN
cana-2486	183	12	of	of	ADP
cana-2486	183	13	m	m	PROPN
cana-2486	183	14	,	,	PUNCT
cana-2486	183	15	if	if	SCONJ
cana-2486	183	16	p	p	PRON
cana-2486	183	17	≠	≠	PROPN
cana-2486	183	18	1	1	NUM
cana-2486	183	19	m	m	NOUN
cana-2486	183	20	and	and	CCONJ
cana-2486	183	21	whenever	whenever	SCONJ
cana-2486	183	22	rn	rn	PROPN
cana-2486	183	23	≤	≤	NOUN
cana-2486	183	24	p	p	X
cana-2486	183	25	,	,	PUNCT
cana-2486	183	26	for	for	ADP
cana-2486	183	27	some	some	DET
cana-2486	183	28	n	n	PRON
cana-2486	183	29	∈	∈	NOUN
cana-2486	183	30	m	m	NOUN
cana-2486	183	31	and	and	CCONJ
cana-2486	183	32	r	r	NOUN
cana-2486	183	33	∈	∈	PROPN
cana-2486	183	34	l	l	NOUN
cana-2486	183	35	,	,	PUNCT
cana-2486	183	36	then	then	ADV
cana-2486	183	37	n	n	CCONJ
cana-2486	183	38	≤	≤	NOUN
cana-2486	183	39	p	p	NOUN
cana-2486	183	40	or	or	CCONJ
cana-2486	183	41	r	r	NOUN
cana-2486	183	42	≤	≤	NUM
cana-2486	183	43	(	(	PUNCT
cana-2486	183	44	p	p	X
cana-2486	183	45	:	:	PUNCT
cana-2486	183	46	1	1	NUM
cana-2486	183	47	m	m	NOUN
cana-2486	183	48	)	)	PUNCT
cana-2486	183	49	.	.	PUNCT
cana-2486	184	1	the	the	DET
cana-2486	184	2	m	m	PROPN
cana-2486	184	3	-radical	-radical	PROPN
cana-2486	184	4	,	,	PUNCT
cana-2486	184	5	rad	rad	PROPN
cana-2486	184	6	n	n	CCONJ
cana-2486	184	7	,	,	PUNCT
cana-2486	184	8	of	of	ADP
cana-2486	184	9	an	an	DET
cana-2486	184	10	element	element	NOUN
cana-2486	184	11	n	n	PROPN
cana-2486	184	12	of	of	ADP
cana-2486	184	13	m	m	PROPN
cana-2486	184	14	is	be	AUX
cana-2486	184	15	defined	define	VERB
cana-2486	184	16	as	as	ADP
cana-2486	184	17	the	the	DET
cana-2486	184	18	meet	meet	NOUN
cana-2486	184	19	of	of	ADP
cana-2486	184	20	all	all	DET
cana-2486	184	21	prime	prime	ADJ
cana-2486	184	22	elements	element	NOUN
cana-2486	184	23	of	of	ADP
cana-2486	184	24	m	m	AUX
cana-2486	184	25	containing	contain	VERB
cana-2486	184	26	n.	n.	NOUN
cana-2486	184	27	if	if	SCONJ
cana-2486	184	28	a	a	PRON
cana-2486	184	29	is	be	AUX
cana-2486	184	30	an	an	DET
cana-2486	184	31	element	element	NOUN
cana-2486	184	32	of	of	ADP
cana-2486	184	33	l	l	NOUN
cana-2486	184	34	,	,	PUNCT
cana-2486	184	35	then	then	ADV
cana-2486	184	36	√𝑎	√𝑎	NOUN
cana-2486	184	37	is	be	AUX
cana-2486	184	38	defined	define	VERB
cana-2486	184	39	as	as	ADP
cana-2486	184	40	the	the	DET
cana-2486	184	41	meet	meet	NOUN
cana-2486	184	42	of	of	ADP
cana-2486	184	43	all	all	DET
cana-2486	184	44	prime	prime	ADJ
cana-2486	184	45	elements	element	NOUN
cana-2486	184	46	of	of	ADP
cana-2486	184	47	l	l	NOUN
cana-2486	184	48	containing	contain	VERB
cana-2486	184	49	a.	a.	NOUN
cana-2486	184	50	if	if	SCONJ
cana-2486	184	51	a	a	PRON
cana-2486	184	52	is	be	AUX
cana-2486	184	53	a	a	DET
cana-2486	184	54	pure	pure	ADJ
cana-2486	184	55	(	(	PUNCT
cana-2486	184	56	and	and	CCONJ
cana-2486	184	57	hence	hence	ADV
cana-2486	184	58	idempotent	idempotent	ADJ
cana-2486	184	59	)	)	PUNCT
cana-2486	184	60	element	element	NOUN
cana-2486	184	61	of	of	ADP
cana-2486	184	62	l	l	NOUN
cana-2486	184	63	,	,	PUNCT
cana-2486	184	64	then	then	ADV
cana-2486	184	65	a	a	DET
cana-2486	184	66	=	=	PUNCT
cana-2486	184	67	a√𝑎.	a√𝑎.	NOUN
cana-2486	184	68	lemma	lemma	PROPN
cana-2486	184	69	2.14	2.14	NUM
cana-2486	184	70	.	.	PUNCT
cana-2486	185	1	let	let	VERB
cana-2486	185	2	n	n	PRON
cana-2486	185	3	be	be	AUX
cana-2486	185	4	a	a	DET
cana-2486	185	5	element	element	NOUN
cana-2486	185	6	of	of	ADP
cana-2486	185	7	an	an	DET
cana-2486	185	8	l	l	NOUN
cana-2486	185	9	-	-	PUNCT
cana-2486	185	10	module	module	NOUN
cana-2486	185	11	m.	m.	NOUN
cana-2486	185	12	then	then	ADV
cana-2486	185	13	√(𝑁	√(𝑁	PROPN
cana-2486	185	14	∶	∶	NOUN
cana-2486	185	15	1𝑀	1𝑀	NOUN
cana-2486	185	16	)	)	PUNCT
cana-2486	185	17	1	1	NUM
cana-2486	185	18	m	m	NOUN
cana-2486	185	19	≤	≤	NOUN
cana-2486	185	20	radn	radn	NOUN
cana-2486	185	21	.	.	PUNCT
cana-2486	186	1	proof	proof	NOUN
cana-2486	186	2	.	.	PUNCT
cana-2486	187	1	if	if	SCONJ
cana-2486	187	2	radn	radn	NOUN
cana-2486	187	3	=	=	SYM
cana-2486	187	4	1	1	NUM
cana-2486	187	5	m	m	NOUN
cana-2486	187	6	,	,	PUNCT
cana-2486	187	7	the	the	DET
cana-2486	187	8	result	result	NOUN
cana-2486	187	9	is	be	AUX
cana-2486	187	10	clear	clear	ADJ
cana-2486	187	11	.	.	PUNCT
cana-2486	188	1	otherwise	otherwise	ADV
cana-2486	188	2	,	,	PUNCT
cana-2486	188	3	if	if	SCONJ
cana-2486	188	4	p	p	NOUN
cana-2486	188	5	is	be	AUX
cana-2486	188	6	any	any	DET
cana-2486	188	7	prime	prime	ADJ
cana-2486	188	8	element	element	NOUN
cana-2486	188	9	of	of	ADP
cana-2486	188	10	m	m	PRON
cana-2486	188	11	which	which	PRON
cana-2486	188	12	contains	contain	VERB
cana-2486	188	13	n	n	PRON
cana-2486	188	14	,	,	PUNCT
cana-2486	188	15	then	then	ADV
cana-2486	188	16	(	(	PUNCT
cana-2486	188	17	n	n	X
cana-2486	188	18	:	:	PUNCT
cana-2486	188	19	1m)≤(p	1m)≤(p	NUM
cana-2486	188	20	:	:	PUNCT
cana-2486	188	21	1	1	NUM
cana-2486	188	22	m	m	NOUN
cana-2486	188	23	)	)	PUNCT
cana-2486	188	24	.	.	PUNCT
cana-2486	189	1	as	as	SCONJ
cana-2486	189	2	p	p	PRON
cana-2486	189	3	is	be	AUX
cana-2486	189	4	a	a	DET
cana-2486	189	5	prime	prime	ADJ
cana-2486	189	6	element	element	NOUN
cana-2486	189	7	of	of	ADP
cana-2486	189	8	m	m	PROPN
cana-2486	189	9	,	,	PUNCT
cana-2486	189	10	so	so	ADV
cana-2486	189	11	(	(	PUNCT
cana-2486	189	12	p	p	X
cana-2486	189	13	:	:	PUNCT
cana-2486	189	14	1	1	NUM
cana-2486	189	15	m	m	VERB
cana-2486	189	16	)	)	PUNCT
cana-2486	189	17	is	be	AUX
cana-2486	189	18	a	a	DET
cana-2486	189	19	prime	prime	ADJ
cana-2486	189	20	element	element	NOUN
cana-2486	189	21	of	of	ADP
cana-2486	189	22	l.	l.	PROPN
cana-2486	189	23	hence	hence	ADV
cana-2486	189	24	√(𝑁	√(𝑁	PROPN
cana-2486	189	25	∶	∶	PROPN
cana-2486	189	26	1𝑀	1𝑀	NOUN
cana-2486	189	27	)	)	PUNCT
cana-2486	189	28	≤	≤	NOUN
cana-2486	189	29	(	(	PUNCT
cana-2486	189	30	p	p	X
cana-2486	189	31	:	:	PUNCT
cana-2486	189	32	1	1	NUM
cana-2486	189	33	m	m	NOUN
cana-2486	189	34	)	)	PUNCT
cana-2486	189	35	and	and	CCONJ
cana-2486	189	36	thus	thus	ADV
cana-2486	189	37	√(𝑁	√(𝑁	PROPN
cana-2486	189	38	∶	∶	NOUN
cana-2486	189	39	1𝑀)1	1𝑀)1	NUM
cana-2486	189	40	m	m	NOUN
cana-2486	189	41	≤	≤	NOUN
cana-2486	190	1	(	(	PUNCT
cana-2486	190	2	p	p	X
cana-2486	190	3	:	:	PUNCT
cana-2486	190	4	1	1	NUM
cana-2486	190	5	m	m	NOUN
cana-2486	190	6	)	)	PUNCT
cana-2486	190	7	1	1	NUM
cana-2486	190	8	m	m	NOUN
cana-2486	190	9	=	=	NOUN
cana-2486	191	1	p.	p.	NOUN
cana-2486	191	2	since	since	SCONJ
cana-2486	191	3	p	p	NOUN
cana-2486	191	4	is	be	AUX
cana-2486	191	5	an	an	DET
cana-2486	191	6	arbitrary	arbitrary	ADJ
cana-2486	191	7	element	element	NOUN
cana-2486	191	8	containing	contain	VERB
cana-2486	191	9	n	n	CCONJ
cana-2486	191	10	,	,	PUNCT
cana-2486	191	11	we	we	PRON
cana-2486	191	12	have	have	VERB
cana-2486	191	13	√(𝑁	√(𝑁	PROPN
cana-2486	191	14	∶	∶	NOUN
cana-2486	191	15	1𝑀)1	1𝑀)1	NUM
cana-2486	191	16	m	m	PROPN
cana-2486	191	17	≤	≤	NOUN
cana-2486	191	18	radn	radn	NOUN
cana-2486	191	19	.	.	PUNCT
cana-2486	192	1	proposition	proposition	NOUN
cana-2486	192	2	2.15	2.15	NUM
cana-2486	192	3	.	.	PUNCT
cana-2486	193	1	[	[	X
cana-2486	193	2	4	4	X
cana-2486	193	3	]	]	X
cana-2486	193	4	let	let	AUX
cana-2486	193	5	l	l	NOUN
cana-2486	193	6	be	be	AUX
cana-2486	193	7	a	a	DET
cana-2486	193	8	multiplicative	multiplicative	ADJ
cana-2486	193	9	pg	pg	NOUN
cana-2486	193	10	-	-	PUNCT
cana-2486	193	11	lattice	lattice	NOUN
cana-2486	193	12	.	.	PUNCT
cana-2486	194	1	let	let	VERB
cana-2486	194	2	m	m	PRON
cana-2486	194	3	be	be	AUX
cana-2486	194	4	a	a	DET
cana-2486	194	5	multiplication	multiplication	NOUN
cana-2486	194	6	lmodule	lmodule	NOUN
cana-2486	194	7	and	and	CCONJ
cana-2486	194	8	ann(m	ann(m	PROPN
cana-2486	194	9	)	)	PUNCT
cana-2486	195	1	≤	≤	NUM
cana-2486	195	2	b	b	NOUN
cana-2486	195	3	for	for	ADP
cana-2486	195	4	some	some	DET
cana-2486	195	5	prime	prime	ADJ
cana-2486	195	6	element	element	NOUN
cana-2486	195	7	b	b	PROPN
cana-2486	195	8	∈	∈	PROPN
cana-2486	195	9	l.	l.	NOUN
cana-2486	195	10	if	if	SCONJ
cana-2486	195	11	a1	a1	PROPN
cana-2486	195	12	m	m	VERB
cana-2486	195	13	≤	≤	NOUN
cana-2486	195	14	b1	b1	NOUN
cana-2486	195	15	m	m	PROPN
cana-2486	195	16	for	for	ADP
cana-2486	195	17	some	some	DET
cana-2486	195	18	a	a	DET
cana-2486	195	19	∈	∈	PROPN
cana-2486	195	20	l	l	NOUN
cana-2486	195	21	,	,	PUNCT
cana-2486	195	22	then	then	ADV
cana-2486	195	23	a	a	DET
cana-2486	195	24	≤	≤	PROPN
cana-2486	195	25	b	b	NOUN
cana-2486	195	26	or	or	CCONJ
cana-2486	195	27	b1	b1	NOUN
cana-2486	195	28	m	m	NOUN
cana-2486	195	29	=	=	SYM
cana-2486	195	30	1	1	NUM
cana-2486	195	31	m.	m.	NOUN
cana-2486	195	32	lemma	lemma	PROPN
cana-2486	195	33	2.16	2.16	NUM
cana-2486	195	34	.	.	PUNCT
cana-2486	196	1	let	let	VERB
cana-2486	196	2	l	l	NOUN
cana-2486	196	3	be	be	AUX
cana-2486	196	4	a	a	DET
cana-2486	196	5	multiplicative	multiplicative	ADJ
cana-2486	196	6	pg	pg	NOUN
cana-2486	196	7	-	-	PUNCT
cana-2486	196	8	lattice	lattice	NOUN
cana-2486	196	9	.	.	PUNCT
cana-2486	197	1	let	let	VERB
cana-2486	197	2	m	m	PRON
cana-2486	197	3	be	be	AUX
cana-2486	197	4	a	a	DET
cana-2486	197	5	multiplication	multiplication	NOUN
cana-2486	197	6	l	l	NOUN
cana-2486	197	7	-	-	NOUN
cana-2486	197	8	module	module	NOUN
cana-2486	197	9	such	such	ADJ
cana-2486	197	10	that	that	SCONJ
cana-2486	197	11	communications	communication	NOUN
cana-2486	197	12	on	on	ADP
cana-2486	197	13	applied	apply	VERB
cana-2486	197	14	nonlinear	nonlinear	ADJ
cana-2486	197	15	analysis	analysis	NOUN
cana-2486	197	16	issn	issn	NOUN
cana-2486	197	17	:	:	PUNCT
cana-2486	197	18	1074	1074	NUM
cana-2486	197	19	-	-	PUNCT
cana-2486	197	20	133x	133x	NUM
cana-2486	197	21	vol	vol	NOUN
cana-2486	197	22	.	.	PROPN
cana-2486	198	1	32	32	NUM
cana-2486	198	2	no	no	INTJ
cana-2486	198	3	.	.	PUNCT
cana-2486	199	1	2s	2s	NUM
cana-2486	199	2	(	(	PUNCT
cana-2486	199	3	2024	2024	NUM
cana-2486	199	4	)	)	PUNCT
cana-2486	199	5	543	543	NUM
cana-2486	199	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2486	199	7	1	1	NUM
cana-2486	199	8	m	m	NOUN
cana-2486	199	9	compact	compact	ADJ
cana-2486	199	10	and	and	CCONJ
cana-2486	199	11	ann(m	ann(m	PROPN
cana-2486	199	12	)	)	PUNCT
cana-2486	199	13	≤	≤	NOUN
cana-2486	199	14	a	a	PRON
cana-2486	199	15	for	for	ADP
cana-2486	199	16	prime	prime	ADJ
cana-2486	199	17	element	element	NOUN
cana-2486	199	18	a	a	DET
cana-2486	199	19	∈	∈	PROPN
cana-2486	199	20	l.	l.	NOUN
cana-2486	199	21	then	then	ADV
cana-2486	199	22	a1	a1	PROPN
cana-2486	199	23	m	m	PROPN
cana-2486	199	24	is	be	AUX
cana-2486	199	25	a	a	DET
cana-2486	199	26	prime	prime	ADJ
cana-2486	199	27	element	element	NOUN
cana-2486	199	28	of	of	ADP
cana-2486	199	29	m.	m.	NOUN
cana-2486	199	30	proof	proof	NOUN
cana-2486	199	31	.	.	PUNCT
cana-2486	200	1	note	note	VERB
cana-2486	200	2	that	that	SCONJ
cana-2486	200	3	a1	a1	PROPN
cana-2486	200	4	m	m	NOUN
cana-2486	200	5	≠	≠	PROPN
cana-2486	200	6	1	1	NUM
cana-2486	200	7	m	m	NOUN
cana-2486	200	8	and	and	CCONJ
cana-2486	200	9	for	for	ADP
cana-2486	200	10	b	b	PROPN
cana-2486	200	11	∈	∈	PROPN
cana-2486	200	12	l	l	NOUN
cana-2486	200	13	and	and	CCONJ
cana-2486	200	14	n	n	CCONJ
cana-2486	200	15	∈	∈	PROPN
cana-2486	200	16	m	m	PROPN
cana-2486	200	17	,	,	PUNCT
cana-2486	200	18	suppose	suppose	VERB
cana-2486	200	19	that	that	SCONJ
cana-2486	200	20	bn	bn	ADJ
cana-2486	200	21	≤	≤	NUM
cana-2486	200	22	a1	a1	NOUN
cana-2486	200	23	m	m	NOUN
cana-2486	200	24	.	.	PUNCT
cana-2486	201	1	as	as	SCONJ
cana-2486	201	2	m	m	PROPN
cana-2486	201	3	is	be	AUX
cana-2486	201	4	a	a	DET
cana-2486	201	5	multiplication	multiplication	NOUN
cana-2486	201	6	l	l	NOUN
cana-2486	201	7	-	-	NOUN
cana-2486	201	8	module	module	NOUN
cana-2486	201	9	,	,	PUNCT
cana-2486	201	10	we	we	PRON
cana-2486	201	11	have	have	VERB
cana-2486	201	12	n	n	NOUN
cana-2486	201	13	=	=	PROPN
cana-2486	201	14	c1	c1	PROPN
cana-2486	201	15	m	m	PROPN
cana-2486	201	16	,	,	PUNCT
cana-2486	201	17	for	for	ADP
cana-2486	201	18	c	c	PROPN
cana-2486	201	19	∈	∈	PROPN
cana-2486	201	20	l	l	NOUN
cana-2486	201	21	,	,	PUNCT
cana-2486	201	22	so	so	ADV
cana-2486	201	23	bn	bn	PROPN
cana-2486	201	24	=	=	PUNCT
cana-2486	201	25	b(c1	b(c1	PROPN
cana-2486	201	26	m	m	NOUN
cana-2486	201	27	)	)	PUNCT
cana-2486	201	28	≤	≤	NUM
cana-2486	201	29	a1	a1	NOUN
cana-2486	201	30	m.	m.	NOUN
cana-2486	201	31	proposition	proposition	NOUN
cana-2486	201	32	2.15	2.15	NUM
cana-2486	201	33	implies	imply	VERB
cana-2486	201	34	that	that	PRON
cana-2486	201	35	bc	bc	PROPN
cana-2486	201	36	≤	≤	PROPN
cana-2486	201	37	a	a	PRON
cana-2486	201	38	,	,	PUNCT
cana-2486	201	39	hence	hence	ADV
cana-2486	201	40	b	b	NOUN
cana-2486	201	41	≤	≤	NOUN
cana-2486	201	42	a	a	PRON
cana-2486	201	43	or	or	CCONJ
cana-2486	201	44	c	c	NOUN
cana-2486	201	45	≤	≤	NOUN
cana-2486	201	46	a	a	DET
cana-2486	201	47	=	=	PUNCT
cana-2486	201	48	(	(	PUNCT
cana-2486	201	49	a1	a1	PROPN
cana-2486	201	50	m	m	NOUN
cana-2486	201	51	:	:	PUNCT
cana-2486	201	52	1	1	NUM
cana-2486	201	53	m	m	NOUN
cana-2486	201	54	)	)	PUNCT
cana-2486	201	55	,	,	PUNCT
cana-2486	201	56	then	then	ADV
cana-2486	201	57	n	n	PROPN
cana-2486	201	58	=	=	PROPN
cana-2486	201	59	c1	c1	PROPN
cana-2486	201	60	m	m	NOUN
cana-2486	201	61	≤	≤	NOUN
cana-2486	201	62	a1	a1	NOUN
cana-2486	201	63	m	m	NOUN
cana-2486	201	64	and	and	CCONJ
cana-2486	201	65	the	the	DET
cana-2486	201	66	proof	proof	NOUN
cana-2486	201	67	is	be	AUX
cana-2486	201	68	complete	complete	ADJ
cana-2486	201	69	.	.	PUNCT
cana-2486	202	1	theorem	theorem	VERB
cana-2486	202	2	2.17	2.17	NUM
cana-2486	202	3	.	.	PUNCT
cana-2486	203	1	let	let	VERB
cana-2486	203	2	l	l	NOUN
cana-2486	203	3	be	be	AUX
cana-2486	203	4	a	a	DET
cana-2486	203	5	multiplicative	multiplicative	ADJ
cana-2486	203	6	pg	pg	NOUN
cana-2486	203	7	-	-	PUNCT
cana-2486	203	8	lattice	lattice	NOUN
cana-2486	203	9	.	.	PUNCT
cana-2486	204	1	let	let	VERB
cana-2486	204	2	m	m	PRON
cana-2486	204	3	be	be	AUX
cana-2486	204	4	a	a	DET
cana-2486	204	5	multiplication	multiplication	NOUN
cana-2486	204	6	l	l	NOUN
cana-2486	204	7	-	-	NOUN
cana-2486	204	8	module	module	NOUN
cana-2486	204	9	such	such	ADJ
cana-2486	204	10	that	that	DET
cana-2486	204	11	1	1	NUM
cana-2486	204	12	m	m	NOUN
cana-2486	204	13	compact	compact	ADJ
cana-2486	204	14	and	and	CCONJ
cana-2486	204	15	let	let	VERB
cana-2486	204	16	b	b	X
cana-2486	204	17	be	be	AUX
cana-2486	204	18	a	a	DET
cana-2486	204	19	element	element	NOUN
cana-2486	204	20	of	of	ADP
cana-2486	204	21	m.	m.	NOUN
cana-2486	204	22	then	then	ADV
cana-2486	204	23	radb	radb	NOUN
cana-2486	204	24	=	=	PUNCT
cana-2486	204	25	√(𝐵	√(𝐵	PROPN
cana-2486	204	26	∶	∶	NOUN
cana-2486	204	27	1𝑀)1	1𝑀)1	NUM
cana-2486	204	28	m.	m.	NOUN
cana-2486	204	29	proof	proof	NOUN
cana-2486	204	30	.	.	PUNCT
cana-2486	205	1	by	by	ADP
cana-2486	205	2	lemma	lemma	PROPN
cana-2486	205	3	2.14	2.14	NUM
cana-2486	205	4	,	,	PUNCT
cana-2486	205	5	√(𝐵	√(𝐵	PROPN
cana-2486	205	6	∶	∶	NOUN
cana-2486	205	7	1𝑀)1	1𝑀)1	NUM
cana-2486	205	8	m	m	PROPN
cana-2486	205	9	≤	≤	NOUN
cana-2486	205	10	radb	radb	NOUN
cana-2486	205	11	.	.	PUNCT
cana-2486	206	1	since	since	SCONJ
cana-2486	206	2	m	m	PROPN
cana-2486	206	3	is	be	AUX
cana-2486	206	4	a	a	DET
cana-2486	206	5	multiplication	multiplication	NOUN
cana-2486	206	6	l	l	NOUN
cana-2486	206	7	-	-	NOUN
cana-2486	206	8	module	module	NOUN
cana-2486	206	9	,	,	PUNCT
cana-2486	206	10	radb	radb	NOUN
cana-2486	206	11	=	=	SYM
cana-2486	206	12	(	(	PUNCT
cana-2486	206	13	radb	radb	NOUN
cana-2486	206	14	:	:	PUNCT
cana-2486	206	15	1	1	NUM
cana-2486	206	16	m	m	NOUN
cana-2486	206	17	)	)	PUNCT
cana-2486	206	18	1	1	NUM
cana-2486	206	19	m.	m.	NOUN
cana-2486	206	20	it	it	PRON
cana-2486	206	21	suffices	suffice	VERB
cana-2486	206	22	then	then	ADV
cana-2486	206	23	to	to	PART
cana-2486	206	24	show	show	VERB
cana-2486	206	25	that	that	SCONJ
cana-2486	206	26	(	(	PUNCT
cana-2486	206	27	radb	radb	NOUN
cana-2486	206	28	:	:	PUNCT
cana-2486	206	29	1	1	NUM
cana-2486	206	30	m	m	NOUN
cana-2486	206	31	)	)	PUNCT
cana-2486	206	32	≤√(𝐵	≤√(𝐵	PROPN
cana-2486	206	33	∶	∶	NOUN
cana-2486	206	34	1𝑀	1𝑀	NOUN
cana-2486	206	35	)	)	PUNCT
cana-2486	206	36	.	.	PUNCT
cana-2486	207	1	let	let	VERB
cana-2486	207	2	a	a	DET
cana-2486	207	3	be	be	AUX
cana-2486	207	4	any	any	DET
cana-2486	207	5	prime	prime	ADJ
cana-2486	207	6	element	element	NOUN
cana-2486	207	7	such	such	ADJ
cana-2486	207	8	that	that	SCONJ
cana-2486	207	9	(	(	PUNCT
cana-2486	207	10	b	b	X
cana-2486	207	11	:	:	PUNCT
cana-2486	207	12	1	1	NUM
cana-2486	207	13	m	m	NOUN
cana-2486	207	14	)	)	PUNCT
cana-2486	207	15	≤	≤	NUM
cana-2486	207	16	a.	a.	NOUN
cana-2486	207	17	since	since	SCONJ
cana-2486	207	18	a	a	PRON
cana-2486	207	19	is	be	AUX
cana-2486	207	20	a	a	DET
cana-2486	207	21	prime	prime	ADJ
cana-2486	207	22	element	element	NOUN
cana-2486	207	23	containing	contain	VERB
cana-2486	207	24	annm	annm	NOUN
cana-2486	207	25	,	,	PUNCT
cana-2486	207	26	then	then	ADV
cana-2486	207	27	a1	a1	PROPN
cana-2486	207	28	m	m	PROPN
cana-2486	207	29	is	be	AUX
cana-2486	207	30	a	a	DET
cana-2486	207	31	prime	prime	ADJ
cana-2486	207	32	element	element	NOUN
cana-2486	207	33	of	of	ADP
cana-2486	207	34	m	m	AUX
cana-2486	207	35	containing	contain	VERB
cana-2486	207	36	b	b	NOUN
cana-2486	207	37	=	=	PUNCT
cana-2486	207	38	(	(	PUNCT
cana-2486	207	39	b	b	NOUN
cana-2486	207	40	:	:	PUNCT
cana-2486	207	41	1	1	NUM
cana-2486	207	42	m	m	NOUN
cana-2486	207	43	)	)	PUNCT
cana-2486	207	44	1	1	NUM
cana-2486	207	45	m	m	NOUN
cana-2486	207	46	.	.	PUNCT
cana-2486	208	1	hence	hence	ADV
cana-2486	208	2	,	,	PUNCT
cana-2486	208	3	(	(	PUNCT
cana-2486	208	4	radb	radb	NOUN
cana-2486	208	5	:	:	PUNCT
cana-2486	208	6	m	m	X
cana-2486	208	7	)	)	PUNCT
cana-2486	208	8	1	1	NUM
cana-2486	208	9	m	m	NOUN
cana-2486	208	10	=	=	VERB
cana-2486	208	11	radb	radb	NOUN
cana-2486	208	12	≤	≤	NUM
cana-2486	208	13	a1	a1	NOUN
cana-2486	208	14	m	m	NOUN
cana-2486	208	15	,	,	PUNCT
cana-2486	208	16	so	so	SCONJ
cana-2486	208	17	that	that	SCONJ
cana-2486	208	18	(	(	PUNCT
cana-2486	208	19	radb	radb	NOUN
cana-2486	208	20	:	:	PUNCT
cana-2486	208	21	1	1	NUM
cana-2486	208	22	m	m	NOUN
cana-2486	208	23	)	)	PUNCT
cana-2486	208	24	≤	≤	NUM
cana-2486	208	25	a.	a.	NOUN
cana-2486	208	26	consequently	consequently	ADV
cana-2486	208	27	,	,	PUNCT
cana-2486	208	28	(	(	PUNCT
cana-2486	208	29	radb	radb	NOUN
cana-2486	208	30	:	:	PUNCT
cana-2486	208	31	1	1	NUM
cana-2486	208	32	m	m	NOUN
cana-2486	208	33	)	)	PUNCT
cana-2486	208	34	≤	≤	PUNCT
cana-2486	208	35	√(𝐵	√(𝐵	PROPN
cana-2486	208	36	∶	∶	NOUN
cana-2486	208	37	1𝑀	1𝑀	NOUN
cana-2486	208	38	)	)	PUNCT
cana-2486	208	39	.	.	PUNCT
cana-2486	209	1	the	the	DET
cana-2486	209	2	next	next	ADJ
cana-2486	209	3	result	result	NOUN
cana-2486	209	4	generalizes	generalize	VERB
cana-2486	209	5	the	the	DET
cana-2486	209	6	above	above	ADJ
cana-2486	209	7	facts	fact	NOUN
cana-2486	209	8	to	to	ADP
cana-2486	209	9	pure	pure	ADJ
cana-2486	209	10	element	element	NOUN
cana-2486	209	11	of	of	ADP
cana-2486	209	12	multiplication	multiplication	NOUN
cana-2486	209	13	l	l	NOUN
cana-2486	209	14	-	-	NOUN
cana-2486	209	15	module	module	NOUN
cana-2486	209	16	.	.	PUNCT
cana-2486	210	1	proposition	proposition	NOUN
cana-2486	210	2	2.18	2.18	NUM
cana-2486	210	3	.	.	PUNCT
cana-2486	211	1	let	let	VERB
cana-2486	211	2	l	l	NOUN
cana-2486	211	3	be	be	AUX
cana-2486	211	4	a	a	DET
cana-2486	211	5	cg	cg	NOUN
cana-2486	211	6	-	-	PUNCT
cana-2486	211	7	multiplicative	multiplicative	ADJ
cana-2486	211	8	lattice	lattice	NOUN
cana-2486	211	9	and	and	CCONJ
cana-2486	211	10	m	m	VERB
cana-2486	211	11	a	a	DET
cana-2486	211	12	faithful	faithful	ADJ
cana-2486	211	13	multiplication	multiplication	NOUN
cana-2486	211	14	l	l	NOUN
cana-2486	211	15	module	module	NOUN
cana-2486	211	16	.	.	PUNCT
cana-2486	212	1	let	let	VERB
cana-2486	212	2	n	n	PRON
cana-2486	212	3	be	be	AUX
cana-2486	212	4	a	a	DET
cana-2486	212	5	pure	pure	ADJ
cana-2486	212	6	element	element	NOUN
cana-2486	212	7	of	of	ADP
cana-2486	212	8	m.	m.	NOUN
cana-2486	212	9	then	then	ADV
cana-2486	212	10	1	1	X
cana-2486	212	11	.	.	PUNCT
cana-2486	213	1	n	n	NOUN
cana-2486	213	2	=	=	SYM
cana-2486	213	3	√(𝑁	√(𝑁	PROPN
cana-2486	213	4	∶	∶	NOUN
cana-2486	213	5	1𝑀)n	1𝑀)n	NUM
cana-2486	213	6	,	,	PUNCT
cana-2486	213	7	2	2	NUM
cana-2486	213	8	.	.	PUNCT
cana-2486	213	9	(	(	PUNCT
cana-2486	213	10	n	n	X
cana-2486	213	11	:	:	PUNCT
cana-2486	213	12	1	1	NUM
cana-2486	213	13	m	m	NOUN
cana-2486	213	14	)	)	PUNCT
cana-2486	213	15	radn	radn	NOUN
cana-2486	213	16	=	=	SYM
cana-2486	213	17	n	n	NOUN
cana-2486	213	18	=	=	PUNCT
cana-2486	213	19	(	(	PUNCT
cana-2486	213	20	radn	radn	NOUN
cana-2486	213	21	:	:	PUNCT
cana-2486	213	22	1	1	NUM
cana-2486	213	23	m	m	NOUN
cana-2486	213	24	)	)	PUNCT
cana-2486	213	25	n.	n.	NOUN
cana-2486	213	26	proof	proof	NOUN
cana-2486	213	27	.	.	PUNCT
cana-2486	214	1	1	1	NUM
cana-2486	214	2	:	:	PUNCT
cana-2486	214	3	⇒	⇒	NOUN
cana-2486	214	4	let	let	AUX
cana-2486	214	5	be	be	AUX
cana-2486	214	6	the	the	DET
cana-2486	214	7	collection	collection	NOUN
cana-2486	214	8	of	of	ADP
cana-2486	214	9	all	all	DET
cana-2486	214	10	prime	prime	ADJ
cana-2486	214	11	elements	element	NOUN
cana-2486	214	12	a	a	PRON
cana-2486	214	13	of	of	ADP
cana-2486	214	14	l	l	NOUN
cana-2486	214	15	contains	contain	VERB
cana-2486	214	16	(	(	PUNCT
cana-2486	214	17	n	n	NOUN
cana-2486	214	18	:	:	PUNCT
cana-2486	214	19	1	1	NUM
cana-2486	214	20	m	m	NOUN
cana-2486	214	21	)	)	PUNCT
cana-2486	214	22	.	.	PUNCT
cana-2486	215	1	then	then	ADV
cana-2486	215	2	√(𝑁	√(𝑁	PROPN
cana-2486	215	3	∶	∶	NOUN
cana-2486	215	4	1𝑀)=	1𝑀)=	NUM
cana-2486	215	5	∧	∧	PROPN
cana-2486	215	6	a∈λa	a∈λa	PROPN
cana-2486	215	7	,	,	PUNCT
cana-2486	215	8	and	and	CCONJ
cana-2486	215	9	so	so	ADV
cana-2486	215	10	,	,	PUNCT
cana-2486	215	11	√(𝑁	√(𝑁	PROPN
cana-2486	215	12	∶	∶	NOUN
cana-2486	215	13	1𝑀	1𝑀	NOUN
cana-2486	215	14	)	)	PUNCT
cana-2486	215	15	n=	n=	NOUN
cana-2486	215	16	(	(	PUNCT
cana-2486	215	17	∧	∧	NOUN
cana-2486	215	18	a∈λa)n=	a∈λa)n=	NOUN
cana-2486	215	19	∧	∧	PROPN
cana-2486	215	20	a∈λan	a∈λan	PROPN
cana-2486	215	21	.	.	PUNCT
cana-2486	216	1	for	for	ADP
cana-2486	216	2	each	each	DET
cana-2486	216	3	a	a	DET
cana-2486	216	4	∈	∈	PROPN
cana-2486	216	5	λ	λ	NOUN
cana-2486	216	6	,	,	PUNCT
cana-2486	216	7	n	n	NOUN
cana-2486	216	8	=	=	SYM
cana-2486	216	9	(	(	PUNCT
cana-2486	216	10	n	n	NOUN
cana-2486	216	11	:	:	PUNCT
cana-2486	216	12	1	1	NUM
cana-2486	216	13	m	m	NOUN
cana-2486	216	14	)	)	PUNCT
cana-2486	217	1	n	n	NOUN
cana-2486	217	2	≤	≤	NOUN
cana-2486	217	3	an	an	DET
cana-2486	217	4	≤	≤	NUM
cana-2486	217	5	n	n	NOUN
cana-2486	217	6	so	so	SCONJ
cana-2486	217	7	that	that	SCONJ
cana-2486	217	8	n	n	NOUN
cana-2486	217	9	=	=	SYM
cana-2486	217	10	an	an	PROPN
cana-2486	217	11	,	,	PUNCT
cana-2486	217	12	and	and	CCONJ
cana-2486	217	13	hence	hence	ADV
cana-2486	217	14	n	n	NOUN
cana-2486	217	15	=	=	SYM
cana-2486	217	16	∧	∧	PROPN
cana-2486	217	17	a∈λa	a∈λa	PROPN
cana-2486	217	18	n=√(𝑁	n=√(𝑁	NOUN
cana-2486	217	19	∶	∶	NOUN
cana-2486	217	20	1𝑀)n	1𝑀)n	NUM
cana-2486	217	21	.	.	PROPN
cana-2486	218	1	2	2	NUM
cana-2486	218	2	:	:	PUNCT
cana-2486	218	3	⇒	⇒	NOUN
cana-2486	218	4	it	it	PRON
cana-2486	218	5	follows	follow	VERB
cana-2486	218	6	from	from	ADP
cana-2486	218	7	(	(	PUNCT
cana-2486	218	8	1	1	NUM
cana-2486	218	9	)	)	PUNCT
cana-2486	218	10	,	,	PUNCT
cana-2486	218	11	and	and	CCONJ
cana-2486	218	12	theorem	theorem	VERB
cana-2486	218	13	2.17	2.17	NUM
cana-2486	218	14	,	,	PUNCT
cana-2486	218	15	that	that	PRON
cana-2486	218	16	n	n	NOUN
cana-2486	218	17	=	=	SYM
cana-2486	218	18	√(𝑁	√(𝑁	PROPN
cana-2486	218	19	∶	∶	NOUN
cana-2486	218	20	1𝑀	1𝑀	NOUN
cana-2486	218	21	)	)	PUNCT
cana-2486	218	22	n	n	NOUN
cana-2486	218	23	=	=	SYM
cana-2486	218	24	√(𝑁	√(𝑁	PROPN
cana-2486	218	25	∶	∶	NOUN
cana-2486	218	26	1𝑀	1𝑀	NOUN
cana-2486	218	27	)	)	PUNCT
cana-2486	218	28	(	(	PUNCT
cana-2486	218	29	n	n	X
cana-2486	218	30	:	:	PUNCT
cana-2486	218	31	1	1	NUM
cana-2486	218	32	m	m	NOUN
cana-2486	218	33	)	)	PUNCT
cana-2486	218	34	1	1	NUM
cana-2486	218	35	m	m	NOUN
cana-2486	218	36	=	=	PUNCT
cana-2486	218	37	(	(	PUNCT
cana-2486	218	38	n	n	NOUN
cana-2486	218	39	:	:	PUNCT
cana-2486	218	40	1	1	NUM
cana-2486	218	41	m	m	NOUN
cana-2486	218	42	)	)	PUNCT
cana-2486	218	43	radn	radn	NOUN
cana-2486	218	44	.	.	PUNCT
cana-2486	219	1	but	but	CCONJ
cana-2486	219	2	radn	radn	VERB
cana-2486	219	3	≤	≤	NUM
cana-2486	219	4	1	1	NUM
cana-2486	219	5	m	m	NOUN
cana-2486	219	6	and	and	CCONJ
cana-2486	219	7	m	m	VERB
cana-2486	219	8	is	be	AUX
cana-2486	219	9	a	a	DET
cana-2486	219	10	multiplication	multiplication	NOUN
cana-2486	219	11	l	l	NOUN
cana-2486	219	12	-	-	NOUN
cana-2486	219	13	module	module	NOUN
cana-2486	219	14	.	.	PUNCT
cana-2486	220	1	thus	thus	ADV
cana-2486	220	2	radn	radn	VERB
cana-2486	220	3	=	=	SYM
cana-2486	220	4	(	(	PUNCT
cana-2486	220	5	radn	radn	NOUN
cana-2486	220	6	:	:	PUNCT
cana-2486	220	7	1	1	NUM
cana-2486	220	8	m	m	NOUN
cana-2486	220	9	)	)	PUNCT
cana-2486	220	10	1	1	NUM
cana-2486	220	11	m	m	NOUN
cana-2486	220	12	,	,	PUNCT
cana-2486	220	13	and	and	CCONJ
cana-2486	220	14	hence	hence	ADV
cana-2486	220	15	(	(	PUNCT
cana-2486	220	16	n	n	X
cana-2486	220	17	:	:	PUNCT
cana-2486	220	18	1	1	NUM
cana-2486	220	19	m	m	NOUN
cana-2486	220	20	)	)	PUNCT
cana-2486	220	21	radn	radn	NOUN
cana-2486	220	22	=	=	SYM
cana-2486	220	23	(	(	PUNCT
cana-2486	220	24	n	n	X
cana-2486	220	25	:	:	PUNCT
cana-2486	220	26	1	1	NUM
cana-2486	220	27	m	m	NOUN
cana-2486	220	28	)	)	PUNCT
cana-2486	220	29	(	(	PUNCT
cana-2486	220	30	radn	radn	NOUN
cana-2486	220	31	:	:	PUNCT
cana-2486	220	32	1	1	NUM
cana-2486	220	33	m	m	NOUN
cana-2486	220	34	)	)	PUNCT
cana-2486	221	1	1	1	NUM
cana-2486	221	2	m	m	NOUN
cana-2486	221	3	=	=	PUNCT
cana-2486	221	4	(	(	PUNCT
cana-2486	221	5	radn	radn	NOUN
cana-2486	221	6	:	:	PUNCT
cana-2486	221	7	1	1	NUM
cana-2486	221	8	m	m	NOUN
cana-2486	221	9	)	)	PUNCT
cana-2486	221	10	n.	n.	PROPN
cana-2486	221	11	iii	iii	PROPN
cana-2486	222	1	.	.	PUNCT
cana-2486	223	1	weakly	weakly	ADJ
cana-2486	223	2	pure	pure	ADJ
cana-2486	223	3	element	element	NOUN
cana-2486	223	4	in	in	ADP
cana-2486	223	5	this	this	DET
cana-2486	223	6	section	section	NOUN
cana-2486	223	7	we	we	PRON
cana-2486	223	8	give	give	VERB
cana-2486	223	9	basic	basic	ADJ
cana-2486	223	10	definition	definition	NOUN
cana-2486	223	11	of	of	ADP
cana-2486	223	12	weakly	weakly	ADJ
cana-2486	223	13	pure	pure	ADJ
cana-2486	223	14	element	element	NOUN
cana-2486	223	15	of	of	ADP
cana-2486	223	16	multiplication	multiplication	NOUN
cana-2486	223	17	l	l	NOUN
cana-2486	223	18	-	-	NOUN
cana-2486	223	19	module	module	NOUN
cana-2486	223	20	,	,	PUNCT
cana-2486	223	21	and	and	CCONJ
cana-2486	223	22	prove	prove	VERB
cana-2486	223	23	some	some	DET
cana-2486	223	24	results	result	NOUN
cana-2486	223	25	related	relate	VERB
cana-2486	223	26	to	to	ADP
cana-2486	223	27	weakly	weakly	ADJ
cana-2486	223	28	pure	pure	ADJ
cana-2486	223	29	element	element	NOUN
cana-2486	223	30	.	.	PUNCT
cana-2486	224	1	we	we	PRON
cana-2486	224	2	begin	begin	VERB
cana-2486	224	3	with	with	ADP
cana-2486	224	4	following	follow	VERB
cana-2486	224	5	definition	definition	NOUN
cana-2486	224	6	.	.	PUNCT
cana-2486	225	1	definition	definition	NOUN
cana-2486	225	2	3.1	3.1	NUM
cana-2486	225	3	.	.	PUNCT
cana-2486	226	1	a	a	DET
cana-2486	226	2	proper	proper	ADJ
cana-2486	226	3	element	element	NOUN
cana-2486	226	4	n	n	PROPN
cana-2486	226	5	of	of	ADP
cana-2486	226	6	l	l	NOUN
cana-2486	226	7	-	-	NOUN
cana-2486	226	8	module	module	NOUN
cana-2486	226	9	m	m	NOUN
cana-2486	226	10	is	be	AUX
cana-2486	226	11	called	call	VERB
cana-2486	226	12	weakly	weakly	ADV
cana-2486	226	13	pure	pure	ADJ
cana-2486	226	14	,	,	PUNCT
cana-2486	226	15	if	if	SCONJ
cana-2486	226	16	an	an	DET
cana-2486	226	17	=	=	NOUN
cana-2486	226	18	n	n	X
cana-2486	226	19	∧	∧	PROPN
cana-2486	226	20	a1	a1	PROPN
cana-2486	226	21	m	m	PROPN
cana-2486	226	22	,	,	PUNCT
cana-2486	226	23	for	for	ADP
cana-2486	226	24	every	every	DET
cana-2486	226	25	idempotent	idempotent	ADJ
cana-2486	226	26	element	element	NOUN
cana-2486	226	27	a	a	PRON
cana-2486	226	28	of	of	ADP
cana-2486	226	29	l.	l.	PROPN
cana-2486	226	30	lemma	lemma	PROPN
cana-2486	226	31	3.2	3.2	NUM
cana-2486	226	32	.	.	PUNCT
cana-2486	227	1	let	let	VERB
cana-2486	227	2	m	m	PRON
cana-2486	227	3	be	be	AUX
cana-2486	227	4	a	a	DET
cana-2486	227	5	faithful	faithful	ADJ
cana-2486	227	6	multiplication	multiplication	NOUN
cana-2486	227	7	l−module	l−module	NOUN
cana-2486	227	8	.	.	PUNCT
cana-2486	228	1	if	if	SCONJ
cana-2486	228	2	n	n	PRON
cana-2486	228	3	is	be	AUX
cana-2486	228	4	a	a	DET
cana-2486	228	5	weakly	weakly	ADJ
cana-2486	228	6	pure	pure	ADJ
cana-2486	228	7	element	element	NOUN
cana-2486	228	8	of	of	ADP
cana-2486	228	9	m	m	PRON
cana-2486	228	10	,	,	PUNCT
cana-2486	228	11	then	then	ADV
cana-2486	228	12	a(n	a(n	ADV
cana-2486	228	13	:	:	PUNCT
cana-2486	228	14	1	1	NUM
cana-2486	228	15	m	m	NOUN
cana-2486	228	16	)	)	PUNCT
cana-2486	228	17	=	=	PUNCT
cana-2486	228	18	a	a	DET
cana-2486	228	19	∧	∧	PROPN
cana-2486	228	20	(	(	PUNCT
cana-2486	228	21	n	n	NOUN
cana-2486	228	22	:	:	PUNCT
cana-2486	228	23	1	1	NUM
cana-2486	228	24	m	m	NOUN
cana-2486	228	25	)	)	PUNCT
cana-2486	228	26	,	,	PUNCT
cana-2486	228	27	for	for	ADP
cana-2486	228	28	every	every	DET
cana-2486	228	29	idempotent	idempotent	ADJ
cana-2486	228	30	element	element	NOUN
cana-2486	228	31	a	a	PRON
cana-2486	228	32	of	of	ADP
cana-2486	228	33	l.	l.	PROPN
cana-2486	228	34	proof	proof	NOUN
cana-2486	228	35	.	.	PUNCT
cana-2486	229	1	proof	proof	NOUN
cana-2486	229	2	follows	follow	VERB
cana-2486	229	3	by	by	ADP
cana-2486	229	4	lemma	lemma	PROPN
cana-2486	229	5	2.9	2.9	NUM
cana-2486	229	6	.	.	PUNCT
cana-2486	230	1	proposition	proposition	NOUN
cana-2486	230	2	3.3	3.3	NUM
cana-2486	230	3	.	.	PUNCT
cana-2486	231	1	let	let	VERB
cana-2486	231	2	m	m	PRON
cana-2486	231	3	be	be	AUX
cana-2486	231	4	a	a	DET
cana-2486	231	5	faithful	faithful	ADJ
cana-2486	231	6	multiplication	multiplication	NOUN
cana-2486	231	7	l−module	l−module	NOUN
cana-2486	231	8	.	.	PUNCT
cana-2486	232	1	if	if	SCONJ
cana-2486	232	2	n	n	PRON
cana-2486	232	3	is	be	AUX
cana-2486	232	4	a	a	DET
cana-2486	232	5	weakly	weakly	ADJ
cana-2486	232	6	pure	pure	ADJ
cana-2486	232	7	element	element	NOUN
cana-2486	232	8	of	of	ADP
cana-2486	232	9	m	m	PROPN
cana-2486	232	10	,	,	PUNCT
cana-2486	232	11	then	then	ADV
cana-2486	232	12	(	(	PUNCT
cana-2486	232	13	n	n	X
cana-2486	232	14	:	:	PUNCT
cana-2486	232	15	1	1	NUM
cana-2486	232	16	m	m	NOUN
cana-2486	232	17	)	)	PUNCT
cana-2486	232	18	is	be	AUX
cana-2486	232	19	idempotent	idempotent	ADJ
cana-2486	232	20	.	.	PUNCT
cana-2486	233	1	proof	proof	NOUN
cana-2486	233	2	.	.	PUNCT
cana-2486	234	1	by	by	ADP
cana-2486	234	2	lemma	lemma	PROPN
cana-2486	234	3	3.2	3.2	NUM
cana-2486	234	4	,	,	PUNCT
cana-2486	234	5	we	we	PRON
cana-2486	234	6	have	have	VERB
cana-2486	234	7	(	(	PUNCT
cana-2486	234	8	n	n	X
cana-2486	234	9	:	:	PUNCT
cana-2486	234	10	1	1	NUM
cana-2486	234	11	m	m	NOUN
cana-2486	234	12	)	)	PUNCT
cana-2486	234	13	2	2	NUM
cana-2486	234	14	=	=	SYM
cana-2486	234	15	(	(	PUNCT
cana-2486	234	16	n	n	NOUN
cana-2486	234	17	:	:	PUNCT
cana-2486	234	18	1	1	NUM
cana-2486	234	19	m	m	NOUN
cana-2486	234	20	)	)	PUNCT
cana-2486	234	21	∧	∧	PROPN
cana-2486	234	22	(	(	PUNCT
cana-2486	234	23	n	n	NOUN
cana-2486	234	24	:	:	PUNCT
cana-2486	234	25	1	1	NUM
cana-2486	234	26	m	m	NOUN
cana-2486	234	27	)	)	PUNCT
cana-2486	235	1	=	=	SYM
cana-2486	235	2	(	(	PUNCT
cana-2486	235	3	n	n	X
cana-2486	235	4	:	:	PUNCT
cana-2486	235	5	1	1	NUM
cana-2486	235	6	m	m	NOUN
cana-2486	235	7	)	)	PUNCT
cana-2486	235	8	.	.	PUNCT
cana-2486	236	1	theorem	theorem	VERB
cana-2486	236	2	3.4	3.4	NUM
cana-2486	236	3	.	.	PUNCT
cana-2486	237	1	let	let	VERB
cana-2486	237	2	m	m	PRON
cana-2486	237	3	be	be	AUX
cana-2486	237	4	a	a	DET
cana-2486	237	5	faithful	faithful	ADJ
cana-2486	237	6	multiplication	multiplication	NOUN
cana-2486	237	7	l−module	l−module	NOUN
cana-2486	237	8	,	,	PUNCT
cana-2486	237	9	and	and	CCONJ
cana-2486	237	10	n	n	PRON
cana-2486	237	11	is	be	AUX
cana-2486	237	12	a	a	DET
cana-2486	237	13	weakly	weakly	ADJ
cana-2486	237	14	pure	pure	ADJ
cana-2486	237	15	element	element	NOUN
cana-2486	237	16	communications	communication	NOUN
cana-2486	237	17	on	on	ADP
cana-2486	237	18	applied	apply	VERB
cana-2486	237	19	nonlinear	nonlinear	ADJ
cana-2486	237	20	analysis	analysis	NOUN
cana-2486	237	21	issn	issn	NOUN
cana-2486	237	22	:	:	PUNCT
cana-2486	237	23	1074	1074	NUM
cana-2486	237	24	-	-	PUNCT
cana-2486	237	25	133x	133x	NUM
cana-2486	237	26	vol	vol	NOUN
cana-2486	237	27	.	.	PROPN
cana-2486	238	1	32	32	NUM
cana-2486	238	2	no	no	INTJ
cana-2486	238	3	.	.	PUNCT
cana-2486	239	1	2s	2s	NUM
cana-2486	239	2	(	(	PUNCT
cana-2486	239	3	2024	2024	NUM
cana-2486	239	4	)	)	PUNCT
cana-2486	239	5	544	544	NUM
cana-2486	239	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2486	239	7	/	/	SYM
cana-2486	239	8	of	of	ADP
cana-2486	239	9	m.	m.	NOUN
cana-2486	240	1	then	then	ADV
cana-2486	240	2	n	n	PRON
cana-2486	240	3	is	be	AUX
cana-2486	240	4	primary	primary	ADJ
cana-2486	240	5	element	element	NOUN
cana-2486	240	6	of	of	ADP
cana-2486	240	7	m	m	PROPN
cana-2486	240	8	if	if	SCONJ
cana-2486	241	1	and	and	CCONJ
cana-2486	241	2	only	only	ADV
cana-2486	241	3	if	if	SCONJ
cana-2486	241	4	it	it	PRON
cana-2486	241	5	is	be	AUX
cana-2486	241	6	weakly	weakly	ADJ
cana-2486	241	7	primary	primary	ADJ
cana-2486	241	8	element	element	NOUN
cana-2486	241	9	of	of	ADP
cana-2486	241	10	m.	m.	NOUN
cana-2486	241	11	proof	proof	NOUN
cana-2486	241	12	.	.	PUNCT
cana-2486	242	1	it	it	PRON
cana-2486	242	2	is	be	AUX
cana-2486	242	3	enough	enough	ADJ
cana-2486	242	4	to	to	PART
cana-2486	242	5	show	show	VERB
cana-2486	242	6	that	that	SCONJ
cana-2486	242	7	,	,	PUNCT
cana-2486	242	8	if	if	SCONJ
cana-2486	242	9	n	n	PRON
cana-2486	242	10	is	be	AUX
cana-2486	242	11	weakly	weakly	ADV
cana-2486	242	12	primary	primary	ADJ
cana-2486	242	13	,	,	PUNCT
cana-2486	242	14	then	then	ADV
cana-2486	242	15	n	n	PROPN
cana-2486	242	16	is	be	AUX
cana-2486	242	17	primary	primary	ADJ
cana-2486	242	18	.	.	PUNCT
cana-2486	243	1	assume	assume	VERB
cana-2486	243	2	that	that	SCONJ
cana-2486	243	3	0	0	NUM
cana-2486	243	4	m	m	NOUN
cana-2486	243	5	≠	≠	NOUN
cana-2486	243	6	n	n	NUM
cana-2486	243	7	is	be	AUX
cana-2486	243	8	a	a	DET
cana-2486	243	9	weakly	weakly	ADJ
cana-2486	243	10	primary	primary	ADJ
cana-2486	243	11	element	element	NOUN
cana-2486	243	12	of	of	ADP
cana-2486	243	13	m	m	PRON
cana-2486	243	14	that	that	PRON
cana-2486	243	15	is	be	AUX
cana-2486	243	16	not	not	PART
cana-2486	243	17	primary	primary	ADJ
cana-2486	243	18	.	.	PUNCT
cana-2486	244	1	then	then	ADV
cana-2486	244	2	by	by	ADP
cana-2486	244	3	proposition	proposition	NOUN
cana-2486	244	4	3.3	3.3	NUM
cana-2486	244	5	,	,	PUNCT
cana-2486	244	6	we	we	PRON
cana-2486	244	7	have	have	VERB
cana-2486	244	8	n	n	NOUN
cana-2486	244	9	=	=	SYM
cana-2486	244	10	(	(	PUNCT
cana-2486	244	11	n	n	NOUN
cana-2486	244	12	:	:	PUNCT
cana-2486	244	13	1	1	NUM
cana-2486	244	14	m	m	NOUN
cana-2486	244	15	)	)	PUNCT
cana-2486	244	16	1	1	NUM
cana-2486	244	17	m	m	NOUN
cana-2486	244	18	=	=	PUNCT
cana-2486	244	19	(	(	PUNCT
cana-2486	244	20	n	n	NOUN
cana-2486	244	21	:	:	PUNCT
cana-2486	244	22	1	1	NUM
cana-2486	244	23	m	m	NOUN
cana-2486	244	24	)	)	PUNCT
cana-2486	244	25	21	21	NUM
cana-2486	244	26	m	m	NOUN
cana-2486	244	27	=	=	PUNCT
cana-2486	244	28	(	(	PUNCT
cana-2486	244	29	n	n	NOUN
cana-2486	244	30	:	:	PUNCT
cana-2486	244	31	1	1	NUM
cana-2486	244	32	m	m	NOUN
cana-2486	244	33	)	)	PUNCT
cana-2486	244	34	n	n	NOUN
cana-2486	244	35	=	=	SYM
cana-2486	244	36	0	0	NUM
cana-2486	244	37	m	m	NOUN
cana-2486	244	38	,	,	PUNCT
cana-2486	244	39	which	which	PRON
cana-2486	244	40	is	be	AUX
cana-2486	244	41	a	a	DET
cana-2486	244	42	contradiction	contradiction	NOUN
cana-2486	244	43	.	.	PUNCT
cana-2486	245	1	thus	thus	ADV
cana-2486	245	2	n	n	ADV
cana-2486	245	3	is	be	AUX
cana-2486	245	4	primary	primary	ADJ
cana-2486	245	5	.	.	PUNCT
cana-2486	246	1	proposition	proposition	NOUN
cana-2486	246	2	3.5	3.5	NUM
cana-2486	246	3	.	.	PUNCT
cana-2486	247	1	let	let	VERB
cana-2486	247	2	m	m	PRON
cana-2486	247	3	be	be	AUX
cana-2486	247	4	a	a	DET
cana-2486	247	5	prime	prime	ADJ
cana-2486	247	6	multiplication	multiplication	NOUN
cana-2486	247	7	faithful	faithful	ADJ
cana-2486	247	8	l	l	NOUN
cana-2486	247	9	-	-	NOUN
cana-2486	247	10	module	module	NOUN
cana-2486	247	11	and	and	CCONJ
cana-2486	247	12	0	0	NUM
cana-2486	247	13	m	m	NOUN
cana-2486	247	14	≠	≠	NOUN
cana-2486	247	15	n	n	VERB
cana-2486	247	16	be	be	VERB
cana-2486	247	17	a	a	DET
cana-2486	247	18	proper	proper	ADJ
cana-2486	247	19	weakly	weakly	ADJ
cana-2486	247	20	pure	pure	ADJ
cana-2486	247	21	element	element	NOUN
cana-2486	247	22	of	of	ADP
cana-2486	247	23	m.	m.	NOUN
cana-2486	247	24	then	then	ADV
cana-2486	247	25	ann(n	ann(n	PROPN
cana-2486	247	26	:	:	PUNCT
cana-2486	247	27	1	1	NUM
cana-2486	247	28	m	m	NOUN
cana-2486	247	29	)	)	PUNCT
cana-2486	248	1	=	=	SYM
cana-2486	248	2	0l	0l	NOUN
cana-2486	248	3	.	.	PUNCT
cana-2486	249	1	proof	proof	NOUN
cana-2486	249	2	.	.	PUNCT
cana-2486	250	1	for	for	ADP
cana-2486	250	2	every	every	DET
cana-2486	250	3	a	a	DET
cana-2486	250	4	≤	≤	ADJ
cana-2486	250	5	ann(n	ann(n	PROPN
cana-2486	250	6	:	:	PUNCT
cana-2486	250	7	1	1	NUM
cana-2486	250	8	m	m	NOUN
cana-2486	250	9	)	)	PUNCT
cana-2486	250	10	,	,	PUNCT
cana-2486	250	11	we	we	PRON
cana-2486	250	12	have	have	VERB
cana-2486	250	13	a(n	a(n	ADV
cana-2486	250	14	:	:	PUNCT
cana-2486	250	15	1	1	NUM
cana-2486	250	16	m	m	NOUN
cana-2486	250	17	)	)	PUNCT
cana-2486	251	1	=	=	SYM
cana-2486	251	2	0l	0l	NOUN
cana-2486	251	3	,	,	PUNCT
cana-2486	251	4	hence	hence	ADV
cana-2486	251	5	an	an	DET
cana-2486	251	6	=	=	X
cana-2486	251	7	a(n	a(n	NOUN
cana-2486	251	8	:	:	PUNCT
cana-2486	251	9	1	1	NUM
cana-2486	251	10	m	m	NOUN
cana-2486	251	11	)	)	PUNCT
cana-2486	251	12	n	n	NOUN
cana-2486	251	13	=	=	SYM
cana-2486	251	14	0	0	NUM
cana-2486	251	15	m	m	NOUN
cana-2486	251	16	,	,	PUNCT
cana-2486	251	17	so	so	SCONJ
cana-2486	251	18	that	that	SCONJ
cana-2486	251	19	a	a	DET
cana-2486	251	20	≤	≤	NUM
cana-2486	251	21	annn	annn	NOUN
cana-2486	251	22	=	=	NOUN
cana-2486	251	23	annm	annm	NOUN
cana-2486	251	24	=	=	SYM
cana-2486	251	25	0l	0l	NOUN
cana-2486	251	26	,	,	PUNCT
cana-2486	251	27	since	since	SCONJ
cana-2486	251	28	m	m	PROPN
cana-2486	251	29	is	be	AUX
cana-2486	251	30	prime	prime	ADJ
cana-2486	251	31	.	.	PUNCT
cana-2486	252	1	hence	hence	ADV
cana-2486	252	2	a	a	DET
cana-2486	252	3	=	=	SYM
cana-2486	252	4	0l	0l	NOUN
cana-2486	252	5	,	,	PUNCT
cana-2486	252	6	so	so	SCONJ
cana-2486	252	7	ann(n	ann(n	PROPN
cana-2486	252	8	:	:	PUNCT
cana-2486	252	9	1	1	NUM
cana-2486	252	10	m	m	NOUN
cana-2486	252	11	)	)	PUNCT
cana-2486	253	1	=	=	SYM
cana-2486	253	2	0l	0l	X
cana-2486	253	3	.	.	PUNCT
cana-2486	254	1	proposition	proposition	NOUN
cana-2486	254	2	3.6	3.6	NUM
cana-2486	254	3	.	.	PUNCT
cana-2486	255	1	let	let	VERB
cana-2486	255	2	l	l	NOUN
cana-2486	255	3	be	be	AUX
cana-2486	255	4	a	a	DET
cana-2486	255	5	noetherian	noetherian	ADJ
cana-2486	255	6	multiplicative	multiplicative	ADJ
cana-2486	255	7	lattice	lattice	NOUN
cana-2486	255	8	with	with	ADP
cana-2486	255	9	jacobson	jacobson	PROPN
cana-2486	255	10	radical	radical	PROPN
cana-2486	255	11	r∗,and	r∗,and	PROPN
cana-2486	255	12	m	m	PROPN
cana-2486	255	13	a	a	DET
cana-2486	255	14	multiplication	multiplication	NOUN
cana-2486	255	15	l	l	NOUN
cana-2486	255	16	-	-	NOUN
cana-2486	255	17	module	module	NOUN
cana-2486	255	18	and	and	CCONJ
cana-2486	255	19	n	n	NOUN
cana-2486	255	20	is	be	AUX
cana-2486	255	21	a	a	DET
cana-2486	255	22	weakly	weakly	ADJ
cana-2486	255	23	pure	pure	ADJ
cana-2486	255	24	element	element	NOUN
cana-2486	255	25	of	of	ADP
cana-2486	255	26	m.	m.	NOUN
cana-2486	255	27	then	then	ADV
cana-2486	255	28	there	there	PRON
cana-2486	255	29	is	be	VERB
cana-2486	255	30	a	a	DET
cana-2486	255	31	maximal	maximal	ADJ
cana-2486	255	32	element	element	NOUN
cana-2486	255	33	r	r	NOUN
cana-2486	255	34	of	of	ADP
cana-2486	255	35	l	l	NOUN
cana-2486	255	36	such	such	ADJ
cana-2486	255	37	that	that	PRON
cana-2486	255	38	(	(	PUNCT
cana-2486	255	39	n	n	X
cana-2486	255	40	:	:	PUNCT
cana-2486	255	41	1	1	NUM
cana-2486	255	42	m	m	NOUN
cana-2486	255	43	)	)	PUNCT
cana-2486	256	1	≰	≰	PROPN
cana-2486	256	2	r.	r.	NOUN
cana-2486	256	3	proof	proof	NOUN
cana-2486	256	4	.	.	PUNCT
cana-2486	257	1	otherwise	otherwise	ADV
cana-2486	257	2	,	,	PUNCT
cana-2486	257	3	(	(	PUNCT
cana-2486	257	4	n	n	X
cana-2486	257	5	:	:	PUNCT
cana-2486	257	6	1	1	NUM
cana-2486	257	7	m	m	NOUN
cana-2486	257	8	)	)	PUNCT
cana-2486	257	9	≤	≤	PUNCT
cana-2486	258	1	r∗	r∗	PROPN
cana-2486	258	2	,	,	PUNCT
cana-2486	258	3	so	so	ADV
cana-2486	258	4	(	(	PUNCT
cana-2486	258	5	n	n	X
cana-2486	258	6	:	:	PUNCT
cana-2486	258	7	1	1	NUM
cana-2486	258	8	m	m	NOUN
cana-2486	258	9	)	)	PUNCT
cana-2486	259	1	=	=	VERB
cana-2486	259	2	∧𝑖=1	∧𝑖=1	NOUN
cana-2486	259	3	∞	∞	NUM
cana-2486	259	4	(	(	PUNCT
cana-2486	259	5	n	n	NOUN
cana-2486	259	6	:	:	PUNCT
cana-2486	259	7	1	1	NUM
cana-2486	259	8	m	m	NOUN
cana-2486	259	9	)	)	PUNCT
cana-2486	260	1	i	i	PRON
cana-2486	260	2	=	=	SYM
cana-2486	260	3	0l	0l	NOUN
cana-2486	260	4	,	,	PUNCT
cana-2486	260	5	by	by	ADP
cana-2486	260	6	proposition	proposition	NOUN
cana-2486	260	7	3.3	3.3	NUM
cana-2486	260	8	,	,	PUNCT
cana-2486	260	9	hence	hence	ADV
cana-2486	260	10	n	n	NOUN
cana-2486	260	11	=	=	SYM
cana-2486	260	12	(	(	PUNCT
cana-2486	260	13	n	n	X
cana-2486	260	14	:	:	PUNCT
cana-2486	260	15	1	1	NUM
cana-2486	260	16	m	m	NOUN
cana-2486	260	17	)	)	PUNCT
cana-2486	260	18	1	1	NUM
cana-2486	260	19	m	m	NOUN
cana-2486	260	20	=	=	SYM
cana-2486	260	21	0	0	NUM
cana-2486	260	22	m	m	VERB
cana-2486	260	23	,	,	PUNCT
cana-2486	260	24	which	which	PRON
cana-2486	260	25	is	be	AUX
cana-2486	260	26	a	a	DET
cana-2486	260	27	contradiction	contradiction	NOUN
cana-2486	260	28	.	.	PUNCT
cana-2486	261	1	hence	hence	ADV
cana-2486	261	2	there	there	PRON
cana-2486	261	3	is	be	VERB
cana-2486	261	4	a	a	DET
cana-2486	261	5	maximal	maximal	ADJ
cana-2486	261	6	element	element	NOUN
cana-2486	261	7	r	r	NOUN
cana-2486	261	8	of	of	ADP
cana-2486	261	9	l	l	NOUN
cana-2486	261	10	such	such	ADJ
cana-2486	261	11	that	that	PRON
cana-2486	261	12	(	(	PUNCT
cana-2486	261	13	n	n	X
cana-2486	261	14	:	:	PUNCT
cana-2486	261	15	1	1	NUM
cana-2486	261	16	m	m	NOUN
cana-2486	261	17	)	)	PUNCT
cana-2486	262	1	≰	≰	PROPN
cana-2486	262	2	r.	r.	PROPN
cana-2486	262	3	references	reference	NOUN
cana-2486	262	4	[	[	X
cana-2486	262	5	1	1	NUM
cana-2486	262	6	]	]	PUNCT
cana-2486	262	7	m.	m.	NOUN
cana-2486	262	8	m.	m.	PROPN
cana-2486	262	9	ali	ali	PROPN
cana-2486	262	10	,	,	PUNCT
cana-2486	262	11	idempotent	idempotent	ADJ
cana-2486	262	12	and	and	CCONJ
cana-2486	262	13	nilpotent	nilpotent	ADJ
cana-2486	262	14	submodules	submodule	NOUN
cana-2486	262	15	of	of	ADP
cana-2486	262	16	multiplication	multiplication	NOUN
cana-2486	262	17	modules	module	NOUN
cana-2486	262	18	,	,	PUNCT
cana-2486	262	19	comm	comm	NOUN
cana-2486	262	20	.	.	PUNCT
cana-2486	263	1	algebra	algebra	PROPN
cana-2486	263	2	,	,	PUNCT
cana-2486	263	3	36(12	36(12	NUM
cana-2486	263	4	)	)	PUNCT
cana-2486	263	5	(	(	PUNCT
cana-2486	263	6	2008	2008	NUM
cana-2486	263	7	)	)	PUNCT
cana-2486	263	8	,	,	PUNCT
cana-2486	263	9	4620	4620	NUM
cana-2486	263	10	-	-	SYM
cana-2486	263	11	4642	4642	NUM
cana-2486	263	12	.	.	PUNCT
cana-2486	264	1	[	[	X
cana-2486	264	2	2	2	NUM
cana-2486	264	3	]	]	PUNCT
cana-2486	264	4	m.	m.	NOUN
cana-2486	264	5	m.	m.	PROPN
cana-2486	264	6	ali	ali	PROPN
cana-2486	264	7	and	and	CCONJ
cana-2486	264	8	d.	d.	PROPN
cana-2486	264	9	j.	j.	PROPN
cana-2486	264	10	smith	smith	PROPN
cana-2486	264	11	,	,	PUNCT
cana-2486	264	12	pure	pure	ADJ
cana-2486	264	13	submodules	submodule	NOUN
cana-2486	264	14	of	of	ADP
cana-2486	264	15	multiplication	multiplication	NOUN
cana-2486	264	16	modules	module	NOUN
cana-2486	264	17	,	,	PUNCT
cana-2486	264	18	beitrage	beitrage	NOUN
cana-2486	264	19	zur	zur	NOUN
cana-2486	264	20	algebra	algebra	NOUN
cana-2486	264	21	and	and	CCONJ
cana-2486	264	22	geometrie	geometrie	NOUN
cana-2486	264	23	,	,	PUNCT
cana-2486	264	24	45(1	45(1	NOUN
cana-2486	264	25	)	)	PUNCT
cana-2486	264	26	(	(	PUNCT
cana-2486	264	27	2004	2004	NUM
cana-2486	264	28	)	)	PUNCT
cana-2486	264	29	,	,	PUNCT
cana-2486	264	30	61	61	NUM
cana-2486	264	31	-	-	SYM
cana-2486	264	32	74	74	NUM
cana-2486	264	33	.	.	PUNCT
cana-2486	265	1	[	[	X
cana-2486	265	2	3	3	X
cana-2486	265	3	]	]	PUNCT
cana-2486	265	4	v.	v.	ADP
cana-2486	265	5	borkar	borkar	PROPN
cana-2486	265	6	,	,	PUNCT
cana-2486	265	7	p.	p.	NOUN
cana-2486	265	8	girase	girase	PROPN
cana-2486	265	9	and	and	CCONJ
cana-2486	265	10	n.	n.	PROPN
cana-2486	265	11	phadatare	phadatare	NOUN
cana-2486	265	12	,	,	PUNCT
cana-2486	265	13	zariski	zariski	VERB
cana-2486	265	14	second	second	ADJ
cana-2486	265	15	radical	radical	ADJ
cana-2486	265	16	elements	element	NOUN
cana-2486	265	17	of	of	ADP
cana-2486	265	18	lattice	lattice	NOUN
cana-2486	265	19	modules	module	NOUN
cana-2486	265	20	,	,	PUNCT
cana-2486	265	21	asian	asian	ADJ
cana-2486	265	22	-	-	PUNCT
cana-2486	265	23	eur	eur	NOUN
cana-2486	265	24	.	.	PUNCT
cana-2486	266	1	j.	j.	PROPN
cana-2486	266	2	math	math	PROPN
cana-2486	266	3	.	.	PROPN
cana-2486	266	4	,	,	PUNCT
cana-2486	266	5	2150055	2150055	NUM
cana-2486	266	6	(	(	PUNCT
cana-2486	266	7	2021	2021	NUM
cana-2486	266	8	)	)	PUNCT
cana-2486	266	9	,	,	PUNCT
cana-2486	266	10	11	11	NUM
cana-2486	266	11	-	-	PUNCT
cana-2486	266	12	pages	page	NOUN
cana-2486	266	13	doi:10.1142	doi:10.1142	NOUN
cana-2486	266	14	/	/	SYM
cana-2486	266	15	s1793557121500558	s1793557121500558	NOUN
cana-2486	266	16	.	.	PUNCT
cana-2486	267	1	[	[	X
cana-2486	267	2	4	4	X
cana-2486	267	3	]	]	PUNCT
cana-2486	267	4	f.	f.	PROPN
cana-2486	267	5	callialp	callialp	PROPN
cana-2486	267	6	and	and	CCONJ
cana-2486	267	7	u.	u.	PROPN
cana-2486	267	8	tekir	tekir	PROPN
cana-2486	267	9	,	,	PUNCT
cana-2486	267	10	multiplication	multiplication	NOUN
cana-2486	267	11	lattice	lattice	NOUN
cana-2486	267	12	modules	module	NOUN
cana-2486	267	13	,	,	PUNCT
cana-2486	267	14	iran	iran	PROPN
cana-2486	267	15	.	.	PUNCT
cana-2486	268	1	j.	j.	PROPN
cana-2486	268	2	sci	sci	PROPN
cana-2486	268	3	.	.	PROPN
cana-2486	268	4	technol	technol	PROPN
cana-2486	268	5	.	.	PROPN
cana-2486	268	6	,	,	PUNCT
cana-2486	268	7	35(4	35(4	NUM
cana-2486	268	8	)	)	PUNCT
cana-2486	268	9	(	(	PUNCT
cana-2486	268	10	2011	2011	NUM
cana-2486	268	11	)	)	PUNCT
cana-2486	268	12	,	,	PUNCT
cana-2486	268	13	309	309	NUM
cana-2486	268	14	-	-	SYM
cana-2486	268	15	313	313	NUM
cana-2486	268	16	.	.	PUNCT
cana-2486	269	1	[	[	X
cana-2486	269	2	5	5	NUM
cana-2486	269	3	]	]	X
cana-2486	269	4	f.callialp	f.callialp	X
cana-2486	269	5	,	,	PUNCT
cana-2486	269	6	u.	u.	NOUN
cana-2486	269	7	tekir	tekir	PROPN
cana-2486	269	8	and	and	CCONJ
cana-2486	269	9	e.	e.	PROPN
cana-2486	269	10	aslankarayigit	aslankarayigit	PROPN
cana-2486	269	11	,	,	PUNCT
cana-2486	269	12	on	on	ADP
cana-2486	269	13	multiplication	multiplication	NOUN
cana-2486	269	14	lattice	lattice	NOUN
cana-2486	269	15	modules	module	NOUN
cana-2486	269	16	,	,	PUNCT
cana-2486	269	17	hacet	hacet	NOUN
cana-2486	269	18	.	.	PUNCT
cana-2486	270	1	j.	j.	PROPN
cana-2486	270	2	math	math	PROPN
cana-2486	270	3	.	.	PUNCT
cana-2486	271	1	stat	stat	PROPN
cana-2486	271	2	.	.	PUNCT
cana-2486	271	3	,	,	PUNCT
cana-2486	271	4	43(4	43(4	NOUN
cana-2486	271	5	)	)	PUNCT
cana-2486	271	6	(	(	PUNCT
cana-2486	271	7	2014	2014	NUM
cana-2486	271	8	)	)	PUNCT
cana-2486	271	9	,	,	PUNCT
cana-2486	271	10	571	571	NUM
cana-2486	271	11	-	-	SYM
cana-2486	271	12	579	579	NUM
cana-2486	271	13	.	.	PUNCT
cana-2486	272	1	[	[	X
cana-2486	272	2	6	6	NUM
cana-2486	272	3	]	]	PUNCT
cana-2486	272	4	f.	f.	PROPN
cana-2486	272	5	callialp	callialp	PROPN
cana-2486	272	6	,	,	PUNCT
cana-2486	272	7	u.	u.	PROPN
cana-2486	272	8	tekir	tekir	PROPN
cana-2486	272	9	,	,	PUNCT
cana-2486	272	10	e.	e.	PROPN
cana-2486	272	11	a.	a.	PROPN
cana-2486	272	12	ugurlu	ugurlu	PROPN
cana-2486	272	13	and	and	CCONJ
cana-2486	272	14	k.	k.	PROPN
cana-2486	272	15	h.	h.	PROPN
cana-2486	272	16	oral	oral	PROPN
cana-2486	272	17	,	,	PUNCT
cana-2486	272	18	second	second	ADJ
cana-2486	272	19	and	and	CCONJ
cana-2486	272	20	secondary	secondary	ADJ
cana-2486	272	21	lattice	lattice	NOUN
cana-2486	272	22	modules	module	NOUN
cana-2486	272	23	,	,	PUNCT
cana-2486	272	24	the	the	DET
cana-2486	272	25	scientific	scientific	ADJ
cana-2486	272	26	world	world	NOUN
cana-2486	272	27	journal	journal	NOUN
cana-2486	272	28	,	,	PUNCT
cana-2486	272	29	i	i	PROPN
cana-2486	272	30	d	d	PROPN
cana-2486	272	31	291924	291924	NUM
cana-2486	272	32	(	(	PUNCT
cana-2486	272	33	2014	2014	NUM
cana-2486	272	34	)	)	PUNCT
cana-2486	272	35	,	,	PUNCT
cana-2486	272	36	4pages	4pages	PROPN
cana-2486	272	37	.	.	PUNCT
cana-2486	273	1	[	[	X
cana-2486	273	2	7	7	X
cana-2486	273	3	]	]	X
cana-2486	273	4	j.	j.	PROPN
cana-2486	273	5	jenkins	jenkins	PROPN
cana-2486	273	6	and	and	CCONJ
cana-2486	273	7	p.	p.	PROPN
cana-2486	273	8	f.	f.	PROPN
cana-2486	273	9	smith	smith	PROPN
cana-2486	273	10	,	,	PUNCT
cana-2486	273	11	on	on	ADP
cana-2486	273	12	the	the	DET
cana-2486	273	13	prime	prime	ADJ
cana-2486	273	14	radical	radical	NOUN
cana-2486	273	15	of	of	ADP
cana-2486	273	16	a	a	DET
cana-2486	273	17	module	module	NOUN
cana-2486	273	18	over	over	ADP
cana-2486	273	19	a	a	DET
cana-2486	273	20	commutative	commutative	ADJ
cana-2486	273	21	ring	ring	NOUN
cana-2486	273	22	,	,	PUNCT
cana-2486	273	23	comm	comm	NOUN
cana-2486	273	24	.	.	PUNCT
cana-2486	274	1	algebra	algebra	NOUN
cana-2486	274	2	,	,	PUNCT
cana-2486	274	3	20(12	20(12	NUM
cana-2486	274	4	)	)	PUNCT
cana-2486	274	5	(	(	PUNCT
cana-2486	274	6	1992	1992	NUM
cana-2486	274	7	)	)	PUNCT
cana-2486	274	8	,	,	PUNCT
cana-2486	274	9	3593	3593	NUM
cana-2486	274	10	-	-	SYM
cana-2486	274	11	3602	3602	NUM
cana-2486	274	12	.	.	PUNCT
cana-2486	275	1	[	[	X
cana-2486	275	2	8	8	NUM
cana-2486	275	3	]	]	X
cana-2486	275	4	c.	c.	PROPN
cana-2486	275	5	p.	p.	PROPN
cana-2486	275	6	lu	lu	PROPN
cana-2486	275	7	,	,	PUNCT
cana-2486	275	8	m	m	NOUN
cana-2486	275	9	-	-	NOUN
cana-2486	275	10	radicals	radical	NOUN
cana-2486	275	11	of	of	ADP
cana-2486	275	12	submodules	submodule	NOUN
cana-2486	275	13	in	in	ADP
cana-2486	275	14	modules	module	NOUN
cana-2486	275	15	,	,	PUNCT
cana-2486	275	16	math	math	NOUN
cana-2486	275	17	.	.	PUNCT
cana-2486	276	1	japonica	japonica	PROPN
cana-2486	276	2	,	,	PUNCT
cana-2486	276	3	34(2	34(2	NUM
cana-2486	276	4	)	)	PUNCT
cana-2486	276	5	(	(	PUNCT
cana-2486	276	6	1989	1989	NUM
cana-2486	276	7	)	)	PUNCT
cana-2486	276	8	,	,	PUNCT
cana-2486	276	9	211	211	NUM
cana-2486	276	10	-	-	SYM
cana-2486	276	11	219	219	NUM
cana-2486	276	12	.	.	PUNCT
cana-2486	277	1	[	[	X
cana-2486	277	2	9	9	NUM
cana-2486	277	3	]	]	PUNCT
cana-2486	277	4	m.	m.	PROPN
cana-2486	277	5	e.	e.	PROPN
cana-2486	277	6	moor	moor	PROPN
cana-2486	277	7	and	and	CCONJ
cana-2486	277	8	r.	r.	PROPN
cana-2486	277	9	l.	l.	PROPN
cana-2486	277	10	mccasland	mccasland	PROPN
cana-2486	277	11	,	,	PUNCT
cana-2486	277	12	on	on	ADP
cana-2486	277	13	radicals	radical	NOUN
cana-2486	277	14	of	of	ADP
cana-2486	277	15	submodules	submodule	NOUN
cana-2486	277	16	of	of	ADP
cana-2486	277	17	finitely	finitely	ADV
cana-2486	277	18	generated	generate	VERB
cana-2486	277	19	modules	module	NOUN
cana-2486	277	20	,	,	PUNCT
cana-2486	277	21	canad	canad	PROPN
cana-2486	277	22	.	.	PUNCT
cana-2486	278	1	math	math	NOUN
cana-2486	278	2	.	.	PUNCT
cana-2486	279	1	bull	bull	PROPN
cana-2486	279	2	.	.	PUNCT
cana-2486	279	3	,	,	PUNCT
cana-2486	279	4	29(1	29(1	NUM
cana-2486	279	5	)	)	PUNCT
cana-2486	279	6	(	(	PUNCT
cana-2486	279	7	1986	1986	NUM
cana-2486	279	8	)	)	PUNCT
cana-2486	279	9	,	,	PUNCT
cana-2486	279	10	37	37	NUM
cana-2486	279	11	-	-	SYM
cana-2486	279	12	39	39	NUM
cana-2486	279	13	.	.	PUNCT
cana-2486	280	1	[	[	X
cana-2486	280	2	10	10	NUM
cana-2486	280	3	]	]	X
cana-2486	280	4	p.	p.	NOUN
cana-2486	280	5	girase	girase	NOUN
cana-2486	280	6	,	,	PUNCT
cana-2486	280	7	v.	v.	ADP
cana-2486	280	8	borkar	borkar	PROPN
cana-2486	280	9	and	and	CCONJ
cana-2486	280	10	n.	n.	PROPN
cana-2486	280	11	phadatare	phadatare	NOUN
cana-2486	280	12	,	,	PUNCT
cana-2486	280	13	zariski	zariski	VERB
cana-2486	280	14	prime	prime	ADJ
cana-2486	280	15	radical	radical	ADJ
cana-2486	280	16	elements	element	NOUN
cana-2486	280	17	of	of	ADP
cana-2486	280	18	lattice	lattice	NOUN
cana-2486	280	19	modules	module	NOUN
cana-2486	280	20	,	,	PUNCT
cana-2486	280	21	southeast	southeast	ADJ
cana-2486	280	22	asian	asian	ADJ
cana-2486	280	23	bull	bull	NOUN
cana-2486	280	24	.	.	PUNCT
cana-2486	281	1	math	math	NOUN
cana-2486	281	2	.	.	PUNCT
cana-2486	281	3	,	,	PUNCT
cana-2486	281	4	44(3	44(3	NUM
cana-2486	281	5	)	)	PUNCT
cana-2486	281	6	(	(	PUNCT
cana-2486	281	7	2020	2020	NUM
cana-2486	281	8	)	)	PUNCT
cana-2486	281	9	,	,	PUNCT
cana-2486	281	10	335	335	NUM
cana-2486	281	11	-	-	SYM
cana-2486	281	12	344	344	NUM
cana-2486	281	13	.	.	PUNCT