id	sid	tid	token	lemma	pos
easat-4034	1	1	edelweiss	edelweiss	PROPN
easat-4034	1	2	applied	apply	VERB
easat-4034	1	3	science	science	NOUN
easat-4034	1	4	and	and	CCONJ
easat-4034	1	5	technology	technology	NOUN
easat-4034	1	6	issn	issn	PROPN
easat-4034	1	7	:	:	PUNCT
easat-4034	1	8	2576	2576	NUM
easat-4034	1	9	-	-	SYM
easat-4034	1	10	8484	8484	NUM
easat-4034	1	11	vol	vol	NOUN
easat-4034	1	12	.	.	PROPN
easat-4034	1	13	8	8	NUM
easat-4034	1	14	,	,	PUNCT
easat-4034	1	15	no	no	INTJ
easat-4034	1	16	.	.	NOUN
easat-4034	1	17	6	6	NUM
easat-4034	1	18	,	,	PUNCT
easat-4034	1	19	9544	9544	NUM
easat-4034	1	20	-	-	SYM
easat-4034	1	21	9554	9554	NUM
easat-4034	1	22	2024	2024	NUM
easat-4034	1	23	publisher	publisher	NOUN
easat-4034	1	24	:	:	PUNCT
easat-4034	1	25	learning	learn	VERB
easat-4034	1	26	gate	gate	NOUN
easat-4034	1	27	doi	doi	PROPN
easat-4034	1	28	:	:	PUNCT
easat-4034	1	29	10.55214/25768484.v8i6.4034	10.55214/25768484.v8i6.4034	NUM
easat-4034	1	30	©	©	ADP
easat-4034	1	31	2024	2024	NUM
easat-4034	1	32	by	by	ADP
easat-4034	1	33	the	the	DET
easat-4034	1	34	authors	author	NOUN
easat-4034	1	35	;	;	PUNCT
easat-4034	1	36	licensee	licensee	PROPN
easat-4034	1	37	learning	learning	NOUN
easat-4034	1	38	gate	gate	NOUN
easat-4034	1	39	©	©	PROPN
easat-4034	1	40	2024	2024	NUM
easat-4034	1	41	by	by	ADP
easat-4034	1	42	the	the	DET
easat-4034	1	43	authors	author	NOUN
easat-4034	1	44	;	;	PUNCT
easat-4034	1	45	licensee	licensee	PROPN
easat-4034	1	46	learning	learn	VERB
easat-4034	1	47	gate	gate	NOUN
easat-4034	1	48	*	*	PUNCT
easat-4034	1	49	correspondence	correspondence	NOUN
easat-4034	1	50	:	:	PUNCT
easat-4034	1	51	gregoria.ariyanti@ukwms.ac.id	gregoria.ariyanti@ukwms.ac.id	NUM
easat-4034	1	52	notes	note	NOUN
easat-4034	1	53	on	on	ADP
easat-4034	1	54	matrix	matrix	NOUN
easat-4034	1	55	inverse	inverse	NOUN
easat-4034	1	56	over	over	ADP
easat-4034	1	57	min	min	PROPN
easat-4034	1	58	-	-	PUNCT
easat-4034	1	59	plus	plus	ADJ
easat-4034	1	60	algebra	algebra	NOUN
easat-4034	1	61	gregoria	gregoria	PROPN
easat-4034	1	62	ariyanti1	ariyanti1	PROPN
easat-4034	1	63	*	*	PROPN
easat-4034	1	64	,	,	PUNCT
easat-4034	1	65	ana	ana	PROPN
easat-4034	1	66	easti	easti	PROPN
easat-4034	1	67	rahayu	rahayu	PROPN
easat-4034	1	68	maya	maya	PROPN
easat-4034	1	69	sari2	sari2	PROPN
easat-4034	1	70	,	,	PUNCT
easat-4034	1	71	christina	christina	PROPN
easat-4034	1	72	manurung3	manurung3	PROPN
easat-4034	2	1	1,2,3department	1,2,3department	NUM
easat-4034	2	2	of	of	ADP
easat-4034	2	3	mathematics	mathematics	PROPN
easat-4034	2	4	education	education	NOUN
easat-4034	2	5	,	,	PUNCT
easat-4034	2	6	widya	widya	PROPN
easat-4034	2	7	mandala	mandala	PROPN
easat-4034	2	8	surabaya	surabaya	PROPN
easat-4034	2	9	catholic	catholic	PROPN
easat-4034	2	10	university	university	PROPN
easat-4034	2	11	;	;	PUNCT
easat-4034	2	12	gregoria.ariyanti@ukwms.ac.id	gregoria.ariyanti@ukwms.ac.id	NUM
easat-4034	2	13	(	(	PUNCT
easat-4034	2	14	g.a	g.a	PROPN
easat-4034	2	15	.	.	PUNCT
easat-4034	2	16	)	)	PUNCT
easat-4034	3	1	abstract	abstract	NOUN
easat-4034	3	2	:	:	PUNCT
easat-4034	3	3	one	one	NUM
easat-4034	3	4	of	of	ADP
easat-4034	3	5	the	the	DET
easat-4034	3	6	semiring	semire	VERB
easat-4034	3	7	structures	structure	NOUN
easat-4034	3	8	is	be	AUX
easat-4034	3	9	the	the	DET
easat-4034	3	10	max	max	PROPN
easat-4034	3	11	-	-	PUNCT
easat-4034	3	12	plus	plus	CCONJ
easat-4034	3	13	algebra	algebra	NOUN
easat-4034	3	14	,	,	PUNCT
easat-4034	3	15	a	a	DET
easat-4034	3	16	set	set	NOUN
easat-4034	3	17	with	with	ADP
easat-4034	3	18	entries	entry	NOUN
easat-4034	3	19	ℝ𝜀	ℝ𝜀	PROPN
easat-4034	3	20	=	=	SYM
easat-4034	3	21	ℝ	ℝ	PROPN
easat-4034	3	22	∪	∪	ADJ
easat-4034	3	23	{	{	PUNCT
easat-4034	3	24	−∞	−∞	NOUN
easat-4034	3	25	}	}	PUNCT
easat-4034	3	26	equipped	equip	VERB
easat-4034	3	27	with	with	ADP
easat-4034	3	28	the	the	DET
easat-4034	3	29	operation	operation	NOUN
easat-4034	3	30	⊕	⊕	PROPN
easat-4034	3	31	,	,	PUNCT
easat-4034	3	32	which	which	PRON
easat-4034	3	33	represents	represent	VERB
easat-4034	3	34	the	the	DET
easat-4034	3	35	maximum	maximum	ADJ
easat-4034	3	36	value	value	NOUN
easat-4034	3	37	,	,	PUNCT
easat-4034	3	38	and	and	CCONJ
easat-4034	3	39	the	the	DET
easat-4034	3	40	operation	operation	NOUN
easat-4034	3	41	⊗	⊗	PROPN
easat-4034	3	42	,	,	PUNCT
easat-4034	3	43	which	which	PRON
easat-4034	3	44	means	mean	VERB
easat-4034	3	45	addition	addition	NOUN
easat-4034	3	46	.	.	PUNCT
easat-4034	4	1	another	another	DET
easat-4034	4	2	semiring	semire	VERB
easat-4034	4	3	structure	structure	NOUN
easat-4034	4	4	is	be	AUX
easat-4034	4	5	the	the	DET
easat-4034	4	6	min	min	PROPN
easat-4034	4	7	-	-	PUNCT
easat-4034	4	8	plus	plus	ADJ
easat-4034	4	9	algebra	algebra	NOUN
easat-4034	4	10	,	,	PUNCT
easat-4034	4	11	a	a	DET
easat-4034	4	12	set	set	NOUN
easat-4034	4	13	with	with	ADP
easat-4034	4	14	entries	entry	NOUN
easat-4034	4	15	ℝ𝜀	ℝ𝜀	PROPN
easat-4034	4	16	=	=	SYM
easat-4034	4	17	ℝ	ℝ	PROPN
easat-4034	4	18	∪	∪	ADJ
easat-4034	4	19	{	{	PUNCT
easat-4034	4	20	+	+	NOUN
easat-4034	4	21	∞	∞	NOUN
easat-4034	4	22	}	}	PUNCT
easat-4034	4	23	equipped	equip	VERB
easat-4034	4	24	with	with	ADP
easat-4034	4	25	the	the	DET
easat-4034	4	26	operation	operation	NOUN
easat-4034	4	27	⊕	⊕	PROPN
easat-4034	4	28	,	,	PUNCT
easat-4034	4	29	representing	represent	VERB
easat-4034	4	30	the	the	DET
easat-4034	4	31	minimum	minimum	ADJ
easat-4034	4	32	value	value	NOUN
easat-4034	4	33	,	,	PUNCT
easat-4034	4	34	and	and	CCONJ
easat-4034	4	35	the	the	DET
easat-4034	4	36	operation	operation	NOUN
easat-4034	4	37	⊗	⊗	PROPN
easat-4034	4	38	,	,	PUNCT
easat-4034	4	39	which	which	PRON
easat-4034	4	40	means	mean	VERB
easat-4034	4	41	addition	addition	NOUN
easat-4034	4	42	.	.	PUNCT
easat-4034	5	1	matrices	matrix	NOUN
easat-4034	5	2	over	over	ADP
easat-4034	5	3	min	min	NOUN
easat-4034	5	4	-	-	PUNCT
easat-4034	5	5	plus	plus	ADJ
easat-4034	5	6	algebras	algebra	NOUN
easat-4034	5	7	can	can	AUX
easat-4034	5	8	have	have	VERB
easat-4034	5	9	inverses	inverse	NOUN
easat-4034	5	10	determined	determine	VERB
easat-4034	5	11	by	by	ADP
easat-4034	5	12	certain	certain	ADJ
easat-4034	5	13	conditions	condition	NOUN
easat-4034	5	14	.	.	PUNCT
easat-4034	6	1	the	the	DET
easat-4034	6	2	general	general	ADJ
easat-4034	6	3	inverse	inverse	NOUN
easat-4034	6	4	type	type	NOUN
easat-4034	6	5	can	can	AUX
easat-4034	6	6	define	define	VERB
easat-4034	6	7	the	the	DET
easat-4034	6	8	inverse	inverse	NOUN
easat-4034	6	9	of	of	ADP
easat-4034	6	10	matrices	matrix	NOUN
easat-4034	6	11	over	over	ADP
easat-4034	6	12	min	min	NOUN
easat-4034	6	13	-	-	PUNCT
easat-4034	6	14	plus	plus	ADJ
easat-4034	6	15	algebras	algebra	NOUN
easat-4034	6	16	.	.	PUNCT
easat-4034	7	1	in	in	ADP
easat-4034	7	2	this	this	DET
easat-4034	7	3	paper	paper	NOUN
easat-4034	7	4	,	,	PUNCT
easat-4034	7	5	we	we	PRON
easat-4034	7	6	will	will	AUX
easat-4034	7	7	develop	develop	VERB
easat-4034	7	8	the	the	DET
easat-4034	7	9	characteristics	characteristic	NOUN
easat-4034	7	10	of	of	ADP
easat-4034	7	11	general	general	ADJ
easat-4034	7	12	inverse	inverse	NOUN
easat-4034	7	13	matrices	matrix	NOUN
easat-4034	7	14	over	over	ADP
easat-4034	7	15	min	min	NOUN
easat-4034	7	16	-	-	PUNCT
easat-4034	7	17	plus	plus	CCONJ
easat-4034	7	18	algebras	algebra	NOUN
easat-4034	7	19	.	.	PUNCT
easat-4034	8	1	the	the	DET
easat-4034	8	2	research	research	NOUN
easat-4034	8	3	method	method	NOUN
easat-4034	8	4	used	use	VERB
easat-4034	8	5	is	be	AUX
easat-4034	8	6	the	the	DET
easat-4034	8	7	literature	literature	NOUN
easat-4034	8	8	study	study	NOUN
easat-4034	8	9	method	method	NOUN
easat-4034	8	10	sourced	source	VERB
easat-4034	8	11	from	from	ADP
easat-4034	8	12	books	book	NOUN
easat-4034	8	13	and	and	CCONJ
easat-4034	8	14	journal	journal	NOUN
easat-4034	8	15	articles	article	NOUN
easat-4034	8	16	.	.	PUNCT
easat-4034	9	1	the	the	DET
easat-4034	9	2	main	main	ADJ
easat-4034	9	3	result	result	NOUN
easat-4034	9	4	of	of	ADP
easat-4034	9	5	this	this	DET
easat-4034	9	6	study	study	NOUN
easat-4034	9	7	is	be	AUX
easat-4034	9	8	that	that	SCONJ
easat-4034	9	9	the	the	DET
easat-4034	9	10	generalized	generalized	ADJ
easat-4034	9	11	inverse	inverse	NOUN
easat-4034	9	12	of	of	ADP
easat-4034	9	13	the	the	DET
easat-4034	9	14	matrix	matrix	NOUN
easat-4034	9	15	𝐴	𝐴	NOUN
easat-4034	9	16	∈	∈	PROPN
easat-4034	9	17	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	9	18	𝑛×𝑛	𝑛×𝑛	PROPN
easat-4034	9	19	can	can	AUX
easat-4034	9	20	be	be	AUX
easat-4034	9	21	obtained	obtain	VERB
easat-4034	9	22	by	by	ADP
easat-4034	9	23	determining	determine	VERB
easat-4034	9	24	the	the	DET
easat-4034	9	25	matrix	matrix	NOUN
easat-4034	9	26	𝑋	𝑋	NOUN
easat-4034	9	27	with	with	ADP
easat-4034	9	28	entry	entry	NOUN
easat-4034	9	29	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	9	30	=	=	SYM
easat-4034	9	31	𝑛	𝑛	PRON
easat-4034	9	32	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
easat-4034	9	33	𝑖	𝑖	NOUN
easat-4034	9	34	=	=	SYM
easat-4034	9	35	1	1	NUM
easat-4034	9	36	𝑛	𝑛	PRON
easat-4034	9	37	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
easat-4034	9	38	𝑗	𝑗	NOUN
easat-4034	9	39	=	=	SYM
easat-4034	9	40	1	1	NUM
easat-4034	9	41	(	(	PUNCT
easat-4034	9	42	−𝑎𝑖𝑘	−𝑎𝑖𝑘	NOUN
easat-4034	9	43	+	+	CCONJ
easat-4034	9	44	𝑎𝑖𝑗	𝑎𝑖𝑗	NOUN
easat-4034	9	45	−	−	PROPN
easat-4034	9	46	𝑎𝑙𝑗	𝑎𝑙𝑗	PROPN
easat-4034	9	47	)	)	PUNCT
easat-4034	9	48	which	which	PRON
easat-4034	9	49	satisfies	satisfy	VERB
easat-4034	9	50	𝐴	𝐴	PROPN
easat-4034	9	51	⊗	⊗	PROPN
easat-4034	9	52	𝑋	𝑋	PROPN
easat-4034	9	53	⊗	⊗	PROPN
easat-4034	9	54	𝐴	𝐴	PROPN
easat-4034	9	55	=	=	SYM
easat-4034	9	56	𝐴.	𝐴.	PROPN
easat-4034	9	57	keywords	keyword	NOUN
easat-4034	9	58	:	:	PUNCT
easat-4034	9	59	generalized	generalize	VERB
easat-4034	9	60	,	,	PUNCT
easat-4034	9	61	inverse	inverse	NOUN
easat-4034	9	62	,	,	PUNCT
easat-4034	9	63	min	min	NOUN
easat-4034	9	64	-	-	ADJ
easat-4034	9	65	plus	plus	ADJ
easat-4034	9	66	algebra	algebra	NOUN
easat-4034	9	67	.	.	PUNCT
easat-4034	10	1	1	1	X
easat-4034	10	2	.	.	X
easat-4034	10	3	introduction	introduction	NOUN
easat-4034	10	4	semiring	semire	VERB
easat-4034	10	5	is	be	AUX
easat-4034	10	6	an	an	DET
easat-4034	10	7	algebraic	algebraic	ADJ
easat-4034	10	8	structure	structure	NOUN
easat-4034	10	9	obtained	obtain	VERB
easat-4034	10	10	from	from	ADP
easat-4034	10	11	rings	ring	NOUN
easat-4034	10	12	with	with	ADP
easat-4034	10	13	the	the	DET
easat-4034	10	14	condition	condition	NOUN
easat-4034	10	15	that	that	SCONJ
easat-4034	10	16	a	a	DET
easat-4034	10	17	ring	ring	NOUN
easat-4034	10	18	is	be	AUX
easat-4034	10	19	weakened	weaken	VERB
easat-4034	10	20	by	by	ADP
easat-4034	10	21	eliminating	eliminate	VERB
easat-4034	10	22	several	several	ADJ
easat-4034	10	23	ring	ring	NOUN
easat-4034	10	24	conditions	condition	NOUN
easat-4034	10	25	.	.	PUNCT
easat-4034	11	1	other	other	ADJ
easat-4034	11	2	algebraic	algebraic	ADJ
easat-4034	11	3	structures	structure	NOUN
easat-4034	11	4	known	know	VERB
easat-4034	11	5	as	as	ADP
easat-4034	11	6	semigroups	semigroup	NOUN
easat-4034	11	7	and	and	CCONJ
easat-4034	11	8	semirings	semiring	NOUN
easat-4034	11	9	will	will	AUX
easat-4034	11	10	emerge	emerge	VERB
easat-4034	11	11	if	if	SCONJ
easat-4034	11	12	some	some	DET
easat-4034	11	13	properties	property	NOUN
easat-4034	11	14	of	of	ADP
easat-4034	11	15	groups	group	NOUN
easat-4034	11	16	and	and	CCONJ
easat-4034	11	17	rings	ring	NOUN
easat-4034	11	18	are	be	AUX
easat-4034	11	19	weakened	weaken	VERB
easat-4034	11	20	.	.	PUNCT
easat-4034	12	1	this	this	PRON
easat-4034	12	2	shows	show	VERB
easat-4034	12	3	that	that	SCONJ
easat-4034	12	4	the	the	DET
easat-4034	12	5	algebraic	algebraic	ADJ
easat-4034	12	6	structures	structure	NOUN
easat-4034	12	7	created	create	VERB
easat-4034	12	8	are	be	AUX
easat-4034	12	9	semigroups	semigroup	NOUN
easat-4034	12	10	and	and	CCONJ
easat-4034	12	11	then	then	ADV
easat-4034	12	12	semirings	semiring	NOUN
easat-4034	12	13	if	if	SCONJ
easat-4034	12	14	some	some	DET
easat-4034	12	15	group	group	NOUN
easat-4034	12	16	or	or	CCONJ
easat-4034	12	17	ring	ring	NOUN
easat-4034	12	18	conditions	condition	NOUN
easat-4034	12	19	are	be	AUX
easat-4034	12	20	removed	remove	VERB
easat-4034	12	21	(	(	PUNCT
easat-4034	12	22	[	[	X
easat-4034	12	23	1	1	NUM
easat-4034	12	24	]	]	PUNCT
easat-4034	12	25	,	,	PUNCT
easat-4034	12	26	[	[	X
easat-4034	12	27	2	2	NUM
easat-4034	12	28	]	]	NUM
easat-4034	12	29	)	)	PUNCT
easat-4034	12	30	.	.	PUNCT
easat-4034	13	1	the	the	DET
easat-4034	13	2	main	main	ADJ
easat-4034	13	3	difference	difference	NOUN
easat-4034	13	4	between	between	ADP
easat-4034	13	5	semiring	semiring	NOUN
easat-4034	13	6	and	and	CCONJ
easat-4034	13	7	ring	ring	NOUN
easat-4034	13	8	structures	structure	NOUN
easat-4034	13	9	can	can	AUX
easat-4034	13	10	be	be	AUX
easat-4034	13	11	seen	see	VERB
easat-4034	13	12	from	from	ADP
easat-4034	13	13	the	the	DET
easat-4034	13	14	existence	existence	NOUN
easat-4034	13	15	of	of	ADP
easat-4034	13	16	an	an	DET
easat-4034	13	17	inverse	inverse	NOUN
easat-4034	13	18	element	element	NOUN
easat-4034	13	19	for	for	ADP
easat-4034	13	20	the	the	DET
easat-4034	13	21	addition	addition	NOUN
easat-4034	13	22	operation	operation	NOUN
easat-4034	13	23	(	(	PUNCT
easat-4034	13	24	[	[	X
easat-4034	13	25	3	3	NUM
easat-4034	13	26	]	]	PUNCT
easat-4034	13	27	,	,	PUNCT
easat-4034	13	28	[	[	X
easat-4034	13	29	4	4	NUM
easat-4034	13	30	]	]	NUM
easat-4034	13	31	)	)	PUNCT
easat-4034	13	32	.	.	PUNCT
easat-4034	14	1	a	a	DET
easat-4034	14	2	structure	structure	NOUN
easat-4034	14	3	(	(	PUNCT
easat-4034	14	4	s,+,×	s,+,×	PROPN
easat-4034	14	5	)	)	PUNCT
easat-4034	14	6	with	with	ADP
easat-4034	14	7	s	s	PROPN
easat-4034	14	8	,	,	PUNCT
easat-4034	14	9	a	a	DET
easat-4034	14	10	non	non	ADJ
easat-4034	14	11	-	-	ADJ
easat-4034	14	12	empty	empty	ADJ
easat-4034	14	13	set	set	NOUN
easat-4034	14	14	,	,	PUNCT
easat-4034	14	15	+	+	CCONJ
easat-4034	14	16	addition	addition	NOUN
easat-4034	14	17	operation	operation	NOUN
easat-4034	14	18	,	,	PUNCT
easat-4034	14	19	and	and	CCONJ
easat-4034	14	20	×	×	NOUN
easat-4034	14	21	multiplication	multiplication	NOUN
easat-4034	14	22	operation	operation	NOUN
easat-4034	14	23	,	,	PUNCT
easat-4034	14	24	is	be	AUX
easat-4034	14	25	semiring	semire	VERB
easat-4034	14	26	if	if	SCONJ
easat-4034	14	27	it	it	PRON
easat-4034	14	28	fulfils	fulfil	VERB
easat-4034	14	29	the	the	DET
easat-4034	14	30	commutative	commutative	ADJ
easat-4034	14	31	and	and	CCONJ
easat-4034	14	32	associative	associative	ADJ
easat-4034	14	33	properties	property	NOUN
easat-4034	14	34	of	of	ADP
easat-4034	14	35	addition	addition	NOUN
easat-4034	14	36	,	,	PUNCT
easat-4034	14	37	multiplication	multiplication	NOUN
easat-4034	14	38	associativity	associativity	NOUN
easat-4034	14	39	,	,	PUNCT
easat-4034	14	40	distributive	distributive	ADJ
easat-4034	14	41	,	,	PUNCT
easat-4034	14	42	has	have	VERB
easat-4034	14	43	a	a	DET
easat-4034	14	44	zero	zero	NUM
easat-4034	14	45	element	element	NOUN
easat-4034	14	46	,	,	PUNCT
easat-4034	14	47	and	and	CCONJ
easat-4034	14	48	a	a	DET
easat-4034	14	49	unit	unit	NOUN
easat-4034	14	50	element	element	NOUN
easat-4034	14	51	(	(	PUNCT
easat-4034	14	52	[	[	X
easat-4034	14	53	4	4	NUM
easat-4034	14	54	]	]	PUNCT
easat-4034	14	55	,	,	PUNCT
easat-4034	14	56	[	[	X
easat-4034	14	57	5	5	NUM
easat-4034	14	58	]	]	PUNCT
easat-4034	14	59	)	)	PUNCT
easat-4034	14	60	.	.	PUNCT
easat-4034	15	1	it	it	PRON
easat-4034	15	2	is	be	AUX
easat-4034	15	3	well	well	ADV
easat-4034	15	4	known	know	VERB
easat-4034	15	5	that	that	SCONJ
easat-4034	15	6	a	a	DET
easat-4034	15	7	semigroup	semigroup	NOUN
easat-4034	15	8	is	be	AUX
easat-4034	15	9	formed	form	VERB
easat-4034	15	10	by	by	ADP
easat-4034	15	11	a	a	DET
easat-4034	15	12	non	non	ADJ
easat-4034	15	13	-	-	ADJ
easat-4034	15	14	empty	empty	ADJ
easat-4034	15	15	set	set	NOUN
easat-4034	15	16	s	s	X
easat-4034	15	17	and	and	CCONJ
easat-4034	15	18	the	the	DET
easat-4034	15	19	associative	associative	ADJ
easat-4034	15	20	binary	binary	ADJ
easat-4034	15	21	operation	operation	NOUN
easat-4034	15	22	×.	×.	PRON
easat-4034	15	23	thus	thus	ADV
easat-4034	15	24	,	,	PUNCT
easat-4034	15	25	the	the	DET
easat-4034	15	26	structure	structure	NOUN
easat-4034	15	27	(	(	PUNCT
easat-4034	15	28	s,+,×	s,+,×	PROPN
easat-4034	15	29	)	)	PUNCT
easat-4034	15	30	is	be	AUX
easat-4034	15	31	said	say	VERB
easat-4034	15	32	to	to	PART
easat-4034	15	33	be	be	AUX
easat-4034	15	34	semiring	semire	VERB
easat-4034	15	35	if	if	SCONJ
easat-4034	15	36	(	(	PUNCT
easat-4034	15	37	s,+	s,+	PRON
easat-4034	15	38	)	)	PUNCT
easat-4034	15	39	is	be	AUX
easat-4034	15	40	a	a	DET
easat-4034	15	41	commutative	commutative	ADJ
easat-4034	15	42	semigroup	semigroup	NOUN
easat-4034	15	43	,	,	PUNCT
easat-4034	15	44	(	(	PUNCT
easat-4034	15	45	s,×	s,×	PROPN
easat-4034	15	46	)	)	PUNCT
easat-4034	15	47	is	be	AUX
easat-4034	15	48	a	a	DET
easat-4034	15	49	semigroup	semigroup	NOUN
easat-4034	15	50	,	,	PUNCT
easat-4034	15	51	distributive	distributive	ADJ
easat-4034	15	52	,	,	PUNCT
easat-4034	15	53	has	have	AUX
easat-4034	15	54	element	element	NOUN
easat-4034	15	55	0	0	PUNCT
easat-4034	15	56	and	and	CCONJ
easat-4034	15	57	has	have	VERB
easat-4034	15	58	a	a	DET
easat-4034	15	59	unit	unit	NOUN
easat-4034	15	60	element	element	NOUN
easat-4034	15	61	(	(	PUNCT
easat-4034	15	62	[	[	X
easat-4034	15	63	6	6	NUM
easat-4034	15	64	]	]	PUNCT
easat-4034	15	65	,	,	PUNCT
easat-4034	15	66	[	[	X
easat-4034	15	67	7	7	NUM
easat-4034	15	68	]	]	NUM
easat-4034	15	69	)	)	PUNCT
easat-4034	15	70	.	.	PUNCT
easat-4034	16	1	with	with	ADP
easat-4034	16	2	semiring	semiring	NOUN
easat-4034	16	3	entries	entry	NOUN
easat-4034	16	4	,	,	PUNCT
easat-4034	16	5	a	a	DET
easat-4034	16	6	semiring	semiring	NOUN
easat-4034	16	7	matrix	matrix	NOUN
easat-4034	16	8	can	can	AUX
easat-4034	16	9	be	be	AUX
easat-4034	16	10	developed	develop	VERB
easat-4034	16	11	[	[	X
easat-4034	16	12	1	1	NUM
easat-4034	16	13	]	]	PUNCT
easat-4034	16	14	,	,	PUNCT
easat-4034	16	15	[	[	X
easat-4034	16	16	5	5	NUM
easat-4034	16	17	]	]	PUNCT
easat-4034	16	18	,	,	PUNCT
easat-4034	16	19	[	[	X
easat-4034	16	20	8	8	NUM
easat-4034	16	21	]	]	PUNCT
easat-4034	16	22	.	.	PUNCT
easat-4034	17	1	one	one	NUM
easat-4034	17	2	structure	structure	NOUN
easat-4034	17	3	that	that	PRON
easat-4034	17	4	is	be	AUX
easat-4034	17	5	a	a	DET
easat-4034	17	6	semiring	semiring	NOUN
easat-4034	17	7	is	be	AUX
easat-4034	17	8	max	max	PROPN
easat-4034	17	9	-	-	PUNCT
easat-4034	17	10	plus	plus	NOUN
easat-4034	17	11	algebra	algebra	NOUN
easat-4034	17	12	.	.	PUNCT
easat-4034	18	1	a	a	DET
easat-4034	18	2	structure	structure	NOUN
easat-4034	18	3	(	(	PUNCT
easat-4034	18	4	ℝmax,⊕,⊗	ℝmax,⊕,⊗	NOUN
easat-4034	18	5	)	)	PUNCT
easat-4034	18	6	with	with	ADP
easat-4034	18	7	ℝmax	ℝmax	PROPN
easat-4034	18	8	=	=	SYM
easat-4034	18	9	ℝ	ℝ	PROPN
easat-4034	18	10	∪	∪	ADJ
easat-4034	18	11	{	{	PUNCT
easat-4034	18	12	−∞	−∞	NOUN
easat-4034	18	13	}	}	PUNCT
easat-4034	18	14	is	be	AUX
easat-4034	18	15	said	say	VERB
easat-4034	18	16	to	to	PART
easat-4034	18	17	be	be	AUX
easat-4034	18	18	a	a	DET
easat-4034	18	19	max	max	PROPN
easat-4034	18	20	-	-	PUNCT
easat-4034	18	21	plus	plus	NOUN
easat-4034	18	22	algebra	algebra	NOUN
easat-4034	18	23	with	with	ADP
easat-4034	18	24	a	a	DET
easat-4034	18	25	maximum	maximum	PROPN
easat-4034	18	26	⊕	⊕	PROPN
easat-4034	18	27	operation	operation	NOUN
easat-4034	18	28	and	and	CCONJ
easat-4034	18	29	an	an	DET
easat-4034	18	30	addition	addition	NOUN
easat-4034	18	31	⊗	⊗	PROPN
easat-4034	18	32	operation	operation	NOUN
easat-4034	18	33	.	.	PUNCT
easat-4034	19	1	in	in	ADP
easat-4034	19	2	another	another	DET
easat-4034	19	3	section	section	NOUN
easat-4034	19	4	,	,	PUNCT
easat-4034	19	5	a	a	DET
easat-4034	19	6	semiring	semire	VERB
easat-4034	19	7	other	other	ADJ
easat-4034	19	8	than	than	ADP
easat-4034	19	9	max	max	PROPN
easat-4034	19	10	-	-	PUNCT
easat-4034	19	11	plus	plus	CCONJ
easat-4034	19	12	algebra	algebra	NOUN
easat-4034	19	13	is	be	AUX
easat-4034	19	14	min	min	ADJ
easat-4034	19	15	-	-	ADJ
easat-4034	19	16	plus	plus	ADJ
easat-4034	19	17	algebra	algebra	NOUN
easat-4034	19	18	.	.	PUNCT
easat-4034	20	1	min	min	ADJ
easat-4034	20	2	-	-	PUNCT
easat-4034	20	3	plus	plus	ADJ
easat-4034	20	4	algebra	algebra	NOUN
easat-4034	20	5	rmin	rmin	NOUN
easat-4034	20	6	=	=	NOUN
easat-4034	20	7	r	r	NOUN
easat-4034	20	8	∪	∪	X
easat-4034	20	9	{	{	PUNCT
easat-4034	20	10	+	+	NOUN
easat-4034	20	11	∞	∞	NOUN
easat-4034	20	12	}	}	PUNCT
easat-4034	20	13	with	with	ADP
easat-4034	20	14	minimum	minimum	NOUN
easat-4034	20	15	(	(	PUNCT
easat-4034	20	16	⊕	⊕	NOUN
easat-4034	20	17	′	′	NUM
easat-4034	20	18	)	)	PUNCT
easat-4034	20	19	and	and	CCONJ
easat-4034	20	20	addition	addition	NOUN
easat-4034	20	21	(	(	PUNCT
easat-4034	20	22	⊗	⊗	ADJ
easat-4034	20	23	)	)	PUNCT
easat-4034	20	24	operations	operation	NOUN
easat-4034	20	25	with	with	ADP
easat-4034	20	26	identity	identity	NOUN
easat-4034	20	27	elements	element	NOUN
easat-4034	20	28	with	with	ADP
easat-4034	20	29	respect	respect	NOUN
easat-4034	20	30	to	to	ADP
easat-4034	20	31	⊕	⊕	PROPN
easat-4034	20	32	′	′	NUM
easat-4034	20	33	are	be	AUX
easat-4034	20	34	𝜀′	𝜀′	X
easat-4034	20	35	=	=	PUNCT
easat-4034	21	1	+	+	NOUN
easat-4034	21	2	∞	∞	NUM
easat-4034	21	3	and	and	CCONJ
easat-4034	21	4	𝑒	𝑒	PROPN
easat-4034	21	5	=	=	SYM
easat-4034	21	6	0	0	NUM
easat-4034	21	7	.	.	PUNCT
easat-4034	21	8	max	max	PROPN
easat-4034	21	9	-	-	PUNCT
easat-4034	21	10	plus	plus	CCONJ
easat-4034	21	11	algebra	algebra	NOUN
easat-4034	21	12	and	and	CCONJ
easat-4034	21	13	min	min	NOUN
easat-4034	21	14	-	-	PUNCT
easat-4034	21	15	plus	plus	ADJ
easat-4034	21	16	algebra	algebra	NOUN
easat-4034	21	17	are	be	AUX
easat-4034	21	18	isomorphic	isomorphic	ADJ
easat-4034	21	19	because	because	SCONJ
easat-4034	21	20	of	of	ADP
easat-4034	21	21	their	their	PRON
easat-4034	21	22	similar	similar	ADJ
easat-4034	21	23	structure	structure	NOUN
easat-4034	21	24	.	.	PUNCT
easat-4034	22	1	it	it	PRON
easat-4034	22	2	is	be	AUX
easat-4034	22	3	possible	possible	ADJ
easat-4034	22	4	to	to	PART
easat-4034	22	5	convert	convert	VERB
easat-4034	22	6	the	the	DET
easat-4034	22	7	idea	idea	NOUN
easat-4034	22	8	of	of	ADP
easat-4034	22	9	max	max	PROPN
easat-4034	22	10	-	-	PUNCT
easat-4034	22	11	plus	plus	CCONJ
easat-4034	22	12	algebra	algebra	NOUN
easat-4034	22	13	into	into	ADP
easat-4034	22	14	min	min	NOUN
easat-4034	22	15	-	-	PUNCT
easat-4034	22	16	plus	plus	ADJ
easat-4034	22	17	algebra	algebra	NOUN
easat-4034	22	18	[	[	X
easat-4034	22	19	5	5	NUM
easat-4034	22	20	]	]	PUNCT
easat-4034	22	21	.	.	PUNCT
easat-4034	23	1	the	the	DET
easat-4034	23	2	semiring	semiring	NOUN
easat-4034	23	3	element	element	NOUN
easat-4034	23	4	has	have	VERB
easat-4034	23	5	an	an	DET
easat-4034	23	6	inverse	inverse	NOUN
easat-4034	23	7	to	to	ADP
easat-4034	23	8	the	the	DET
easat-4034	23	9	addition	addition	NOUN
easat-4034	23	10	operation	operation	NOUN
easat-4034	23	11	so	so	SCONJ
easat-4034	23	12	that	that	SCONJ
easat-4034	23	13	the	the	DET
easat-4034	23	14	determinant	determinant	NOUN
easat-4034	23	15	of	of	ADP
easat-4034	23	16	a	a	DET
easat-4034	23	17	matrix	matrix	NOUN
easat-4034	23	18	over	over	ADP
easat-4034	23	19	the	the	DET
easat-4034	23	20	semiring	semiring	NOUN
easat-4034	23	21	can	can	AUX
easat-4034	23	22	be	be	AUX
easat-4034	23	23	defined	define	VERB
easat-4034	23	24	(	(	PUNCT
easat-4034	23	25	[	[	X
easat-4034	23	26	3	3	NUM
easat-4034	23	27	]	]	PUNCT
easat-4034	23	28	,	,	PUNCT
easat-4034	24	1	[	[	X
easat-4034	24	2	6	6	NUM
easat-4034	24	3	]	]	NUM
easat-4034	24	4	)	)	PUNCT
easat-4034	24	5	.	.	PUNCT
easat-4034	25	1	the	the	DET
easat-4034	25	2	inverse	inverse	NOUN
easat-4034	25	3	of	of	ADP
easat-4034	25	4	the	the	DET
easat-4034	25	5	semiring	semiring	NOUN
easat-4034	25	6	matrix	matrix	NOUN
easat-4034	25	7	can	can	AUX
easat-4034	25	8	be	be	AUX
easat-4034	25	9	determined	determine	VERB
easat-4034	25	10	by	by	ADP
easat-4034	25	11	determining	determine	VERB
easat-4034	25	12	the	the	DET
easat-4034	25	13	determinant	determinant	NOUN
easat-4034	25	14	of	of	ADP
easat-4034	25	15	the	the	DET
easat-4034	25	16	semiring	semiring	NOUN
easat-4034	25	17	matrix	matrix	NOUN
easat-4034	25	18	.	.	PUNCT
easat-4034	26	1	looking	look	VERB
easat-4034	26	2	at	at	ADP
easat-4034	26	3	the	the	DET
easat-4034	26	4	characteristics	characteristic	NOUN
easat-4034	26	5	of	of	ADP
easat-4034	26	6	the	the	DET
easat-4034	26	7	semiring	semiring	NOUN
easat-4034	26	8	,	,	PUNCT
easat-4034	26	9	we	we	PRON
easat-4034	26	10	will	will	AUX
easat-4034	26	11	specifically	specifically	ADV
easat-4034	26	12	look	look	VERB
easat-4034	26	13	at	at	ADP
easat-4034	26	14	the	the	DET
easat-4034	26	15	characteristics	characteristic	NOUN
easat-4034	26	16	of	of	ADP
easat-4034	26	17	min	min	NOUN
easat-4034	26	18	-	-	ADJ
easat-4034	26	19	plus	plus	ADJ
easat-4034	26	20	algebra	algebra	NOUN
easat-4034	26	21	.	.	PUNCT
easat-4034	27	1	it	it	PRON
easat-4034	27	2	is	be	AUX
easat-4034	27	3	done	do	VERB
easat-4034	27	4	because	because	SCONJ
easat-4034	27	5	not	not	PART
easat-4034	27	6	all	all	PRON
easat-4034	27	7	semiring	semire	VERB
easat-4034	27	8	properties	property	NOUN
easat-4034	27	9	also	also	ADV
easat-4034	27	10	apply	apply	VERB
easat-4034	27	11	to	to	ADP
easat-4034	27	12	min	min	ADJ
easat-4034	27	13	-	-	PUNCT
easat-4034	27	14	plus	plus	ADJ
easat-4034	27	15	algebra	algebra	NOUN
easat-4034	27	16	.	.	PUNCT
easat-4034	28	1	as	as	ADP
easat-4034	28	2	with	with	ADP
easat-4034	28	3	group	group	NOUN
easat-4034	28	4	and	and	CCONJ
easat-4034	28	5	ring	ring	NOUN
easat-4034	28	6	structures	structure	NOUN
easat-4034	28	7	,	,	PUNCT
easat-4034	28	8	the	the	DET
easat-4034	28	9	commutative	commutative	ADJ
easat-4034	28	10	characteristic	characteristic	NOUN
easat-4034	28	11	applies	apply	VERB
easat-4034	28	12	to	to	ADP
easat-4034	28	13	certain	certain	ADJ
easat-4034	28	14	semirings	semiring	NOUN
easat-4034	28	15	[	[	X
easat-4034	28	16	6	6	NUM
easat-4034	28	17	]	]	PUNCT
easat-4034	28	18	,	,	PUNCT
easat-4034	29	1	[	[	X
easat-4034	29	2	9	9	NUM
easat-4034	29	3	]	]	PUNCT
easat-4034	29	4	,	,	PUNCT
easat-4034	29	5	[	[	X
easat-4034	29	6	10	10	NUM
easat-4034	29	7	]	]	PUNCT
easat-4034	29	8	.	.	PUNCT
easat-4034	30	1	a	a	DET
easat-4034	30	2	particular	particular	ADJ
easat-4034	30	3	semiring	semiring	NOUN
easat-4034	30	4	owns	own	VERB
easat-4034	30	5	the	the	DET
easat-4034	30	6	existence	existence	NOUN
easat-4034	30	7	of	of	ADP
easat-4034	30	8	an	an	DET
easat-4034	30	9	inverse	inverse	NOUN
easat-4034	30	10	element	element	NOUN
easat-4034	30	11	for	for	ADP
easat-4034	30	12	addition	addition	NOUN
easat-4034	30	13	on	on	ADP
easat-4034	30	14	a	a	DET
easat-4034	30	15	semiring	semiring	NOUN
easat-4034	30	16	.	.	PUNCT
easat-4034	31	1	in	in	ADP
easat-4034	31	2	9545	9545	NUM
easat-4034	31	3	edelweiss	edelweiss	PROPN
easat-4034	31	4	applied	apply	VERB
easat-4034	31	5	science	science	NOUN
easat-4034	31	6	and	and	CCONJ
easat-4034	31	7	technology	technology	NOUN
easat-4034	31	8	issn	issn	PROPN
easat-4034	31	9	:	:	PUNCT
easat-4034	31	10	2576	2576	NUM
easat-4034	31	11	-	-	SYM
easat-4034	31	12	8484	8484	NUM
easat-4034	31	13	vol	vol	NOUN
easat-4034	31	14	.	.	PROPN
easat-4034	31	15	8	8	NUM
easat-4034	31	16	,	,	PUNCT
easat-4034	31	17	no	no	INTJ
easat-4034	31	18	.	.	NOUN
easat-4034	32	1	6	6	NUM
easat-4034	32	2	:	:	PUNCT
easat-4034	32	3	9544	9544	NUM
easat-4034	32	4	-	-	SYM
easat-4034	32	5	9554	9554	NUM
easat-4034	32	6	,	,	PUNCT
easat-4034	32	7	2024	2024	NUM
easat-4034	32	8	doi	doi	NOUN
easat-4034	32	9	:	:	PUNCT
easat-4034	32	10	10.55214/25768484.v8i6.4034	10.55214/25768484.v8i6.4034	NUM
easat-4034	32	11	©	©	ADP
easat-4034	32	12	2024	2024	NUM
easat-4034	32	13	by	by	ADP
easat-4034	32	14	the	the	DET
easat-4034	32	15	authors	author	NOUN
easat-4034	32	16	;	;	PUNCT
easat-4034	32	17	licensee	licensee	PROPN
easat-4034	32	18	learning	learning	NOUN
easat-4034	32	19	gate	gate	VERB
easat-4034	32	20	a	a	DET
easat-4034	32	21	semiring	semiring	NOUN
easat-4034	32	22	,	,	PUNCT
easat-4034	32	23	the	the	DET
easat-4034	32	24	entry	entry	NOUN
easat-4034	32	25	on	on	ADP
easat-4034	32	26	the	the	DET
easat-4034	32	27	semiring	semiring	NOUN
easat-4034	32	28	has	have	VERB
easat-4034	32	29	the	the	DET
easat-4034	32	30	inverse	inverse	NOUN
easat-4034	32	31	of	of	ADP
easat-4034	32	32	the	the	DET
easat-4034	32	33	+	+	NUM
easat-4034	32	34	operation	operation	NOUN
easat-4034	32	35	so	so	SCONJ
easat-4034	32	36	that	that	SCONJ
easat-4034	32	37	the	the	DET
easat-4034	32	38	determinant	determinant	NOUN
easat-4034	32	39	of	of	ADP
easat-4034	32	40	the	the	DET
easat-4034	32	41	matrix	matrix	NOUN
easat-4034	32	42	on	on	ADP
easat-4034	32	43	the	the	DET
easat-4034	32	44	semiring	semiring	NOUN
easat-4034	32	45	can	can	AUX
easat-4034	32	46	be	be	AUX
easat-4034	32	47	defined	define	VERB
easat-4034	32	48	.	.	PUNCT
easat-4034	33	1	this	this	DET
easat-4034	33	2	study	study	NOUN
easat-4034	33	3	aims	aim	VERB
easat-4034	33	4	to	to	PART
easat-4034	33	5	develop	develop	VERB
easat-4034	33	6	the	the	DET
easat-4034	33	7	characteristics	characteristic	NOUN
easat-4034	33	8	of	of	ADP
easat-4034	33	9	the	the	DET
easat-4034	33	10	generalized	generalized	ADJ
easat-4034	33	11	inverse	inverse	NOUN
easat-4034	33	12	matrix	matrix	NOUN
easat-4034	33	13	over	over	ADP
easat-4034	33	14	a	a	DET
easat-4034	33	15	min	min	NOUN
easat-4034	33	16	-	-	PUNCT
easat-4034	33	17	plus	plus	ADJ
easat-4034	33	18	algebra	algebra	NOUN
easat-4034	33	19	.	.	PUNCT
easat-4034	34	1	2	2	X
easat-4034	34	2	.	.	X
easat-4034	34	3	materials	material	NOUN
easat-4034	34	4	and	and	CCONJ
easat-4034	34	5	methods	method	NOUN
easat-4034	34	6	reducing	reduce	VERB
easat-4034	34	7	several	several	ADJ
easat-4034	34	8	properties	property	NOUN
easat-4034	34	9	will	will	AUX
easat-4034	34	10	form	form	VERB
easat-4034	34	11	a	a	DET
easat-4034	34	12	new	new	ADJ
easat-4034	34	13	algebraic	algebraic	ADJ
easat-4034	34	14	structure	structure	NOUN
easat-4034	34	15	.	.	PUNCT
easat-4034	35	1	not	not	PART
easat-4034	35	2	all	all	DET
easat-4034	35	3	properties	property	NOUN
easat-4034	35	4	of	of	ADP
easat-4034	35	5	the	the	DET
easat-4034	35	6	complete	complete	ADJ
easat-4034	35	7	structure	structure	NOUN
easat-4034	35	8	will	will	AUX
easat-4034	35	9	also	also	ADV
easat-4034	35	10	be	be	AUX
easat-4034	35	11	reduced	reduce	VERB
easat-4034	35	12	to	to	ADP
easat-4034	35	13	the	the	DET
easat-4034	35	14	new	new	ADJ
easat-4034	35	15	algebraic	algebraic	ADJ
easat-4034	35	16	structure	structure	NOUN
easat-4034	35	17	.	.	PUNCT
easat-4034	36	1	the	the	DET
easat-4034	36	2	research	research	NOUN
easat-4034	36	3	uses	use	VERB
easat-4034	36	4	a	a	DET
easat-4034	36	5	literature	literature	NOUN
easat-4034	36	6	study	study	NOUN
easat-4034	36	7	method	method	NOUN
easat-4034	36	8	sourced	source	VERB
easat-4034	36	9	from	from	ADP
easat-4034	36	10	books	book	NOUN
easat-4034	36	11	and	and	CCONJ
easat-4034	36	12	journal	journal	NOUN
easat-4034	36	13	articles	article	NOUN
easat-4034	36	14	.	.	PUNCT
easat-4034	37	1	the	the	DET
easat-4034	37	2	steps	step	NOUN
easat-4034	37	3	for	for	ADP
easat-4034	37	4	developing	develop	VERB
easat-4034	37	5	ideas	idea	NOUN
easat-4034	37	6	in	in	ADP
easat-4034	37	7	this	this	DET
easat-4034	37	8	study	study	NOUN
easat-4034	37	9	are	be	AUX
easat-4034	37	10	shown	show	VERB
easat-4034	37	11	in	in	ADP
easat-4034	37	12	figure	figure	NOUN
easat-4034	37	13	1	1	NUM
easat-4034	37	14	.	.	PUNCT
easat-4034	37	15	figure	figure	NOUN
easat-4034	37	16	1	1	NUM
easat-4034	37	17	.	.	PUNCT
easat-4034	37	18	procedure	procedure	NOUN
easat-4034	37	19	for	for	ADP
easat-4034	37	20	the	the	DET
easat-4034	37	21	characteristic	characteristic	ADJ
easat-4034	37	22	study	study	NOUN
easat-4034	37	23	of	of	ADP
easat-4034	37	24	min	min	NOUN
easat-4034	37	25	-	-	PUNCT
easat-4034	37	26	plus	plus	ADJ
easat-4034	37	27	algebra	algebra	NOUN
easat-4034	37	28	.	.	PUNCT
easat-4034	38	1	2.1	2.1	NUM
easat-4034	38	2	.	.	PUNCT
easat-4034	39	1	min	min	NOUN
easat-4034	39	2	-	-	PUNCT
easat-4034	39	3	plus	plus	NOUN
easat-4034	39	4	algebra	algebra	VERB
easat-4034	39	5	the	the	DET
easat-4034	39	6	properties	property	NOUN
easat-4034	39	7	of	of	ADP
easat-4034	39	8	a	a	DET
easat-4034	39	9	max	max	PROPN
easat-4034	39	10	-	-	PUNCT
easat-4034	39	11	plus	plus	CCONJ
easat-4034	39	12	algebra	algebra	NOUN
easat-4034	39	13	can	can	AUX
easat-4034	39	14	be	be	AUX
easat-4034	39	15	used	use	VERB
easat-4034	39	16	to	to	PART
easat-4034	39	17	create	create	VERB
easat-4034	39	18	a	a	DET
easat-4034	39	19	min	min	NOUN
easat-4034	39	20	-	-	PUNCT
easat-4034	39	21	plus	plus	ADJ
easat-4034	39	22	algebra	algebra	NOUN
easat-4034	39	23	.	.	PUNCT
easat-4034	40	1	definition	definition	NOUN
easat-4034	40	2	1	1	NUM
easat-4034	40	3	the	the	DET
easat-4034	40	4	structure	structure	NOUN
easat-4034	40	5	(	(	PUNCT
easat-4034	40	6	ℝ𝑚𝑖𝑛,⊕′,⊗	ℝ𝑚𝑖𝑛,⊕′,⊗	PROPN
easat-4034	40	7	)	)	PUNCT
easat-4034	40	8	is	be	AUX
easat-4034	40	9	said	say	VERB
easat-4034	40	10	to	to	PART
easat-4034	40	11	be	be	AUX
easat-4034	40	12	a	a	DET
easat-4034	40	13	min	min	NOUN
easat-4034	40	14	-	-	PUNCT
easat-4034	40	15	plus	plus	ADJ
easat-4034	40	16	algebra	algebra	NOUN
easat-4034	40	17	with	with	ADP
easat-4034	40	18	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	40	19	=	=	SYM
easat-4034	40	20	ℝ	ℝ	PROPN
easat-4034	40	21	∪	∪	ADJ
easat-4034	40	22	{	{	PUNCT
easat-4034	40	23	+	+	NOUN
easat-4034	40	24	∞	∞	NUM
easat-4034	40	25	}	}	PUNCT
easat-4034	40	26	,	,	PUNCT
easat-4034	40	27	the	the	DET
easat-4034	40	28	binary	binary	PROPN
easat-4034	40	29	operations	operations	PROPN
easat-4034	40	30	⊕	⊕	PROPN
easat-4034	40	31	′	′	NUM
easat-4034	40	32	and	and	CCONJ
easat-4034	40	33	⊗	⊗	PROPN
easat-4034	40	34	defined	define	VERB
easat-4034	40	35	as	as	ADP
easat-4034	40	36	𝑎	𝑎	PROPN
easat-4034	40	37	⊕	⊕	NOUN
easat-4034	40	38	′	′	NOUN
easat-4034	41	1	𝑏	𝑏	NOUN
easat-4034	41	2	=	=	SYM
easat-4034	41	3	𝑚𝑖𝑛	𝑚𝑖𝑛	PROPN
easat-4034	41	4	{	{	PUNCT
easat-4034	41	5	𝑎	𝑎	NOUN
easat-4034	41	6	,	,	PUNCT
easat-4034	41	7	𝑏	𝑏	NOUN
easat-4034	41	8	}	}	PUNCT
easat-4034	41	9	and	and	CCONJ
easat-4034	41	10	𝑎	𝑎	DET
easat-4034	41	11	⊗	⊗	ADJ
easat-4034	41	12	𝑏	𝑏	NOUN
easat-4034	41	13	=	=	SYM
easat-4034	41	14	𝑎	𝑎	PROPN
easat-4034	41	15	+	+	X
easat-4034	41	16	𝑏	𝑏	NOUN
easat-4034	41	17	for	for	ADP
easat-4034	41	18	𝑎	𝑎	NOUN
easat-4034	41	19	,	,	PUNCT
easat-4034	41	20	𝑏	𝑏	PROPN
easat-4034	41	21	∈	∈	PROPN
easat-4034	41	22	ℝ𝑚𝑖𝑛.	ℝ𝑚𝑖𝑛.	PROPN
easat-4034	41	23	theorem	theorem	NOUN
easat-4034	41	24	1	1	NUM
easat-4034	41	25	provides	provide	VERB
easat-4034	41	26	the	the	DET
easat-4034	41	27	algebraic	algebraic	ADJ
easat-4034	41	28	property	property	NOUN
easat-4034	41	29	of	of	ADP
easat-4034	41	30	min	min	NOUN
easat-4034	41	31	-	-	PUNCT
easat-4034	41	32	plus	plus	ADJ
easat-4034	41	33	.	.	PUNCT
easat-4034	42	1	theorem	theorem	NOUN
easat-4034	42	2	1	1	NUM
easat-4034	42	3	for	for	ADP
easat-4034	42	4	an	an	DET
easat-4034	42	5	𝑥	𝑥	PROPN
easat-4034	42	6	,	,	PUNCT
easat-4034	42	7	𝑦	𝑦	NOUN
easat-4034	42	8	,	,	PUNCT
easat-4034	42	9	𝑧	𝑧	PRON
easat-4034	42	10	∈	∈	PROPN
easat-4034	42	11	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	42	12	with	with	ADP
easat-4034	42	13	𝑒	𝑒	PROPN
easat-4034	42	14	≔	≔	NOUN
easat-4034	42	15	0	0	NUM
easat-4034	42	16	and	and	CCONJ
easat-4034	42	17	𝜀′	𝜀′	PUNCT
easat-4034	43	1	=	=	SYM
easat-4034	43	2	+	+	NUM
easat-4034	43	3	∞	∞	PROPN
easat-4034	43	4	applies	apply	VERB
easat-4034	43	5	1	1	NUM
easat-4034	43	6	.	.	NOUN
easat-4034	43	7	associative	associative	ADJ
easat-4034	43	8	,	,	PUNCT
easat-4034	43	9	namely	namely	ADV
easat-4034	43	10	∀𝑥	∀𝑥	NUM
easat-4034	43	11	,	,	PUNCT
easat-4034	43	12	𝑦	𝑦	NOUN
easat-4034	43	13	,	,	PUNCT
easat-4034	43	14	𝑧	𝑧	PRON
easat-4034	43	15	∈	∈	PROPN
easat-4034	43	16	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	43	17	,	,	PUNCT
easat-4034	43	18	𝑥	𝑥	PROPN
easat-4034	43	19	⊕′	⊕′	PROPN
easat-4034	43	20	(	(	PUNCT
easat-4034	43	21	𝑦	𝑦	NOUN
easat-4034	43	22	⊕′	⊕′	PROPN
easat-4034	43	23	𝑧	𝑧	NOUN
easat-4034	43	24	)	)	PUNCT
easat-4034	43	25	=	=	SYM
easat-4034	43	26	(	(	PUNCT
easat-4034	43	27	𝑥	𝑥	PROPN
easat-4034	43	28	⊕′	⊕′	PROPN
easat-4034	43	29	𝑦	𝑦	NOUN
easat-4034	43	30	)	)	PUNCT
easat-4034	43	31	⊕	⊕	PROPN
easat-4034	43	32	′𝑧	′𝑧	PROPN
easat-4034	43	33	and	and	CCONJ
easat-4034	43	34	𝑥	𝑥	PROPN
easat-4034	43	35	⊗	⊗	PROPN
easat-4034	43	36	(	(	PUNCT
easat-4034	43	37	𝑦	𝑦	NOUN
easat-4034	43	38	⊗	⊗	NOUN
easat-4034	43	39	𝑧	𝑧	NOUN
easat-4034	43	40	)	)	PUNCT
easat-4034	44	1	=	=	SYM
easat-4034	44	2	(	(	PUNCT
easat-4034	44	3	𝑥	𝑥	PROPN
easat-4034	44	4	⊗	⊗	PROPN
easat-4034	44	5	𝑦	𝑦	NOUN
easat-4034	44	6	)	)	PUNCT
easat-4034	44	7	⊗	⊗	PROPN
easat-4034	44	8	𝑧	𝑧	PROPN
easat-4034	44	9	2	2	NUM
easat-4034	44	10	.	.	NOUN
easat-4034	44	11	commutative	commutative	ADJ
easat-4034	44	12	,	,	PUNCT
easat-4034	44	13	namely	namely	ADV
easat-4034	44	14	∀𝑥	∀𝑥	NUM
easat-4034	44	15	,	,	PUNCT
easat-4034	45	1	𝑦	𝑦	NOUN
easat-4034	45	2	∈	∈	NOUN
easat-4034	45	3	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	45	4	,	,	PUNCT
easat-4034	45	5	𝑥	𝑥	PROPN
easat-4034	45	6	⊕	⊕	PROPN
easat-4034	45	7	′𝑦	′𝑦	PROPN
easat-4034	45	8	=	=	SYM
easat-4034	45	9	𝑦	𝑦	PROPN
easat-4034	45	10	⊕	⊕	PROPN
easat-4034	45	11	′𝑥	′𝑥	PROPN
easat-4034	45	12	and	and	CCONJ
easat-4034	45	13	𝑥	𝑥	NOUN
easat-4034	45	14	⊗	⊗	PROPN
easat-4034	45	15	𝑦	𝑦	PROPN
easat-4034	45	16	=	=	PUNCT
easat-4034	45	17	𝑦	𝑦	PROPN
easat-4034	45	18	⊗	⊗	NOUN
easat-4034	46	1	𝑥	𝑥	PROPN
easat-4034	46	2	3	3	X
easat-4034	46	3	.	.	X
easat-4034	46	4	distributive	distributive	ADJ
easat-4034	46	5	of	of	ADP
easat-4034	46	6	⊗	⊗	PROPN
easat-4034	46	7	over	over	ADP
easat-4034	46	8	⊕	⊕	PROPN
easat-4034	46	9	′	′	PROPN
easat-4034	46	10	,	,	PUNCT
easat-4034	46	11	namely	namely	ADV
easat-4034	46	12	∀𝑥	∀𝑥	NUM
easat-4034	46	13	,	,	PUNCT
easat-4034	46	14	𝑦	𝑦	NOUN
easat-4034	46	15	,	,	PUNCT
easat-4034	46	16	𝑥	𝑥	PROPN
easat-4034	46	17	∈	∈	PROPN
easat-4034	47	1	ℝ𝑚𝑖𝑛,𝑥	ℝ𝑚𝑖𝑛,𝑥	CCONJ
easat-4034	47	2	⊗	⊗	PROPN
easat-4034	47	3	(	(	PUNCT
easat-4034	47	4	𝑦	𝑦	PROPN
easat-4034	47	5	⊕′	⊕′	PROPN
easat-4034	47	6	𝑧	𝑧	NOUN
easat-4034	47	7	)	)	PUNCT
easat-4034	47	8	=	=	SYM
easat-4034	47	9	(	(	PUNCT
easat-4034	47	10	𝑥	𝑥	PROPN
easat-4034	47	11	⊗	⊗	PROPN
easat-4034	47	12	𝑦	𝑦	NOUN
easat-4034	47	13	)	)	PUNCT
easat-4034	47	14	⊕	⊕	PROPN
easat-4034	47	15	′(𝑥	′(𝑥	VERB
easat-4034	47	16	⊗	⊗	PROPN
easat-4034	47	17	𝑧	𝑧	NOUN
easat-4034	47	18	)	)	PUNCT
easat-4034	47	19	4	4	NUM
easat-4034	47	20	.	.	X
easat-4034	48	1	there	there	PRON
easat-4034	48	2	is	be	VERB
easat-4034	48	3	a	a	DET
easat-4034	48	4	zero	zero	NUM
easat-4034	48	5	element	element	NOUN
easat-4034	48	6	,	,	PUNCT
easat-4034	48	7	namely	namely	ADV
easat-4034	48	8	∀𝑥	∀𝑥	PROPN
easat-4034	48	9	∈	∈	PROPN
easat-4034	48	10	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	48	11	,	,	PUNCT
easat-4034	48	12	𝑥	𝑥	PROPN
easat-4034	48	13	⊕	⊕	PROPN
easat-4034	48	14	′𝜀	′𝜀	PROPN
easat-4034	48	15	=	=	PUNCT
easat-4034	48	16	𝜀	𝜀	PROPN
easat-4034	48	17	⊕	⊕	PROPN
easat-4034	48	18	′𝑥	′𝑥	PROPN
easat-4034	49	1	=	=	SYM
easat-4034	49	2	𝑥	𝑥	DET
easat-4034	49	3	5	5	NUM
easat-4034	49	4	.	.	X
easat-4034	50	1	there	there	PRON
easat-4034	50	2	is	be	VERB
easat-4034	50	3	a	a	DET
easat-4034	50	4	unit	unit	NOUN
easat-4034	50	5	element	element	NOUN
easat-4034	50	6	,	,	PUNCT
easat-4034	50	7	namely	namely	ADV
easat-4034	50	8	∀𝑥	∀𝑥	PROPN
easat-4034	50	9	∈	∈	PROPN
easat-4034	50	10	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	50	11	,	,	PUNCT
easat-4034	50	12	𝑥	𝑥	PROPN
easat-4034	50	13	⊗	⊗	PROPN
easat-4034	50	14	𝑒	𝑒	PROPN
easat-4034	50	15	=	=	PUNCT
easat-4034	50	16	𝑒	𝑒	PROPN
easat-4034	50	17	⊗	⊗	NUM
easat-4034	50	18	𝑥	𝑥	PROPN
easat-4034	50	19	=	=	PUNCT
easat-4034	50	20	𝑥	𝑥	DET
easat-4034	50	21	6	6	NUM
easat-4034	50	22	.	.	PUNCT
easat-4034	51	1	there	there	PRON
easat-4034	51	2	is	be	VERB
easat-4034	51	3	an	an	DET
easat-4034	51	4	absorption	absorption	NOUN
easat-4034	51	5	property	property	NOUN
easat-4034	51	6	by	by	ADP
easat-4034	51	7	the	the	DET
easat-4034	51	8	zero	zero	NUM
easat-4034	51	9	element	element	NOUN
easat-4034	51	10	𝜀′	𝜀′	PROPN
easat-4034	51	11	towards	towards	ADP
easat-4034	51	12	⊗,that	⊗,that	PRON
easat-4034	51	13	is	be	AUX
easat-4034	51	14	∀𝑥	∀𝑥	PROPN
easat-4034	51	15	∈	∈	PROPN
easat-4034	51	16	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	51	17	,	,	PUNCT
easat-4034	52	1	𝑥	𝑥	PROPN
easat-4034	52	2	⊗	⊗	X
easat-4034	52	3	𝜀′	𝜀′	X
easat-4034	52	4	=	=	SYM
easat-4034	52	5	𝜀′	𝜀′	VERB
easat-4034	52	6	⊗	⊗	NUM
easat-4034	52	7	𝑥	𝑥	NOUN
easat-4034	52	8	=	=	SYM
easat-4034	52	9	𝜀′	𝜀′	X
easat-4034	52	10	7	7	X
easat-4034	52	11	.	.	PUNCT
easat-4034	53	1	the	the	DET
easat-4034	53	2	idempotent	idempotent	ADJ
easat-4034	53	3	property	property	NOUN
easat-4034	53	4	of	of	ADP
easat-4034	53	5	⊕	⊕	PROPN
easat-4034	53	6	′	′	PROPN
easat-4034	53	7	,	,	PUNCT
easat-4034	53	8	namely	namely	ADV
easat-4034	53	9	∀𝑥	∀𝑥	PROPN
easat-4034	53	10	∈	∈	PROPN
easat-4034	53	11	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	53	12	,	,	PUNCT
easat-4034	53	13	𝑥	𝑥	PROPN
easat-4034	53	14	⊕	⊕	PROPN
easat-4034	53	15	′𝑥	′𝑥	PROPN
easat-4034	54	1	=	=	SYM
easat-4034	54	2	𝑥	𝑥	PROPN
easat-4034	54	3	meanwhile	meanwhile	ADV
easat-4034	54	4	,	,	PUNCT
easat-4034	54	5	defining	define	VERB
easat-4034	54	6	the	the	DET
easat-4034	54	7	determinant	determinant	ADJ
easat-4034	54	8	uses	use	VERB
easat-4034	54	9	permutation	permutation	NOUN
easat-4034	54	10	characteristics	characteristic	NOUN
easat-4034	54	11	.	.	PUNCT
easat-4034	55	1	the	the	DET
easat-4034	55	2	definition	definition	NOUN
easat-4034	55	3	of	of	ADP
easat-4034	55	4	permutation	permutation	NOUN
easat-4034	55	5	is	be	AUX
easat-4034	55	6	given	give	VERB
easat-4034	55	7	as	as	SCONJ
easat-4034	55	8	follows	follow	VERB
easat-4034	55	9	.	.	PUNCT
easat-4034	56	1	9546	9546	NUM
easat-4034	56	2	edelweiss	edelweiss	PROPN
easat-4034	56	3	applied	apply	VERB
easat-4034	56	4	science	science	NOUN
easat-4034	56	5	and	and	CCONJ
easat-4034	56	6	technology	technology	NOUN
easat-4034	56	7	issn	issn	PROPN
easat-4034	56	8	:	:	PUNCT
easat-4034	56	9	2576	2576	NUM
easat-4034	56	10	-	-	SYM
easat-4034	56	11	8484	8484	NUM
easat-4034	56	12	vol	vol	NOUN
easat-4034	56	13	.	.	PROPN
easat-4034	56	14	8	8	NUM
easat-4034	56	15	,	,	PUNCT
easat-4034	56	16	no	no	INTJ
easat-4034	56	17	.	.	NOUN
easat-4034	57	1	6	6	NUM
easat-4034	57	2	:	:	PUNCT
easat-4034	57	3	9544	9544	NUM
easat-4034	57	4	-	-	SYM
easat-4034	57	5	9554	9554	NUM
easat-4034	57	6	,	,	PUNCT
easat-4034	57	7	2024	2024	NUM
easat-4034	57	8	doi	doi	NOUN
easat-4034	57	9	:	:	PUNCT
easat-4034	57	10	10.55214/25768484.v8i6.4034	10.55214/25768484.v8i6.4034	NUM
easat-4034	57	11	©	©	ADP
easat-4034	57	12	2024	2024	NUM
easat-4034	57	13	by	by	ADP
easat-4034	57	14	the	the	DET
easat-4034	57	15	authors	author	NOUN
easat-4034	57	16	;	;	PUNCT
easat-4034	57	17	licensee	licensee	PROPN
easat-4034	57	18	learning	learning	NOUN
easat-4034	57	19	gate	gate	PROPN
easat-4034	57	20	definition	definition	NOUN
easat-4034	57	21	2	2	NUM
easat-4034	57	22	a	a	DET
easat-4034	57	23	permutation	permutation	NOUN
easat-4034	57	24	matrix	matrix	NOUN
easat-4034	57	25	is	be	AUX
easat-4034	57	26	a	a	DET
easat-4034	57	27	matrix	matrix	NOUN
easat-4034	57	28	with	with	ADP
easat-4034	57	29	exactly	exactly	ADV
easat-4034	57	30	one	one	NUM
easat-4034	57	31	entry	entry	NOUN
easat-4034	57	32	(	(	PUNCT
easat-4034	57	33	𝑒	𝑒	NOUN
easat-4034	57	34	)	)	PUNCT
easat-4034	57	35	and	and	CCONJ
easat-4034	57	36	another	another	DET
easat-4034	57	37	entry	entry	NOUN
easat-4034	57	38	(	(	PUNCT
easat-4034	57	39	𝜀′	𝜀′	NOUN
easat-4034	57	40	)	)	PUNCT
easat-4034	57	41	in	in	ADP
easat-4034	57	42	each	each	PRON
easat-4034	57	43	of	of	ADP
easat-4034	57	44	its	its	PRON
easat-4034	57	45	𝑖	𝑖	SYM
easat-4034	57	46	−th	−th	PROPN
easat-4034	57	47	row	row	NOUN
easat-4034	57	48	and	and	CCONJ
easat-4034	57	49	𝑗	𝑗	PRON
easat-4034	57	50	−th	−th	PROPN
easat-4034	57	51	column	column	NOUN
easat-4034	57	52	.	.	PUNCT
easat-4034	58	1	the	the	DET
easat-4034	58	2	permutation	permutation	NOUN
easat-4034	58	3	matrix	matrix	NOUN
easat-4034	58	4	over	over	ADP
easat-4034	58	5	the	the	DET
easat-4034	58	6	min	min	NOUN
easat-4034	58	7	-	-	PUNCT
easat-4034	58	8	plus	plus	ADJ
easat-4034	58	9	algebra	algebra	NOUN
easat-4034	58	10	can	can	AUX
easat-4034	58	11	be	be	AUX
easat-4034	58	12	described	describe	VERB
easat-4034	58	13	as	as	ADP
easat-4034	58	14	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	58	15	=	=	SYM
easat-4034	59	1	[	[	X
easat-4034	59	2	𝑝𝑖𝑗	𝑝𝑖𝑗	VERB
easat-4034	59	3	]	]	PUNCT
easat-4034	59	4	with	with	ADP
easat-4034	59	5	𝑝𝑖𝑗	𝑝𝑖𝑗	PROPN
easat-4034	59	6	=	=	SYM
easat-4034	59	7	{	{	PUNCT
easat-4034	59	8	𝑒	𝑒	PROPN
easat-4034	59	9	;	;	PUNCT
easat-4034	59	10	𝑖	𝑖	SYM
easat-4034	59	11	=	=	SYM
easat-4034	59	12	𝜎(𝑗	𝜎(𝑗	PROPN
easat-4034	59	13	)	)	PUNCT
easat-4034	59	14	𝜀′	𝜀′	NOUN
easat-4034	59	15	;	;	PUNCT
easat-4034	59	16	𝑖	𝑖	SYM
easat-4034	59	17	≠	≠	PROPN
easat-4034	59	18	𝜎(𝑗	𝜎(𝑗	PROPN
easat-4034	59	19	)	)	PUNCT
easat-4034	59	20	if	if	SCONJ
easat-4034	59	21	𝜎	𝜎	ADJ
easat-4034	59	22	:	:	PUNCT
easat-4034	59	23	{	{	PUNCT
easat-4034	59	24	1	1	NUM
easat-4034	59	25	,	,	PUNCT
easat-4034	59	26	2	2	NUM
easat-4034	59	27	,	,	PUNCT
easat-4034	59	28	…	…	PUNCT
easat-4034	59	29	,	,	PUNCT
easat-4034	59	30	𝑛	𝑛	ADJ
easat-4034	59	31	}	}	PUNCT
easat-4034	59	32	⟶	⟶	NOUN
easat-4034	59	33	{	{	PUNCT
easat-4034	59	34	1	1	NUM
easat-4034	59	35	,	,	PUNCT
easat-4034	59	36	2	2	NUM
easat-4034	59	37	,	,	PUNCT
easat-4034	59	38	…	…	PUNCT
easat-4034	59	39	,	,	PUNCT
easat-4034	59	40	𝑛	𝑛	X
easat-4034	59	41	}	}	PUNCT
easat-4034	59	42	is	be	AUX
easat-4034	59	43	a	a	DET
easat-4034	59	44	permutation	permutation	NOUN
easat-4034	59	45	.	.	PUNCT
easat-4034	60	1	thus	thus	ADV
easat-4034	60	2	,	,	PUNCT
easat-4034	60	3	𝑒	𝑒	PROPN
easat-4034	60	4	appears	appear	VERB
easat-4034	60	5	in	in	ADP
easat-4034	60	6	the	the	DET
easat-4034	60	7	𝜎(𝑗	𝜎(𝑗	PROPN
easat-4034	60	8	−	−	PROPN
easat-4034	60	9	𝑡ℎ	𝑡ℎ	PROPN
easat-4034	60	10	)	)	PUNCT
easat-4034	60	11	row	row	NOUN
easat-4034	60	12	in	in	ADP
easat-4034	60	13	the	the	DET
easat-4034	60	14	𝑗	𝑗	INTJ
easat-4034	60	15	−th	−th	NOUN
easat-4034	60	16	column	column	NOUN
easat-4034	60	17	of	of	ADP
easat-4034	60	18	𝑃𝜎.	𝑃𝜎.	PROPN
easat-4034	60	19	example	example	NOUN
easat-4034	60	20	1	1	NUM
easat-4034	60	21	let	let	VERB
easat-4034	60	22	𝜎	𝜎	PRON
easat-4034	60	23	:	:	PUNCT
easat-4034	60	24	{	{	PUNCT
easat-4034	60	25	1	1	NUM
easat-4034	60	26	,	,	PUNCT
easat-4034	60	27	2	2	NUM
easat-4034	60	28	}	}	PUNCT
easat-4034	60	29	→	→	SYM
easat-4034	60	30	{	{	PUNCT
easat-4034	60	31	1	1	NUM
easat-4034	60	32	,	,	PUNCT
easat-4034	60	33	2	2	NUM
easat-4034	60	34	}	}	PUNCT
easat-4034	60	35	with	with	ADP
easat-4034	60	36	𝜎(1	𝜎(1	PROPN
easat-4034	60	37	)	)	PUNCT
easat-4034	60	38	=	=	SYM
easat-4034	60	39	2	2	NUM
easat-4034	60	40	and	and	CCONJ
easat-4034	60	41	𝜎(2	𝜎(2	NOUN
easat-4034	60	42	)	)	PUNCT
easat-4034	60	43	=	=	SYM
easat-4034	60	44	1	1	NUM
easat-4034	60	45	,	,	PUNCT
easat-4034	60	46	then	then	ADV
easat-4034	60	47	𝑝11	𝑝11	NOUN
easat-4034	60	48	=	=	PUNCT
easat-4034	60	49	{	{	PUNCT
easat-4034	60	50	𝑒	𝑒	PART
easat-4034	60	51	𝑗𝑖𝑘𝑎	𝑗𝑖𝑘𝑎	VERB
easat-4034	60	52	1	1	NUM
easat-4034	60	53	=	=	SYM
easat-4034	60	54	𝜎(1	𝜎(1	PROPN
easat-4034	60	55	)	)	PUNCT
easat-4034	60	56	𝜀′	𝜀′	X
easat-4034	60	57	𝑗𝑖𝑘𝑎	𝑗𝑖𝑘𝑎	VERB
easat-4034	60	58	1	1	NUM
easat-4034	60	59	≠	≠	PROPN
easat-4034	60	60	𝜎(1	𝜎(1	PROPN
easat-4034	60	61	)	)	PUNCT
easat-4034	60	62	,	,	PUNCT
easat-4034	60	63	𝑝11	𝑝11	NOUN
easat-4034	60	64	=	=	SYM
easat-4034	60	65	𝜀′	𝜀′	NOUN
easat-4034	60	66	𝑝12	𝑝12	PROPN
easat-4034	60	67	=	=	X
easat-4034	60	68	{	{	PUNCT
easat-4034	60	69	𝑒	𝑒	PART
easat-4034	60	70	𝑗𝑖𝑘𝑎	𝑗𝑖𝑘𝑎	VERB
easat-4034	60	71	1	1	NUM
easat-4034	60	72	=	=	SYM
easat-4034	60	73	𝜎(2	𝜎(2	NOUN
easat-4034	60	74	)	)	PUNCT
easat-4034	60	75	𝜀′	𝜀′	NOUN
easat-4034	60	76	𝑗𝑖𝑘𝑎	𝑗𝑖𝑘𝑎	VERB
easat-4034	60	77	1	1	NUM
easat-4034	60	78	≠	≠	PROPN
easat-4034	60	79	𝜎(2	𝜎(2	NOUN
easat-4034	60	80	)	)	PUNCT
easat-4034	60	81	,	,	PUNCT
easat-4034	60	82	𝑝12	𝑝12	PROPN
easat-4034	60	83	=	=	PROPN
easat-4034	60	84	𝑒	𝑒	PROPN
easat-4034	60	85	𝑝21	𝑝21	NOUN
easat-4034	60	86	=	=	PUNCT
easat-4034	60	87	{	{	PUNCT
easat-4034	60	88	𝑒	𝑒	PART
easat-4034	60	89	𝑗𝑖𝑘𝑎	𝑗𝑖𝑘𝑎	VERB
easat-4034	60	90	2	2	NUM
easat-4034	60	91	=	=	SYM
easat-4034	60	92	𝜎(1	𝜎(1	PROPN
easat-4034	60	93	)	)	PUNCT
easat-4034	60	94	𝜀′	𝜀′	X
easat-4034	60	95	𝑗𝑖𝑘𝑎	𝑗𝑖𝑘𝑎	VERB
easat-4034	60	96	2	2	NUM
easat-4034	60	97	≠	≠	PROPN
easat-4034	60	98	𝜎(1	𝜎(1	PROPN
easat-4034	60	99	)	)	PUNCT
easat-4034	60	100	,	,	PUNCT
easat-4034	60	101	𝑝21	𝑝21	NOUN
easat-4034	60	102	=	=	SYM
easat-4034	60	103	𝑒	𝑒	NOUN
easat-4034	60	104	𝑝21	𝑝21	NOUN
easat-4034	60	105	=	=	PUNCT
easat-4034	60	106	{	{	PUNCT
easat-4034	60	107	𝑒	𝑒	PART
easat-4034	60	108	𝑗𝑖𝑘𝑎	𝑗𝑖𝑘𝑎	VERB
easat-4034	60	109	2	2	NUM
easat-4034	60	110	=	=	SYM
easat-4034	60	111	𝜎(2	𝜎(2	NOUN
easat-4034	60	112	)	)	PUNCT
easat-4034	60	113	𝜀′	𝜀′	NOUN
easat-4034	60	114	𝑗𝑖𝑘𝑎	𝑗𝑖𝑘𝑎	VERB
easat-4034	60	115	2	2	NUM
easat-4034	60	116	≠	≠	PROPN
easat-4034	60	117	𝜎(2	𝜎(2	NOUN
easat-4034	60	118	)	)	PUNCT
easat-4034	60	119	,	,	PUNCT
easat-4034	61	1	𝑝22	𝑝22	PROPN
easat-4034	61	2	=	=	SYM
easat-4034	61	3	𝜀	𝜀	PROPN
easat-4034	61	4	′	′	NOUN
easat-4034	62	1	the	the	DET
easat-4034	62	2	permutation	permutation	NOUN
easat-4034	62	3	matrix	matrix	NOUN
easat-4034	62	4	is	be	AUX
easat-4034	62	5	[	[	PUNCT
easat-4034	62	6	𝜀′	𝜀′	X
easat-4034	62	7	𝑒	𝑒	X
easat-4034	62	8	𝑒	𝑒	X
easat-4034	62	9	𝜀′	𝜀′	NOUN
easat-4034	62	10	]	]	PUNCT
easat-4034	62	11	.	.	PUNCT
easat-4034	62	12	example	example	NOUN
easat-4034	62	13	2	2	NUM
easat-4034	62	14	let	let	VERB
easat-4034	62	15	𝐴	𝐴	PROPN
easat-4034	62	16	=	=	PUNCT
easat-4034	62	17	[	[	PUNCT
easat-4034	62	18	1	1	NUM
easat-4034	62	19	3	3	NUM
easat-4034	62	20	−2	−2	NOUN
easat-4034	62	21	3	3	NUM
easat-4034	62	22	5	5	NUM
easat-4034	62	23	8	8	NUM
easat-4034	62	24	4	4	NUM
easat-4034	62	25	6	6	NUM
easat-4034	62	26	−1	−1	NOUN
easat-4034	62	27	]	]	PUNCT
easat-4034	62	28	,	,	PUNCT
easat-4034	63	1	𝑃𝜎	𝑃𝜎	NOUN
easat-4034	63	2	=	=	PRON
easat-4034	63	3	[	[	PUNCT
easat-4034	63	4	𝜀′	𝜀′	X
easat-4034	63	5	𝑒	𝑒	X
easat-4034	63	6	𝜀′	𝜀′	NOUN
easat-4034	63	7	𝜀′	𝜀′	X
easat-4034	63	8	𝜀′	𝜀′	X
easat-4034	63	9	𝑒	𝑒	PROPN
easat-4034	63	10	𝑒	𝑒	X
easat-4034	63	11	𝜀′	𝜀′	X
easat-4034	63	12	𝜀′	𝜀′	PROPN
easat-4034	63	13	]	]	X
easat-4034	63	14	we	we	PRON
easat-4034	63	15	have	have	VERB
easat-4034	63	16	𝐴	𝐴	PROPN
easat-4034	63	17	⊗	⊗	NOUN
easat-4034	63	18	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	63	19	=	=	SYM
easat-4034	63	20	[	[	PUNCT
easat-4034	63	21	1	1	NUM
easat-4034	63	22	3	3	NUM
easat-4034	63	23	−2	−2	NOUN
easat-4034	63	24	3	3	NUM
easat-4034	63	25	5	5	NUM
easat-4034	63	26	8	8	NUM
easat-4034	63	27	4	4	NUM
easat-4034	63	28	6	6	NUM
easat-4034	63	29	−1	−1	NOUN
easat-4034	63	30	]	]	PUNCT
easat-4034	64	1	⊗	⊗	PROPN
easat-4034	64	2	[	[	PUNCT
easat-4034	64	3	𝜀′	𝜀′	X
easat-4034	64	4	𝑒	𝑒	X
easat-4034	64	5	𝜀′	𝜀′	NOUN
easat-4034	64	6	𝜀′	𝜀′	X
easat-4034	64	7	𝜀′	𝜀′	X
easat-4034	64	8	𝑒	𝑒	PROPN
easat-4034	64	9	𝑒	𝑒	X
easat-4034	64	10	𝜀′	𝜀′	X
easat-4034	64	11	𝜀′	𝜀′	X
easat-4034	64	12	]	]	PUNCT
easat-4034	64	13	=	=	PUNCT
easat-4034	64	14	[	[	PUNCT
easat-4034	64	15	−2	−2	NOUN
easat-4034	64	16	1	1	NUM
easat-4034	64	17	3	3	NUM
easat-4034	64	18	8	8	NUM
easat-4034	64	19	3	3	NUM
easat-4034	64	20	5	5	NUM
easat-4034	64	21	−1	−1	NOUN
easat-4034	64	22	4	4	NUM
easat-4034	64	23	6	6	NUM
easat-4034	64	24	]	]	PUNCT
easat-4034	64	25	the	the	DET
easat-4034	64	26	right	right	ADJ
easat-4034	64	27	-	-	PUNCT
easat-4034	64	28	hand	hand	NOUN
easat-4034	64	29	multiplication	multiplication	NOUN
easat-4034	64	30	of	of	ADP
easat-4034	64	31	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	64	32	creates	create	VERB
easat-4034	64	33	a	a	DET
easat-4034	64	34	permutation	permutation	NOUN
easat-4034	64	35	of	of	ADP
easat-4034	64	36	the	the	DET
easat-4034	64	37	matrix	matrix	NOUN
easat-4034	64	38	columns	column	NOUN
easat-4034	64	39	so	so	SCONJ
easat-4034	64	40	that	that	SCONJ
easat-4034	64	41	the	the	DET
easat-4034	64	42	𝑖	𝑖	SYM
easat-4034	64	43	−th	−th	PROPN
easat-4034	64	44	column	column	NOUN
easat-4034	64	45	of	of	ADP
easat-4034	64	46	a	a	DET
easat-4034	64	47	appears	appear	NOUN
easat-4034	64	48	as	as	ADP
easat-4034	64	49	the	the	DET
easat-4034	64	50	𝜎(𝑖	𝜎(𝑖	NOUN
easat-4034	64	51	)	)	PUNCT
easat-4034	64	52	−th	−th	PROPN
easat-4034	64	53	column	column	NOUN
easat-4034	64	54	of	of	ADP
easat-4034	64	55	𝐴	𝐴	PROPN
easat-4034	64	56	⊗	⊗	PROPN
easat-4034	64	57	𝑃𝜎.	𝑃𝜎.	PROPN
easat-4034	64	58	the	the	DET
easat-4034	64	59	permutation	permutation	NOUN
easat-4034	64	60	matrix	matrix	NOUN
easat-4034	64	61	𝑃𝜎	𝑃𝜎	NOUN
easat-4034	64	62	has	have	VERB
easat-4034	64	63	an	an	DET
easat-4034	64	64	inverse	inverse	NOUN
easat-4034	64	65	,	,	PUNCT
easat-4034	64	66	namely	namely	ADV
easat-4034	64	67	𝑃𝜎−1	𝑃𝜎−1	ADJ
easat-4034	64	68	where	where	SCONJ
easat-4034	64	69	𝑃𝜎−1	𝑃𝜎−1	PROPN
easat-4034	64	70	is	be	AUX
easat-4034	64	71	the	the	DET
easat-4034	64	72	transpose	transpose	NOUN
easat-4034	64	73	of	of	ADP
easat-4034	64	74	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	64	75	obtained	obtain	VERB
easat-4034	64	76	𝑃𝜎−1	𝑃𝜎−1	NOUN
easat-4034	64	77	=	=	NOUN
easat-4034	64	78	𝑃𝜎𝑇	𝑃𝜎𝑇	NOUN
easat-4034	64	79	so	so	SCONJ
easat-4034	64	80	that	that	SCONJ
easat-4034	64	81	𝑃𝜎	𝑃𝜎	NOUN
easat-4034	64	82	⊗	⊗	ADV
easat-4034	64	83	𝑃𝜎−1	𝑃𝜎−1	NOUN
easat-4034	64	84	=	=	PUNCT
easat-4034	64	85	𝐸.	𝐸.	ADV
easat-4034	64	86	if	if	SCONJ
easat-4034	64	87	𝐴	𝐴	PROPN
easat-4034	64	88	is	be	AUX
easat-4034	64	89	a	a	DET
easat-4034	64	90	matrix	matrix	NOUN
easat-4034	64	91	over	over	ADP
easat-4034	64	92	a	a	DET
easat-4034	64	93	field	field	NOUN
easat-4034	64	94	,	,	PUNCT
easat-4034	64	95	then	then	ADV
easat-4034	64	96	a	a	DET
easat-4034	64	97	single	single	ADJ
easat-4034	64	98	matrix	matrix	NOUN
easat-4034	64	99	𝐵	𝐵	NOUN
easat-4034	64	100	must	must	AUX
easat-4034	64	101	satisfy	satisfy	VERB
easat-4034	64	102	the	the	DET
easat-4034	64	103	property	property	NOUN
easat-4034	64	104	𝐴	𝐴	PROPN
easat-4034	64	105	⊗	⊗	PROPN
easat-4034	64	106	𝐵	𝐵	PROPN
easat-4034	64	107	⊗	⊗	PROPN
easat-4034	64	108	𝐴	𝐴	PROPN
easat-4034	64	109	=	=	SYM
easat-4034	64	110	𝐴.	𝐴.	PROPN
easat-4034	64	111	a	a	DET
easat-4034	64	112	matrix	matrix	NOUN
easat-4034	64	113	𝐵	𝐵	NOUN
easat-4034	64	114	that	that	PRON
easat-4034	64	115	satisfies	satisfy	VERB
easat-4034	64	116	this	this	DET
easat-4034	64	117	property	property	NOUN
easat-4034	64	118	is	be	AUX
easat-4034	64	119	called	call	VERB
easat-4034	64	120	the	the	DET
easat-4034	64	121	generalized	generalized	ADJ
easat-4034	64	122	inverse	inverse	NOUN
easat-4034	64	123	of	of	ADP
easat-4034	64	124	matrix	matrix	NOUN
easat-4034	64	125	𝐴.	𝐴.	PROPN
easat-4034	64	126	in	in	ADP
easat-4034	64	127	min	min	ADJ
easat-4034	64	128	-	-	PUNCT
easat-4034	64	129	plus	plus	ADJ
easat-4034	64	130	algebra	algebra	NOUN
easat-4034	64	131	,	,	PUNCT
easat-4034	64	132	there	there	PRON
easat-4034	64	133	is	be	VERB
easat-4034	64	134	no	no	DET
easat-4034	64	135	guarantee	guarantee	NOUN
easat-4034	64	136	that	that	SCONJ
easat-4034	64	137	every	every	DET
easat-4034	64	138	matrix	matrix	NOUN
easat-4034	64	139	has	have	AUX
easat-4034	64	140	a	a	DET
easat-4034	64	141	generalized	generalized	ADJ
easat-4034	64	142	inverse	inverse	NOUN
easat-4034	64	143	.	.	PUNCT
easat-4034	65	1	if	if	SCONJ
easat-4034	65	2	𝐴	𝐴	PROPN
easat-4034	65	3	has	have	VERB
easat-4034	65	4	a	a	DET
easat-4034	65	5	generalized	generalized	ADJ
easat-4034	65	6	inverse	inverse	NOUN
easat-4034	65	7	,	,	PUNCT
easat-4034	65	8	then	then	ADV
easat-4034	65	9	𝐴	𝐴	PROPN
easat-4034	65	10	is	be	AUX
easat-4034	65	11	considered	consider	VERB
easat-4034	65	12	regular	regular	ADJ
easat-4034	65	13	.	.	PUNCT
easat-4034	66	1	we	we	PRON
easat-4034	66	2	will	will	AUX
easat-4034	66	3	discuss	discuss	VERB
easat-4034	66	4	determining	determine	VERB
easat-4034	66	5	whether	whether	SCONJ
easat-4034	66	6	a	a	DET
easat-4034	66	7	matrix	matrix	NOUN
easat-4034	66	8	𝐴	𝐴	NOUN
easat-4034	66	9	is	be	AUX
easat-4034	66	10	a	a	DET
easat-4034	66	11	regular	regular	ADJ
easat-4034	66	12	min	min	NOUN
easat-4034	66	13	-	-	PUNCT
easat-4034	66	14	plus	plus	ADJ
easat-4034	66	15	algebra	algebra	NOUN
easat-4034	66	16	.	.	PUNCT
easat-4034	67	1	a	a	DET
easat-4034	67	2	min	min	NOUN
easat-4034	67	3	-	-	PUNCT
easat-4034	67	4	plus	plus	ADJ
easat-4034	67	5	algebra	algebra	NOUN
easat-4034	67	6	can	can	AUX
easat-4034	67	7	be	be	AUX
easat-4034	67	8	formed	form	VERB
easat-4034	67	9	based	base	VERB
easat-4034	67	10	on	on	ADP
easat-4034	67	11	the	the	DET
easat-4034	67	12	characteristics	characteristic	NOUN
easat-4034	67	13	of	of	ADP
easat-4034	67	14	a	a	DET
easat-4034	67	15	max	max	PROPN
easat-4034	67	16	-	-	PUNCT
easat-4034	67	17	plus	plus	NOUN
easat-4034	67	18	algebra	algebra	NOUN
easat-4034	67	19	.	.	PUNCT
easat-4034	68	1	as	as	ADP
easat-4034	68	2	an	an	DET
easat-4034	68	3	initial	initial	ADJ
easat-4034	68	4	characteristic	characteristic	NOUN
easat-4034	68	5	,	,	PUNCT
easat-4034	68	6	the	the	DET
easat-4034	68	7	following	follow	VERB
easat-4034	68	8	theorem	theorem	NOUN
easat-4034	68	9	is	be	AUX
easat-4034	68	10	given	give	VERB
easat-4034	68	11	.	.	PUNCT
easat-4034	69	1	theorem	theorem	NOUN
easat-4034	69	2	2	2	NUM
easat-4034	69	3	given	give	VERB
easat-4034	69	4	an	an	DET
easat-4034	69	5	idempotent	idempotent	ADJ
easat-4034	69	6	commutative	commutative	ADJ
easat-4034	69	7	semigroup	semigroup	NOUN
easat-4034	69	8	(	(	PUNCT
easat-4034	69	9	𝑆	𝑆	PROPN
easat-4034	69	10	,	,	PUNCT
easat-4034	69	11	+	+	NOUN
easat-4034	69	12	)	)	PUNCT
easat-4034	69	13	.	.	PUNCT
easat-4034	70	1	if	if	SCONJ
easat-4034	70	2	on	on	ADP
easat-4034	70	3	s	s	VERB
easat-4034	70	4	a	a	DET
easat-4034	70	5	relation	relation	NOUN
easat-4034	70	6	≥	≥	NOUN
easat-4034	70	7	is	be	AUX
easat-4034	70	8	defined	define	VERB
easat-4034	70	9	by	by	ADP
easat-4034	70	10	𝑏	𝑏	PROPN
easat-4034	70	11	≥	≥	NOUN
easat-4034	70	12	𝑎	𝑎	PROPN
easat-4034	70	13	⟺	⟺	NOUN
easat-4034	70	14	𝑎	𝑎	X
easat-4034	70	15	+	+	NOUN
easat-4034	70	16	𝑏	𝑏	NOUN
easat-4034	70	17	=	=	SYM
easat-4034	70	18	𝑏	𝑏	NOUN
easat-4034	70	19	,	,	PUNCT
easat-4034	70	20	then	then	ADV
easat-4034	70	21	the	the	DET
easat-4034	70	22	relation	relation	NOUN
easat-4034	70	23	≤	≤	PROPN
easat-4034	70	24	is	be	AUX
easat-4034	70	25	a	a	DET
easat-4034	70	26	partial	partial	ADJ
easat-4034	70	27	order	order	NOUN
easat-4034	70	28	on	on	ADP
easat-4034	70	29	s.	s.	PROPN
easat-4034	70	30	proof	proof	NOUN
easat-4034	70	31	:	:	PUNCT
easat-4034	70	32	given	give	VERB
easat-4034	70	33	any	any	DET
easat-4034	70	34	𝑎	𝑎	NOUN
easat-4034	70	35	,	,	PUNCT
easat-4034	70	36	𝑏	𝑏	NOUN
easat-4034	70	37	,	,	PUNCT
easat-4034	70	38	𝑐	𝑐	PROPN
easat-4034	70	39	∈	∈	PROPN
easat-4034	70	40	𝑆	𝑆	PROPN
easat-4034	70	41	then	then	ADV
easat-4034	70	42	1	1	X
easat-4034	70	43	.	.	PUNCT
easat-4034	71	1	since	since	SCONJ
easat-4034	71	2	s	s	PROPN
easat-4034	71	3	is	be	AUX
easat-4034	71	4	idempotent	idempotent	ADJ
easat-4034	71	5	,	,	PUNCT
easat-4034	71	6	then	then	ADV
easat-4034	71	7	𝑎	𝑎	X
easat-4034	71	8	+	+	PUNCT
easat-4034	71	9	𝑎	𝑎	NOUN
easat-4034	71	10	=	=	SYM
easat-4034	71	11	𝑎	𝑎	X
easat-4034	71	12	⟺	⟺	NOUN
easat-4034	71	13	𝑎	𝑎	DET
easat-4034	71	14	≥	≥	NOUN
easat-4034	71	15	𝑎	𝑎	PROPN
easat-4034	71	16	9547	9547	NUM
easat-4034	71	17	edelweiss	edelweiss	PROPN
easat-4034	71	18	applied	apply	VERB
easat-4034	71	19	science	science	NOUN
easat-4034	71	20	and	and	CCONJ
easat-4034	71	21	technology	technology	NOUN
easat-4034	71	22	issn	issn	PROPN
easat-4034	71	23	:	:	PUNCT
easat-4034	71	24	2576	2576	NUM
easat-4034	71	25	-	-	SYM
easat-4034	71	26	8484	8484	NUM
easat-4034	71	27	vol	vol	NOUN
easat-4034	71	28	.	.	PROPN
easat-4034	71	29	8	8	NUM
easat-4034	71	30	,	,	PUNCT
easat-4034	71	31	no	no	INTJ
easat-4034	71	32	.	.	NOUN
easat-4034	71	33	6	6	NUM
easat-4034	71	34	:	:	PUNCT
easat-4034	71	35	9544	9544	NUM
easat-4034	71	36	-	-	SYM
easat-4034	71	37	9554	9554	NUM
easat-4034	71	38	,	,	PUNCT
easat-4034	71	39	2024	2024	NUM
easat-4034	71	40	doi	doi	NOUN
easat-4034	71	41	:	:	PUNCT
easat-4034	71	42	10.55214/25768484.v8i6.4034	10.55214/25768484.v8i6.4034	NUM
easat-4034	71	43	©	©	ADP
easat-4034	71	44	2024	2024	NUM
easat-4034	71	45	by	by	ADP
easat-4034	71	46	the	the	DET
easat-4034	71	47	authors	author	NOUN
easat-4034	71	48	;	;	PUNCT
easat-4034	71	49	licensee	licensee	PROPN
easat-4034	71	50	learning	learning	NOUN
easat-4034	71	51	gate	gate	NOUN
easat-4034	71	52	2	2	X
easat-4034	71	53	.	.	PUNCT
easat-4034	72	1	if	if	SCONJ
easat-4034	72	2	𝑏	𝑏	PRON
easat-4034	72	3	≥	≥	VERB
easat-4034	72	4	𝑎	𝑎	NOUN
easat-4034	72	5	and	and	CCONJ
easat-4034	72	6	𝑎	𝑎	DET
easat-4034	72	7	≥	≥	NOUN
easat-4034	72	8	𝑏	𝑏	NOUN
easat-4034	72	9	,	,	PUNCT
easat-4034	72	10	then	then	ADV
easat-4034	72	11	𝑎	𝑎	X
easat-4034	72	12	+	+	X
easat-4034	72	13	𝑏	𝑏	NOUN
easat-4034	72	14	=	=	SYM
easat-4034	72	15	𝑏	𝑏	PROPN
easat-4034	72	16	and	and	CCONJ
easat-4034	72	17	𝑏	𝑏	PROPN
easat-4034	72	18	+	+	CCONJ
easat-4034	72	19	𝑎	𝑎	PROPN
easat-4034	72	20	=	=	X
easat-4034	72	21	𝑎.	𝑎.	NOUN
easat-4034	72	22	since	since	SCONJ
easat-4034	72	23	𝑆	𝑆	PROPN
easat-4034	72	24	is	be	AUX
easat-4034	72	25	commutative	commutative	ADJ
easat-4034	72	26	,	,	PUNCT
easat-4034	72	27	then	then	ADV
easat-4034	72	28	𝑎	𝑎	NOUN
easat-4034	72	29	=	=	SYM
easat-4034	72	30	𝑏	𝑏	PROPN
easat-4034	72	31	3	3	NUM
easat-4034	72	32	.	.	PUNCT
easat-4034	73	1	if	if	SCONJ
easat-4034	73	2	𝑏	𝑏	PROPN
easat-4034	73	3	>	>	X
easat-4034	73	4	𝑎	𝑎	PROPN
easat-4034	73	5	and	and	CCONJ
easat-4034	73	6	𝑎	𝑎	ADP
easat-4034	73	7	>	>	X
easat-4034	73	8	𝑐	𝑐	NOUN
easat-4034	73	9	then	then	ADV
easat-4034	73	10	𝑎	𝑎	X
easat-4034	73	11	+	+	X
easat-4034	73	12	𝑏	𝑏	NOUN
easat-4034	73	13	=	=	SYM
easat-4034	73	14	𝑏	𝑏	PROPN
easat-4034	73	15	and	and	CCONJ
easat-4034	73	16	𝑐	𝑐	PROPN
easat-4034	74	1	+	+	NOUN
easat-4034	74	2	𝑎	𝑎	X
easat-4034	74	3	=	=	SYM
easat-4034	74	4	𝑎	𝑎	NOUN
easat-4034	74	5	then	then	ADV
easat-4034	74	6	𝑏	𝑏	PROPN
easat-4034	75	1	+	+	NOUN
easat-4034	75	2	𝑐	𝑐	NOUN
easat-4034	75	3	=	=	PUNCT
easat-4034	75	4	(	(	PUNCT
easat-4034	75	5	𝑎	𝑎	X
easat-4034	75	6	+	+	X
easat-4034	75	7	𝑏	𝑏	NOUN
easat-4034	75	8	)	)	PUNCT
easat-4034	75	9	+	+	NUM
easat-4034	75	10	𝑐	𝑐	NOUN
easat-4034	75	11	=	=	SYM
easat-4034	75	12	(	(	PUNCT
easat-4034	75	13	𝑏	𝑏	PROPN
easat-4034	75	14	+	+	CCONJ
easat-4034	75	15	𝑎	𝑎	X
easat-4034	75	16	)	)	PUNCT
easat-4034	75	17	+	+	CCONJ
easat-4034	75	18	𝑐	𝑐	NOUN
easat-4034	75	19	=	=	SYM
easat-4034	75	20	𝑏	𝑏	NOUN
easat-4034	75	21	+	+	CCONJ
easat-4034	75	22	(	(	PUNCT
easat-4034	75	23	𝑎	𝑎	X
easat-4034	75	24	+	+	NUM
easat-4034	75	25	𝑐	𝑐	NOUN
easat-4034	75	26	)	)	PUNCT
easat-4034	75	27	=	=	SYM
easat-4034	76	1	𝑏	𝑏	PROPN
easat-4034	76	2	+	+	CCONJ
easat-4034	76	3	𝑎	𝑎	PROPN
easat-4034	76	4	=	=	SYM
easat-4034	76	5	𝑏	𝑏	NOUN
easat-4034	77	1	so	so	ADV
easat-4034	77	2	,	,	PUNCT
easat-4034	77	3	we	we	PRON
easat-4034	77	4	have	have	VERB
easat-4034	77	5	𝑏	𝑏	PRON
easat-4034	77	6	>	>	PUNCT
easat-4034	77	7	𝑐.	𝑐.	ADJ
easat-4034	77	8	definition	definition	NOUN
easat-4034	77	9	3	3	NUM
easat-4034	77	10	in	in	ADP
easat-4034	77	11	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	77	12	,	,	PUNCT
easat-4034	77	13	the	the	DET
easat-4034	77	14	relation	relation	NOUN
easat-4034	77	15	≥𝑚𝑖𝑛	≥𝑚𝑖𝑛	PROPN
easat-4034	77	16	is	be	AUX
easat-4034	77	17	defined	define	VERB
easat-4034	77	18	as	as	ADP
easat-4034	77	19	𝑥	𝑥	PROPN
easat-4034	77	20	≥𝑚𝑖𝑛	≥𝑚𝑖𝑛	PROPN
easat-4034	77	21	𝑦	𝑦	PROPN
easat-4034	77	22	⟺	⟺	NOUN
easat-4034	77	23	𝑥	𝑥	X
easat-4034	77	24	⊕′	⊕′	PROPN
easat-4034	77	25	𝑦	𝑦	NOUN
easat-4034	77	26	=	=	SYM
easat-4034	77	27	𝑦	𝑦	NOUN
easat-4034	77	28	theorem	theorem	NOUN
easat-4034	77	29	3	3	NUM
easat-4034	77	30	the	the	DET
easat-4034	77	31	relation	relation	NOUN
easat-4034	77	32	≥𝑚𝑖𝑛	≥𝑚𝑖𝑛	PROPN
easat-4034	77	33	is	be	AUX
easat-4034	77	34	a	a	DET
easat-4034	77	35	partial	partial	ADJ
easat-4034	77	36	order	order	NOUN
easat-4034	77	37	.	.	PUNCT
easat-4034	78	1	proof	proof	NOUN
easat-4034	78	2	:	:	PUNCT
easat-4034	78	3	given	give	VERB
easat-4034	78	4	𝑎	𝑎	PROPN
easat-4034	78	5	,	,	PUNCT
easat-4034	78	6	𝑏	𝑏	NOUN
easat-4034	78	7	,	,	PUNCT
easat-4034	78	8	𝑐	𝑐	PROPN
easat-4034	78	9	∈	∈	PROPN
easat-4034	78	10	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	78	11	,	,	PUNCT
easat-4034	78	12	then	then	ADV
easat-4034	78	13	1	1	X
easat-4034	78	14	.	.	PUNCT
easat-4034	79	1	since	since	SCONJ
easat-4034	79	2	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	79	3	is	be	AUX
easat-4034	79	4	idempotent	idempotent	ADJ
easat-4034	79	5	then	then	ADV
easat-4034	79	6	𝑎	𝑎	PROPN
easat-4034	79	7	⊕′	⊕′	PROPN
easat-4034	79	8	𝑎	𝑎	NOUN
easat-4034	79	9	=	=	SYM
easat-4034	79	10	min{𝑎	min{𝑎	NOUN
easat-4034	79	11	,	,	PUNCT
easat-4034	79	12	𝑎	𝑎	NOUN
easat-4034	79	13	}	}	PUNCT
easat-4034	79	14	=	=	PUNCT
easat-4034	79	15	𝑎	𝑎	NOUN
easat-4034	79	16	therefore	therefore	ADV
easat-4034	79	17	𝑎	𝑎	PRON
easat-4034	79	18	≥𝑚𝑖𝑛	≥𝑚𝑖𝑛	PROPN
easat-4034	79	19	𝑎	𝑎	PROPN
easat-4034	79	20	2	2	NUM
easat-4034	79	21	.	.	PUNCT
easat-4034	80	1	if	if	SCONJ
easat-4034	80	2	𝑎	𝑎	PRON
easat-4034	80	3	≥𝑚𝑖𝑛	≥𝑚𝑖𝑛	PROPN
easat-4034	80	4	𝑏	𝑏	PROPN
easat-4034	80	5	and	and	CCONJ
easat-4034	80	6	𝑏	𝑏	PROPN
easat-4034	80	7	≥𝑚𝑖𝑛	≥𝑚𝑖𝑛	PROPN
easat-4034	80	8	𝑎	𝑎	PUNCT
easat-4034	80	9	then	then	ADV
easat-4034	80	10	𝑎	𝑎	PROPN
easat-4034	80	11	⊕	⊕	NOUN
easat-4034	80	12	′𝑏	′𝑏	PUNCT
easat-4034	80	13	=	=	PUNCT
easat-4034	80	14	𝑏	𝑏	PROPN
easat-4034	80	15	and	and	CCONJ
easat-4034	80	16	𝑏	𝑏	PROPN
easat-4034	80	17	⊕	⊕	PROPN
easat-4034	80	18	′𝑎	′𝑎	PROPN
easat-4034	80	19	=	=	PUNCT
easat-4034	80	20	𝑎.	𝑎.	NOUN
easat-4034	80	21	since	since	SCONJ
easat-4034	80	22	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	80	23	is	be	AUX
easat-4034	80	24	commutative	commutative	ADJ
easat-4034	80	25	then	then	ADV
easat-4034	80	26	𝑎	𝑎	PROPN
easat-4034	80	27	=	=	SYM
easat-4034	80	28	𝑏	𝑏	PROPN
easat-4034	80	29	3	3	NUM
easat-4034	80	30	.	.	PUNCT
easat-4034	81	1	if	if	SCONJ
easat-4034	81	2	𝑎	𝑎	PRON
easat-4034	81	3	≥𝑚𝑖𝑛	≥𝑚𝑖𝑛	PROPN
easat-4034	81	4	𝑏	𝑏	PROPN
easat-4034	81	5	and	and	CCONJ
easat-4034	81	6	𝑏	𝑏	PROPN
easat-4034	81	7	≥𝑚𝑖𝑛	≥𝑚𝑖𝑛	PROPN
easat-4034	81	8	𝑐	𝑐	PROPN
easat-4034	81	9	then	then	ADV
easat-4034	81	10	𝑎	𝑎	PROPN
easat-4034	81	11	⊕′	⊕′	PROPN
easat-4034	81	12	𝑏	𝑏	NOUN
easat-4034	81	13	=	=	SYM
easat-4034	81	14	𝑏	𝑏	PROPN
easat-4034	81	15	and	and	CCONJ
easat-4034	81	16	𝑏	𝑏	PROPN
easat-4034	81	17	⊕′	⊕′	PROPN
easat-4034	81	18	𝑐	𝑐	NOUN
easat-4034	81	19	=	=	SYM
easat-4034	81	20	𝑐	𝑐	PROPN
easat-4034	81	21	then	then	ADV
easat-4034	81	22	𝑎	𝑎	PROPN
easat-4034	81	23	⊕′	⊕′	PROPN
easat-4034	81	24	𝑐	𝑐	NOUN
easat-4034	81	25	=	=	PUNCT
easat-4034	81	26	𝑎	𝑎	PRON
easat-4034	81	27	⊕′	⊕′	PROPN
easat-4034	81	28	(	(	PUNCT
easat-4034	81	29	𝑏	𝑏	PROPN
easat-4034	81	30	⊕′	⊕′	PROPN
easat-4034	81	31	𝑐	𝑐	NOUN
easat-4034	81	32	)	)	PUNCT
easat-4034	81	33	=	=	PUNCT
easat-4034	82	1	𝑎	𝑎	DET
easat-4034	82	2	⊕′	⊕′	PROPN
easat-4034	82	3	(	(	PUNCT
easat-4034	82	4	𝑏	𝑏	PROPN
easat-4034	82	5	⊕′	⊕′	PROPN
easat-4034	82	6	𝑐	𝑐	NOUN
easat-4034	82	7	)	)	PUNCT
easat-4034	82	8	=	=	NOUN
easat-4034	82	9	(	(	PUNCT
easat-4034	82	10	𝑎	𝑎	PROPN
easat-4034	82	11	⊕′	⊕′	PROPN
easat-4034	82	12	𝑏	𝑏	NOUN
easat-4034	82	13	)	)	PUNCT
easat-4034	82	14	⊕′	⊕′	NOUN
easat-4034	82	15	𝑐	𝑐	NOUN
easat-4034	82	16	=	=	PUNCT
easat-4034	82	17	𝑏	𝑏	PROPN
easat-4034	82	18	⊕′	⊕′	PROPN
easat-4034	82	19	𝑐	𝑐	NOUN
easat-4034	82	20	=	=	SYM
easat-4034	82	21	𝑐	𝑐	PROPN
easat-4034	82	22	so	so	ADV
easat-4034	82	23	,	,	PUNCT
easat-4034	82	24	𝑎	𝑎	DET
easat-4034	82	25	≥𝑚𝑖𝑛	≥𝑚𝑖𝑛	PROPN
easat-4034	82	26	𝑐.	𝑐.	NOUN
easat-4034	82	27	definition	definition	NOUN
easat-4034	82	28	4	4	NUM
easat-4034	82	29	for	for	ADP
easat-4034	82	30	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	82	31	,	,	PUNCT
easat-4034	82	32	we	we	PRON
easat-4034	82	33	use	use	VERB
easat-4034	82	34	the	the	DET
easat-4034	82	35	parsial	parsial	ADJ
easat-4034	82	36	ordered	order	VERB
easat-4034	82	37	≥𝑚𝑖𝑛	≥𝑚𝑖𝑛	PROPN
easat-4034	82	38	,	,	PUNCT
easat-4034	82	39	that	that	PRON
easat-4034	82	40	is	be	AUX
easat-4034	82	41	𝑎	𝑎	PROPN
easat-4034	82	42	≥𝑚𝑖𝑛	≥𝑚𝑖𝑛	PROPN
easat-4034	82	43	𝑏	𝑏	NOUN
easat-4034	82	44	⟺	⟺	NOUN
easat-4034	83	1	𝑎	𝑎	X
easat-4034	83	2	⊕′	⊕′	PROPN
easat-4034	83	3	𝑏	𝑏	NOUN
easat-4034	83	4	=	=	SYM
easat-4034	83	5	min(𝑎	min(𝑎	PROPN
easat-4034	83	6	,	,	PUNCT
easat-4034	83	7	𝑏	𝑏	NOUN
easat-4034	83	8	)	)	PUNCT
easat-4034	83	9	=	=	VERB
easat-4034	83	10	𝑏.	𝑏.	VERB
easat-4034	83	11	the	the	DET
easat-4034	83	12	structure	structure	NOUN
easat-4034	83	13	(	(	PUNCT
easat-4034	83	14	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	83	15	,	,	PUNCT
easat-4034	83	16	≤	≤	NUM
easat-4034	83	17	)	)	PUNCT
easat-4034	83	18	is	be	AUX
easat-4034	83	19	a	a	DET
easat-4034	83	20	partially	partially	ADV
easat-4034	83	21	ordered	order	VERB
easat-4034	83	22	set	set	NOUN
easat-4034	83	23	(	(	PUNCT
easat-4034	83	24	poset	poset	NOUN
easat-4034	83	25	)	)	PUNCT
easat-4034	83	26	.	.	PUNCT
easat-4034	84	1	theorem	theorem	NOUN
easat-4034	84	2	4	4	NUM
easat-4034	84	3	let	let	VERB
easat-4034	84	4	𝐴	𝐴	PROPN
easat-4034	84	5	∈	∈	PROPN
easat-4034	84	6	𝑀𝑛(ℝ𝑚𝑖𝑛	𝑀𝑛(ℝ𝑚𝑖𝑛	PROPN
easat-4034	84	7	)	)	PUNCT
easat-4034	84	8	and	and	CCONJ
easat-4034	84	9	supposed	suppose	VERB
easat-4034	84	10	𝐿𝐴	𝐿𝐴	PROPN
easat-4034	84	11	:	:	PUNCT
easat-4034	85	1	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	85	2	𝑛	𝑛	PROPN
easat-4034	85	3	⟶	⟶	NOUN
easat-4034	85	4	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	85	5	𝑛	𝑛	NOUN
easat-4034	85	6	with	with	ADP
easat-4034	85	7	𝐿𝐴(𝑥	𝐿𝐴(𝑥	NOUN
easat-4034	85	8	)	)	PUNCT
easat-4034	86	1	=	=	SYM
easat-4034	86	2	𝐴	𝐴	PROPN
easat-4034	86	3	⊗	⊗	PROPN
easat-4034	86	4	𝑥.	𝑥.	ADV
easat-4034	86	5	we	we	PRON
easat-4034	86	6	have	have	VERB
easat-4034	86	7	𝐴	𝐴	NOUN
easat-4034	86	8	=	=	SYM
easat-4034	87	1	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	87	2	⊗	⊗	NOUN
easat-4034	87	3	𝐷(𝜆𝑖	𝐷(𝜆𝑖	X
easat-4034	87	4	)	)	PUNCT
easat-4034	87	5	for	for	ADP
easat-4034	87	6	a	a	DET
easat-4034	87	7	permutation	permutation	NOUN
easat-4034	87	8	and	and	CCONJ
easat-4034	87	9	𝜆𝑖	𝜆𝑖	X
easat-4034	87	10	>	>	X
easat-4034	87	11	𝜀	𝜀	X
easat-4034	87	12	if	if	SCONJ
easat-4034	87	13	and	and	CCONJ
easat-4034	87	14	only	only	ADV
easat-4034	87	15	if	if	SCONJ
easat-4034	87	16	𝐿𝐴	𝐿𝐴	PROPN
easat-4034	87	17	injective	injective	VERB
easat-4034	87	18	.	.	PUNCT
easat-4034	88	1	proof	proof	NOUN
easat-4034	88	2	:	:	PUNCT
easat-4034	88	3	(	(	PUNCT
easat-4034	88	4	⟹	⟹	X
easat-4034	88	5	)	)	PUNCT
easat-4034	88	6	𝐿𝐴(𝑥	𝐿𝐴(𝑥	NOUN
easat-4034	88	7	)	)	PUNCT
easat-4034	88	8	=	=	SYM
easat-4034	88	9	𝐿𝐴(𝑥′	𝐿𝐴(𝑥′	PROPN
easat-4034	88	10	)	)	PUNCT
easat-4034	88	11	such	such	ADJ
easat-4034	88	12	as	as	ADP
easat-4034	88	13	𝐴	𝐴	PROPN
easat-4034	88	14	⊗	⊗	NOUN
easat-4034	89	1	𝑥	𝑥	PROPN
easat-4034	90	1	=	=	SYM
easat-4034	90	2	𝐴	𝐴	PROPN
easat-4034	91	1	⊗	⊗	PROPN
easat-4034	92	1	𝑥′	𝑥′	INTJ
easat-4034	93	1	so	so	ADV
easat-4034	93	2	𝑥	𝑥	NOUN
easat-4034	93	3	=	=	SYM
easat-4034	93	4	𝑥′.	𝑥′.	NOUN
easat-4034	93	5	(	(	PUNCT
easat-4034	93	6	⟸	⟸	ADJ
easat-4034	93	7	)	)	PUNCT
easat-4034	93	8	it	it	PRON
easat-4034	93	9	is	be	AUX
easat-4034	93	10	known	know	VERB
easat-4034	93	11	that	that	SCONJ
easat-4034	93	12	𝐿𝐴	𝐿𝐴	PROPN
easat-4034	93	13	is	be	AUX
easat-4034	93	14	injective	injective	ADJ
easat-4034	93	15	.	.	PUNCT
easat-4034	94	1	for	for	ADP
easat-4034	94	2	each	each	DET
easat-4034	94	3	𝑖	𝑖	NOUN
easat-4034	94	4	can	can	AUX
easat-4034	94	5	be	be	AUX
easat-4034	94	6	defined	define	VERB
easat-4034	94	7	𝐹𝑖	𝐹𝑖	PROPN
easat-4034	94	8	=	=	PUNCT
easat-4034	94	9	{	{	PUNCT
easat-4034	94	10	𝑗|𝑎𝑗𝑖	𝑗|𝑎𝑗𝑖	ADV
easat-4034	94	11	>	>	X
easat-4034	94	12	𝜀	𝜀	X
easat-4034	94	13	}	}	PUNCT
easat-4034	94	14	and	and	CCONJ
easat-4034	94	15	𝐺𝑖	𝐺𝑖	VERB
easat-4034	94	16	=	=	PUNCT
easat-4034	94	17	{	{	PUNCT
easat-4034	94	18	𝑗|𝑎𝑗𝑘	𝑗|𝑎𝑗𝑘	PROPN
easat-4034	94	19	>	>	X
easat-4034	94	20	𝜀	𝜀	PROPN
easat-4034	94	21	,	,	PUNCT
easat-4034	94	22	𝑘	𝑘	PROPN
easat-4034	94	23	≠	≠	PROPN
easat-4034	94	24	𝑖	𝑖	PRON
easat-4034	94	25	}	}	PUNCT
easat-4034	94	26	.	.	PUNCT
easat-4034	95	1	we	we	PRON
easat-4034	95	2	called	call	VERB
easat-4034	95	3	𝐹𝑖	𝐹𝑖	PROPN
easat-4034	95	4	⊆	⊆	NUM
easat-4034	95	5	𝐺𝑖	𝐺𝑖	PROPN
easat-4034	95	6	;	;	PUNCT
easat-4034	95	7	the	the	DET
easat-4034	95	8	contradiction	contradiction	NOUN
easat-4034	95	9	assumes	assume	VERB
easat-4034	95	10	that	that	SCONJ
easat-4034	95	11	𝐹𝑖	𝐹𝑖	PROPN
easat-4034	95	12	⊆	⊆	NUM
easat-4034	95	13	𝐺𝑖.	𝐺𝑖.	PROPN
easat-4034	95	14	we	we	PRON
easat-4034	95	15	will	will	AUX
easat-4034	95	16	show	show	VERB
easat-4034	95	17	a	a	DET
easat-4034	95	18	contradiction	contradiction	NOUN
easat-4034	95	19	with	with	ADP
easat-4034	95	20	injective	injective	ADJ
easat-4034	95	21	𝐿𝐴.	𝐿𝐴.	INTJ
easat-4034	95	22	let	let	VERB
easat-4034	95	23	𝑥	𝑥	PRON
easat-4034	95	24	=	=	PUNCT
easat-4034	96	1	[	[	X
easat-4034	96	2	𝑥𝑘	𝑥𝑘	X
easat-4034	96	3	]	]	PUNCT
easat-4034	96	4	with	with	ADP
easat-4034	96	5	𝑥𝑘	𝑥𝑘	X
easat-4034	96	6	=	=	PUNCT
easat-4034	96	7	{	{	PUNCT
easat-4034	96	8	𝑒	𝑒	ADV
easat-4034	96	9	;	;	PUNCT
easat-4034	96	10	𝑘	𝑘	PROPN
easat-4034	96	11	≠	≠	PROPN
easat-4034	96	12	𝑖	𝑖	DET
easat-4034	96	13	𝜀	𝜀	NOUN
easat-4034	96	14	;	;	PUNCT
easat-4034	96	15	𝑘	𝑘	X
easat-4034	96	16	=	=	SYM
easat-4034	96	17	𝑖	𝑖	PROPN
easat-4034	96	18	.	.	PUNCT
easat-4034	97	1	suppose	suppose	VERB
easat-4034	97	2	𝑏	𝑏	NOUN
easat-4034	97	3	=	=	SYM
easat-4034	97	4	𝐴	𝐴	PROPN
easat-4034	97	5	⊗	⊗	NOUN
easat-4034	98	1	𝑥	𝑥	X
easat-4034	98	2	=	=	PUNCT
easat-4034	98	3	⨂	⨂	PROPN
easat-4034	98	4	𝑎∗𝑘𝑘≠𝑖	𝑎∗𝑘𝑘≠𝑖	NOUN
easat-4034	98	5	with	with	ADP
easat-4034	98	6	𝑎∗𝑘	𝑎∗𝑘	PROPN
easat-4034	98	7	defined	define	VERB
easat-4034	98	8	the	the	DET
easat-4034	98	9	𝑘	𝑘	PROPN
easat-4034	98	10	−th	−th	PROPN
easat-4034	98	11	column	column	NOUN
easat-4034	98	12	of	of	ADP
easat-4034	98	13	a.	a.	NOUN
easat-4034	98	14	suppose	suppose	VERB
easat-4034	99	1	𝑗	𝑗	X
easat-4034	99	2	∈	∈	PROPN
easat-4034	99	3	𝐹𝑖	𝐹𝑖	PROPN
easat-4034	99	4	,	,	PUNCT
easat-4034	99	5	then	then	ADV
easat-4034	99	6	𝑗	𝑗	PRON
easat-4034	99	7	∈	∈	NOUN
easat-4034	99	8	𝐺𝑖.	𝐺𝑖.	PROPN
easat-4034	100	1	its	its	PRON
easat-4034	100	2	mean	mean	VERB
easat-4034	100	3	that	that	SCONJ
easat-4034	100	4	𝑘	𝑘	PRON
easat-4034	100	5	≠	≠	PROPN
easat-4034	100	6	𝑖	𝑖	PROPN
easat-4034	100	7	for	for	ADP
easat-4034	100	8	𝑎𝑗𝑘	𝑎𝑗𝑘	NOUN
easat-4034	100	9	>	>	X
easat-4034	100	10	𝜀.	𝜀.	NOUN
easat-4034	100	11	in	in	ADP
easat-4034	100	12	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	100	13	,	,	PUNCT
easat-4034	100	14	we	we	PRON
easat-4034	100	15	can	can	AUX
easat-4034	100	16	complete	complete	VERB
easat-4034	100	17	the	the	DET
easat-4034	100	18	order	order	NOUN
easat-4034	100	19	relation	relation	NOUN
easat-4034	100	20	≤	≤	NOUN
easat-4034	100	21	,	,	PUNCT
easat-4034	100	22	namely	namely	ADV
easat-4034	100	23	𝑎	𝑎	DET
easat-4034	100	24	≤	≤	ADJ
easat-4034	100	25	𝑏	𝑏	NOUN
easat-4034	100	26	if	if	SCONJ
easat-4034	101	1	and	and	CCONJ
easat-4034	101	2	only	only	ADV
easat-4034	101	3	if	if	SCONJ
easat-4034	101	4	𝑎	𝑎	PRON
easat-4034	101	5	⊕	⊕	NOUN
easat-4034	101	6	′𝑏	′𝑏	NOUN
easat-4034	102	1	=	=	PUNCT
easat-4034	102	2	𝑎.	𝑎.	NOUN
easat-4034	103	1	so	so	ADV
easat-4034	103	2	(	(	PUNCT
easat-4034	103	3	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	103	4	,	,	PUNCT
easat-4034	103	5	≤	≤	NUM
easat-4034	103	6	)	)	PUNCT
easat-4034	103	7	is	be	AUX
easat-4034	103	8	a	a	DET
easat-4034	103	9	poset	poset	NOUN
easat-4034	103	10	(	(	PUNCT
easat-4034	103	11	partially	partially	ADV
easat-4034	103	12	ordered	order	VERB
easat-4034	103	13	set	set	NOUN
easat-4034	103	14	)	)	PUNCT
easat-4034	103	15	.	.	PUNCT
easat-4034	104	1	definition	definition	NOUN
easat-4034	104	2	5	5	NUM
easat-4034	104	3	a	a	DET
easat-4034	104	4	mapping	mapping	NOUN
easat-4034	104	5	f	f	NOUN
easat-4034	104	6	on	on	ADP
easat-4034	104	7	a	a	DET
easat-4034	104	8	partially	partially	ADV
easat-4034	104	9	ordered	order	VERB
easat-4034	104	10	set	set	NOUN
easat-4034	104	11	is	be	AUX
easat-4034	104	12	said	say	VERB
easat-4034	104	13	to	to	PART
easat-4034	104	14	be	be	AUX
easat-4034	104	15	isotone	isotone	NOUN
easat-4034	104	16	if	if	SCONJ
easat-4034	104	17	for	for	ADP
easat-4034	104	18	𝑥	𝑥	PROPN
easat-4034	104	19	≤	≤	NUM
easat-4034	104	20	𝑦	𝑦	NOUN
easat-4034	104	21	the	the	DET
easat-4034	104	22	result	result	NOUN
easat-4034	104	23	is	be	AUX
easat-4034	104	24	𝑓(𝑥	𝑓(𝑥	NOUN
easat-4034	104	25	)	)	PUNCT
easat-4034	104	26	≤	≤	NOUN
easat-4034	104	27	𝑓(𝑦	𝑓(𝑦	ADV
easat-4034	104	28	)	)	PUNCT
easat-4034	104	29	.	.	PUNCT
easat-4034	105	1	example	example	NOUN
easat-4034	105	2	3	3	NUM
easat-4034	105	3	given	give	VERB
easat-4034	105	4	𝑓:ℝ𝑚𝑖𝑛	𝑓:ℝ𝑚𝑖𝑛	NOUN
easat-4034	105	5	⟶	⟶	NOUN
easat-4034	105	6	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	105	7	with	with	ADP
easat-4034	105	8	𝑓(𝑥	𝑓(𝑥	NOUN
easat-4034	105	9	)	)	PUNCT
easat-4034	105	10	=	=	SYM
easat-4034	106	1	𝑥	𝑥	PRON
easat-4034	106	2	⊗	⊗	PROPN
easat-4034	106	3	′7	′7	PROPN
easat-4034	106	4	is	be	AUX
easat-4034	106	5	an	an	DET
easat-4034	106	6	isotone	isotone	NOUN
easat-4034	106	7	mapping	mapping	NOUN
easat-4034	106	8	,	,	PUNCT
easat-4034	106	9	namely	namely	ADV
easat-4034	106	10	for	for	ADP
easat-4034	106	11	every	every	DET
easat-4034	106	12	𝑥	𝑥	PROPN
easat-4034	106	13	≤	≤	NOUN
easat-4034	106	14	𝑦	𝑦	NOUN
easat-4034	106	15	results	result	VERB
easat-4034	106	16	in	in	ADP
easat-4034	106	17	𝑥	𝑥	PRON
easat-4034	106	18	−	−	PROPN
easat-4034	106	19	7	7	NUM
easat-4034	106	20	≤	≤	NOUN
easat-4034	106	21	𝑦	𝑦	NOUN
easat-4034	106	22	−	−	PROPN
easat-4034	106	23	7	7	NUM
easat-4034	106	24	results	result	NOUN
easat-4034	106	25	in	in	ADP
easat-4034	106	26	𝑓(𝑥	𝑓(𝑥	NOUN
easat-4034	106	27	)	)	PUNCT
easat-4034	106	28	≤	≤	NOUN
easat-4034	106	29	𝑓(𝑦	𝑓(𝑦	PROPN
easat-4034	106	30	)	)	PUNCT
easat-4034	106	31	.	.	PUNCT
easat-4034	107	1	definition	definition	NOUN
easat-4034	107	2	6	6	NUM
easat-4034	107	3	given	give	VERB
easat-4034	107	4	(	(	PUNCT
easat-4034	107	5	𝐸,≤	𝐸,≤	NOUN
easat-4034	107	6	)	)	PUNCT
easat-4034	107	7	is	be	AUX
easat-4034	107	8	a	a	DET
easat-4034	107	9	poset	poset	NOUN
easat-4034	107	10	and	and	CCONJ
easat-4034	107	11	𝐴	𝐴	PROPN
easat-4034	107	12	⊆	⊆	NUM
easat-4034	107	13	𝐸.	𝐸.	PROPN
easat-4034	107	14	9548	9548	NUM
easat-4034	107	15	edelweiss	edelweiss	PROPN
easat-4034	107	16	applied	apply	VERB
easat-4034	107	17	science	science	NOUN
easat-4034	107	18	and	and	CCONJ
easat-4034	107	19	technology	technology	NOUN
easat-4034	107	20	issn	issn	PROPN
easat-4034	107	21	:	:	PUNCT
easat-4034	107	22	2576	2576	NUM
easat-4034	107	23	-	-	SYM
easat-4034	107	24	8484	8484	NUM
easat-4034	107	25	vol	vol	NOUN
easat-4034	107	26	.	.	PROPN
easat-4034	107	27	8	8	NUM
easat-4034	107	28	,	,	PUNCT
easat-4034	107	29	no	no	INTJ
easat-4034	107	30	.	.	NOUN
easat-4034	108	1	6	6	NUM
easat-4034	108	2	:	:	PUNCT
easat-4034	108	3	9544	9544	NUM
easat-4034	108	4	-	-	SYM
easat-4034	108	5	9554	9554	NUM
easat-4034	108	6	,	,	PUNCT
easat-4034	108	7	2024	2024	NUM
easat-4034	108	8	doi	doi	NOUN
easat-4034	108	9	:	:	PUNCT
easat-4034	108	10	10.55214/25768484.v8i6.4034	10.55214/25768484.v8i6.4034	NUM
easat-4034	108	11	©	©	ADP
easat-4034	108	12	2024	2024	NUM
easat-4034	108	13	by	by	ADP
easat-4034	108	14	the	the	DET
easat-4034	108	15	authors	author	NOUN
easat-4034	108	16	;	;	PUNCT
easat-4034	108	17	licensee	licensee	PROPN
easat-4034	108	18	learning	learning	NOUN
easat-4034	108	19	gate	gate	PROPN
easat-4034	108	20	i	i	PROPN
easat-4034	108	21	)	)	PUNCT
easat-4034	108	22	for	for	ADP
easat-4034	108	23	𝑎	𝑎	PROPN
easat-4034	108	24	∈	∈	PROPN
easat-4034	108	25	𝐴	𝐴	NOUN
easat-4034	108	26	there	there	PRON
easat-4034	108	27	is	be	VERB
easat-4034	108	28	𝑥	𝑥	DET
easat-4034	108	29	∈	∈	PROPN
easat-4034	108	30	𝐴	𝐴	PROPN
easat-4034	108	31	resulting	result	VERB
easat-4034	108	32	in	in	ADP
easat-4034	108	33	𝑎	𝑎	PROPN
easat-4034	108	34	≤	≤	NUM
easat-4034	108	35	𝑥	𝑥	PRON
easat-4034	109	1	so	so	ADV
easat-4034	109	2	𝑎	𝑎	NOUN
easat-4034	109	3	is	be	AUX
easat-4034	109	4	called	call	VERB
easat-4034	109	5	minimum	minimum	PROPN
easat-4034	109	6	ii	ii	NOUN
easat-4034	109	7	)	)	PUNCT
easat-4034	109	8	for	for	ADP
easat-4034	109	9	𝑎	𝑎	PROPN
easat-4034	109	10	∈	∈	PROPN
easat-4034	109	11	𝐴	𝐴	NOUN
easat-4034	109	12	it	it	PRON
easat-4034	109	13	is	be	AUX
easat-4034	109	14	called	call	VERB
easat-4034	109	15	the	the	DET
easat-4034	109	16	minimal	minimal	ADJ
easat-4034	109	17	element	element	NOUN
easat-4034	109	18	of	of	ADP
easat-4034	109	19	a	a	PRON
easat-4034	109	20	if	if	SCONJ
easat-4034	109	21	there	there	PRON
easat-4034	109	22	is	be	VERB
easat-4034	109	23	𝑥	𝑥	DET
easat-4034	109	24	∈	∈	PROPN
easat-4034	109	25	𝐴	𝐴	PROPN
easat-4034	109	26	with	with	ADP
easat-4034	109	27	𝑎	𝑎	PROPN
easat-4034	109	28	≤	≤	NUM
easat-4034	109	29	𝑥	𝑥	NOUN
easat-4034	109	30	then	then	ADV
easat-4034	109	31	𝑎	𝑎	X
easat-4034	109	32	=	=	PUNCT
easat-4034	109	33	𝑥.	𝑥.	ADJ
easat-4034	109	34	definition	definition	NOUN
easat-4034	109	35	7	7	NUM
easat-4034	109	36	an	an	DET
easat-4034	109	37	isotone	isotone	NOUN
easat-4034	109	38	mapping	map	VERB
easat-4034	109	39	𝑓	𝑓	PRON
easat-4034	109	40	:	:	PUNCT
easat-4034	109	41	𝐷	𝐷	PROPN
easat-4034	109	42	⟶	⟶	NOUN
easat-4034	109	43	𝐸	𝐸	PROPN
easat-4034	109	44	with	with	ADP
easat-4034	109	45	d	d	PROPN
easat-4034	109	46	,	,	PUNCT
easat-4034	109	47	e	e	PROPN
easat-4034	109	48	poset	poset	NOUN
easat-4034	109	49	is	be	AUX
easat-4034	109	50	said	say	VERB
easat-4034	109	51	to	to	PART
easat-4034	109	52	be	be	AUX
easat-4034	109	53	a	a	DET
easat-4034	109	54	residual	residual	ADJ
easat-4034	109	55	mapping	mapping	NOUN
easat-4034	109	56	if	if	SCONJ
easat-4034	109	57	for	for	ADP
easat-4034	109	58	all	all	DET
easat-4034	109	59	𝑏	𝑏	PRON
easat-4034	109	60	∈	∈	PROPN
easat-4034	109	61	𝐸	𝐸	PROPN
easat-4034	109	62	then	then	ADV
easat-4034	109	63	{	{	PUNCT
easat-4034	109	64	𝑥|𝑏	𝑥|𝑏	ADV
easat-4034	109	65	≤	≤	NUM
easat-4034	109	66	𝑓(𝑥	𝑓(𝑥	NOUN
easat-4034	109	67	)	)	PUNCT
easat-4034	109	68	}	}	PUNCT
easat-4034	109	69	has	have	VERB
easat-4034	109	70	a	a	DET
easat-4034	109	71	minimum	minimum	ADJ
easat-4034	109	72	element	element	NOUN
easat-4034	109	73	denoted	denote	VERB
easat-4034	109	74	𝑓#(𝑏	𝑓#(𝑏	PROPN
easat-4034	109	75	)	)	PUNCT
easat-4034	109	76	.	.	PUNCT
easat-4034	110	1	the	the	DET
easat-4034	110	2	isotone	isotone	NOUN
easat-4034	110	3	mapping	map	VERB
easat-4034	110	4	𝑓	𝑓	ADV
easat-4034	110	5	#	#	NOUN
easat-4034	110	6	:	:	PUNCT
easat-4034	110	7	𝐸	𝐸	PROPN
easat-4034	110	8	⟶	⟶	NOUN
easat-4034	110	9	𝐷	𝐷	PROPN
easat-4034	110	10	is	be	AUX
easat-4034	110	11	called	call	VERB
easat-4034	110	12	residual	residual	ADJ
easat-4034	110	13	of	of	ADP
easat-4034	110	14	f.	f.	PROPN
easat-4034	110	15	theorem	theorem	VERB
easat-4034	110	16	5	5	NUM
easat-4034	110	17	if	if	SCONJ
easat-4034	110	18	𝑓	𝑓	PRON
easat-4034	110	19	:	:	PUNCT
easat-4034	110	20	𝐷	𝐷	PROPN
easat-4034	110	21	→	→	SYM
easat-4034	110	22	𝐸	𝐸	PROPN
easat-4034	110	23	is	be	AUX
easat-4034	110	24	a	a	DET
easat-4034	110	25	residualized	residualize	VERB
easat-4034	110	26	mapping	mapping	NOUN
easat-4034	110	27	,	,	PUNCT
easat-4034	110	28	then	then	ADV
easat-4034	110	29	the	the	DET
easat-4034	110	30	equation	equation	NOUN
easat-4034	110	31	𝑓(𝑥	𝑓(𝑥	NOUN
easat-4034	110	32	)	)	PUNCT
easat-4034	111	1	=	=	SYM
easat-4034	111	2	𝑏	𝑏	PROPN
easat-4034	111	3	has	have	VERB
easat-4034	111	4	a	a	DET
easat-4034	111	5	solution	solution	NOUN
easat-4034	111	6	if	if	SCONJ
easat-4034	111	7	and	and	CCONJ
easat-4034	111	8	only	only	ADV
easat-4034	111	9	if	if	SCONJ
easat-4034	111	10	𝑓(𝑓	𝑓(𝑓	NOUN
easat-4034	111	11	#	#	NOUN
easat-4034	111	12	(	(	PUNCT
easat-4034	111	13	𝑏	𝑏	NOUN
easat-4034	111	14	)	)	PUNCT
easat-4034	111	15	)	)	PUNCT
easat-4034	112	1	=	=	PUNCT
easat-4034	112	2	𝑏.	𝑏.	ADJ
easat-4034	112	3	proof	proof	NOUN
easat-4034	112	4	:	:	PUNCT
easat-4034	112	5	(	(	PUNCT
easat-4034	112	6	⇒	⇒	NOUN
easat-4034	112	7	)	)	PUNCT
easat-4034	112	8	given	give	VERB
easat-4034	112	9	𝑓(𝑥	𝑓(𝑥	NOUN
easat-4034	112	10	)	)	PUNCT
easat-4034	112	11	=	=	SYM
easat-4034	113	1	𝑏	𝑏	PROPN
easat-4034	113	2	has	have	VERB
easat-4034	113	3	a	a	DET
easat-4034	113	4	solution	solution	NOUN
easat-4034	113	5	,	,	PUNCT
easat-4034	113	6	say	say	VERB
easat-4034	113	7	𝑥1	𝑥1	NOUN
easat-4034	113	8	.	.	PUNCT
easat-4034	114	1	we	we	PRON
easat-4034	114	2	get	get	VERB
easat-4034	114	3	𝑓(𝑥1	𝑓(𝑥1	ADV
easat-4034	114	4	)	)	PUNCT
easat-4034	114	5	=	=	VERB
easat-4034	114	6	𝑏.	𝑏.	ADV
easat-4034	114	7	since	since	SCONJ
easat-4034	114	8	𝑓#(𝑏	𝑓#(𝑏	PROPN
easat-4034	114	9	)	)	PUNCT
easat-4034	114	10	is	be	AUX
easat-4034	114	11	a	a	DET
easat-4034	114	12	minimal	minimal	ADJ
easat-4034	114	13	element	element	NOUN
easat-4034	114	14	in	in	ADP
easat-4034	114	15	{	{	PUNCT
easat-4034	114	16	𝑥|𝑓(𝑥	𝑥|𝑓(𝑥	NOUN
easat-4034	114	17	)	)	PUNCT
easat-4034	114	18	≤	≤	NOUN
easat-4034	115	1	𝑏	𝑏	NOUN
easat-4034	115	2	}	}	PUNCT
easat-4034	115	3	,	,	PUNCT
easat-4034	115	4	then	then	ADV
easat-4034	115	5	𝑥1	𝑥1	NOUN
easat-4034	115	6	≤	≤	PROPN
easat-4034	115	7	𝑓#(𝑏	𝑓#(𝑏	PROPN
easat-4034	115	8	)	)	PUNCT
easat-4034	115	9	.	.	PUNCT
easat-4034	116	1	since	since	SCONJ
easat-4034	116	2	𝑓	𝑓	PRON
easat-4034	116	3	is	be	AUX
easat-4034	116	4	isotone	isotone	NOUN
easat-4034	116	5	then	then	ADV
easat-4034	116	6	𝑓(𝑥1	𝑓(𝑥1	ADV
easat-4034	116	7	)	)	PUNCT
easat-4034	116	8	≤	≤	NOUN
easat-4034	116	9	𝑓(𝑓#(𝑏	𝑓(𝑓#(𝑏	VERB
easat-4034	116	10	)	)	PUNCT
easat-4034	116	11	)	)	PUNCT
easat-4034	116	12	,	,	PUNCT
easat-4034	116	13	according	accord	VERB
easat-4034	116	14	to	to	ADP
easat-4034	116	15	(	(	PUNCT
easat-4034	116	16	*	*	PUNCT
easat-4034	116	17	)	)	PUNCT
easat-4034	116	18	𝑓(𝑓#(𝑏	𝑓(𝑓#(𝑏	ADJ
easat-4034	116	19	)	)	PUNCT
easat-4034	116	20	)	)	PUNCT
easat-4034	117	1	≤	≤	NOUN
easat-4034	117	2	𝑏	𝑏	NOUN
easat-4034	117	3	,	,	PUNCT
easat-4034	117	4	consequently	consequently	ADV
easat-4034	117	5	𝑏	𝑏	NOUN
easat-4034	117	6	=	=	PUNCT
easat-4034	117	7	𝑓(𝑥1	𝑓(𝑥1	ADJ
easat-4034	117	8	)	)	PUNCT
easat-4034	117	9	≤	≤	NUM
easat-4034	117	10	𝑓𝑓	𝑓𝑓	ADP
easat-4034	117	11	#	#	NOUN
easat-4034	117	12	(	(	PUNCT
easat-4034	117	13	𝑏	𝑏	NOUN
easat-4034	117	14	)	)	PUNCT
easat-4034	117	15	≤	≤	NOUN
easat-4034	117	16	𝑏	𝑏	NOUN
easat-4034	117	17	,	,	PUNCT
easat-4034	117	18	namely	namely	ADV
easat-4034	117	19	𝑓(𝑓#(𝑏	𝑓(𝑓#(𝑏	VERB
easat-4034	117	20	)	)	PUNCT
easat-4034	117	21	)	)	PUNCT
easat-4034	118	1	=	=	PUNCT
easat-4034	118	2	𝑏.	𝑏.	NOUN
easat-4034	118	3	(	(	PUNCT
easat-4034	118	4	⇐	⇐	ADJ
easat-4034	118	5	)	)	PUNCT
easat-4034	118	6	given	give	VERB
easat-4034	118	7	𝑓(𝑓^	𝑓(𝑓^	VERB
easat-4034	118	8	#	#	NOUN
easat-4034	118	9	(	(	PUNCT
easat-4034	118	10	𝑏	𝑏	NOUN
easat-4034	118	11	)	)	PUNCT
easat-4034	118	12	)	)	PUNCT
easat-4034	119	1	=	=	SYM
easat-4034	119	2	𝑏	𝑏	NOUN
easat-4034	119	3	,	,	PUNCT
easat-4034	119	4	then	then	ADV
easat-4034	119	5	the	the	DET
easat-4034	119	6	equation	equation	NOUN
easat-4034	119	7	𝑓(𝑥	𝑓(𝑥	NOUN
easat-4034	119	8	)	)	PUNCT
easat-4034	120	1	=	=	SYM
easat-4034	121	1	𝑏	𝑏	PROPN
easat-4034	121	2	has	have	VERB
easat-4034	121	3	a	a	DET
easat-4034	121	4	solution	solution	NOUN
easat-4034	121	5	,	,	PUNCT
easat-4034	121	6	namely	namely	ADV
easat-4034	121	7	𝑥	𝑥	PRON
easat-4034	121	8	=	=	SYM
easat-4034	122	1	𝑓	𝑓	PRON
easat-4034	122	2	#	#	NOUN
easat-4034	122	3	(	(	PUNCT
easat-4034	122	4	𝑏	𝑏	NOUN
easat-4034	122	5	)	)	PUNCT
easat-4034	122	6	.	.	PUNCT
easat-4034	123	1	the	the	DET
easat-4034	123	2	function	function	NOUN
easat-4034	123	3	𝑓	𝑓	NOUN
easat-4034	123	4	is	be	AUX
easat-4034	123	5	residualized	residualize	VERB
easat-4034	123	6	,	,	PUNCT
easat-4034	123	7	because	because	SCONJ
easat-4034	123	8	𝑦	𝑦	PRON
easat-4034	123	9	∈	∈	PROPN
easat-4034	123	10	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	123	11	with	with	ADP
easat-4034	123	12	{	{	PUNCT
easat-4034	123	13	𝑥|𝑦	𝑥|𝑦	NOUN
easat-4034	123	14	≤	≤	NUM
easat-4034	123	15	𝑥	𝑥	PRON
easat-4034	123	16	⊗′	⊗′	PROPN
easat-4034	123	17	7	7	NUM
easat-4034	123	18	=	=	SYM
easat-4034	123	19	𝑓(𝑥	𝑓(𝑥	NOUN
easat-4034	123	20	)	)	PUNCT
easat-4034	123	21	}	}	PUNCT
easat-4034	123	22	is	be	AUX
easat-4034	123	23	a	a	DET
easat-4034	123	24	minimal	minimal	ADJ
easat-4034	123	25	element	element	NOUN
easat-4034	123	26	,	,	PUNCT
easat-4034	123	27	namely	namely	ADV
easat-4034	123	28	𝑥	𝑥	PRON
easat-4034	123	29	=	=	SYM
easat-4034	123	30	𝑓#(𝑦	𝑓#(𝑦	NOUN
easat-4034	123	31	)	)	PUNCT
easat-4034	124	1	=	=	SYM
easat-4034	124	2	𝑦	𝑦	NOUN
easat-4034	124	3	+	+	NOUN
easat-4034	124	4	7	7	NUM
easat-4034	124	5	.	.	PUNCT
easat-4034	124	6	definition	definition	NOUN
easat-4034	124	7	8	8	NUM
easat-4034	124	8	for	for	ADP
easat-4034	124	9	every	every	DET
easat-4034	124	10	𝑏	𝑏	PROPN
easat-4034	124	11	∈	∈	PROPN
easat-4034	124	12	𝐸	𝐸	PROPN
easat-4034	124	13	then	then	ADV
easat-4034	124	14	{	{	PUNCT
easat-4034	124	15	𝑥|𝑏	𝑥|𝑏	ADV
easat-4034	124	16	≤	≤	NUM
easat-4034	124	17	𝑓(𝑥	𝑓(𝑥	NOUN
easat-4034	124	18	)	)	PUNCT
easat-4034	124	19	}	}	PUNCT
easat-4034	124	20	has	have	VERB
easat-4034	124	21	a	a	DET
easat-4034	124	22	minimal	minimal	ADJ
easat-4034	124	23	element	element	NOUN
easat-4034	124	24	denoted	denote	VERB
easat-4034	124	25	𝑓#(𝑏	𝑓#(𝑏	PROPN
easat-4034	124	26	)	)	PUNCT
easat-4034	124	27	.	.	PUNCT
easat-4034	125	1	for	for	ADP
easat-4034	125	2	𝑦	𝑦	NOUN
easat-4034	125	3	∈	∈	PROPN
easat-4034	125	4	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	125	5	{	{	PUNCT
easat-4034	125	6	𝑥|𝑦	𝑥|𝑦	NOUN
easat-4034	125	7	≤	≤	NUM
easat-4034	125	8	𝑥	𝑥	PRON
easat-4034	125	9	⊗′	⊗′	PROPN
easat-4034	125	10	7	7	NUM
easat-4034	125	11	=	=	SYM
easat-4034	125	12	𝑓(𝑥	𝑓(𝑥	NOUN
easat-4034	125	13	)	)	PUNCT
easat-4034	125	14	}	}	PUNCT
easat-4034	125	15	the	the	DET
easat-4034	125	16	minimal	minimal	ADJ
easat-4034	125	17	element	element	NOUN
easat-4034	125	18	is	be	AUX
easat-4034	125	19	𝑥	𝑥	NOUN
easat-4034	125	20	=	=	SYM
easat-4034	125	21	𝑓#(𝑦	𝑓#(𝑦	PROPN
easat-4034	125	22	)	)	PUNCT
easat-4034	125	23	=	=	SYM
easat-4034	125	24	𝑦	𝑦	NOUN
easat-4034	125	25	+	+	NOUN
easat-4034	125	26	7	7	NUM
easat-4034	125	27	.	.	PUNCT
easat-4034	125	28	definition	definition	NOUN
easat-4034	125	29	9	9	NUM
easat-4034	125	30	a	a	DET
easat-4034	125	31	subsolution	subsolution	NOUN
easat-4034	125	32	of	of	ADP
easat-4034	125	33	𝐴	𝐴	PROPN
easat-4034	125	34	⊗	⊗	NOUN
easat-4034	126	1	𝑥	𝑥	PROPN
easat-4034	126	2	=	=	PUNCT
easat-4034	126	3	𝑏	𝑏	PROPN
easat-4034	126	4	is	be	AUX
easat-4034	126	5	𝑥	𝑥	PRON
easat-4034	126	6	that	that	PRON
easat-4034	126	7	satisfies	satisfy	VERB
easat-4034	126	8	𝐴	𝐴	PROPN
easat-4034	126	9	⊗	⊗	PROPN
easat-4034	126	10	𝑥	𝑥	PROPN
easat-4034	126	11	≥	≥	NUM
easat-4034	126	12	𝑏	𝑏	NOUN
easat-4034	126	13	,	,	PUNCT
easat-4034	126	14	a	a	DET
easat-4034	126	15	linear	linear	ADJ
easat-4034	126	16	system	system	NOUN
easat-4034	126	17	for	for	ADP
easat-4034	126	18	obtaining	obtain	VERB
easat-4034	126	19	the	the	DET
easat-4034	126	20	general	general	ADJ
easat-4034	126	21	result	result	NOUN
easat-4034	126	22	of	of	ADP
easat-4034	126	23	the	the	DET
easat-4034	126	24	equation	equation	NOUN
easat-4034	126	25	𝐴	𝐴	PROPN
easat-4034	126	26	⊗	⊗	NOUN
easat-4034	127	1	𝑥	𝑥	X
easat-4034	128	1	=	=	PUNCT
easat-4034	128	2	𝑏.	𝑏.	VERB
easat-4034	128	3	an	an	DET
easat-4034	128	4	ordered	order	VERB
easat-4034	128	5	pair	pair	NOUN
easat-4034	128	6	of	of	ADP
easat-4034	128	7	vectors	vector	NOUN
easat-4034	128	8	is	be	AUX
easat-4034	128	9	defined	define	VERB
easat-4034	128	10	by	by	ADP
easat-4034	128	11	𝑥	𝑥	PRON
easat-4034	128	12	≥	≥	NUM
easat-4034	128	13	𝑦	𝑦	NOUN
easat-4034	128	14	if	if	SCONJ
easat-4034	128	15	𝑥	𝑥	PRON
easat-4034	128	16	⊕′	⊕′	PROPN
easat-4034	128	17	𝑦	𝑦	NOUN
easat-4034	128	18	=	=	X
easat-4034	128	19	𝑦.	𝑦.	PROPN
easat-4034	128	20	since	since	SCONJ
easat-4034	128	21	𝐴	𝐴	PROPN
easat-4034	128	22	∈	∈	PROPN
easat-4034	128	23	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	128	24	𝑛×𝑛	𝑛×𝑛	PROPN
easat-4034	128	25	and	and	CCONJ
easat-4034	128	26	𝑋	𝑋	PROPN
easat-4034	128	27	∈	∈	PROPN
easat-4034	128	28	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	128	29	𝑛×𝑛	𝑛×𝑛	PROPN
easat-4034	128	30	then	then	ADV
easat-4034	128	31	𝐴	𝐴	PROPN
easat-4034	128	32	⊗	⊗	NOUN
easat-4034	129	1	𝑥	𝑥	PROPN
easat-4034	130	1	=	=	PUNCT
easat-4034	130	2	[	[	PUNCT
easat-4034	130	3	𝑎11	𝑎11	VERB
easat-4034	130	4	𝑎12	𝑎12	PROPN
easat-4034	130	5	𝑎21	𝑎21	PROPN
easat-4034	130	6	𝑎22	𝑎22	NOUN
easat-4034	130	7	⋯	⋯	VERB
easat-4034	130	8	𝑎1𝑛	𝑎1𝑛	NOUN
easat-4034	130	9	⋯	⋯	NOUN
easat-4034	130	10	𝑎2𝑛	𝑎2𝑛	PROPN
easat-4034	130	11	⋮	⋮	NOUN
easat-4034	130	12	⋮	⋮	NOUN
easat-4034	130	13	𝑎𝑛1	𝑎𝑛1	VERB
easat-4034	130	14	𝑎𝑛2	𝑎𝑛2	X
easat-4034	130	15	⋱	⋱	PUNCT
easat-4034	130	16	⋮	⋮	NOUN
easat-4034	130	17	⋯	⋯	VERB
easat-4034	130	18	𝑎𝑛𝑛	𝑎𝑛𝑛	NOUN
easat-4034	130	19	]	]	PUNCT
easat-4034	131	1	⊗	⊗	PROPN
easat-4034	131	2	[	[	PUNCT
easat-4034	131	3	𝑥1	𝑥1	NOUN
easat-4034	131	4	𝑥2	𝑥2	NOUN
easat-4034	131	5	⋮	⋮	NOUN
easat-4034	131	6	𝑥𝑛	𝑥𝑛	VERB
easat-4034	131	7	]	]	PUNCT
easat-4034	131	8	=	=	PUNCT
easat-4034	132	1	[	[	PUNCT
easat-4034	132	2	𝑎11	𝑎11	ADJ
easat-4034	132	3	+	+	NUM
easat-4034	132	4	𝑥1	𝑥1	NOUN
easat-4034	132	5	⊕′	⊕′	PROPN
easat-4034	132	6	𝑎12	𝑎12	NOUN
easat-4034	132	7	+	+	CCONJ
easat-4034	132	8	𝑥2	𝑥2	NOUN
easat-4034	132	9	⊕′	⊕′	PROPN
easat-4034	132	10	⋯⊕′	⋯⊕′	NOUN
easat-4034	132	11	𝑎1𝑛	𝑎1𝑛	VERB
easat-4034	132	12	+	+	CCONJ
easat-4034	132	13	𝑥𝑛	𝑥𝑛	PROPN
easat-4034	132	14	𝑎21	𝑎21	NOUN
easat-4034	132	15	+	+	CCONJ
easat-4034	132	16	𝑥1	𝑥1	PROPN
easat-4034	132	17	⊕′	⊕′	PROPN
easat-4034	132	18	𝑎22	𝑎22	NOUN
easat-4034	132	19	+	+	CCONJ
easat-4034	132	20	𝑥2	𝑥2	NOUN
easat-4034	132	21	⊕′	⊕′	PROPN
easat-4034	132	22	⋯⊕′	⋯⊕′	NOUN
easat-4034	132	23	𝑎2𝑛	𝑎2𝑛	PROPN
easat-4034	133	1	+	+	CCONJ
easat-4034	133	2	𝑥𝑛	𝑥𝑛	PROPN
easat-4034	133	3	⋮	⋮	NOUN
easat-4034	133	4	𝑎𝑛1	𝑎𝑛1	VERB
easat-4034	133	5	+	+	CCONJ
easat-4034	133	6	𝑥1	𝑥1	NOUN
easat-4034	133	7	⊕′	⊕′	PROPN
easat-4034	133	8	𝑎𝑛2	𝑎𝑛2	NOUN
easat-4034	133	9	+	+	CCONJ
easat-4034	133	10	𝑥2	𝑥2	NOUN
easat-4034	133	11	⊕′	⊕′	PROPN
easat-4034	133	12	⋯⊕′	⋯⊕′	NOUN
easat-4034	133	13	𝑎𝑛𝑛	𝑎𝑛𝑛	VERB
easat-4034	133	14	+	+	CCONJ
easat-4034	133	15	𝑥𝑛	𝑥𝑛	X
easat-4034	133	16	]	]	PUNCT
easat-4034	133	17	=	=	PUNCT
easat-4034	133	18	[	[	PUNCT
easat-4034	133	19	⊕	⊕	PROPN
easat-4034	133	20	′𝑎1𝑗	′𝑎1𝑗	PROPN
easat-4034	133	21	+	+	CCONJ
easat-4034	133	22	𝑥𝑗	𝑥𝑗	PROPN
easat-4034	133	23	⊕	⊕	PROPN
easat-4034	133	24	′𝑎2𝑗	′𝑎2𝑗	PROPN
easat-4034	133	25	+	+	CCONJ
easat-4034	133	26	𝑥𝑗	𝑥𝑗	PROPN
easat-4034	133	27	⋮	⋮	NOUN
easat-4034	133	28	⊕	⊕	PROPN
easat-4034	133	29	′𝑎𝑛𝑗	′𝑎𝑛𝑗	NOUN
easat-4034	134	1	+	+	CCONJ
easat-4034	134	2	𝑥𝑗	𝑥𝑗	PROPN
easat-4034	134	3	]	]	X
easat-4034	134	4	,	,	PUNCT
easat-4034	134	5	𝑗	𝑗	NOUN
easat-4034	134	6	=	=	SYM
easat-4034	134	7	1,2	1,2	NUM
easat-4034	134	8	,	,	PUNCT
easat-4034	134	9	…	…	PUNCT
easat-4034	134	10	,	,	PUNCT
easat-4034	134	11	𝑛	𝑛	X
easat-4034	134	12	with	with	ADP
easat-4034	134	13	⊕	⊕	PROPN
easat-4034	134	14	′𝑎1𝑗	′𝑎1𝑗	PROPN
easat-4034	134	15	+	+	CCONJ
easat-4034	134	16	𝑥𝑗	𝑥𝑗	PROPN
easat-4034	134	17	=	=	PUNCT
easat-4034	134	18	𝑚𝑖𝑛{𝑎11	𝑚𝑖𝑛{𝑎11	NOUN
easat-4034	134	19	+	+	CCONJ
easat-4034	134	20	𝑥1	𝑥1	NOUN
easat-4034	134	21	,	,	PUNCT
easat-4034	134	22	𝑎12	𝑎12	NOUN
easat-4034	134	23	+	+	CCONJ
easat-4034	134	24	𝑥2	𝑥2	NOUN
easat-4034	134	25	,	,	PUNCT
easat-4034	134	26	⋯	⋯	NOUN
easat-4034	134	27	,	,	PUNCT
easat-4034	134	28	𝑎1𝑛	𝑎1𝑛	ADJ
easat-4034	134	29	+	+	CCONJ
easat-4034	134	30	𝑥𝑛	𝑥𝑛	NOUN
easat-4034	134	31	}	}	PUNCT
easat-4034	134	32	9549	9549	NUM
easat-4034	134	33	edelweiss	edelweiss	PROPN
easat-4034	134	34	applied	apply	VERB
easat-4034	134	35	science	science	NOUN
easat-4034	134	36	and	and	CCONJ
easat-4034	134	37	technology	technology	NOUN
easat-4034	134	38	issn	issn	PROPN
easat-4034	134	39	:	:	PUNCT
easat-4034	134	40	2576	2576	NUM
easat-4034	134	41	-	-	SYM
easat-4034	134	42	8484	8484	NUM
easat-4034	134	43	vol	vol	NOUN
easat-4034	134	44	.	.	PROPN
easat-4034	134	45	8	8	NUM
easat-4034	134	46	,	,	PUNCT
easat-4034	134	47	no	no	INTJ
easat-4034	134	48	.	.	NOUN
easat-4034	135	1	6	6	NUM
easat-4034	135	2	:	:	PUNCT
easat-4034	135	3	9544	9544	NUM
easat-4034	135	4	-	-	SYM
easat-4034	135	5	9554	9554	NUM
easat-4034	135	6	,	,	PUNCT
easat-4034	135	7	2024	2024	NUM
easat-4034	135	8	doi	doi	NOUN
easat-4034	135	9	:	:	PUNCT
easat-4034	135	10	10.55214/25768484.v8i6.4034	10.55214/25768484.v8i6.4034	NUM
easat-4034	135	11	©	©	ADP
easat-4034	135	12	2024	2024	NUM
easat-4034	135	13	by	by	ADP
easat-4034	135	14	the	the	DET
easat-4034	135	15	authors	author	NOUN
easat-4034	135	16	;	;	PUNCT
easat-4034	135	17	licensee	licensee	PROPN
easat-4034	135	18	learning	learning	NOUN
easat-4034	135	19	gate	gate	NOUN
easat-4034	135	20	form	form	NOUN
easat-4034	135	21	𝑓𝑗(𝑥𝑗	𝑓𝑗(𝑥𝑗	PROPN
easat-4034	135	22	)	)	PUNCT
easat-4034	136	1	=	=	PUNCT
easat-4034	136	2	[	[	PUNCT
easat-4034	136	3	𝑎1𝑗	𝑎1𝑗	X
easat-4034	136	4	+	+	CCONJ
easat-4034	136	5	𝑥1	𝑥1	PROPN
easat-4034	136	6	𝑎2𝑗	𝑎2𝑗	ADP
easat-4034	136	7	+	+	NUM
easat-4034	136	8	𝑥2	𝑥2	NOUN
easat-4034	136	9	⋮	⋮	NOUN
easat-4034	136	10	𝑎𝑛𝑗	𝑎𝑛𝑗	NOUN
easat-4034	137	1	+	+	CCONJ
easat-4034	137	2	𝑥𝑛	𝑥𝑛	PROPN
easat-4034	137	3	]	]	PUNCT
easat-4034	137	4	so	so	SCONJ
easat-4034	137	5	that	that	SCONJ
easat-4034	137	6	𝐴	𝐴	PROPN
easat-4034	137	7	⊗	⊗	NOUN
easat-4034	137	8	𝑥	𝑥	PROPN
easat-4034	137	9	=	=	PUNCT
easat-4034	137	10	𝑛	𝑛	PRON
easat-4034	137	11	⊕′	⊕′	NOUN
easat-4034	137	12	𝑗	𝑗	NOUN
easat-4034	137	13	=	=	SYM
easat-4034	137	14	1	1	NUM
easat-4034	137	15	𝑓𝑗(𝑥𝑗	𝑓𝑗(𝑥𝑗	NOUN
easat-4034	137	16	)	)	PUNCT
easat-4034	137	17	.	.	PUNCT
easat-4034	138	1	so	so	ADV
easat-4034	138	2	,	,	PUNCT
easat-4034	138	3	𝐴	𝐴	PROPN
easat-4034	138	4	⊗	⊗	NOUN
easat-4034	138	5	𝑥	𝑥	X
easat-4034	138	6	=	=	SYM
easat-4034	138	7	𝑓1(𝑥1	𝑓1(𝑥1	NOUN
easat-4034	138	8	)	)	PUNCT
easat-4034	138	9	⊕′	⊕′	PROPN
easat-4034	138	10	𝑓2(𝑥2	𝑓2(𝑥2	NOUN
easat-4034	138	11	)	)	PUNCT
easat-4034	138	12	⊕′	⊕′	PROPN
easat-4034	138	13	⋯⊕′	⋯⊕′	NOUN
easat-4034	138	14	𝑓𝑛(𝑥𝑛	𝑓𝑛(𝑥𝑛	NUM
easat-4034	138	15	)	)	PUNCT
easat-4034	138	16	.	.	PUNCT
easat-4034	139	1	for	for	ADP
easat-4034	139	2	each	each	DET
easat-4034	139	3	𝑗	𝑗	PROPN
easat-4034	139	4	,	,	PUNCT
easat-4034	139	5	if	if	SCONJ
easat-4034	139	6	𝑥𝑗ℎ	𝑥𝑗ℎ	VERB
easat-4034	139	7	≤	≤	NUM
easat-4034	139	8	𝑥𝑗𝑘	𝑥𝑗𝑘	NOUN
easat-4034	139	9	⟹	⟹	NOUN
easat-4034	139	10	𝑓𝑗(𝑥𝑗ℎ	𝑓𝑗(𝑥𝑗ℎ	NOUN
easat-4034	139	11	)	)	PUNCT
easat-4034	139	12	≤	≤	NUM
easat-4034	139	13	𝑓𝑗(𝑥𝑗𝑘	𝑓𝑗(𝑥𝑗𝑘	NOUN
easat-4034	139	14	)	)	PUNCT
easat-4034	139	15	.	.	PUNCT
easat-4034	140	1	according	accord	VERB
easat-4034	140	2	definition	definition	NOUN
easat-4034	140	3	5	5	NUM
easat-4034	140	4	and	and	CCONJ
easat-4034	140	5	theorem	theorem	VERB
easat-4034	140	6	5	5	NUM
easat-4034	140	7	,	,	PUNCT
easat-4034	140	8	so	so	CCONJ
easat-4034	140	9	{	{	PUNCT
easat-4034	140	10	𝑥|𝐴	𝑥|𝐴	PROPN
easat-4034	140	11	⊗	⊗	PROPN
easat-4034	140	12	𝑥	𝑥	PROPN
easat-4034	140	13	≥	≥	NOUN
easat-4034	140	14	𝑏	𝑏	NOUN
easat-4034	140	15	}	}	PUNCT
easat-4034	140	16	has	have	VERB
easat-4034	140	17	a	a	DET
easat-4034	140	18	minimum	minimum	ADJ
easat-4034	140	19	element	element	NOUN
easat-4034	140	20	denoted	denote	VERB
easat-4034	140	21	𝐴#(𝑏	𝐴#(𝑏	PROPN
easat-4034	140	22	)	)	PUNCT
easat-4034	140	23	.	.	PUNCT
easat-4034	141	1	therefore	therefore	ADV
easat-4034	141	2	,	,	PUNCT
easat-4034	141	3	to	to	PART
easat-4034	141	4	determine	determine	VERB
easat-4034	141	5	the	the	DET
easat-4034	141	6	solution	solution	NOUN
easat-4034	141	7	of	of	ADP
easat-4034	141	8	the	the	DET
easat-4034	141	9	equation	equation	NOUN
easat-4034	141	10	𝐴𝑥	𝐴𝑥	NOUN
easat-4034	141	11	=	=	SYM
easat-4034	141	12	𝑏	𝑏	NOUN
easat-4034	141	13	,	,	PUNCT
easat-4034	141	14	check	check	VERB
easat-4034	141	15	wether	wether	PROPN
easat-4034	141	16	𝐴	𝐴	PROPN
easat-4034	141	17	(	(	PUNCT
easat-4034	141	18	𝐴#(𝑏	𝐴#(𝑏	PROPN
easat-4034	141	19	)	)	PUNCT
easat-4034	141	20	)	)	PUNCT
easat-4034	142	1	=	=	VERB
easat-4034	142	2	𝑏.	𝑏.	VERB
easat-4034	142	3	the	the	DET
easat-4034	142	4	following	follow	VERB
easat-4034	142	5	is	be	AUX
easat-4034	142	6	a	a	DET
easat-4034	142	7	theorem	theorem	NOUN
easat-4034	142	8	that	that	PRON
easat-4034	142	9	states	state	VERB
easat-4034	142	10	this	this	DET
easat-4034	142	11	characteristic	characteristic	NOUN
easat-4034	142	12	.	.	PUNCT
easat-4034	143	1	theorem	theorem	VERB
easat-4034	143	2	6	6	NUM
easat-4034	143	3	if	if	SCONJ
easat-4034	143	4	𝐴	𝐴	PROPN
easat-4034	143	5	∈	∈	PROPN
easat-4034	143	6	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	143	7	𝑛×𝑛	𝑛×𝑛	PROPN
easat-4034	143	8	and	and	CCONJ
easat-4034	143	9	𝑏	𝑏	PRON
easat-4034	143	10	∈	∈	PROPN
easat-4034	143	11	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	143	12	𝑛	𝑛	PROPN
easat-4034	143	13	,	,	PUNCT
easat-4034	143	14	then	then	ADV
easat-4034	143	15	the	the	DET
easat-4034	143	16	equation	equation	NOUN
easat-4034	143	17	𝐴	𝐴	PROPN
easat-4034	143	18	⊗	⊗	NOUN
easat-4034	144	1	𝑥	𝑥	PROPN
easat-4034	144	2	=	=	PUNCT
easat-4034	144	3	𝑏	𝑏	PROPN
easat-4034	144	4	has	have	VERB
easat-4034	144	5	a	a	DET
easat-4034	144	6	solution	solution	NOUN
easat-4034	144	7	if	if	SCONJ
easat-4034	144	8	and	and	CCONJ
easat-4034	144	9	only	only	ADV
easat-4034	144	10	if	if	SCONJ
easat-4034	144	11	𝐴	𝐴	PROPN
easat-4034	144	12	(	(	PUNCT
easat-4034	144	13	𝐴#(𝑏	𝐴#(𝑏	PROPN
easat-4034	144	14	)	)	PUNCT
easat-4034	144	15	)	)	PUNCT
easat-4034	144	16	=	=	PUNCT
easat-4034	144	17	𝑏.	𝑏.	ADV
easat-4034	144	18	in	in	ADP
easat-4034	144	19	other	other	ADJ
easat-4034	144	20	words	word	NOUN
easat-4034	144	21	,	,	PUNCT
easat-4034	144	22	the	the	DET
easat-4034	144	23	solution	solution	NOUN
easat-4034	144	24	is	be	AUX
easat-4034	144	25	𝑥	𝑥	PROPN
easat-4034	144	26	=	=	SYM
easat-4034	144	27	𝐴#(𝑏	𝐴#(𝑏	PROPN
easat-4034	144	28	)	)	PUNCT
easat-4034	144	29	.	.	PUNCT
easat-4034	145	1	proof	proof	NOUN
easat-4034	145	2	:	:	PUNCT
easat-4034	145	3	(	(	PUNCT
easat-4034	145	4	⟹	⟹	X
easat-4034	145	5	)	)	PUNCT
easat-4034	145	6	it	it	PRON
easat-4034	145	7	is	be	AUX
easat-4034	145	8	known	know	VERB
easat-4034	145	9	that	that	SCONJ
easat-4034	145	10	the	the	DET
easat-4034	145	11	equation	equation	NOUN
easat-4034	145	12	𝐴	𝐴	PROPN
easat-4034	145	13	⊗	⊗	NOUN
easat-4034	146	1	𝑥	𝑥	PROPN
easat-4034	146	2	=	=	PUNCT
easat-4034	146	3	𝑏	𝑏	PROPN
easat-4034	146	4	has	have	VERB
easat-4034	146	5	a	a	DET
easat-4034	146	6	solution	solution	NOUN
easat-4034	146	7	𝑥∗	𝑥∗	PROPN
easat-4034	146	8	,	,	PUNCT
easat-4034	146	9	namely	namely	ADV
easat-4034	146	10	𝐴	𝐴	PROPN
easat-4034	147	1	⊗	⊗	PROPN
easat-4034	147	2	𝑥∗	𝑥∗	PROPN
easat-4034	147	3	=	=	SYM
easat-4034	147	4	𝑏	𝑏	NOUN
easat-4034	147	5	,	,	PUNCT
easat-4034	147	6	so	so	SCONJ
easat-4034	147	7	that	that	SCONJ
easat-4034	147	8	𝐴	𝐴	PROPN
easat-4034	147	9	⊗	⊗	PROPN
easat-4034	147	10	𝑥∗	𝑥∗	PROPN
easat-4034	147	11	≥	≥	PRON
easat-4034	147	12	𝑏.	𝑏.	VERB
easat-4034	147	13	since	since	SCONJ
easat-4034	147	14	𝐴#𝑏	𝐴#𝑏	PROPN
easat-4034	147	15	is	be	AUX
easat-4034	147	16	the	the	DET
easat-4034	147	17	minimal	minimal	ADJ
easat-4034	147	18	element	element	NOUN
easat-4034	147	19	in	in	ADP
easat-4034	147	20	{	{	PUNCT
easat-4034	147	21	𝑥|𝐴	𝑥|𝐴	NOUN
easat-4034	147	22	⊗	⊗	PROPN
easat-4034	147	23	𝑥	𝑥	PROPN
easat-4034	147	24	≥	≥	NUM
easat-4034	147	25	𝑏	𝑏	NOUN
easat-4034	147	26	}	}	PUNCT
easat-4034	147	27	,	,	PUNCT
easat-4034	147	28	then	then	ADV
easat-4034	147	29	𝑥∗	𝑥∗	PROPN
easat-4034	147	30	≥	≥	PRON
easat-4034	148	1	𝐴#𝑏.	𝐴#𝑏.	VERB
easat-4034	148	2	it	it	PRON
easat-4034	148	3	is	be	AUX
easat-4034	148	4	obtained	obtain	VERB
easat-4034	148	5	𝐴	𝐴	PROPN
easat-4034	148	6	⊗	⊗	PROPN
easat-4034	148	7	𝑥∗	𝑥∗	PROPN
easat-4034	148	8	≤	≤	PROPN
easat-4034	148	9	𝐴(𝐴#𝑏	𝐴(𝐴#𝑏	PROPN
easat-4034	148	10	)	)	PUNCT
easat-4034	149	1	⟺	⟺	NOUN
easat-4034	150	1	𝑏	𝑏	NOUN
easat-4034	150	2	=	=	SYM
easat-4034	150	3	𝐴	𝐴	PROPN
easat-4034	150	4	⊗	⊗	PROPN
easat-4034	150	5	𝑥∗	𝑥∗	PROPN
easat-4034	150	6	≥	≥	PROPN
easat-4034	150	7	𝐴(𝐴#𝑏	𝐴(𝐴#𝑏	PROPN
easat-4034	150	8	)	)	PUNCT
easat-4034	150	9	…	…	PUNCT
easat-4034	150	10	……	……	X
easat-4034	150	11	.	.	PUNCT
easat-4034	150	12	.	.	PUNCT
easat-4034	151	1	(	(	PUNCT
easat-4034	151	2	∗	∗	NOUN
easat-4034	151	3	)	)	PUNCT
easat-4034	151	4	furthermore	furthermore	ADV
easat-4034	151	5	,	,	PUNCT
easat-4034	151	6	according	accord	VERB
easat-4034	151	7	to	to	ADP
easat-4034	151	8	theorem	theorem	ADJ
easat-4034	151	9	5	5	NUM
easat-4034	151	10	𝐴(𝐴#𝑏	𝐴(𝐴#𝑏	NOUN
easat-4034	151	11	)	)	PUNCT
easat-4034	151	12	≥	≥	NOUN
easat-4034	151	13	𝑏	𝑏	NOUN
easat-4034	151	14	…	…	PUNCT
easat-4034	151	15	…	…	SYM
easat-4034	151	16	……	……	NOUN
easat-4034	151	17	(	(	PUNCT
easat-4034	151	18	∗∗	∗∗	NOUN
easat-4034	151	19	)	)	PUNCT
easat-4034	151	20	from	from	ADP
easat-4034	151	21	(	(	PUNCT
easat-4034	151	22	∗	∗	NOUN
easat-4034	151	23	)	)	PUNCT
easat-4034	151	24	and	and	CCONJ
easat-4034	151	25	(	(	PUNCT
easat-4034	151	26	∗∗	∗∗	X
easat-4034	151	27	)	)	PUNCT
easat-4034	152	1	it	it	PRON
easat-4034	152	2	is	be	AUX
easat-4034	152	3	obtained	obtain	VERB
easat-4034	152	4	𝐴	𝐴	PROPN
easat-4034	152	5	(	(	PUNCT
easat-4034	152	6	𝐴#(𝑏	𝐴#(𝑏	PROPN
easat-4034	152	7	)	)	PUNCT
easat-4034	152	8	)	)	PUNCT
easat-4034	153	1	=	=	PUNCT
easat-4034	153	2	𝑏.	𝑏.	NOUN
easat-4034	153	3	(	(	PUNCT
easat-4034	153	4	⟸	⟸	X
easat-4034	153	5	)	)	PUNCT
easat-4034	153	6	it	it	PRON
easat-4034	153	7	is	be	AUX
easat-4034	153	8	known	know	VERB
easat-4034	153	9	that	that	DET
easat-4034	153	10	𝐴(𝐴#𝑏	𝐴(𝐴#𝑏	NOUN
easat-4034	153	11	)	)	PUNCT
easat-4034	153	12	=	=	PUNCT
easat-4034	154	1	𝑏.	𝑏.	VERB
easat-4034	155	1	so	so	SCONJ
easat-4034	155	2	the	the	DET
easat-4034	155	3	equation	equation	NOUN
easat-4034	155	4	𝐴	𝐴	PROPN
easat-4034	155	5	⊗	⊗	NOUN
easat-4034	156	1	𝑥	𝑥	PROPN
easat-4034	156	2	=	=	PUNCT
easat-4034	156	3	𝑏	𝑏	PROPN
easat-4034	156	4	has	have	VERB
easat-4034	156	5	a	a	DET
easat-4034	156	6	solution	solution	NOUN
easat-4034	156	7	,	,	PUNCT
easat-4034	156	8	namely	namely	ADV
easat-4034	156	9	𝑥	𝑥	PRON
easat-4034	156	10	=	=	PUNCT
easat-4034	157	1	𝐴#𝑏.	𝐴#𝑏.	PROPN
easat-4034	157	2	therefore	therefore	ADV
easat-4034	157	3	,	,	PUNCT
easat-4034	157	4	𝐴	𝐴	PROPN
easat-4034	157	5	⊗	⊗	NOUN
easat-4034	158	1	𝑥	𝑥	PROPN
easat-4034	158	2	=	=	PUNCT
easat-4034	158	3	𝑏	𝑏	PROPN
easat-4034	158	4	has	have	VERB
easat-4034	158	5	a	a	DET
easat-4034	158	6	solution	solution	NOUN
easat-4034	158	7	𝑥	𝑥	X
easat-4034	158	8	=	=	SYM
easat-4034	158	9	𝐴#(𝑏	𝐴#(𝑏	PROPN
easat-4034	158	10	)	)	PUNCT
easat-4034	158	11	.	.	PUNCT
easat-4034	159	1	it	it	PRON
easat-4034	159	2	means	mean	VERB
easat-4034	159	3	that	that	SCONJ
easat-4034	159	4	𝐴	𝐴	PROPN
easat-4034	159	5	(	(	PUNCT
easat-4034	159	6	𝐴#(𝑏	𝐴#(𝑏	PROPN
easat-4034	159	7	)	)	PUNCT
easat-4034	159	8	)	)	PUNCT
easat-4034	160	1	=	=	PUNCT
easat-4034	160	2	𝑏.	𝑏.	VERB
easat-4034	160	3	for	for	ADP
easat-4034	160	4	example	example	NOUN
easat-4034	160	5	,	,	PUNCT
easat-4034	160	6	if	if	SCONJ
easat-4034	160	7	𝐴	𝐴	PROPN
easat-4034	160	8	⊗	⊗	VERB
easat-4034	160	9	𝑥	𝑥	PROPN
easat-4034	161	1	=	=	PUNCT
easat-4034	161	2	𝑏	𝑏	PROPN
easat-4034	161	3	has	have	VERB
easat-4034	161	4	solution	solution	NOUN
easat-4034	161	5	𝑥𝑗	𝑥𝑗	PROPN
easat-4034	161	6	,	,	PUNCT
easat-4034	161	7	then	then	ADV
easat-4034	161	8	there	there	PRON
easat-4034	161	9	is	be	VERB
easat-4034	161	10	a	a	DET
easat-4034	161	11	smallest	small	ADJ
easat-4034	161	12	subsolution	subsolution	NOUN
easat-4034	161	13	𝐴	𝐴	NOUN
easat-4034	161	14	⊗	⊗	NOUN
easat-4034	162	1	𝑥	𝑥	PROPN
easat-4034	162	2	=	=	PUNCT
easat-4034	162	3	𝑏.	𝑏.	NOUN
easat-4034	162	4	𝐴	𝐴	PROPN
easat-4034	162	5	⊗	⊗	PROPN
easat-4034	163	1	𝑥	𝑥	PROPN
easat-4034	163	2	≥	≥	NOUN
easat-4034	163	3	𝑏	𝑏	NOUN
easat-4034	163	4	⟺	⟺	NOUN
easat-4034	163	5	⊕′	⊕′	PROPN
easat-4034	163	6	𝑗	𝑗	NOUN
easat-4034	164	1	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	164	2	⊗	⊗	PROPN
easat-4034	164	3	𝑥𝑗	𝑥𝑗	PROPN
easat-4034	164	4	≥	≥	NOUN
easat-4034	164	5	𝑏𝑖	𝑏𝑖	ADP
easat-4034	164	6	,	,	PUNCT
easat-4034	164	7	∀𝑖	∀𝑖	PROPN
easat-4034	164	8	then	then	ADV
easat-4034	164	9	𝐴𝑖1	𝐴𝑖1	VERB
easat-4034	164	10	⊗	⊗	PROPN
easat-4034	164	11	𝑥1	𝑥1	PROPN
easat-4034	164	12	⊕′	⊕′	PROPN
easat-4034	164	13	𝐴𝑖2	𝐴𝑖2	NOUN
easat-4034	164	14	⊗	⊗	PROPN
easat-4034	164	15	𝑥2	𝑥2	PROPN
easat-4034	164	16	⊕′	⊕′	PROPN
easat-4034	164	17	…	…	PUNCT
easat-4034	164	18	⊕′	⊕′	PROPN
easat-4034	164	19	𝐴𝑖𝑛	𝐴𝑖𝑛	PROPN
easat-4034	164	20	⊗	⊗	PROPN
easat-4034	164	21	𝑥𝑛	𝑥𝑛	PROPN
easat-4034	164	22	≥	≥	NOUN
easat-4034	164	23	𝑏𝑖	𝑏𝑖	ADP
easat-4034	164	24	𝑖	𝑖	NOUN
easat-4034	164	25	=	=	NOUN
easat-4034	164	26	1	1	NUM
easat-4034	164	27	⟹	⟹	NUM
easat-4034	164	28	𝐴11	𝐴11	PROPN
easat-4034	164	29	⊗	⊗	PROPN
easat-4034	164	30	𝑥1	𝑥1	PROPN
easat-4034	164	31	⊕′	⊕′	PROPN
easat-4034	164	32	𝐴12	𝐴12	PROPN
easat-4034	164	33	⊗	⊗	ADJ
easat-4034	164	34	𝑥2	𝑥2	PROPN
easat-4034	164	35	⊕′	⊕′	PROPN
easat-4034	164	36	…	…	PUNCT
easat-4034	164	37	⊕′	⊕′	PROPN
easat-4034	164	38	𝐴1𝑛	𝐴1𝑛	PROPN
easat-4034	164	39	⊗	⊗	PROPN
easat-4034	164	40	𝑥𝑛	𝑥𝑛	PROPN
easat-4034	164	41	≥	≥	NOUN
easat-4034	164	42	𝑏1	𝑏1	VERB
easat-4034	164	43	𝑖	𝑖	SYM
easat-4034	164	44	=	=	SYM
easat-4034	164	45	2	2	NUM
easat-4034	164	46	⟹	⟹	NUM
easat-4034	164	47	𝐴21	𝐴21	PROPN
easat-4034	164	48	⊗	⊗	PROPN
easat-4034	164	49	𝑥1	𝑥1	PROPN
easat-4034	164	50	⊕′	⊕′	PROPN
easat-4034	164	51	𝐴22	𝐴22	PROPN
easat-4034	164	52	⊗	⊗	PROPN
easat-4034	164	53	𝑥2	𝑥2	PROPN
easat-4034	164	54	⊕′	⊕′	PROPN
easat-4034	164	55	…	…	PUNCT
easat-4034	164	56	⊕′	⊕′	PROPN
easat-4034	164	57	𝐴2𝑛	𝐴2𝑛	PROPN
easat-4034	164	58	⊗	⊗	PROPN
easat-4034	164	59	𝑥𝑛	𝑥𝑛	PROPN
easat-4034	164	60	≥	≥	NOUN
easat-4034	164	61	𝑏2	𝑏2	PROPN
easat-4034	164	62	⋮	⋮	NOUN
easat-4034	164	63	𝑖	𝑖	PROPN
easat-4034	164	64	=	=	SYM
easat-4034	164	65	𝑛	𝑛	PRON
easat-4034	164	66	⟹	⟹	NUM
easat-4034	164	67	𝐴𝑛1	𝐴𝑛1	NOUN
easat-4034	164	68	⊗	⊗	PROPN
easat-4034	164	69	𝑥1	𝑥1	PROPN
easat-4034	164	70	⊕′	⊕′	PROPN
easat-4034	164	71	𝐴𝑛2	𝐴𝑛2	NOUN
easat-4034	164	72	⊗	⊗	PROPN
easat-4034	164	73	𝑥2	𝑥2	PROPN
easat-4034	165	1	⊕′	⊕′	PROPN
easat-4034	165	2	…	…	PUNCT
easat-4034	165	3	⊕′	⊕′	PROPN
easat-4034	165	4	𝐴𝑛𝑛	𝐴𝑛𝑛	PROPN
easat-4034	165	5	⊗	⊗	PROPN
easat-4034	165	6	𝑥𝑛	𝑥𝑛	PROPN
easat-4034	165	7	≥	≥	NOUN
easat-4034	165	8	𝑏𝑛	𝑏𝑛	ADP
easat-4034	165	9	for	for	ADP
easat-4034	165	10	𝑖	𝑖	SYM
easat-4034	165	11	,	,	PUNCT
easat-4034	165	12	min{𝐴𝑖1	min{𝐴𝑖1	PROPN
easat-4034	165	13	+	+	PROPN
easat-4034	165	14	𝑥1	𝑥1	NOUN
easat-4034	165	15	,	,	PUNCT
easat-4034	165	16	𝐴𝑖2	𝐴𝑖2	NOUN
easat-4034	165	17	+	+	CCONJ
easat-4034	165	18	𝑥2	𝑥2	NOUN
easat-4034	165	19	,	,	PUNCT
easat-4034	165	20	…	…	PUNCT
easat-4034	165	21	,	,	PUNCT
easat-4034	165	22	𝐴𝑖𝑛	𝐴𝑖𝑛	PROPN
easat-4034	165	23	+	+	CCONJ
easat-4034	165	24	𝑥𝑛	𝑥𝑛	PROPN
easat-4034	165	25	}	}	PUNCT
easat-4034	165	26	≥	≥	PRON
easat-4034	165	27	𝑏𝑖.	𝑏𝑖.	VERB
easat-4034	165	28	for	for	ADP
easat-4034	165	29	𝑖	𝑖	SYM
easat-4034	165	30	,	,	PUNCT
easat-4034	165	31	𝑗	𝑗	INTJ
easat-4034	165	32	,	,	PUNCT
easat-4034	165	33	we	we	PRON
easat-4034	165	34	have	have	VERB
easat-4034	165	35	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	165	36	+	+	CCONJ
easat-4034	165	37	𝑥𝑗	𝑥𝑗	PROPN
easat-4034	165	38	≥	≥	PRON
easat-4034	165	39	𝑏𝑖	𝑏𝑖	ADP
easat-4034	165	40	,	,	PUNCT
easat-4034	165	41	which	which	PRON
easat-4034	165	42	results	result	VERB
easat-4034	165	43	in	in	ADP
easat-4034	165	44	𝑥𝑗	𝑥𝑗	PROPN
easat-4034	165	45	≥	≥	NOUN
easat-4034	165	46	𝑏𝑖	𝑏𝑖	ADP
easat-4034	165	47	−	−	PROPN
easat-4034	165	48	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	165	49	.	.	PUNCT
easat-4034	166	1	obtain	obtain	VERB
easat-4034	166	2	𝑥𝑗	𝑥𝑗	PRON
easat-4034	166	3	≥	≥	NUM
easat-4034	166	4	𝑚𝑎𝑥{𝑏𝑖	𝑚𝑎𝑥{𝑏𝑖	PUNCT
easat-4034	166	5	−	−	PROPN
easat-4034	167	1	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	167	2	}	}	PUNCT
easat-4034	167	3	for	for	ADP
easat-4034	167	4	each	each	DET
easat-4034	167	5	𝑖.	𝑖.	ADJ
easat-4034	167	6	next	next	ADV
easat-4034	167	7	,	,	PUNCT
easat-4034	167	8	−𝑥𝑗	−𝑥𝑗	PROPN
easat-4034	167	9	≤	≤	NUM
easat-4034	167	10	𝑚𝑖𝑛{−𝑏𝑖	𝑚𝑖𝑛{−𝑏𝑖	NOUN
easat-4034	167	11	+	+	X
easat-4034	167	12	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	167	13	}	}	PUNCT
easat-4034	167	14	for	for	ADP
easat-4034	167	15	each	each	DET
easat-4034	167	16	𝑖.	𝑖.	ADJ
easat-4034	167	17	in	in	ADP
easat-4034	167	18	other	other	ADJ
easat-4034	167	19	words	word	NOUN
easat-4034	167	20	−𝑥𝑗	−𝑥𝑗	PROPN
easat-4034	167	21	≤	≤	NUM
easat-4034	167	22	𝑚𝑖𝑛{−𝑏𝑖	𝑚𝑖𝑛{−𝑏𝑖	NOUN
easat-4034	167	23	⊗	⊗	PROPN
easat-4034	167	24	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	167	25	}	}	PUNCT
easat-4034	167	26	for	for	ADP
easat-4034	167	27	each	each	DET
easat-4034	167	28	𝑖.	𝑖.	NOUN
easat-4034	167	29	so	so	SCONJ
easat-4034	167	30	that	that	DET
easat-4034	167	31	−𝑥𝑗	−𝑥𝑗	NOUN
easat-4034	167	32	=	=	SYM
easat-4034	167	33	𝑚𝑖𝑛{−𝑏𝑖	𝑚𝑖𝑛{−𝑏𝑖	PROPN
easat-4034	167	34	⊗	⊗	PROPN
easat-4034	167	35	𝐴𝑖𝑗}subsolution	𝐴𝑖𝑗}subsolution	PROPN
easat-4034	167	36	of	of	ADP
easat-4034	167	37	𝐴	𝐴	PROPN
easat-4034	167	38	⊗	⊗	NOUN
easat-4034	168	1	𝑥	𝑥	PROPN
easat-4034	168	2	=	=	PUNCT
easat-4034	168	3	𝑏	𝑏	PROPN
easat-4034	168	4	or	or	CCONJ
easat-4034	168	5	expressed	express	VERB
easat-4034	168	6	−𝑥𝑗	−𝑥𝑗	PROPN
easat-4034	168	7	=	=	SYM
easat-4034	168	8	𝑚𝑖𝑛	𝑚𝑖𝑛	PROPN
easat-4034	168	9	{	{	PUNCT
easat-4034	168	10	(	(	PUNCT
easat-4034	168	11	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	168	12	)	)	PUNCT
easat-4034	168	13	𝑡	𝑡	PROPN
easat-4034	168	14	⊗	⊗	PROPN
easat-4034	168	15	(	(	PUNCT
easat-4034	168	16	−𝑏𝑗	−𝑏𝑗	NOUN
easat-4034	168	17	)	)	PUNCT
easat-4034	168	18	}	}	PUNCT
easat-4034	168	19	.	.	PUNCT
easat-4034	169	1	example	example	NOUN
easat-4034	169	2	4	4	NUM
easat-4034	169	3	given	give	VERB
easat-4034	169	4	a	a	DET
easat-4034	169	5	system	system	NOUN
easat-4034	169	6	of	of	ADP
easat-4034	169	7	linear	linear	PROPN
easat-4034	169	8	equations	equation	NOUN
easat-4034	169	9	over	over	ADP
easat-4034	169	10	min	min	NOUN
easat-4034	169	11	-	-	PUNCT
easat-4034	169	12	plus	plus	ADJ
easat-4034	169	13	algebra	algebra	NOUN
easat-4034	169	14	9550	9550	NUM
easat-4034	169	15	edelweiss	edelweiss	PROPN
easat-4034	169	16	applied	apply	VERB
easat-4034	169	17	science	science	NOUN
easat-4034	169	18	and	and	CCONJ
easat-4034	169	19	technology	technology	NOUN
easat-4034	169	20	issn	issn	PROPN
easat-4034	169	21	:	:	PUNCT
easat-4034	169	22	2576	2576	NUM
easat-4034	169	23	-	-	SYM
easat-4034	169	24	8484	8484	NUM
easat-4034	169	25	vol	vol	NOUN
easat-4034	169	26	.	.	PROPN
easat-4034	169	27	8	8	NUM
easat-4034	169	28	,	,	PUNCT
easat-4034	169	29	no	no	INTJ
easat-4034	169	30	.	.	NOUN
easat-4034	170	1	6	6	NUM
easat-4034	170	2	:	:	PUNCT
easat-4034	170	3	9544	9544	NUM
easat-4034	170	4	-	-	SYM
easat-4034	170	5	9554	9554	NUM
easat-4034	170	6	,	,	PUNCT
easat-4034	170	7	2024	2024	NUM
easat-4034	170	8	doi	doi	NOUN
easat-4034	170	9	:	:	PUNCT
easat-4034	170	10	10.55214/25768484.v8i6.4034	10.55214/25768484.v8i6.4034	NUM
easat-4034	170	11	©	©	ADP
easat-4034	170	12	2024	2024	NUM
easat-4034	170	13	by	by	ADP
easat-4034	170	14	the	the	DET
easat-4034	170	15	authors	author	NOUN
easat-4034	170	16	;	;	PUNCT
easat-4034	170	17	licensee	licensee	PROPN
easat-4034	170	18	learning	learning	NOUN
easat-4034	170	19	gate	gate	NOUN
easat-4034	170	20	[	[	PUNCT
easat-4034	170	21	2	2	NUM
easat-4034	170	22	3	3	NUM
easat-4034	170	23	4	4	NUM
easat-4034	170	24	5	5	NUM
easat-4034	170	25	]	]	PUNCT
easat-4034	171	1	⊗	⊗	PROPN
easat-4034	171	2	[	[	PUNCT
easat-4034	171	3	𝑥1	𝑥1	NOUN
easat-4034	171	4	𝑥2	𝑥2	NOUN
easat-4034	171	5	]	]	PUNCT
easat-4034	172	1	=	=	PUNCT
easat-4034	172	2	[	[	PUNCT
easat-4034	172	3	6	6	NUM
easat-4034	172	4	7	7	NUM
easat-4034	172	5	]	]	PUNCT
easat-4034	172	6	we	we	PRON
easat-4034	172	7	get	get	VERB
easat-4034	172	8	−𝑥𝑗	−𝑥𝑗	NOUN
easat-4034	172	9	=	=	SYM
easat-4034	172	10	𝑚𝑖𝑛{−𝑏𝑖	𝑚𝑖𝑛{−𝑏𝑖	PROPN
easat-4034	172	11	⊗	⊗	PROPN
easat-4034	172	12	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	172	13	}	}	PUNCT
easat-4034	172	14	=	=	PUNCT
easat-4034	173	1	[	[	X
easat-4034	173	2	−6	−6	X
easat-4034	173	3	−7	−7	X
easat-4034	173	4	]	]	X
easat-4034	173	5	⊗	⊗	PROPN
easat-4034	173	6	[	[	PUNCT
easat-4034	173	7	2	2	NUM
easat-4034	173	8	3	3	NUM
easat-4034	173	9	4	4	NUM
easat-4034	173	10	5	5	NUM
easat-4034	173	11	]	]	PUNCT
easat-4034	173	12	=	=	PUNCT
easat-4034	174	1	[	[	X
easat-4034	174	2	−4	−4	X
easat-4034	174	3	⊕′−	⊕′−	NUM
easat-4034	174	4	3	3	NUM
easat-4034	174	5	−3	−3	NOUN
easat-4034	174	6	⊕′−	⊕′−	NUM
easat-4034	174	7	2	2	NUM
easat-4034	174	8	]	]	PUNCT
easat-4034	174	9	=	=	PUNCT
easat-4034	175	1	[	[	X
easat-4034	175	2	−4	−4	X
easat-4034	175	3	−3	−3	X
easat-4034	175	4	]	]	X
easat-4034	176	1	so	so	ADV
easat-4034	176	2	,	,	PUNCT
easat-4034	176	3	𝑥𝑗	𝑥𝑗	PROPN
easat-4034	176	4	=	=	PUNCT
easat-4034	177	1	[	[	X
easat-4034	177	2	4	4	NUM
easat-4034	177	3	3	3	NUM
easat-4034	177	4	]	]	PUNCT
easat-4034	177	5	.	.	PUNCT
easat-4034	178	1	or	or	CCONJ
easat-4034	178	2	,	,	PUNCT
easat-4034	178	3	−𝑥𝑗	−𝑥𝑗	PROPN
easat-4034	178	4	=	=	PUNCT
easat-4034	178	5	𝑚𝑖𝑛	𝑚𝑖𝑛	PROPN
easat-4034	178	6	{	{	PUNCT
easat-4034	178	7	(	(	PUNCT
easat-4034	178	8	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	178	9	)	)	PUNCT
easat-4034	178	10	𝑡	𝑡	PROPN
easat-4034	178	11	⊗	⊗	PROPN
easat-4034	178	12	(	(	PUNCT
easat-4034	178	13	−𝑏𝑗	−𝑏𝑗	NOUN
easat-4034	178	14	)	)	PUNCT
easat-4034	178	15	}	}	PUNCT
easat-4034	179	1	=	=	PUNCT
easat-4034	179	2	[	[	PUNCT
easat-4034	179	3	2	2	NUM
easat-4034	179	4	4	4	NUM
easat-4034	179	5	3	3	NUM
easat-4034	179	6	5	5	NUM
easat-4034	179	7	]	]	PUNCT
easat-4034	180	1	⊗	⊗	PROPN
easat-4034	180	2	[	[	PUNCT
easat-4034	180	3	−6	−6	INTJ
easat-4034	180	4	−7	−7	NOUN
easat-4034	180	5	]	]	PUNCT
easat-4034	180	6	=	=	PUNCT
easat-4034	180	7	[	[	PUNCT
easat-4034	180	8	−4	−4	X
easat-4034	180	9	⊕′−	⊕′−	NUM
easat-4034	180	10	3	3	NUM
easat-4034	181	1	−3	−3	NOUN
easat-4034	181	2	⊕′−	⊕′−	NUM
easat-4034	181	3	2	2	NUM
easat-4034	181	4	]	]	PUNCT
easat-4034	181	5	=	=	PUNCT
easat-4034	182	1	[	[	PUNCT
easat-4034	182	2	−4	−4	X
easat-4034	182	3	3	3	NUM
easat-4034	182	4	]	]	PUNCT
easat-4034	182	5	it	it	PRON
easat-4034	182	6	can	can	AUX
easat-4034	182	7	be	be	AUX
easat-4034	182	8	proven	prove	VERB
easat-4034	182	9	that	that	SCONJ
easat-4034	182	10	[	[	PUNCT
easat-4034	182	11	2	2	NUM
easat-4034	182	12	3	3	NUM
easat-4034	182	13	4	4	NUM
easat-4034	182	14	5	5	NUM
easat-4034	182	15	]	]	PUNCT
easat-4034	183	1	⊗	⊗	PROPN
easat-4034	183	2	[	[	PUNCT
easat-4034	183	3	𝑥1	𝑥1	NOUN
easat-4034	183	4	𝑥2	𝑥2	NOUN
easat-4034	183	5	]	]	PUNCT
easat-4034	184	1	=	=	PUNCT
easat-4034	184	2	[	[	PUNCT
easat-4034	184	3	2	2	NUM
easat-4034	184	4	3	3	NUM
easat-4034	184	5	4	4	NUM
easat-4034	184	6	5	5	NUM
easat-4034	184	7	]	]	PUNCT
easat-4034	184	8	⊗	⊗	PROPN
easat-4034	184	9	[	[	PUNCT
easat-4034	184	10	4	4	NUM
easat-4034	184	11	3	3	NUM
easat-4034	184	12	]	]	PUNCT
easat-4034	184	13	=	=	PUNCT
easat-4034	185	1	[	[	PUNCT
easat-4034	185	2	6	6	NUM
easat-4034	185	3	⊕′	⊕′	NOUN
easat-4034	185	4	6	6	NUM
easat-4034	185	5	8	8	NUM
easat-4034	185	6	⊕′	⊕′	PROPN
easat-4034	185	7	8	8	NUM
easat-4034	185	8	]	]	PUNCT
easat-4034	185	9	=	=	PUNCT
easat-4034	185	10	[	[	PUNCT
easat-4034	185	11	6	6	NUM
easat-4034	185	12	8	8	NUM
easat-4034	185	13	]	]	PUNCT
easat-4034	185	14	≥	≥	NOUN
easat-4034	185	15	[	[	PUNCT
easat-4034	185	16	6	6	NUM
easat-4034	185	17	7	7	NUM
easat-4034	185	18	]	]	SYM
easat-4034	185	19	3	3	X
easat-4034	185	20	.	.	NOUN
easat-4034	185	21	results	result	NOUN
easat-4034	185	22	and	and	CCONJ
easat-4034	185	23	discussion	discussion	NOUN
easat-4034	185	24	based	base	VERB
easat-4034	185	25	on	on	ADP
easat-4034	185	26	the	the	DET
easat-4034	185	27	explanation	explanation	NOUN
easat-4034	185	28	above	above	ADV
easat-4034	185	29	,	,	PUNCT
easat-4034	185	30	the	the	DET
easat-4034	185	31	following	follow	VERB
easat-4034	185	32	properties	property	NOUN
easat-4034	185	33	are	be	AUX
easat-4034	185	34	obtained	obtain	VERB
easat-4034	185	35	.	.	PUNCT
easat-4034	186	1	definition	definition	NOUN
easat-4034	186	2	7	7	NUM
easat-4034	186	3	a	a	DET
easat-4034	186	4	matrix	matrix	NOUN
easat-4034	186	5	𝐴	𝐴	NOUN
easat-4034	186	6	∈	∈	PROPN
easat-4034	186	7	𝑀𝑛(𝑅	𝑀𝑛(𝑅	X
easat-4034	186	8	)	)	PUNCT
easat-4034	186	9	is	be	AUX
easat-4034	186	10	said	say	VERB
easat-4034	186	11	to	to	PART
easat-4034	186	12	be	be	AUX
easat-4034	186	13	invertible	invertible	ADJ
easat-4034	186	14	over	over	ADP
easat-4034	186	15	the	the	DET
easat-4034	186	16	min	min	NOUN
easat-4034	186	17	-	-	PUNCT
easat-4034	186	18	plus	plus	ADJ
easat-4034	186	19	algebra	algebra	NOUN
easat-4034	186	20	if	if	SCONJ
easat-4034	186	21	there	there	PRON
easat-4034	186	22	is	be	VERB
easat-4034	186	23	a	a	DET
easat-4034	186	24	matrix	matrix	NOUN
easat-4034	186	25	b	b	NOUN
easat-4034	186	26	such	such	ADJ
easat-4034	186	27	that	that	SCONJ
easat-4034	186	28	𝐴	𝐴	PROPN
easat-4034	186	29	⊗	⊗	PROPN
easat-4034	186	30	𝐵	𝐵	PROPN
easat-4034	186	31	=	=	PUNCT
easat-4034	186	32	𝐸	𝐸	PROPN
easat-4034	186	33	with	with	ADP
easat-4034	186	34	e	e	NOUN
easat-4034	186	35	the	the	DET
easat-4034	186	36	identity	identity	NOUN
easat-4034	186	37	matrix	matrix	NOUN
easat-4034	186	38	over	over	ADP
easat-4034	186	39	the	the	DET
easat-4034	186	40	min	min	NOUN
easat-4034	186	41	-	-	PUNCT
easat-4034	186	42	plus	plus	ADJ
easat-4034	186	43	algebra	algebra	NOUN
easat-4034	186	44	and	and	CCONJ
easat-4034	186	45	is	be	AUX
easat-4034	186	46	denoted	denote	VERB
easat-4034	186	47	𝐵	𝐵	NOUN
easat-4034	186	48	=	=	SYM
easat-4034	186	49	𝐴⊗−1	𝐴⊗−1	PROPN
easat-4034	186	50	.	.	PUNCT
easat-4034	187	1	to	to	PART
easat-4034	187	2	determine	determine	VERB
easat-4034	187	3	the	the	DET
easat-4034	187	4	inverse	inverse	NOUN
easat-4034	187	5	matrix	matrix	NOUN
easat-4034	187	6	over	over	ADP
easat-4034	187	7	min	min	ADJ
easat-4034	187	8	-	-	PUNCT
easat-4034	187	9	plus	plus	ADJ
easat-4034	187	10	algebra	algebra	NOUN
easat-4034	187	11	,	,	PUNCT
easat-4034	187	12	permutation	permutation	NOUN
easat-4034	187	13	is	be	AUX
easat-4034	187	14	required	require	VERB
easat-4034	187	15	.	.	PUNCT
easat-4034	188	1	definition	definition	NOUN
easat-4034	188	2	8	8	NUM
easat-4034	188	3	if	if	SCONJ
easat-4034	188	4	𝜆1	𝜆1	VERB
easat-4034	188	5	,	,	PUNCT
easat-4034	188	6	𝜆2	𝜆2	PROPN
easat-4034	188	7	,	,	PUNCT
easat-4034	188	8	…	…	PUNCT
easat-4034	188	9	,	,	PUNCT
easat-4034	188	10	𝜆𝑛	𝜆𝑛	ADP
easat-4034	188	11	∈	∈	PROPN
easat-4034	188	12	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	188	13	,	,	PUNCT
easat-4034	188	14	𝜆𝑖	𝜆𝑖	PROPN
easat-4034	188	15	≠	≠	PROPN
easat-4034	188	16	𝜀	𝜀	PROPN
easat-4034	188	17	then	then	ADV
easat-4034	188	18	the	the	DET
easat-4034	188	19	diagonal	diagonal	ADJ
easat-4034	188	20	matrix	matrix	NOUN
easat-4034	188	21	is	be	AUX
easat-4034	188	22	defined	define	VERB
easat-4034	188	23	as	as	SCONJ
easat-4034	188	24	follows	follow	VERB
easat-4034	188	25	.	.	PUNCT
easat-4034	189	1	𝐷(𝜆𝑖	𝐷(𝜆𝑖	X
easat-4034	189	2	)	)	PUNCT
easat-4034	190	1	=	=	PUNCT
easat-4034	190	2	[	[	PUNCT
easat-4034	190	3	𝜆1	𝜆1	NOUN
easat-4034	190	4	𝜀	𝜀	PROPN
easat-4034	190	5	𝜀	𝜀	X
easat-4034	190	6	𝜆2	𝜆2	PROPN
easat-4034	190	7	⋯	⋯	PROPN
easat-4034	190	8	𝜀	𝜀	PROPN
easat-4034	190	9	⋮	⋮	PROPN
easat-4034	190	10	𝜀	𝜀	PROPN
easat-4034	190	11	⋯	⋯	PROPN
easat-4034	190	12	⋯	⋯	PROPN
easat-4034	190	13	𝜀	𝜀	PROPN
easat-4034	190	14	𝜀	𝜀	PROPN
easat-4034	190	15	⋱	⋱	PUNCT
easat-4034	190	16	⋯	⋯	NOUN
easat-4034	190	17	⋯	⋯	NOUN
easat-4034	190	18	𝜆𝑛	𝜆𝑛	NOUN
easat-4034	190	19	]	]	PUNCT
easat-4034	190	20	theorem	theorem	VERB
easat-4034	190	21	7	7	NUM
easat-4034	190	22	given	give	VERB
easat-4034	190	23	𝐴	𝐴	PROPN
easat-4034	190	24	∈	∈	PROPN
easat-4034	190	25	𝑀𝑛(𝑅min	𝑀𝑛(𝑅min	NOUN
easat-4034	190	26	)	)	PUNCT
easat-4034	190	27	.	.	PUNCT
easat-4034	191	1	if	if	SCONJ
easat-4034	191	2	and	and	CCONJ
easat-4034	191	3	only	only	ADV
easat-4034	191	4	if	if	SCONJ
easat-4034	191	5	there	there	PRON
easat-4034	191	6	is	be	VERB
easat-4034	191	7	a	a	DET
easat-4034	191	8	permutation	permutation	NOUN
easat-4034	191	9	𝜎	𝜎	NOUN
easat-4034	191	10	and	and	CCONJ
easat-4034	191	11	values	value	NOUN
easat-4034	191	12	𝜆𝑖	𝜆𝑖	VERB
easat-4034	191	13	>	>	X
easat-4034	191	14	𝜀	𝜀	PROPN
easat-4034	191	15	,	,	PUNCT
easat-4034	191	16	𝑖	𝑖	SYM
easat-4034	191	17	∈	∈	PROPN
easat-4034	191	18	{	{	PUNCT
easat-4034	191	19	1,2	1,2	NUM
easat-4034	191	20	,	,	PUNCT
easat-4034	191	21	…	…	PUNCT
easat-4034	191	22	,	,	PUNCT
easat-4034	191	23	𝑛	𝑛	X
easat-4034	191	24	}	}	PUNCT
easat-4034	191	25	such	such	ADJ
easat-4034	191	26	that	that	SCONJ
easat-4034	191	27	𝐴	𝐴	PROPN
easat-4034	191	28	=	=	PUNCT
easat-4034	191	29	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	191	30	⊗	⊗	PROPN
easat-4034	191	31	𝐷(𝜆𝑖	𝐷(𝜆𝑖	PROPN
easat-4034	191	32	)	)	PUNCT
easat-4034	191	33	,	,	PUNCT
easat-4034	191	34	then	then	ADV
easat-4034	191	35	𝐴	𝐴	PROPN
easat-4034	191	36	∈	∈	PROPN
easat-4034	191	37	𝑀𝑛(𝑅min	𝑀𝑛(𝑅min	PROPN
easat-4034	191	38	)	)	PUNCT
easat-4034	191	39	has	have	VERB
easat-4034	191	40	a	a	DET
easat-4034	191	41	left	left	ADJ
easat-4034	191	42	inverse	inverse	NOUN
easat-4034	191	43	.	.	PUNCT
easat-4034	192	1	proof	proof	NOUN
easat-4034	192	2	:	:	PUNCT
easat-4034	192	3	(	(	PUNCT
easat-4034	192	4	⟹	⟹	X
easat-4034	192	5	)	)	PUNCT
easat-4034	192	6	given	give	VERB
easat-4034	192	7	𝐴	𝐴	PROPN
easat-4034	192	8	∈	∈	PROPN
easat-4034	193	1	𝑀𝑛	𝑀𝑛	PROPN
easat-4034	193	2	(	(	PUNCT
easat-4034	193	3	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	193	4	)	)	PUNCT
easat-4034	193	5	,	,	PUNCT
easat-4034	193	6	there	there	PRON
easat-4034	193	7	exist	exist	VERB
easat-4034	193	8	𝐵	𝐵	NOUN
easat-4034	193	9	so	so	SCONJ
easat-4034	193	10	that	that	SCONJ
easat-4034	193	11	it	it	PRON
easat-4034	193	12	satisfies	satisfy	VERB
easat-4034	193	13	the	the	DET
easat-4034	193	14	equation	equation	NOUN
easat-4034	193	15	𝐴	𝐴	PROPN
easat-4034	193	16	⊗	⊗	PROPN
easat-4034	193	17	𝐵	𝐵	PROPN
easat-4034	193	18	=	=	PROPN
easat-4034	193	19	𝐸	𝐸	PROPN
easat-4034	193	20	,	,	PUNCT
easat-4034	193	21	meaning	mean	VERB
easat-4034	193	22	(	(	PUNCT
easat-4034	193	23	1	1	NUM
easat-4034	193	24	)	)	PUNCT
easat-4034	193	25	𝑚𝑖𝑛𝑘(𝑎𝑖𝑘	𝑚𝑖𝑛𝑘(𝑎𝑖𝑘	NOUN
easat-4034	193	26	+	+	CCONJ
easat-4034	193	27	𝑏𝑖𝑘	𝑏𝑖𝑘	NOUN
easat-4034	193	28	)	)	PUNCT
easat-4034	193	29	=	=	PUNCT
easat-4034	193	30	𝑒	𝑒	PROPN
easat-4034	193	31	=	=	SYM
easat-4034	193	32	0	0	NUM
easat-4034	193	33	for	for	ADP
easat-4034	193	34	every	every	DET
easat-4034	193	35	𝑘	𝑘	NOUN
easat-4034	193	36	there	there	PRON
easat-4034	193	37	is	be	VERB
easat-4034	193	38	𝑖	𝑖	PRON
easat-4034	193	39	so	so	SCONJ
easat-4034	193	40	that	that	SCONJ
easat-4034	193	41	𝑎𝑖𝑘	𝑎𝑖𝑘	PROPN
easat-4034	193	42	+	+	CCONJ
easat-4034	193	43	𝑏𝑘𝑖	𝑏𝑘𝑖	PROPN
easat-4034	193	44	=	=	SYM
easat-4034	193	45	𝑒	𝑒	PROPN
easat-4034	193	46	,	,	PUNCT
easat-4034	193	47	we	we	PRON
easat-4034	193	48	have	have	VERB
easat-4034	193	49	the	the	DET
easat-4034	193	50	function	function	NOUN
easat-4034	193	51	𝑖	𝑖	NOUN
easat-4034	193	52	=	=	PUNCT
easat-4034	193	53	𝜃(𝑘	𝜃(𝑘	X
easat-4034	193	54	)	)	PUNCT
easat-4034	193	55	with	with	ADP
easat-4034	193	56	𝑎𝑖𝜃(𝑖	𝑎𝑖𝜃(𝑖	PROPN
easat-4034	193	57	)	)	PUNCT
easat-4034	193	58	>	>	PUNCT
easat-4034	193	59	𝜀	𝜀	PROPN
easat-4034	193	60	and	and	CCONJ
easat-4034	193	61	𝑏𝜃(𝑖)𝑖	𝑏𝜃(𝑖)𝑖	PROPN
easat-4034	193	62	>	>	X
easat-4034	193	63	𝜀.	𝜀.	NOUN
easat-4034	193	64	(	(	PUNCT
easat-4034	193	65	2	2	X
easat-4034	193	66	)	)	PUNCT
easat-4034	193	67	𝑚𝑖𝑛𝑘(𝑎𝑖𝑘	𝑚𝑖𝑛𝑘(𝑎𝑖𝑘	NOUN
easat-4034	193	68	+	+	CCONJ
easat-4034	193	69	𝑏𝑘𝑗	𝑏𝑘𝑗	NOUN
easat-4034	193	70	)	)	PUNCT
easat-4034	193	71	=	=	SYM
easat-4034	193	72	𝜀′	𝜀′	PUNCT
easat-4034	193	73	=	=	SYM
easat-4034	193	74	∞	∞	PROPN
easat-4034	193	75	for	for	ADP
easat-4034	193	76	all	all	PRON
easat-4034	193	77	𝑖	𝑖	PRON
easat-4034	193	78	≠	≠	PROPN
easat-4034	193	79	𝑗	𝑗	PRON
easat-4034	193	80	based	base	VERB
easat-4034	193	81	on	on	ADP
easat-4034	193	82	(	(	PUNCT
easat-4034	193	83	2	2	NUM
easat-4034	193	84	)	)	PUNCT
easat-4034	193	85	,	,	PUNCT
easat-4034	193	86	it	it	PRON
easat-4034	193	87	is	be	AUX
easat-4034	193	88	obtained	obtain	VERB
easat-4034	193	89	(	(	PUNCT
easat-4034	193	90	3	3	NUM
easat-4034	193	91	)	)	PUNCT
easat-4034	193	92	𝑎𝑖𝜃(𝑗	𝑎𝑖𝜃(𝑗	PROPN
easat-4034	193	93	)	)	PUNCT
easat-4034	193	94	=	=	SYM
easat-4034	193	95	𝜀′	𝜀′	PUNCT
easat-4034	193	96	for	for	ADP
easat-4034	193	97	all	all	DET
easat-4034	193	98	𝑖	𝑖	DET
easat-4034	193	99	≠	≠	PROPN
easat-4034	193	100	𝑗.	𝑗.	NOUN
easat-4034	193	101	since	since	SCONJ
easat-4034	193	102	𝑎𝑖𝜃(𝑖	𝑎𝑖𝜃(𝑖	PROPN
easat-4034	193	103	)	)	PUNCT
easat-4034	193	104	>	>	PUNCT
easat-4034	193	105	𝜀′	𝜀′	PUNCT
easat-4034	193	106	=	=	SYM
easat-4034	193	107	𝑎𝑖𝜃(𝑗	𝑎𝑖𝜃(𝑗	PROPN
easat-4034	193	108	)	)	PUNCT
easat-4034	193	109	for	for	ADP
easat-4034	193	110	all	all	PRON
easat-4034	193	111	𝑖	𝑖	ADP
easat-4034	193	112	≠	≠	NOUN
easat-4034	193	113	𝑗	𝑗	INTJ
easat-4034	193	114	then	then	ADV
easat-4034	193	115	𝜃	𝜃	NOUN
easat-4034	193	116	is	be	AUX
easat-4034	193	117	an	an	DET
easat-4034	193	118	injection	injection	NOUN
easat-4034	193	119	and	and	CCONJ
easat-4034	193	120	permutation	permutation	NOUN
easat-4034	193	121	function	function	NOUN
easat-4034	193	122	.	.	PUNCT
easat-4034	194	1	meanwhile	meanwhile	ADV
easat-4034	194	2	,	,	PUNCT
easat-4034	194	3	𝑎𝑖𝜃(𝑖	𝑎𝑖𝜃(𝑖	PROPN
easat-4034	194	4	)	)	PUNCT
easat-4034	194	5	is	be	AUX
easat-4034	194	6	a	a	DET
easat-4034	194	7	single	single	ADJ
easat-4034	194	8	entry	entry	NOUN
easat-4034	194	9	in	in	ADP
easat-4034	194	10	the	the	DET
easat-4034	194	11	𝜃(𝑖	𝜃(𝑖	NOUN
easat-4034	194	12	)	)	PUNCT
easat-4034	194	13	−th	−th	PROPN
easat-4034	194	14	column	column	NOUN
easat-4034	194	15	of	of	ADP
easat-4034	194	16	𝐴	𝐴	PROPN
easat-4034	194	17	,	,	PUNCT
easat-4034	194	18	which	which	PRON
easat-4034	194	19	is	be	AUX
easat-4034	194	20	not	not	PART
easat-4034	194	21	𝜀′.	𝜀′.	PROPN
easat-4034	194	22	for	for	ADP
easat-4034	194	23	example	example	NOUN
easat-4034	194	24	,	,	PUNCT
easat-4034	194	25	�	�	PROPN
easat-4034	194	26	̂	̂	VERB
easat-4034	194	27	�	�	NOUN
easat-4034	194	28	=	=	SYM
easat-4034	195	1	𝑃𝜃	𝑃𝜃	PROPN
easat-4034	195	2	⊗	⊗	PROPN
easat-4034	195	3	𝐴.	𝐴.	PROPN
easat-4034	195	4	the	the	DET
easat-4034	195	5	𝜃(𝑖	𝜃(𝑖	NOUN
easat-4034	195	6	)	)	PUNCT
easat-4034	195	7	−th	−th	PROPN
easat-4034	195	8	row	row	NOUN
easat-4034	195	9	of	of	ADP
easat-4034	195	10	�	�	PROPN
easat-4034	195	11	̂	̂	SYM
easat-4034	195	12	�	�	NOUN
easat-4034	195	13	is	be	AUX
easat-4034	195	14	the	the	DET
easat-4034	195	15	𝑖	𝑖	SYM
easat-4034	195	16	−th	−th	PROPN
easat-4034	195	17	row	row	NOUN
easat-4034	195	18	of	of	ADP
easat-4034	195	19	𝐴	𝐴	PROPN
easat-4034	195	20	,	,	PUNCT
easat-4034	195	21	which	which	PRON
easat-4034	195	22	has	have	VERB
easat-4034	195	23	a	a	DET
easat-4034	195	24	larger	large	ADJ
easat-4034	195	25	entry	entry	NOUN
easat-4034	195	26	then	then	ADV
easat-4034	195	27	𝜀′	𝜀′	VERB
easat-4034	195	28	in	in	ADP
easat-4034	195	29	the	the	DET
easat-4034	195	30	𝜃(𝑖	𝜃(𝑖	NOUN
easat-4034	195	31	)	)	PUNCT
easat-4034	195	32	−th	−th	PROPN
easat-4034	195	33	column	column	NOUN
easat-4034	195	34	.	.	PUNCT
easat-4034	196	1	thus	thus	ADV
easat-4034	196	2	,	,	PUNCT
easat-4034	196	3	all	all	DET
easat-4034	196	4	larger	large	ADJ
easat-4034	196	5	�	�	NOUN
easat-4034	196	6	̂	̂	SYM
easat-4034	196	7	�	�	NOUN
easat-4034	196	8	diagonal	diagonal	ADJ
easat-4034	196	9	entries	entry	NOUN
easat-4034	196	10	become	become	VERB
easat-4034	196	11	𝜀′.	𝜀′.	PRON
easat-4034	196	12	a	a	PRON
easat-4034	196	13	has	have	VERB
easat-4034	196	14	only	only	ADV
easat-4034	196	15	one	one	NUM
easat-4034	196	16	non−𝜀′	non−𝜀′	ADJ
easat-4034	196	17	entry	entry	NOUN
easat-4034	196	18	in	in	ADP
easat-4034	196	19	each	each	DET
easat-4034	196	20	column	column	NOUN
easat-4034	196	21	,	,	PUNCT
easat-4034	196	22	which	which	PRON
easat-4034	196	23	is	be	AUX
easat-4034	196	24	also	also	ADV
easat-4034	196	25	true	true	ADJ
easat-4034	196	26	for	for	ADP
easat-4034	196	27	�	�	PROPN
easat-4034	196	28	̂	̂	NOUN
easat-4034	196	29	�	�	PROPN
easat-4034	196	30	.	.	PUNCT
easat-4034	196	31	9551	9551	NUM
easat-4034	196	32	edelweiss	edelweiss	PROPN
easat-4034	196	33	applied	apply	VERB
easat-4034	196	34	science	science	NOUN
easat-4034	196	35	and	and	CCONJ
easat-4034	196	36	technology	technology	NOUN
easat-4034	196	37	issn	issn	PROPN
easat-4034	196	38	:	:	PUNCT
easat-4034	196	39	2576	2576	NUM
easat-4034	196	40	-	-	SYM
easat-4034	196	41	8484	8484	NUM
easat-4034	196	42	vol	vol	NOUN
easat-4034	196	43	.	.	PROPN
easat-4034	197	1	8	8	NUM
easat-4034	197	2	,	,	PUNCT
easat-4034	197	3	no	no	INTJ
easat-4034	197	4	.	.	NOUN
easat-4034	198	1	6	6	NUM
easat-4034	198	2	:	:	PUNCT
easat-4034	198	3	9544	9544	NUM
easat-4034	198	4	-	-	SYM
easat-4034	198	5	9554	9554	NUM
easat-4034	198	6	,	,	PUNCT
easat-4034	198	7	2024	2024	NUM
easat-4034	198	8	doi	doi	NOUN
easat-4034	198	9	:	:	PUNCT
easat-4034	198	10	10.55214/25768484.v8i6.4034	10.55214/25768484.v8i6.4034	NUM
easat-4034	198	11	©	©	ADP
easat-4034	198	12	2024	2024	NUM
easat-4034	198	13	by	by	ADP
easat-4034	198	14	the	the	DET
easat-4034	198	15	authors	author	NOUN
easat-4034	198	16	;	;	PUNCT
easat-4034	198	17	licensee	licensee	PROPN
easat-4034	198	18	learning	learning	NOUN
easat-4034	198	19	gate	gate	NOUN
easat-4034	199	1	so	so	SCONJ
easat-4034	199	2	we	we	PRON
easat-4034	199	3	get	get	VERB
easat-4034	199	4	𝑃𝜃	𝑃𝜃	ADJ
easat-4034	199	5	⊗	⊗	ADJ
easat-4034	199	6	𝐴	𝐴	PROPN
easat-4034	199	7	=	=	SYM
easat-4034	199	8	�	�	PROPN
easat-4034	199	9	̂	̂	NOUN
easat-4034	199	10	�	�	NOUN
easat-4034	199	11	=	=	SYM
easat-4034	199	12	𝐷(𝜆𝑖	𝐷(𝜆𝑖	X
easat-4034	199	13	)	)	PUNCT
easat-4034	199	14	with	with	ADP
easat-4034	199	15	𝜆𝑖	𝜆𝑖	PROPN
easat-4034	199	16	=	=	PUNCT
easat-4034	199	17	𝑎𝜃−1(𝑖)𝑖	𝑎𝜃−1(𝑖)𝑖	PROPN
easat-4034	199	18	>	>	X
easat-4034	199	19	𝜀′.	𝜀′.	PROPN
easat-4034	199	20	suppose	suppose	VERB
easat-4034	199	21	,	,	PUNCT
easat-4034	199	22	𝜎	𝜎	PROPN
easat-4034	199	23	=	=	SYM
easat-4034	199	24	𝜃−1	𝜃−1	PROPN
easat-4034	199	25	,	,	PUNCT
easat-4034	199	26	because	because	SCONJ
easat-4034	199	27	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	199	28	⊗	⊗	NOUN
easat-4034	199	29	𝑃𝜃	𝑃𝜃	NOUN
easat-4034	199	30	=	=	PUNCT
easat-4034	199	31	𝑃𝜃−1	𝑃𝜃−1	NOUN
easat-4034	199	32	⊗	⊗	ADJ
easat-4034	199	33	𝑃𝜃	𝑃𝜃	PROPN
easat-4034	199	34	=	=	SYM
easat-4034	199	35	𝐸	𝐸	PROPN
easat-4034	199	36	,	,	PUNCT
easat-4034	199	37	then	then	ADV
easat-4034	199	38	𝐴	𝐴	PROPN
easat-4034	199	39	=	=	PUNCT
easat-4034	200	1	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	200	2	⊗	⊗	NOUN
easat-4034	200	3	𝐷(𝜆𝑖	𝐷(𝜆𝑖	PROPN
easat-4034	200	4	)	)	PUNCT
easat-4034	200	5	.	.	PUNCT
easat-4034	201	1	so	so	ADV
easat-4034	201	2	,	,	PUNCT
easat-4034	201	3	it	it	PRON
easat-4034	201	4	is	be	AUX
easat-4034	201	5	proven	prove	VERB
easat-4034	201	6	that	that	SCONJ
easat-4034	201	7	𝐴	𝐴	PROPN
easat-4034	201	8	=	=	PUNCT
easat-4034	202	1	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	202	2	⊗	⊗	NOUN
easat-4034	202	3	𝐷(𝜆𝑖	𝐷(𝜆𝑖	PROPN
easat-4034	202	4	)	)	PUNCT
easat-4034	202	5	.	.	PUNCT
easat-4034	203	1	(	(	PUNCT
easat-4034	203	2	⟸	⟸	ADJ
easat-4034	203	3	)	)	PUNCT
easat-4034	203	4	assume	assume	VERB
easat-4034	203	5	𝐴	𝐴	NOUN
easat-4034	203	6	=	=	PUNCT
easat-4034	203	7	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	203	8	⊗	⊗	NOUN
easat-4034	203	9	𝐷(𝜆𝑖	𝐷(𝜆𝑖	PROPN
easat-4034	203	10	)	)	PUNCT
easat-4034	203	11	with	with	ADP
easat-4034	203	12	𝜆	𝜆	DET
easat-4034	203	13	∈	∈	PROPN
easat-4034	203	14	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	203	15	and	and	CCONJ
easat-4034	203	16	𝜆𝑖	𝜆𝑖	X
easat-4034	203	17	>	>	X
easat-4034	203	18	𝜀.	𝜀.	VERB
easat-4034	203	19	if	if	SCONJ
easat-4034	203	20	the	the	DET
easat-4034	203	21	statement	statement	NOUN
easat-4034	203	22	is	be	AUX
easat-4034	203	23	true	true	ADJ
easat-4034	203	24	then	then	ADV
easat-4034	203	25	for	for	ADP
easat-4034	203	26	example	example	NOUN
easat-4034	203	27	𝐵	𝐵	NOUN
easat-4034	203	28	=	=	SYM
easat-4034	203	29	𝑃𝜎−1	𝑃𝜎−1	PROPN
easat-4034	203	30	⊗	⊗	PROPN
easat-4034	203	31	𝐷(−𝜆𝑖	𝐷(−𝜆𝑖	NOUN
easat-4034	203	32	)	)	PUNCT
easat-4034	203	33	,	,	PUNCT
easat-4034	203	34	with	with	ADP
easat-4034	203	35	−𝜆𝑖	−𝜆𝑖	X
easat-4034	203	36	=	=	SYM
easat-4034	203	37	𝜆𝑖	𝜆𝑖	PROPN
easat-4034	203	38	⊗−1	⊗−1	PROPN
easat-4034	203	39	.	.	PUNCT
easat-4034	204	1	so	so	ADV
easat-4034	204	2	we	we	PRON
easat-4034	204	3	have	have	VERB
easat-4034	204	4	𝐴	𝐴	PROPN
easat-4034	204	5	⊗	⊗	PROPN
easat-4034	204	6	𝐵	𝐵	NOUN
easat-4034	204	7	=	=	PUNCT
easat-4034	204	8	(	(	PUNCT
easat-4034	204	9	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	204	10	⊗	⊗	PROPN
easat-4034	204	11	𝐷(𝜆𝑖	𝐷(𝜆𝑖	X
easat-4034	204	12	)	)	PUNCT
easat-4034	204	13	)	)	PUNCT
easat-4034	205	1	⊗	⊗	PROPN
easat-4034	205	2	(	(	PUNCT
easat-4034	205	3	𝑃𝜎−1	𝑃𝜎−1	PROPN
easat-4034	205	4	⊗	⊗	PROPN
easat-4034	205	5	𝐷(−𝜆𝑖	𝐷(−𝜆𝑖	PROPN
easat-4034	205	6	)	)	PUNCT
easat-4034	205	7	)	)	PUNCT
easat-4034	206	1	=	=	PUNCT
easat-4034	206	2	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	206	3	⊗	⊗	PROPN
easat-4034	206	4	(	(	PUNCT
easat-4034	206	5	𝐷(𝜆𝑖	𝐷(𝜆𝑖	PROPN
easat-4034	206	6	)	)	PUNCT
easat-4034	206	7	⊗	⊗	NUM
easat-4034	206	8	𝐷(−𝜆𝑖	𝐷(−𝜆𝑖	PROPN
easat-4034	206	9	)	)	PUNCT
easat-4034	206	10	)	)	PUNCT
easat-4034	207	1	⊗	⊗	PROPN
easat-4034	207	2	𝑃𝜎−1	𝑃𝜎−1	NOUN
easat-4034	207	3	=	=	PUNCT
easat-4034	208	1	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	208	2	⊗	⊗	ADJ
easat-4034	208	3	𝐸	𝐸	PROPN
easat-4034	208	4	⊗	⊗	ADJ
easat-4034	208	5	𝑃𝜎−1	𝑃𝜎−1	NOUN
easat-4034	208	6	=	=	PUNCT
easat-4034	209	1	𝑃𝜎	𝑃𝜎	VERB
easat-4034	209	2	⊗	⊗	ADJ
easat-4034	209	3	𝑃𝜎−1	𝑃𝜎−1	NOUN
easat-4034	209	4	=	=	SYM
easat-4034	209	5	𝐸	𝐸	PROPN
easat-4034	209	6	and	and	CCONJ
easat-4034	209	7	,	,	PUNCT
easat-4034	209	8	𝐴	𝐴	PROPN
easat-4034	209	9	⊗	⊗	PROPN
easat-4034	209	10	𝐵	𝐵	PROPN
easat-4034	209	11	=	=	PUNCT
easat-4034	209	12	𝐸	𝐸	PROPN
easat-4034	209	13	and	and	CCONJ
easat-4034	209	14	𝐵	𝐵	PROPN
easat-4034	209	15	is	be	AUX
easat-4034	209	16	the	the	DET
easat-4034	209	17	right	right	ADJ
easat-4034	209	18	inverse	inverse	NOUN
easat-4034	209	19	of	of	ADP
easat-4034	209	20	𝐴.	𝐴.	PROPN
easat-4034	209	21	from	from	ADP
easat-4034	209	22	the	the	DET
easat-4034	209	23	theorem	theorem	NOUN
easat-4034	209	24	above	above	ADV
easat-4034	209	25	,	,	PUNCT
easat-4034	209	26	we	we	PRON
easat-4034	209	27	get	get	VERB
easat-4034	209	28	the	the	DET
easat-4034	209	29	necessary	necessary	ADJ
easat-4034	209	30	and	and	CCONJ
easat-4034	209	31	sufficient	sufficient	ADJ
easat-4034	209	32	conditions	condition	NOUN
easat-4034	209	33	for	for	ADP
easat-4034	209	34	matrix	matrix	NOUN
easat-4034	209	35	a	a	PRON
easat-4034	209	36	to	to	PART
easat-4034	209	37	be	be	AUX
easat-4034	209	38	invertible	invertible	ADJ
easat-4034	209	39	over	over	ADP
easat-4034	209	40	min	min	ADJ
easat-4034	209	41	-	-	PUNCT
easat-4034	209	42	plus	plus	ADJ
easat-4034	209	43	algebra	algebra	NOUN
easat-4034	209	44	,	,	PUNCT
easat-4034	209	45	namely	namely	ADV
easat-4034	209	46	matrix	matrix	NOUN
easat-4034	209	47	a	a	PRON
easat-4034	209	48	is	be	AUX
easat-4034	209	49	invertible	invertible	ADJ
easat-4034	209	50	if	if	SCONJ
easat-4034	209	51	and	and	CCONJ
easat-4034	209	52	only	only	ADV
easat-4034	209	53	if	if	SCONJ
easat-4034	209	54	matrix	matrix	NOUN
easat-4034	209	55	a	a	PRON
easat-4034	209	56	is	be	AUX
easat-4034	209	57	a	a	DET
easat-4034	209	58	permuted	permuted	ADJ
easat-4034	209	59	diagonal	diagonal	ADJ
easat-4034	209	60	matrix	matrix	NOUN
easat-4034	209	61	with	with	ADP
easat-4034	209	62	𝐴	𝐴	PROPN
easat-4034	210	1	=	=	PUNCT
easat-4034	210	2	𝑃𝜎	𝑃𝜎	PROPN
easat-4034	210	3	⊗	⊗	NOUN
easat-4034	210	4	𝐷(𝜆𝑖	𝐷(𝜆𝑖	PROPN
easat-4034	210	5	)	)	PUNCT
easat-4034	210	6	.	.	PUNCT
easat-4034	211	1	the	the	DET
easat-4034	211	2	purpose	purpose	NOUN
easat-4034	211	3	of	of	ADP
easat-4034	211	4	finding	find	VERB
easat-4034	211	5	the	the	DET
easat-4034	211	6	generalized	generalized	ADJ
easat-4034	211	7	inverse	inverse	NOUN
easat-4034	211	8	is	be	AUX
easat-4034	211	9	to	to	PART
easat-4034	211	10	determine	determine	VERB
easat-4034	211	11	the	the	DET
easat-4034	211	12	solution	solution	NOUN
easat-4034	211	13	of	of	ADP
easat-4034	211	14	the	the	DET
easat-4034	211	15	linear	linear	ADJ
easat-4034	211	16	equation	equation	NOUN
easat-4034	211	17	system	system	NOUN
easat-4034	211	18	𝐴𝑋𝐵	𝐴𝑋𝐵	PROPN
easat-4034	211	19	=	=	SYM
easat-4034	211	20	𝐶.	𝐶.	PROPN
easat-4034	211	21	a	a	DET
easat-4034	211	22	matrix	matrix	NOUN
easat-4034	211	23	𝐴	𝐴	NOUN
easat-4034	211	24	∈	∈	PROPN
easat-4034	211	25	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	211	26	𝑛×𝑛	𝑛×𝑛	PROPN
easat-4034	211	27	has	have	VERB
easat-4034	211	28	a	a	DET
easat-4034	211	29	generalized	generalized	ADJ
easat-4034	211	30	inverse	inverse	NOUN
easat-4034	211	31	matrix	matrix	NOUN
easat-4034	211	32	𝑋	𝑋	NOUN
easat-4034	211	33	∈	∈	PROPN
easat-4034	211	34	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	211	35	𝑛×𝑛	𝑛×𝑛	PROPN
easat-4034	212	1	if	if	SCONJ
easat-4034	212	2	𝐴	𝐴	PROPN
easat-4034	212	3	⊗	⊗	PROPN
easat-4034	212	4	𝑋	𝑋	PROPN
easat-4034	212	5	⊗	⊗	PROPN
easat-4034	212	6	𝐴	𝐴	PROPN
easat-4034	212	7	=	=	SYM
easat-4034	212	8	𝐴.	𝐴.	PROPN
easat-4034	212	9	so	so	ADV
easat-4034	212	10	,	,	PUNCT
easat-4034	212	11	it	it	PRON
easat-4034	212	12	can	can	AUX
easat-4034	212	13	be	be	AUX
easat-4034	212	14	said	say	VERB
easat-4034	212	15	that	that	SCONJ
easat-4034	212	16	the	the	DET
easat-4034	212	17	generalized	generalized	ADJ
easat-4034	212	18	inverse	inverse	NOUN
easat-4034	212	19	is	be	AUX
easat-4034	212	20	the	the	DET
easat-4034	212	21	smallest	small	ADJ
easat-4034	212	22	subsolution	subsolution	NOUN
easat-4034	212	23	of	of	ADP
easat-4034	212	24	the	the	DET
easat-4034	212	25	equation	equation	NOUN
easat-4034	213	1	𝐴	𝐴	PROPN
easat-4034	213	2	⊗	⊗	PROPN
easat-4034	213	3	𝑋	𝑋	PROPN
easat-4034	213	4	⊗	⊗	PROPN
easat-4034	213	5	𝐴	𝐴	PROPN
easat-4034	213	6	=	=	SYM
easat-4034	213	7	𝐴.	𝐴.	NOUN
easat-4034	213	8	several	several	ADJ
easat-4034	213	9	steps	step	NOUN
easat-4034	213	10	are	be	AUX
easat-4034	213	11	required	require	VERB
easat-4034	213	12	to	to	PART
easat-4034	213	13	determine	determine	VERB
easat-4034	213	14	matrix	matrix	NOUN
easat-4034	213	15	𝑋	𝑋	NOUN
easat-4034	213	16	as	as	ADP
easat-4034	213	17	the	the	DET
easat-4034	213	18	generalized	generalized	ADJ
easat-4034	213	19	inverse	inverse	NOUN
easat-4034	213	20	of	of	ADP
easat-4034	213	21	equation	equation	NOUN
easat-4034	213	22	𝐴	𝐴	PROPN
easat-4034	213	23	⊗	⊗	PROPN
easat-4034	213	24	𝑋	𝑋	PROPN
easat-4034	213	25	⊗	⊗	PROPN
easat-4034	213	26	𝐴	𝐴	PROPN
easat-4034	213	27	=	=	SYM
easat-4034	213	28	𝐴.	𝐴.	PROPN
easat-4034	213	29	definition	definition	NOUN
easat-4034	213	30	9	9	NUM
easat-4034	213	31	for	for	ADP
easat-4034	213	32	a	a	DET
easat-4034	213	33	matrix	matrix	NOUN
easat-4034	213	34	𝐴	𝐴	NOUN
easat-4034	213	35	∈	∈	PROPN
easat-4034	213	36	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	213	37	𝑛×𝑛	𝑛×𝑛	PROPN
easat-4034	213	38	,	,	PUNCT
easat-4034	213	39	then	then	ADV
easat-4034	213	40	the	the	DET
easat-4034	213	41	matrix	matrix	NOUN
easat-4034	213	42	𝐵	𝐵	NOUN
easat-4034	213	43	∈	∈	PROPN
easat-4034	213	44	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	213	45	𝑛×𝑛	𝑛×𝑛	PROPN
easat-4034	213	46	is	be	AUX
easat-4034	213	47	said	say	VERB
easat-4034	213	48	to	to	PART
easat-4034	213	49	be	be	AUX
easat-4034	213	50	the	the	DET
easat-4034	213	51	generalized	generalized	ADJ
easat-4034	213	52	inverse	inverse	NOUN
easat-4034	213	53	of	of	ADP
easat-4034	213	54	the	the	DET
easat-4034	213	55	matrix	matrix	NOUN
easat-4034	214	1	a	a	PRON
easat-4034	214	2	if	if	SCONJ
easat-4034	214	3	𝐴	𝐴	PROPN
easat-4034	214	4	⊗	⊗	PROPN
easat-4034	214	5	𝐵	𝐵	PROPN
easat-4034	214	6	⊗	⊗	PROPN
easat-4034	214	7	𝐴	𝐴	PROPN
easat-4034	214	8	=	=	PROPN
easat-4034	214	9	𝐴	𝐴	PROPN
easat-4034	214	10	is	be	AUX
easat-4034	214	11	satisfied	satisfied	ADJ
easat-4034	214	12	.	.	PUNCT
easat-4034	215	1	to	to	PART
easat-4034	215	2	determine	determine	VERB
easat-4034	215	3	whether	whether	SCONJ
easat-4034	215	4	or	or	CCONJ
easat-4034	215	5	not	not	PART
easat-4034	215	6	there	there	PRON
easat-4034	215	7	is	be	VERB
easat-4034	215	8	a	a	DET
easat-4034	215	9	matrix	matrix	NOUN
easat-4034	215	10	𝐵	𝐵	NOUN
easat-4034	215	11	that	that	PRON
easat-4034	215	12	satisfies	satisfy	VERB
easat-4034	215	13	𝐴	𝐴	PROPN
easat-4034	215	14	⊗	⊗	PROPN
easat-4034	215	15	𝐵	𝐵	PROPN
easat-4034	215	16	⊗	⊗	PROPN
easat-4034	215	17	𝐴	𝐴	PROPN
easat-4034	215	18	=	=	PROPN
easat-4034	215	19	𝐴	𝐴	PROPN
easat-4034	215	20	,	,	PUNCT
easat-4034	215	21	is	be	AUX
easat-4034	215	22	equivalent	equivalent	ADJ
easat-4034	215	23	to	to	ADP
easat-4034	215	24	determining	determine	VERB
easat-4034	215	25	whether	whether	SCONJ
easat-4034	215	26	or	or	CCONJ
easat-4034	215	27	not	not	PART
easat-4034	215	28	there	there	PRON
easat-4034	215	29	is	be	VERB
easat-4034	215	30	a	a	DET
easat-4034	215	31	solution	solution	NOUN
easat-4034	215	32	to	to	ADP
easat-4034	215	33	the	the	DET
easat-4034	215	34	equation	equation	NOUN
easat-4034	215	35	𝐴	𝐴	PROPN
easat-4034	215	36	⊗	⊗	PROPN
easat-4034	215	37	𝑋	𝑋	PROPN
easat-4034	215	38	⊗	⊗	PROPN
easat-4034	215	39	𝐴	𝐴	PROPN
easat-4034	215	40	=	=	PROPN
easat-4034	215	41	𝐴	𝐴	PROPN
easat-4034	215	42	with	with	ADP
easat-4034	215	43	𝐴	𝐴	PROPN
easat-4034	215	44	∈	∈	PROPN
easat-4034	215	45	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	215	46	𝑛×𝑛.	𝑛×𝑛.	PROPN
easat-4034	215	47	1	1	NUM
easat-4034	215	48	.	.	PUNCT
easat-4034	216	1	bring	bring	VERB
easat-4034	216	2	the	the	DET
easat-4034	216	3	equation	equation	NOUN
easat-4034	216	4	𝐴	𝐴	PROPN
easat-4034	216	5	⊗	⊗	PROPN
easat-4034	216	6	𝑋	𝑋	PROPN
easat-4034	216	7	⊗	⊗	PROPN
easat-4034	216	8	𝐴	𝐴	PROPN
easat-4034	216	9	=	=	PROPN
easat-4034	216	10	𝐴	𝐴	PROPN
easat-4034	216	11	to	to	ADP
easat-4034	216	12	the	the	DET
easat-4034	216	13	form	form	NOUN
easat-4034	216	14	𝐴𝑥	𝐴𝑥	NOUN
easat-4034	216	15	=	=	PUNCT
easat-4034	216	16	𝑏.	𝑏.	NOUN
easat-4034	217	1	2	2	X
easat-4034	217	2	.	.	PUNCT
easat-4034	217	3	determine	determine	VERB
easat-4034	217	4	the	the	DET
easat-4034	217	5	matrix	matrix	NOUN
easat-4034	217	6	𝑋	𝑋	NOUN
easat-4034	217	7	3	3	NUM
easat-4034	217	8	.	.	PUNCT
easat-4034	217	9	prove	prove	VERB
easat-4034	217	10	that	that	SCONJ
easat-4034	217	11	the	the	DET
easat-4034	217	12	matrix	matrix	NOUN
easat-4034	217	13	x	x	PUNCT
easat-4034	217	14	is	be	AUX
easat-4034	217	15	a	a	DET
easat-4034	217	16	generalized	generalized	ADJ
easat-4034	217	17	inverse	inverse	NOUN
easat-4034	217	18	by	by	ADP
easat-4034	217	19	substituting	substitute	VERB
easat-4034	217	20	it	it	PRON
easat-4034	217	21	into	into	ADP
easat-4034	217	22	the	the	DET
easat-4034	217	23	equation	equation	NOUN
easat-4034	217	24	𝐴	𝐴	PROPN
easat-4034	217	25	⊗	⊗	PROPN
easat-4034	217	26	𝑋	𝑋	PROPN
easat-4034	217	27	⊗	⊗	PROPN
easat-4034	217	28	𝐴	𝐴	PROPN
easat-4034	217	29	=	=	SYM
easat-4034	217	30	𝐴.	𝐴.	PROPN
easat-4034	217	31	given	give	VERB
easat-4034	217	32	𝐴	𝐴	PROPN
easat-4034	217	33	∈	∈	PROPN
easat-4034	217	34	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	217	35	𝑛×𝑛	𝑛×𝑛	PROPN
easat-4034	217	36	with	with	ADP
easat-4034	217	37	operations	operation	NOUN
easat-4034	217	38	⊕	⊕	PROPN
easat-4034	217	39	′	′	NUM
easat-4034	217	40	and	and	CCONJ
easat-4034	217	41	⊗.	⊗.	NOUN
easat-4034	217	42	the	the	DET
easat-4034	217	43	𝑖𝑗	𝑖𝑗	X
easat-4034	217	44	−th	−th	PROPN
easat-4034	217	45	element	element	NOUN
easat-4034	217	46	in	in	ADP
easat-4034	217	47	𝐴	𝐴	PROPN
easat-4034	217	48	⊗	⊗	PROPN
easat-4034	217	49	𝑋	𝑋	PROPN
easat-4034	217	50	⊗	⊗	PROPN
easat-4034	217	51	𝐴	𝐴	PROPN
easat-4034	217	52	with	with	ADP
easat-4034	217	53	1	1	NUM
easat-4034	217	54	≤	≤	NUM
easat-4034	217	55	𝑖	𝑖	NOUN
easat-4034	217	56	,	,	PUNCT
easat-4034	217	57	𝑗	𝑗	PROPN
easat-4034	217	58	≤	≤	X
easat-4034	217	59	𝑛	𝑛	DET
easat-4034	217	60	in	in	ADP
easat-4034	217	61	𝐴	𝐴	PROPN
easat-4034	217	62	⊗	⊗	PROPN
easat-4034	217	63	𝑋	𝑋	PROPN
easat-4034	217	64	⊗	⊗	PROPN
easat-4034	217	65	𝐴	𝐴	PROPN
easat-4034	217	66	is	be	AUX
easat-4034	218	1	[	[	X
easat-4034	218	2	𝐴	𝐴	PROPN
easat-4034	218	3	⊗	⊗	PROPN
easat-4034	218	4	𝑋	𝑋	PROPN
easat-4034	218	5	⊗	⊗	PROPN
easat-4034	218	6	𝐴]𝑖𝑗	𝐴]𝑖𝑗	PROPN
easat-4034	218	7	=	=	PUNCT
easat-4034	219	1	𝐴𝑖𝑗	𝐴𝑖𝑗	NUM
easat-4034	219	2	⟺	⟺	NOUN
easat-4034	220	1	[	[	X
easat-4034	220	2	𝐴	𝐴	PROPN
easat-4034	220	3	⊗	⊗	PROPN
easat-4034	220	4	𝑋]𝑖𝑙	𝑋]𝑖𝑙	PROPN
easat-4034	221	1	⊗	⊗	PROPN
easat-4034	221	2	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	221	3	=	=	PUNCT
easat-4034	221	4	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	221	5	1	1	NUM
easat-4034	221	6	≤	≤	NOUN
easat-4034	221	7	𝑙	𝑙	PRON
easat-4034	221	8	≤	≤	NUM
easat-4034	221	9	𝑛	𝑛	PRON
easat-4034	221	10	⟺	⟺	NOUN
easat-4034	222	1	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	222	2	⊗	⊗	NOUN
easat-4034	223	1	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	223	2	⊗	⊗	PROPN
easat-4034	223	3	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	223	4	=	=	PUNCT
easat-4034	223	5	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	223	6	1	1	NUM
easat-4034	223	7	≤	≤	NUM
easat-4034	223	8	𝑘	𝑘	ADP
easat-4034	223	9	,	,	PUNCT
easat-4034	223	10	𝑙	𝑙	PROPN
easat-4034	223	11	≤	≤	NUM
easat-4034	223	12	𝑛	𝑛	PRON
easat-4034	223	13	⟺	⟺	NOUN
easat-4034	224	1	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	224	2	⊗	⊗	NOUN
easat-4034	224	3	𝑛	𝑛	PROPN
easat-4034	224	4	⊕	⊕	PROPN
easat-4034	224	5	′	′	NOUN
easat-4034	225	1	𝑙	𝑙	X
easat-4034	225	2	=	=	SYM
easat-4034	225	3	1	1	NUM
easat-4034	225	4	𝑥𝑘𝑙	𝑥𝑘𝑙	NOUN
easat-4034	225	5	⊗	⊗	PROPN
easat-4034	225	6	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	225	7	=	=	PUNCT
easat-4034	225	8	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	225	9	1	1	NUM
easat-4034	225	10	≤	≤	NOUN
easat-4034	225	11	𝑘	𝑘	DET
easat-4034	225	12	≤	≤	NUM
easat-4034	225	13	𝑛	𝑛	PRON
easat-4034	225	14	⟺	⟺	NOUN
easat-4034	225	15	𝑛	𝑛	PRON
easat-4034	225	16	⊕	⊕	PROPN
easat-4034	225	17	′	′	VERB
easat-4034	226	1	𝑘	𝑘	PRON
easat-4034	226	2	=	=	SYM
easat-4034	226	3	1	1	NUM
easat-4034	226	4	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	226	5	⊗	⊗	NOUN
easat-4034	226	6	𝑛	𝑛	PROPN
easat-4034	226	7	⊕	⊕	PROPN
easat-4034	226	8	′	′	NOUN
easat-4034	227	1	𝑙	𝑙	X
easat-4034	227	2	=	=	SYM
easat-4034	227	3	1	1	NUM
easat-4034	227	4	𝑥𝑘𝑙	𝑥𝑘𝑙	NOUN
easat-4034	227	5	⊗	⊗	PROPN
easat-4034	227	6	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	227	7	=	=	PUNCT
easat-4034	227	8	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	227	9	⟺	⟺	NOUN
easat-4034	227	10	𝑛	𝑛	PRON
easat-4034	227	11	⊕	⊕	NOUN
easat-4034	227	12	′	′	VERB
easat-4034	228	1	𝑘	𝑘	PRON
easat-4034	228	2	=	=	SYM
easat-4034	228	3	1	1	NUM
easat-4034	228	4	𝑛	𝑛	PRON
easat-4034	228	5	⊕	⊕	NOUN
easat-4034	228	6	′	′	NOUN
easat-4034	229	1	𝑙	𝑙	NOUN
easat-4034	229	2	=	=	SYM
easat-4034	229	3	1	1	NUM
easat-4034	229	4	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	229	5	⊗	⊗	NUM
easat-4034	229	6	𝑥𝑘𝑙	𝑥𝑘𝑙	NOUN
easat-4034	229	7	⊗	⊗	PROPN
easat-4034	229	8	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	229	9	=	=	SYM
easat-4034	229	10	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	229	11	9552	9552	NUM
easat-4034	229	12	edelweiss	edelweiss	PROPN
easat-4034	229	13	applied	apply	VERB
easat-4034	229	14	science	science	NOUN
easat-4034	229	15	and	and	CCONJ
easat-4034	229	16	technology	technology	NOUN
easat-4034	229	17	issn	issn	PROPN
easat-4034	229	18	:	:	PUNCT
easat-4034	229	19	2576	2576	NUM
easat-4034	229	20	-	-	SYM
easat-4034	229	21	8484	8484	NUM
easat-4034	229	22	vol	vol	NOUN
easat-4034	229	23	.	.	PROPN
easat-4034	229	24	8	8	NUM
easat-4034	229	25	,	,	PUNCT
easat-4034	229	26	no	no	INTJ
easat-4034	229	27	.	.	NOUN
easat-4034	230	1	6	6	NUM
easat-4034	230	2	:	:	PUNCT
easat-4034	230	3	9544	9544	NUM
easat-4034	230	4	-	-	SYM
easat-4034	230	5	9554	9554	NUM
easat-4034	230	6	,	,	PUNCT
easat-4034	230	7	2024	2024	NUM
easat-4034	230	8	doi	doi	NOUN
easat-4034	230	9	:	:	PUNCT
easat-4034	230	10	10.55214/25768484.v8i6.4034	10.55214/25768484.v8i6.4034	NUM
easat-4034	230	11	©	©	ADP
easat-4034	230	12	2024	2024	NUM
easat-4034	230	13	by	by	ADP
easat-4034	230	14	the	the	DET
easat-4034	230	15	authors	author	NOUN
easat-4034	230	16	;	;	PUNCT
easat-4034	231	1	licensee	licensee	PROPN
easat-4034	231	2	learning	learning	NOUN
easat-4034	231	3	gate	gate	PROPN
easat-4034	231	4	⟺	⟺	NOUN
easat-4034	231	5	𝑛	𝑛	PRON
easat-4034	231	6	⊕′	⊕′	NOUN
easat-4034	231	7	𝑘	𝑘	NOUN
easat-4034	231	8	=	=	SYM
easat-4034	231	9	1	1	NUM
easat-4034	231	10	[	[	PUNCT
easat-4034	231	11	𝑛	𝑛	PROPN
easat-4034	231	12	⊕′	⊕′	PROPN
easat-4034	231	13	𝑙	𝑙	NOUN
easat-4034	231	14	=	=	SYM
easat-4034	231	15	1	1	NUM
easat-4034	231	16	(	(	PUNCT
easat-4034	231	17	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	231	18	⊗	⊗	PROPN
easat-4034	231	19	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	231	20	⊗	⊗	PROPN
easat-4034	231	21	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	231	22	)	)	PUNCT
easat-4034	231	23	]	]	PUNCT
easat-4034	232	1	=	=	PUNCT
easat-4034	232	2	𝐴𝑖𝑗	𝐴𝑖𝑗	NOUN
easat-4034	232	3	so	so	ADV
easat-4034	232	4	we	we	PRON
easat-4034	232	5	get	get	VERB
easat-4034	232	6	𝑛	𝑛	PRON
easat-4034	232	7	⊕′	⊕′	NOUN
easat-4034	232	8	𝑖	𝑖	SYM
easat-4034	232	9	=	=	NOUN
easat-4034	232	10	1	1	NUM
easat-4034	232	11	𝑛	𝑛	DET
easat-4034	232	12	⊕′	⊕′	NOUN
easat-4034	232	13	𝑗	𝑗	NOUN
easat-4034	232	14	=	=	SYM
easat-4034	232	15	1	1	NUM
easat-4034	232	16	𝑓𝑖𝑗(𝑋𝑘𝑙	𝑓𝑖𝑗(𝑋𝑘𝑙	NOUN
easat-4034	232	17	)	)	PUNCT
easat-4034	232	18	=	=	PUNCT
easat-4034	233	1	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	233	2	.	.	PUNCT
easat-4034	234	1	the	the	DET
easat-4034	234	2	generalized	generalized	ADJ
easat-4034	234	3	inverse	inverse	NOUN
easat-4034	234	4	can	can	AUX
easat-4034	234	5	be	be	AUX
easat-4034	234	6	found	find	VERB
easat-4034	234	7	by	by	ADP
easat-4034	234	8	solving	solve	VERB
easat-4034	234	9	the	the	DET
easat-4034	234	10	equation	equation	NOUN
easat-4034	234	11	𝐴	𝐴	NOUN
easat-4034	234	12	⊗	⊗	NOUN
easat-4034	235	1	𝑥	𝑥	PROPN
easat-4034	235	2	=	=	PUNCT
easat-4034	235	3	𝑏	𝑏	PROPN
easat-4034	235	4	in	in	ADP
easat-4034	235	5	min	min	ADJ
easat-4034	235	6	-	-	PUNCT
easat-4034	235	7	plus	plus	ADJ
easat-4034	235	8	algebra	algebra	NOUN
easat-4034	235	9	.	.	PUNCT
easat-4034	236	1	for	for	ADP
easat-4034	236	2	the	the	DET
easat-4034	236	3	generalized	generalized	ADJ
easat-4034	236	4	inverse	inverse	NOUN
easat-4034	236	5	,	,	PUNCT
easat-4034	236	6	denoted	denote	VERB
easat-4034	236	7	𝐴⊗−1	𝐴⊗−1	PROPN
easat-4034	236	8	,	,	PUNCT
easat-4034	236	9	the	the	DET
easat-4034	236	10	form	form	NOUN
easat-4034	236	11	𝐴	𝐴	PROPN
easat-4034	236	12	⊗	⊗	PROPN
easat-4034	236	13	𝑋	𝑋	PROPN
easat-4034	236	14	⊗	⊗	PROPN
easat-4034	236	15	𝐴	𝐴	PROPN
easat-4034	236	16	=	=	PROPN
easat-4034	236	17	𝐴	𝐴	PROPN
easat-4034	236	18	is	be	AUX
easat-4034	236	19	brought	bring	VERB
easat-4034	236	20	to	to	ADP
easat-4034	236	21	the	the	DET
easat-4034	236	22	form	form	NOUN
easat-4034	236	23	𝐴	𝐴	PROPN
easat-4034	236	24	⊗	⊗	NOUN
easat-4034	237	1	𝑥	𝑥	X
easat-4034	237	2	=	=	PUNCT
easat-4034	237	3	𝑏.	𝑏.	NOUN
easat-4034	237	4	for	for	ADP
easat-4034	237	5	the	the	DET
easat-4034	237	6	𝑖𝑗	𝑖𝑗	PROPN
easat-4034	237	7	−th	−th	PROPN
easat-4034	237	8	element	element	NOUN
easat-4034	237	9	,	,	PUNCT
easat-4034	237	10	it	it	PRON
easat-4034	237	11	is	be	AUX
easat-4034	237	12	obtained	obtain	VERB
easat-4034	237	13	as	as	SCONJ
easat-4034	237	14	follows	follow	VERB
easat-4034	237	15	:	:	PUNCT
easat-4034	237	16	⟺	⟺	PROPN
easat-4034	238	1	[	[	X
easat-4034	238	2	𝐴	𝐴	PROPN
easat-4034	238	3	⊗	⊗	PROPN
easat-4034	238	4	𝑋	𝑋	PROPN
easat-4034	238	5	⊗	⊗	PROPN
easat-4034	238	6	𝐴]𝑖𝑗	𝐴]𝑖𝑗	PROPN
easat-4034	238	7	=	=	PUNCT
easat-4034	239	1	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	239	2	⟺	⟺	NOUN
easat-4034	240	1	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	240	2	⊗	⊗	NOUN
easat-4034	241	1	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	241	2	⊗	⊗	PROPN
easat-4034	241	3	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	241	4	=	=	PUNCT
easat-4034	242	1	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	242	2	⟺	⟺	NOUN
easat-4034	243	1	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	243	2	⊗	⊗	NOUN
easat-4034	243	3	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	243	4	⊗	⊗	PROPN
easat-4034	243	5	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	244	1	⊗	⊗	PROPN
easat-4034	244	2	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	244	3	⊗−1	⊗−1	PROPN
easat-4034	244	4	=	=	SYM
easat-4034	245	1	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	245	2	⊗	⊗	PROPN
easat-4034	245	3	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	245	4	⊗−1	⊗−1	PROPN
easat-4034	245	5	⟺	⟺	NOUN
easat-4034	245	6	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	245	7	⊗	⊗	NOUN
easat-4034	246	1	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	246	2	⊗	⊗	PROPN
easat-4034	246	3	𝐸	𝐸	PROPN
easat-4034	246	4	=	=	PUNCT
easat-4034	246	5	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	246	6	−	−	PROPN
easat-4034	246	7	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	246	8	if	if	SCONJ
easat-4034	246	9	suppose	suppose	VERB
easat-4034	246	10	𝑏	𝑏	PROPN
easat-4034	246	11	=	=	SYM
easat-4034	246	12	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	246	13	−	−	PROPN
easat-4034	246	14	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	246	15	then	then	ADV
easat-4034	246	16	we	we	PRON
easat-4034	246	17	write	write	VERB
easat-4034	246	18	⟺	⟺	NOUN
easat-4034	246	19	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	246	20	⊗	⊗	NOUN
easat-4034	247	1	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	247	2	=	=	PUNCT
easat-4034	247	3	𝑏	𝑏	PROPN
easat-4034	247	4	⟺	⟺	NOUN
easat-4034	247	5	𝐸	𝐸	NOUN
easat-4034	247	6	⊗	⊗	NOUN
easat-4034	248	1	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	248	2	=	=	PUNCT
easat-4034	248	3	−𝐴𝑖𝑘	−𝐴𝑖𝑘	PROPN
easat-4034	248	4	⊗	⊗	PROPN
easat-4034	249	1	𝑏	𝑏	PROPN
easat-4034	249	2	⟺	⟺	NOUN
easat-4034	249	3	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	249	4	=	=	SYM
easat-4034	249	5	−𝐴𝑖𝑘	−𝐴𝑖𝑘	PROPN
easat-4034	249	6	⊗	⊗	PROPN
easat-4034	249	7	𝑏	𝑏	PROPN
easat-4034	250	1	⟺	⟺	NOUN
easat-4034	250	2	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	250	3	=	=	SYM
easat-4034	250	4	−	−	PROPN
easat-4034	250	5	[	[	PUNCT
easat-4034	250	6	𝑎11	𝑎11	ADJ
easat-4034	250	7	𝑎12	𝑎12	PROPN
easat-4034	250	8	𝑎21	𝑎21	PROPN
easat-4034	250	9	𝑎22	𝑎22	NOUN
easat-4034	250	10	⋯	⋯	PROPN
easat-4034	250	11	𝑎1𝑘	𝑎1𝑘	PROPN
easat-4034	250	12	⋯	⋯	NOUN
easat-4034	250	13	𝑎2𝑘	𝑎2𝑘	NOUN
easat-4034	250	14	⋮	⋮	NOUN
easat-4034	250	15	⋮	⋮	NOUN
easat-4034	250	16	𝑎𝑖1	𝑎𝑖1	NOUN
easat-4034	250	17	𝑎𝑖1	𝑎𝑖1	NOUN
easat-4034	250	18	⋱	⋱	PUNCT
easat-4034	250	19	⋮	⋮	NOUN
easat-4034	250	20	⋯	⋯	PROPN
easat-4034	250	21	𝑎𝑖𝑘	𝑎𝑖𝑘	PROPN
easat-4034	250	22	]	]	PUNCT
easat-4034	251	1	⊗	⊗	PROPN
easat-4034	251	2	[	[	PUNCT
easat-4034	251	3	𝑏11	𝑏11	PROPN
easat-4034	251	4	𝑏12	𝑏12	VERB
easat-4034	251	5	𝑏21	𝑏21	NOUN
easat-4034	251	6	𝑏22	𝑏22	NOUN
easat-4034	251	7	⋯	⋯	NOUN
easat-4034	251	8	𝑏1𝑛	𝑏1𝑛	PROPN
easat-4034	251	9	⋯	⋯	VERB
easat-4034	251	10	𝑏2𝑛	𝑏2𝑛	PROPN
easat-4034	251	11	⋮	⋮	PROPN
easat-4034	251	12	⋮	⋮	PROPN
easat-4034	251	13	𝑏𝑛1	𝑏𝑛1	PROPN
easat-4034	251	14	𝑏𝑛2	𝑏𝑛2	ADJ
easat-4034	251	15	⋱	⋱	PUNCT
easat-4034	251	16	⋮	⋮	NOUN
easat-4034	251	17	⋯	⋯	NOUN
easat-4034	251	18	𝑏𝑛𝑛	𝑏𝑛𝑛	NOUN
easat-4034	251	19	]	]	PUNCT
easat-4034	251	20	⟺	⟺	PRON
easat-4034	251	21	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	251	22	=	=	PUNCT
easat-4034	252	1	[	[	PUNCT
easat-4034	252	2	−𝑎11	−𝑎11	X
easat-4034	252	3	−𝑎12	−𝑎12	X
easat-4034	252	4	−𝑎21	−𝑎21	NUM
easat-4034	252	5	−𝑎22	−𝑎22	NOUN
easat-4034	252	6	⋯	⋯	PROPN
easat-4034	252	7	−𝑎1𝑘	−𝑎1𝑘	SYM
easat-4034	252	8	⋯	⋯	PROPN
easat-4034	252	9	−𝑎2𝑘	−𝑎2𝑘	PROPN
easat-4034	252	10	⋮	⋮	NOUN
easat-4034	252	11	⋮	⋮	NOUN
easat-4034	252	12	−𝑎𝑖1	−𝑎𝑖1	X
easat-4034	252	13	−𝑎𝑖1	−𝑎𝑖1	PROPN
easat-4034	252	14	⋱	⋱	PUNCT
easat-4034	252	15	⋮	⋮	NOUN
easat-4034	252	16	⋯	⋯	VERB
easat-4034	252	17	−𝑎𝑖𝑘	−𝑎𝑖𝑘	NOUN
easat-4034	252	18	]	]	PUNCT
easat-4034	253	1	⊗	⊗	PROPN
easat-4034	253	2	[	[	PUNCT
easat-4034	253	3	𝑏11	𝑏11	PROPN
easat-4034	253	4	𝑏12	𝑏12	VERB
easat-4034	253	5	𝑏21	𝑏21	NOUN
easat-4034	253	6	𝑏22	𝑏22	NOUN
easat-4034	253	7	⋯	⋯	NOUN
easat-4034	253	8	𝑏1𝑛	𝑏1𝑛	PROPN
easat-4034	253	9	⋯	⋯	VERB
easat-4034	253	10	𝑏2𝑛	𝑏2𝑛	PROPN
easat-4034	253	11	⋮	⋮	PROPN
easat-4034	253	12	⋮	⋮	PROPN
easat-4034	253	13	𝑏𝑛1	𝑏𝑛1	PROPN
easat-4034	253	14	𝑏𝑛2	𝑏𝑛2	ADJ
easat-4034	253	15	⋱	⋱	PUNCT
easat-4034	253	16	⋮	⋮	NOUN
easat-4034	253	17	⋯	⋯	NOUN
easat-4034	253	18	𝑏𝑛𝑛	𝑏𝑛𝑛	NOUN
easat-4034	253	19	]	]	PUNCT
easat-4034	253	20	for	for	ADP
easat-4034	253	21	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	253	22	=	=	PUNCT
easat-4034	253	23	[	[	PUNCT
easat-4034	253	24	𝑥11	𝑥11	ADV
easat-4034	253	25	𝑥12	𝑥12	VERB
easat-4034	253	26	𝑥21	𝑥21	NOUN
easat-4034	253	27	𝑥22	𝑥22	NOUN
easat-4034	253	28	⋯	⋯	PROPN
easat-4034	253	29	𝑥1𝑙	𝑥1𝑙	PROPN
easat-4034	253	30	⋯	⋯	PROPN
easat-4034	253	31	𝑥2𝑙	𝑥2𝑙	PROPN
easat-4034	253	32	⋮	⋮	PROPN
easat-4034	253	33	⋮	⋮	PROPN
easat-4034	253	34	𝑥𝑘1	𝑥𝑘1	PROPN
easat-4034	253	35	𝑥𝑘2	𝑥𝑘2	X
easat-4034	253	36	⋱	⋱	PUNCT
easat-4034	253	37	⋮	⋮	NOUN
easat-4034	253	38	⋯	⋯	VERB
easat-4034	253	39	𝑥𝑘𝑙	𝑥𝑘𝑙	NOUN
easat-4034	253	40	]	]	PUNCT
easat-4034	253	41	the	the	DET
easat-4034	253	42	result	result	NOUN
easat-4034	253	43	,	,	PUNCT
easat-4034	253	44	𝑥11	𝑥11	ADV
easat-4034	253	45	=	=	SYM
easat-4034	253	46	min{−𝑎11	min{−𝑎11	PROPN
easat-4034	254	1	+	+	CCONJ
easat-4034	254	2	𝑏11	𝑏11	ADJ
easat-4034	254	3	,	,	PUNCT
easat-4034	254	4	−𝑎12	−𝑎12	NUM
easat-4034	255	1	+	+	NUM
easat-4034	255	2	𝑏21	𝑏21	PROPN
easat-4034	255	3	,	,	PUNCT
easat-4034	255	4	…	…	PUNCT
easat-4034	255	5	,	,	PUNCT
easat-4034	255	6	−𝑎1𝑘	−𝑎1𝑘	PUNCT
easat-4034	256	1	+	+	CCONJ
easat-4034	256	2	𝑏𝑛1	𝑏𝑛1	NOUN
easat-4034	256	3	}	}	PUNCT
easat-4034	256	4	𝑥12	𝑥12	VERB
easat-4034	256	5	=	=	SYM
easat-4034	256	6	min{−𝑎11	min{−𝑎11	NOUN
easat-4034	256	7	+	+	CCONJ
easat-4034	256	8	𝑏12	𝑏12	VERB
easat-4034	256	9	,	,	PUNCT
easat-4034	256	10	−𝑎12	−𝑎12	X
easat-4034	257	1	+	+	NUM
easat-4034	257	2	𝑏22	𝑏22	NOUN
easat-4034	257	3	,	,	PUNCT
easat-4034	257	4	…	…	PUNCT
easat-4034	257	5	,	,	PUNCT
easat-4034	257	6	−𝑎1𝑘	−𝑎1𝑘	PUNCT
easat-4034	258	1	+	+	CCONJ
easat-4034	258	2	𝑏𝑛2	𝑏𝑛2	ADJ
easat-4034	258	3	}	}	PUNCT
easat-4034	258	4	⋮	⋮	NOUN
easat-4034	258	5	𝑥1𝑙	𝑥1𝑙	NOUN
easat-4034	258	6	=	=	SYM
easat-4034	258	7	min{−𝑎11	min{−𝑎11	NOUN
easat-4034	258	8	+	+	CCONJ
easat-4034	258	9	𝑏1𝑛	𝑏1𝑛	NOUN
easat-4034	258	10	,	,	PUNCT
easat-4034	258	11	−𝑎12	−𝑎12	PROPN
easat-4034	259	1	+	+	CCONJ
easat-4034	259	2	𝑏2𝑛	𝑏2𝑛	ADJ
easat-4034	259	3	,	,	PUNCT
easat-4034	259	4	…	…	PUNCT
easat-4034	259	5	,	,	PUNCT
easat-4034	259	6	−𝑎1𝑘	−𝑎1𝑘	PUNCT
easat-4034	260	1	+	+	CCONJ
easat-4034	260	2	𝑏𝑛𝑛	𝑏𝑛𝑛	NOUN
easat-4034	260	3	}	}	PUNCT
easat-4034	260	4	𝑥21	𝑥21	NOUN
easat-4034	260	5	=	=	SYM
easat-4034	260	6	min{−𝑎21	min{−𝑎21	PROPN
easat-4034	260	7	+	+	CCONJ
easat-4034	260	8	𝑏11	𝑏11	ADJ
easat-4034	260	9	,	,	PUNCT
easat-4034	260	10	−𝑎22	−𝑎22	PROPN
easat-4034	260	11	+	+	NUM
easat-4034	260	12	𝑏21	𝑏21	PROPN
easat-4034	260	13	,	,	PUNCT
easat-4034	260	14	…	…	PUNCT
easat-4034	260	15	,	,	PUNCT
easat-4034	260	16	−𝑎2𝑘	−𝑎2𝑘	X
easat-4034	260	17	+	+	CCONJ
easat-4034	260	18	𝑏𝑛1	𝑏𝑛1	NOUN
easat-4034	260	19	}	}	PUNCT
easat-4034	260	20	𝑥22	𝑥22	NOUN
easat-4034	260	21	=	=	SYM
easat-4034	260	22	min{−𝑎21	min{−𝑎21	PROPN
easat-4034	260	23	+	+	CCONJ
easat-4034	260	24	𝑏12	𝑏12	VERB
easat-4034	260	25	,	,	PUNCT
easat-4034	260	26	−𝑎22	−𝑎22	PROPN
easat-4034	260	27	+	+	CCONJ
easat-4034	260	28	𝑏22	𝑏22	PROPN
easat-4034	260	29	,	,	PUNCT
easat-4034	260	30	…	…	PUNCT
easat-4034	260	31	,	,	PUNCT
easat-4034	260	32	−𝑎2𝑘	−𝑎2𝑘	X
easat-4034	260	33	+	+	CCONJ
easat-4034	260	34	𝑏𝑛2	𝑏𝑛2	ADJ
easat-4034	260	35	}	}	PUNCT
easat-4034	260	36	⋮	⋮	NOUN
easat-4034	260	37	𝑥2𝑙	𝑥2𝑙	X
easat-4034	260	38	=	=	SYM
easat-4034	260	39	min{−𝑎21	min{−𝑎21	PROPN
easat-4034	260	40	+	+	CCONJ
easat-4034	260	41	𝑏1𝑛	𝑏1𝑛	NOUN
easat-4034	260	42	,	,	PUNCT
easat-4034	260	43	−𝑎22	−𝑎22	NOUN
easat-4034	260	44	+	+	CCONJ
easat-4034	260	45	𝑏2𝑛	𝑏2𝑛	ADJ
easat-4034	260	46	,	,	PUNCT
easat-4034	260	47	…	…	PUNCT
easat-4034	260	48	,	,	PUNCT
easat-4034	260	49	−𝑎2𝑘	−𝑎2𝑘	X
easat-4034	260	50	+	+	CCONJ
easat-4034	260	51	𝑏𝑛𝑛	𝑏𝑛𝑛	NOUN
easat-4034	260	52	}	}	PUNCT
easat-4034	260	53	⋮	⋮	NOUN
easat-4034	260	54	𝑥𝑘1	𝑥𝑘1	NOUN
easat-4034	260	55	=	=	SYM
easat-4034	260	56	min{−𝑎𝑖1	min{−𝑎𝑖1	PROPN
easat-4034	260	57	+	+	CCONJ
easat-4034	260	58	𝑏11	𝑏11	ADJ
easat-4034	260	59	,	,	PUNCT
easat-4034	260	60	−𝑎𝑖2	−𝑎𝑖2	PROPN
easat-4034	260	61	+	+	NUM
easat-4034	260	62	𝑏21	𝑏21	NOUN
easat-4034	260	63	,	,	PUNCT
easat-4034	260	64	…	…	PUNCT
easat-4034	260	65	,	,	PUNCT
easat-4034	260	66	−𝑎𝑖𝑘	−𝑎𝑖𝑘	NOUN
easat-4034	260	67	+	+	CCONJ
easat-4034	261	1	𝑏𝑛1	𝑏𝑛1	NOUN
easat-4034	261	2	}	}	PUNCT
easat-4034	261	3	𝑥𝑘2	𝑥𝑘2	PROPN
easat-4034	261	4	=	=	SYM
easat-4034	261	5	min{−𝑎𝑖1	min{−𝑎𝑖1	PROPN
easat-4034	261	6	+	+	CCONJ
easat-4034	261	7	𝑏12	𝑏12	ADJ
easat-4034	261	8	,	,	PUNCT
easat-4034	261	9	−𝑎𝑖2	−𝑎𝑖2	VERB
easat-4034	261	10	+	+	CCONJ
easat-4034	261	11	𝑏22	𝑏22	NOUN
easat-4034	261	12	,	,	PUNCT
easat-4034	261	13	…	…	PUNCT
easat-4034	261	14	,	,	PUNCT
easat-4034	261	15	−𝑎𝑖𝑘	−𝑎𝑖𝑘	NOUN
easat-4034	261	16	+	+	CCONJ
easat-4034	261	17	𝑏𝑛2	𝑏𝑛2	ADJ
easat-4034	261	18	}	}	PUNCT
easat-4034	261	19	⋮	⋮	ADJ
easat-4034	261	20	𝑥𝑘𝑙	𝑥𝑘𝑙	NOUN
easat-4034	261	21	=	=	SYM
easat-4034	261	22	min{−𝑎𝑖1	min{−𝑎𝑖1	PROPN
easat-4034	261	23	+	+	CCONJ
easat-4034	261	24	𝑏1𝑛	𝑏1𝑛	NOUN
easat-4034	261	25	,	,	PUNCT
easat-4034	261	26	−𝑎𝑖2	−𝑎𝑖2	VERB
easat-4034	261	27	+	+	CCONJ
easat-4034	261	28	𝑏2𝑛	𝑏2𝑛	ADJ
easat-4034	261	29	,	,	PUNCT
easat-4034	261	30	…	…	PUNCT
easat-4034	261	31	,	,	PUNCT
easat-4034	261	32	−𝑎𝑖𝑘	−𝑎𝑖𝑘	NOUN
easat-4034	261	33	+	+	CCONJ
easat-4034	261	34	𝑏𝑛𝑛	𝑏𝑛𝑛	NOUN
easat-4034	261	35	}	}	PUNCT
easat-4034	261	36	so	so	ADV
easat-4034	261	37	,	,	PUNCT
easat-4034	261	38	it	it	PRON
easat-4034	261	39	can	can	AUX
easat-4034	261	40	be	be	AUX
easat-4034	261	41	written	write	VERB
easat-4034	261	42	as	as	ADP
easat-4034	261	43	9553	9553	NUM
easat-4034	261	44	edelweiss	edelweiss	PROPN
easat-4034	261	45	applied	apply	VERB
easat-4034	261	46	science	science	NOUN
easat-4034	261	47	and	and	CCONJ
easat-4034	261	48	technology	technology	NOUN
easat-4034	261	49	issn	issn	PROPN
easat-4034	261	50	:	:	PUNCT
easat-4034	261	51	2576	2576	NUM
easat-4034	261	52	-	-	SYM
easat-4034	261	53	8484	8484	NUM
easat-4034	261	54	vol	vol	NOUN
easat-4034	261	55	.	.	PROPN
easat-4034	262	1	8	8	NUM
easat-4034	262	2	,	,	PUNCT
easat-4034	262	3	no	no	INTJ
easat-4034	262	4	.	.	NOUN
easat-4034	263	1	6	6	NUM
easat-4034	263	2	:	:	PUNCT
easat-4034	263	3	9544	9544	NUM
easat-4034	263	4	-	-	SYM
easat-4034	263	5	9554	9554	NUM
easat-4034	263	6	,	,	PUNCT
easat-4034	263	7	2024	2024	NUM
easat-4034	263	8	doi	doi	NOUN
easat-4034	263	9	:	:	PUNCT
easat-4034	263	10	10.55214/25768484.v8i6.4034	10.55214/25768484.v8i6.4034	NUM
easat-4034	263	11	©	©	ADP
easat-4034	263	12	2024	2024	NUM
easat-4034	263	13	by	by	ADP
easat-4034	263	14	the	the	DET
easat-4034	263	15	authors	author	NOUN
easat-4034	263	16	;	;	PUNCT
easat-4034	263	17	licensee	licensee	PROPN
easat-4034	263	18	learning	learning	NOUN
easat-4034	263	19	gate	gate	NOUN
easat-4034	263	20	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	264	1	=	=	PUNCT
easat-4034	265	1	[	[	PUNCT
easat-4034	265	2	min{−𝑎1𝑘	min{−𝑎1𝑘	NOUN
easat-4034	265	3	+	+	CCONJ
easat-4034	265	4	𝑏𝑛1	𝑏𝑛1	NOUN
easat-4034	265	5	}	}	PUNCT
easat-4034	265	6	min{−𝑎1𝑘	min{−𝑎1𝑘	PROPN
easat-4034	265	7	+	+	CCONJ
easat-4034	265	8	𝑏𝑛2	𝑏𝑛2	ADJ
easat-4034	265	9	}	}	PUNCT
easat-4034	265	10	min{−𝑎2𝑘	min{−𝑎2𝑘	NOUN
easat-4034	265	11	+	+	CCONJ
easat-4034	265	12	𝑏𝑛1	𝑏𝑛1	NOUN
easat-4034	265	13	}	}	PUNCT
easat-4034	265	14	min{−𝑎2𝑘	min{−𝑎2𝑘	NOUN
easat-4034	265	15	+	+	CCONJ
easat-4034	265	16	𝑏𝑛2	𝑏𝑛2	ADJ
easat-4034	265	17	}	}	PUNCT
easat-4034	265	18	⋯	⋯	ADP
easat-4034	265	19	min{−𝑎1𝑘	min{−𝑎1𝑘	NOUN
easat-4034	265	20	+	+	CCONJ
easat-4034	265	21	𝑏𝑛𝑛	𝑏𝑛𝑛	NOUN
easat-4034	265	22	}	}	PUNCT
easat-4034	265	23	⋯	⋯	ADP
easat-4034	265	24	min{−𝑎2𝑘	min{−𝑎2𝑘	NOUN
easat-4034	265	25	+	+	CCONJ
easat-4034	265	26	𝑏𝑛𝑛	𝑏𝑛𝑛	NOUN
easat-4034	265	27	}	}	PUNCT
easat-4034	265	28	⋮	⋮	NOUN
easat-4034	265	29	⋮	⋮	NOUN
easat-4034	265	30	min{−𝑎𝑖𝑘	min{−𝑎𝑖𝑘	PROPN
easat-4034	266	1	+	+	CCONJ
easat-4034	266	2	𝑏𝑛1	𝑏𝑛1	NOUN
easat-4034	266	3	}	}	PUNCT
easat-4034	266	4	min{−𝑎𝑖𝑘	min{−𝑎𝑖𝑘	PROPN
easat-4034	266	5	+	+	CCONJ
easat-4034	266	6	𝑏𝑛2	𝑏𝑛2	ADJ
easat-4034	266	7	}	}	PUNCT
easat-4034	266	8	⋱	⋱	SYM
easat-4034	266	9	⋮	⋮	NOUN
easat-4034	266	10	⋯	⋯	PROPN
easat-4034	267	1	min{−𝑎𝑖𝑘	min{−𝑎𝑖𝑘	PROPN
easat-4034	267	2	+	+	CCONJ
easat-4034	267	3	𝑏𝑛𝑛	𝑏𝑛𝑛	NOUN
easat-4034	267	4	}	}	PUNCT
easat-4034	267	5	]	]	PUNCT
easat-4034	267	6	suppose	suppose	VERB
easat-4034	268	1	𝑏	𝑏	PROPN
easat-4034	268	2	=	=	SYM
easat-4034	268	3	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	268	4	−	−	PROPN
easat-4034	268	5	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	268	6	,	,	PUNCT
easat-4034	268	7	we	we	PRON
easat-4034	268	8	have	have	VERB
easat-4034	268	9	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	268	10	=	=	SYM
easat-4034	268	11	𝑛	𝑛	PRON
easat-4034	268	12	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
easat-4034	268	13	𝑖	𝑖	NOUN
easat-4034	268	14	=	=	SYM
easat-4034	268	15	1	1	NUM
easat-4034	268	16	𝑛	𝑛	PRON
easat-4034	268	17	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
easat-4034	268	18	𝑗	𝑗	NOUN
easat-4034	268	19	=	=	SYM
easat-4034	268	20	1	1	NUM
easat-4034	268	21	(	(	PUNCT
easat-4034	268	22	−𝑎𝑖𝑘	−𝑎𝑖𝑘	NOUN
easat-4034	268	23	+	+	CCONJ
easat-4034	268	24	𝑎𝑖𝑗	𝑎𝑖𝑗	NOUN
easat-4034	268	25	−	−	PROPN
easat-4034	268	26	𝑎𝑙𝑗	𝑎𝑙𝑗	PROPN
easat-4034	268	27	)	)	PUNCT
easat-4034	268	28	from	from	ADP
easat-4034	268	29	the	the	DET
easat-4034	268	30	form	form	NOUN
easat-4034	268	31	𝑥𝑘𝑙	𝑥𝑘𝑙	NOUN
easat-4034	268	32	,	,	PUNCT
easat-4034	268	33	we	we	PRON
easat-4034	268	34	obtain	obtain	VERB
easat-4034	268	35	a	a	DET
easat-4034	268	36	form	form	NOUN
easat-4034	268	37	that	that	PRON
easat-4034	268	38	can	can	AUX
easat-4034	268	39	be	be	AUX
easat-4034	268	40	expressed	express	VERB
easat-4034	268	41	𝑚	𝑚	ADP
easat-4034	268	42	⊕	⊕	PROPN
easat-4034	268	43	𝑘	𝑘	NOUN
easat-4034	268	44	=	=	SYM
easat-4034	268	45	1	1	NUM
easat-4034	268	46	[	[	PUNCT
easat-4034	268	47	𝑚	𝑚	PROPN
easat-4034	268	48	⊕	⊕	PROPN
easat-4034	268	49	𝑙	𝑙	NOUN
easat-4034	269	1	=	=	SYM
easat-4034	269	2	1	1	NUM
easat-4034	269	3	(	(	PUNCT
easat-4034	269	4	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	269	5	+	+	CCONJ
easat-4034	269	6	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	269	7	+	+	CCONJ
easat-4034	269	8	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	269	9	)	)	PUNCT
easat-4034	269	10	]	]	PUNCT
easat-4034	270	1	=	=	PUNCT
easat-4034	270	2	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	270	3	.	.	PUNCT
easat-4034	271	1	by	by	ADP
easat-4034	271	2	determining	determine	VERB
easat-4034	271	3	whether	whether	SCONJ
easat-4034	271	4	or	or	CCONJ
easat-4034	271	5	not	not	PART
easat-4034	271	6	there	there	PRON
easat-4034	271	7	is	be	VERB
easat-4034	271	8	a	a	DET
easat-4034	271	9	solution	solution	NOUN
easat-4034	271	10	to	to	ADP
easat-4034	271	11	the	the	DET
easat-4034	271	12	equation	equation	NOUN
easat-4034	271	13	𝐴	𝐴	PROPN
easat-4034	271	14	⊗	⊗	PROPN
easat-4034	271	15	𝑋	𝑋	PROPN
easat-4034	271	16	⊗	⊗	PROPN
easat-4034	271	17	𝐴	𝐴	PROPN
easat-4034	271	18	=	=	PROPN
easat-4034	271	19	𝐴	𝐴	PROPN
easat-4034	271	20	with	with	ADP
easat-4034	271	21	𝐴	𝐴	PROPN
easat-4034	271	22	∈	∈	PROPN
easat-4034	271	23	ℝ𝑚𝑖𝑛	ℝ𝑚𝑖𝑛	PROPN
easat-4034	271	24	𝑛×𝑚.	𝑛×𝑚.	NOUN
easat-4034	271	25	the	the	DET
easat-4034	271	26	𝑖𝑗	𝑖𝑗	X
easat-4034	271	27	−th	−th	PROPN
easat-4034	271	28	element	element	NOUN
easat-4034	271	29	in	in	ADP
easat-4034	271	30	𝐴	𝐴	PROPN
easat-4034	271	31	⊗	⊗	PROPN
easat-4034	271	32	𝑋	𝑋	PROPN
easat-4034	271	33	⊗	⊗	PROPN
easat-4034	271	34	𝐴	𝐴	PROPN
easat-4034	271	35	is	be	AUX
easat-4034	271	36	[	[	X
easat-4034	271	37	𝐴	𝐴	PROPN
easat-4034	271	38	⊗	⊗	PROPN
easat-4034	271	39	𝑋	𝑋	PROPN
easat-4034	271	40	⊗	⊗	PROPN
easat-4034	272	1	𝐴]𝑖𝑗	𝐴]𝑖𝑗	PROPN
easat-4034	272	2	=	=	PUNCT
easat-4034	273	1	𝑚	𝑚	PROPN
easat-4034	273	2	⨁′	⨁′	NOUN
easat-4034	273	3	𝑘	𝑘	X
easat-4034	273	4	=	=	SYM
easat-4034	273	5	1	1	NUM
easat-4034	273	6	𝑛	𝑛	DET
easat-4034	273	7	⨁′	⨁′	PROPN
easat-4034	273	8	𝑙	𝑙	NOUN
easat-4034	273	9	=	=	SYM
easat-4034	273	10	1	1	NUM
easat-4034	273	11	(	(	PUNCT
easat-4034	273	12	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	273	13	+	+	CCONJ
easat-4034	273	14	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	273	15	+	+	CCONJ
easat-4034	273	16	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	273	17	)	)	PUNCT
easat-4034	273	18	.	.	PUNCT
easat-4034	274	1	so	so	ADV
easat-4034	274	2	,	,	PUNCT
easat-4034	274	3	we	we	PRON
easat-4034	274	4	get	get	VERB
easat-4034	274	5	the	the	DET
easat-4034	274	6	equation	equation	NOUN
easat-4034	274	7	𝑚	𝑚	PROPN
easat-4034	274	8	⨁′	⨁′	PROPN
easat-4034	274	9	𝑘	𝑘	PROPN
easat-4034	274	10	=	=	SYM
easat-4034	274	11	1	1	NUM
easat-4034	274	12	[	[	PUNCT
easat-4034	274	13	𝑛	𝑛	PROPN
easat-4034	274	14	⨁′	⨁′	NOUN
easat-4034	274	15	𝑙	𝑙	PROPN
easat-4034	274	16	=	=	SYM
easat-4034	274	17	1	1	NUM
easat-4034	274	18	(	(	PUNCT
easat-4034	274	19	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	274	20	+	+	CCONJ
easat-4034	274	21	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	274	22	+	+	CCONJ
easat-4034	274	23	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	274	24	)	)	PUNCT
easat-4034	274	25	]	]	PUNCT
easat-4034	275	1	=	=	PUNCT
easat-4034	275	2	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	275	3	…	…	PUNCT
easat-4034	275	4	…	…	PUNCT
easat-4034	275	5	…	…	PUNCT
easat-4034	275	6	…	…	PUNCT
easat-4034	275	7	…	…	PUNCT
easat-4034	275	8	(	(	PUNCT
easat-4034	275	9	3	3	NUM
easat-4034	275	10	)	)	PUNCT
easat-4034	275	11	if	if	SCONJ
easat-4034	275	12	for	for	ADP
easat-4034	275	13	each	each	DET
easat-4034	275	14	𝑘	𝑘	NOUN
easat-4034	275	15	,	,	PUNCT
easat-4034	275	16	𝑙	𝑙	PROPN
easat-4034	275	17	is	be	AUX
easat-4034	275	18	formed	form	VERB
easat-4034	275	19	𝑓𝑖𝑗(𝑋𝑘𝑙	𝑓𝑖𝑗(𝑋𝑘𝑙	PRON
easat-4034	275	20	)	)	PUNCT
easat-4034	275	21	=	=	PUNCT
easat-4034	276	1	𝐴𝑖𝑘	𝐴𝑖𝑘	PROPN
easat-4034	276	2	+	+	NUM
easat-4034	276	3	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	276	4	+	+	CCONJ
easat-4034	276	5	𝐴𝑙𝑗	𝐴𝑙𝑗	PROPN
easat-4034	276	6	…	…	PUNCT
easat-4034	276	7	…	…	PUNCT
easat-4034	276	8	..	..	PUNCT
easat-4034	276	9	(	(	PUNCT
easat-4034	276	10	4	4	X
easat-4034	276	11	)	)	PUNCT
easat-4034	276	12	then	then	ADV
easat-4034	276	13	equation	equation	NOUN
easat-4034	276	14	(	(	PUNCT
easat-4034	276	15	3	3	X
easat-4034	276	16	)	)	PUNCT
easat-4034	276	17	becomes	become	VERB
easat-4034	276	18	𝑛	𝑛	PRON
easat-4034	276	19	⨁	⨁	PROPN
easat-4034	276	20	𝑖	𝑖	SYM
easat-4034	276	21	=	=	NOUN
easat-4034	276	22	1	1	NUM
easat-4034	276	23	𝑚	𝑚	PROPN
easat-4034	276	24	⨁	⨁	PROPN
easat-4034	276	25	𝑖	𝑖	SYM
easat-4034	276	26	=	=	SYM
easat-4034	276	27	1	1	NUM
easat-4034	276	28	𝑓𝑖𝑗(𝑋𝑘𝑙	𝑓𝑖𝑗(𝑋𝑘𝑙	NOUN
easat-4034	276	29	)	)	PUNCT
easat-4034	276	30	=	=	PUNCT
easat-4034	277	1	𝐴𝑖𝑗	𝐴𝑖𝑗	PROPN
easat-4034	277	2	…	…	PUNCT
easat-4034	277	3	…	…	PUNCT
easat-4034	277	4	…	…	PUNCT
easat-4034	277	5	(	(	PUNCT
easat-4034	277	6	5	5	NUM
easat-4034	277	7	)	)	PUNCT
easat-4034	277	8	it	it	PRON
easat-4034	277	9	is	be	AUX
easat-4034	277	10	obtained	obtain	VERB
easat-4034	277	11	that	that	SCONJ
easat-4034	277	12	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	277	13	=	=	SYM
easat-4034	277	14	𝑛	𝑛	PRON
easat-4034	277	15	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
easat-4034	277	16	𝑖	𝑖	NOUN
easat-4034	277	17	=	=	SYM
easat-4034	277	18	1	1	NUM
easat-4034	277	19	𝑛	𝑛	PRON
easat-4034	277	20	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
easat-4034	277	21	𝑗	𝑗	NOUN
easat-4034	277	22	=	=	SYM
easat-4034	277	23	1	1	NUM
easat-4034	277	24	(	(	PUNCT
easat-4034	277	25	−𝑎𝑖𝑘	−𝑎𝑖𝑘	NOUN
easat-4034	277	26	+	+	CCONJ
easat-4034	277	27	𝑎𝑖𝑗	𝑎𝑖𝑗	NOUN
easat-4034	277	28	−	−	PROPN
easat-4034	277	29	𝑎𝑙𝑗	𝑎𝑙𝑗	PROPN
easat-4034	277	30	)	)	PUNCT
easat-4034	277	31	corresponds	correspond	VERB
easat-4034	277	32	if	if	SCONJ
easat-4034	277	33	substituted	substitute	VERB
easat-4034	277	34	into	into	ADP
easat-4034	277	35	the	the	DET
easat-4034	277	36	equation	equation	NOUN
easat-4034	277	37	𝐴	𝐴	PROPN
easat-4034	277	38	⊗	⊗	PROPN
easat-4034	277	39	𝑋	𝑋	PROPN
easat-4034	277	40	⊗	⊗	PROPN
easat-4034	277	41	𝐴	𝐴	PROPN
easat-4034	277	42	=	=	SYM
easat-4034	278	1	𝐴.	𝐴.	PROPN
easat-4034	278	2	4	4	NUM
easat-4034	278	3	.	.	PUNCT
easat-4034	278	4	conclusions	conclusion	NOUN
easat-4034	278	5	in	in	ADP
easat-4034	278	6	this	this	DET
easat-4034	278	7	conclusion	conclusion	NOUN
easat-4034	278	8	,	,	PUNCT
easat-4034	278	9	the	the	DET
easat-4034	278	10	property	property	NOUN
easat-4034	278	11	is	be	AUX
easat-4034	278	12	obtained	obtain	VERB
easat-4034	278	13	that	that	DET
easat-4034	278	14	permutation	permutation	NOUN
easat-4034	278	15	is	be	AUX
easat-4034	278	16	required	require	VERB
easat-4034	278	17	determining	determine	VERB
easat-4034	278	18	the	the	DET
easat-4034	278	19	inverse	inverse	NOUN
easat-4034	278	20	matrix	matrix	NOUN
easat-4034	278	21	over	over	ADP
easat-4034	278	22	min	min	ADJ
easat-4034	278	23	-	-	PUNCT
easat-4034	278	24	plus	plus	ADJ
easat-4034	278	25	algebra	algebra	NOUN
easat-4034	278	26	.	.	PUNCT
easat-4034	279	1	we	we	PRON
easat-4034	279	2	obtain	obtain	VERB
easat-4034	279	3	several	several	ADJ
easat-4034	279	4	theorems	theorem	NOUN
easat-4034	279	5	that	that	PRON
easat-4034	279	6	show	show	VERB
easat-4034	279	7	the	the	DET
easat-4034	279	8	characteristics	characteristic	NOUN
easat-4034	279	9	of	of	ADP
easat-4034	279	10	the	the	DET
easat-4034	279	11	inverse	inverse	NOUN
easat-4034	279	12	of	of	ADP
easat-4034	279	13	matrices	matrix	NOUN
easat-4034	279	14	in	in	ADP
easat-4034	279	15	min	min	ADJ
easat-4034	279	16	-	-	PUNCT
easat-4034	279	17	plus	plus	ADJ
easat-4034	279	18	algebra	algebra	NOUN
easat-4034	279	19	,	,	PUNCT
easat-4034	279	20	especially	especially	ADV
easat-4034	279	21	generalized	generalize	VERB
easat-4034	279	22	matrices	matrix	NOUN
easat-4034	279	23	.	.	PUNCT
easat-4034	280	1	the	the	DET
easat-4034	280	2	characterization	characterization	NOUN
easat-4034	280	3	is	be	AUX
easat-4034	280	4	obtained	obtain	VERB
easat-4034	280	5	by	by	ADP
easat-4034	280	6	considering	consider	VERB
easat-4034	280	7	the	the	DET
easat-4034	280	8	characteristics	characteristic	NOUN
easat-4034	280	9	of	of	ADP
easat-4034	280	10	the	the	DET
easat-4034	280	11	solutions	solution	NOUN
easat-4034	280	12	linear	linear	PROPN
easat-4034	280	13	equations	equation	NOUN
easat-4034	280	14	system	system	NOUN
easat-4034	280	15	over	over	ADP
easat-4034	280	16	min	min	ADJ
easat-4034	280	17	-	-	PUNCT
easat-4034	280	18	plus	plus	ADJ
easat-4034	280	19	algebra	algebra	NOUN
easat-4034	280	20	.	.	PUNCT
easat-4034	281	1	the	the	DET
easat-4034	281	2	generalized	generalized	ADJ
easat-4034	281	3	inverse	inverse	NOUN
easat-4034	281	4	of	of	ADP
easat-4034	281	5	the	the	DET
easat-4034	281	6	matrix	matrix	NOUN
easat-4034	281	7	𝐴	𝐴	NOUN
easat-4034	281	8	∈	∈	PROPN
easat-4034	281	9	𝑅𝑚𝑖𝑛	𝑅𝑚𝑖𝑛	PROPN
easat-4034	281	10	𝑛×𝑛	𝑛×𝑛	PROPN
easat-4034	281	11	can	can	AUX
easat-4034	281	12	be	be	AUX
easat-4034	281	13	obtained	obtain	VERB
easat-4034	281	14	by	by	ADP
easat-4034	281	15	determining	determine	VERB
easat-4034	281	16	the	the	DET
easat-4034	281	17	matrix	matrix	NOUN
easat-4034	281	18	𝑋	𝑋	NOUN
easat-4034	281	19	with	with	ADP
easat-4034	281	20	entry	entry	NOUN
easat-4034	281	21	𝑋𝑘𝑙	𝑋𝑘𝑙	PROPN
easat-4034	281	22	=	=	SYM
easat-4034	281	23	𝑛	𝑛	PRON
easat-4034	281	24	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
easat-4034	281	25	𝑖	𝑖	NOUN
easat-4034	281	26	=	=	SYM
easat-4034	281	27	1	1	NUM
easat-4034	281	28	𝑛	𝑛	PRON
easat-4034	281	29	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
easat-4034	281	30	𝑗	𝑗	NOUN
easat-4034	281	31	=	=	SYM
easat-4034	281	32	1	1	NUM
easat-4034	281	33	(	(	PUNCT
easat-4034	281	34	−𝑎𝑖𝑘	−𝑎𝑖𝑘	NOUN
easat-4034	281	35	+	+	CCONJ
easat-4034	281	36	𝑎𝑖𝑗	𝑎𝑖𝑗	NOUN
easat-4034	281	37	−	−	PROPN
easat-4034	281	38	𝑎𝑙𝑗	𝑎𝑙𝑗	PROPN
easat-4034	281	39	)	)	PUNCT
easat-4034	281	40	which	which	PRON
easat-4034	281	41	satisfies	satisfy	VERB
easat-4034	281	42	𝐴	𝐴	PROPN
easat-4034	281	43	⊗	⊗	PROPN
easat-4034	281	44	𝑋	𝑋	PROPN
easat-4034	281	45	⊗	⊗	PROPN
easat-4034	281	46	𝐴	𝐴	PROPN
easat-4034	281	47	=	=	SYM
easat-4034	282	1	𝐴.	𝐴.	PROPN
easat-4034	282	2	copyright	copyright	NOUN
easat-4034	282	3	:	:	PUNCT
easat-4034	282	4	©	©	PROPN
easat-4034	282	5	2024	2024	NUM
easat-4034	282	6	by	by	ADP
easat-4034	282	7	the	the	DET
easat-4034	282	8	authors	author	NOUN
easat-4034	282	9	.	.	PUNCT
easat-4034	283	1	this	this	DET
easat-4034	283	2	article	article	NOUN
easat-4034	283	3	is	be	AUX
easat-4034	283	4	an	an	DET
easat-4034	283	5	open	open	ADJ
easat-4034	283	6	access	access	NOUN
easat-4034	283	7	article	article	NOUN
easat-4034	283	8	distributed	distribute	VERB
easat-4034	283	9	under	under	ADP
easat-4034	283	10	the	the	DET
easat-4034	283	11	terms	term	NOUN
easat-4034	283	12	and	and	CCONJ
easat-4034	283	13	conditions	condition	NOUN
easat-4034	283	14	of	of	ADP
easat-4034	283	15	the	the	DET
easat-4034	283	16	creative	creative	ADJ
easat-4034	283	17	commons	common	NOUN
easat-4034	283	18	attribution	attribution	NOUN
easat-4034	283	19	(	(	PUNCT
easat-4034	283	20	cc	cc	NOUN
easat-4034	283	21	by	by	ADP
easat-4034	283	22	)	)	PUNCT
easat-4034	283	23	license	license	NOUN
easat-4034	283	24	(	(	PUNCT
easat-4034	283	25	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-4034	283	26	)	)	PUNCT
easat-4034	283	27	.	.	PUNCT
easat-4034	284	1	references	reference	NOUN
easat-4034	284	2	[	[	X
easat-4034	284	3	1	1	X
easat-4034	284	4	]	]	PUNCT
easat-4034	284	5	s.	s.	PROPN
easat-4034	284	6	r.	r.	PROPN
easat-4034	284	7	lisapaly	lisapaly	PROPN
easat-4034	284	8	and	and	CCONJ
easat-4034	284	9	e.	e.	PROPN
easat-4034	284	10	r.	r.	PROPN
easat-4034	284	11	persulessy	persulessy	PROPN
easat-4034	284	12	,	,	PUNCT
easat-4034	284	13	“	"	PUNCT
easat-4034	284	14	semiring	semire	VERB
easat-4034	284	15	,	,	PUNCT
easat-4034	284	16	”	"	PUNCT
easat-4034	284	17	barekeng	barekeng	PROPN
easat-4034	284	18	j.	j.	PROPN
easat-4034	284	19	ilmu	ilmu	PROPN
easat-4034	284	20	mat	mat	PROPN
easat-4034	284	21	.	.	PUNCT
easat-4034	285	1	dan	dan	PROPN
easat-4034	285	2	terap	terap	PROPN
easat-4034	285	3	.	.	PUNCT
easat-4034	285	4	,	,	PUNCT
easat-4034	285	5	vol	vol	NOUN
easat-4034	285	6	.	.	PROPN
easat-4034	285	7	5	5	NUM
easat-4034	285	8	,	,	PUNCT
easat-4034	285	9	no	no	INTJ
easat-4034	285	10	.	.	NOUN
easat-4034	285	11	2	2	NUM
easat-4034	285	12	,	,	PUNCT
easat-4034	285	13	pp	pp	ADJ
easat-4034	285	14	.	.	PUNCT
easat-4034	286	1	45–47	45–47	NUM
easat-4034	286	2	,	,	PUNCT
easat-4034	286	3	2011	2011	NUM
easat-4034	286	4	,	,	PUNCT
easat-4034	286	5	doi	doi	NOUN
easat-4034	286	6	:	:	PUNCT
easat-4034	286	7	10.30598	10.30598	NUM
easat-4034	286	8	/	/	SYM
easat-4034	286	9	barekengvol5iss2pp45	barekengvol5iss2pp45	PROPN
easat-4034	286	10	-	-	SYM
easat-4034	286	11	47	47	NUM
easat-4034	286	12	.	.	PUNCT
easat-4034	287	1	[	[	X
easat-4034	287	2	2	2	NUM
easat-4034	287	3	]	]	X
easat-4034	287	4	g.	g.	PROPN
easat-4034	287	5	ariyanti	ariyanti	PROPN
easat-4034	287	6	,	,	PUNCT
easat-4034	287	7	“	"	PUNCT
easat-4034	287	8	a	a	DET
easat-4034	287	9	note	note	NOUN
easat-4034	287	10	of	of	ADP
easat-4034	287	11	the	the	DET
easat-4034	287	12	linear	linear	ADJ
easat-4034	287	13	equation	equation	NOUN
easat-4034	287	14	ax	ax	NOUN
easat-4034	287	15	=	=	SYM
easat-4034	287	16	b	b	NOUN
easat-4034	287	17	with	with	ADP
easat-4034	287	18	multiplicatively	multiplicatively	ADV
easat-4034	287	19	-	-	PUNCT
easat-4034	287	20	reguler	reguler	NOUN
easat-4034	287	21	matrix	matrix	NOUN
easat-4034	287	22	a	a	PRON
easat-4034	287	23	in	in	ADP
easat-4034	287	24	semiring	semiring	NOUN
easat-4034	287	25	,	,	PUNCT
easat-4034	287	26	”	"	PUNCT
easat-4034	287	27	j.	j.	PROPN
easat-4034	287	28	phys	phys	PROPN
easat-4034	287	29	.	.	PUNCT
easat-4034	287	30	conf	conf	PROPN
easat-4034	287	31	.	.	PUNCT
easat-4034	288	1	ser	ser	PROPN
easat-4034	288	2	.	.	PROPN
easat-4034	288	3	,	,	PUNCT
easat-4034	288	4	vol	vol	NOUN
easat-4034	288	5	.	.	PUNCT
easat-4034	289	1	1366	1366	NUM
easat-4034	289	2	,	,	PUNCT
easat-4034	289	3	no	no	INTJ
easat-4034	289	4	.	.	NOUN
easat-4034	289	5	1	1	NUM
easat-4034	289	6	,	,	PUNCT
easat-4034	289	7	2019	2019	NUM
easat-4034	289	8	,	,	PUNCT
easat-4034	289	9	doi	doi	NOUN
easat-4034	289	10	:	:	PUNCT
easat-4034	289	11	10.1088/1742	10.1088/1742	NUM
easat-4034	289	12	-	-	SYM
easat-4034	289	13	6596/1366/1/012063	6596/1366/1/012063	NUM
easat-4034	289	14	.	.	PUNCT
easat-4034	290	1	[	[	X
easat-4034	290	2	3	3	X
easat-4034	290	3	]	]	X
easat-4034	290	4	g.	g.	PROPN
easat-4034	290	5	ariyanti	ariyanti	PROPN
easat-4034	290	6	,	,	PUNCT
easat-4034	290	7	a.	a.	NOUN
easat-4034	290	8	suparwanto	suparwanto	NOUN
easat-4034	290	9	,	,	PUNCT
easat-4034	290	10	and	and	CCONJ
easat-4034	290	11	b.	b.	PROPN
easat-4034	290	12	surodjo	surodjo	PROPN
easat-4034	290	13	,	,	PUNCT
easat-4034	290	14	“	"	PUNCT
easat-4034	290	15	necessary	necessary	ADJ
easat-4034	290	16	and	and	CCONJ
easat-4034	290	17	sufficient	sufficient	ADJ
easat-4034	290	18	conditions	condition	NOUN
easat-4034	290	19	for	for	ADP
easat-4034	290	20	the	the	DET
easat-4034	290	21	solution	solution	NOUN
easat-4034	290	22	of	of	ADP
easat-4034	290	23	the	the	DET
easat-4034	290	24	linear	linear	ADJ
easat-4034	290	25	balanced	balanced	ADJ
easat-4034	290	26	systems	system	NOUN
easat-4034	290	27	in	in	ADP
easat-4034	290	28	the	the	DET
easat-4034	290	29	symmetrized	symmetrize	VERB
easat-4034	290	30	max	max	PROPN
easat-4034	290	31	plus	plus	CCONJ
easat-4034	290	32	algebra	algebra	PROPN
easat-4034	290	33	,	,	PUNCT
easat-4034	290	34	”	"	PUNCT
easat-4034	290	35	far	far	ADV
easat-4034	290	36	east	east	PROPN
easat-4034	290	37	j.	j.	PROPN
easat-4034	290	38	math	math	PROPN
easat-4034	290	39	.	.	PUNCT
easat-4034	291	1	sci	sci	PROPN
easat-4034	291	2	.	.	PROPN
easat-4034	291	3	,	,	PUNCT
easat-4034	291	4	vol	vol	NOUN
easat-4034	291	5	.	.	PROPN
easat-4034	292	1	97	97	NUM
easat-4034	292	2	,	,	PUNCT
easat-4034	292	3	no	no	INTJ
easat-4034	292	4	.	.	NOUN
easat-4034	292	5	2	2	NUM
easat-4034	292	6	,	,	PUNCT
easat-4034	292	7	pp	pp	ADJ
easat-4034	292	8	.	.	PUNCT
easat-4034	293	1	253–266	253–266	NUM
easat-4034	293	2	,	,	PUNCT
easat-4034	293	3	2015	2015	NUM
easat-4034	293	4	.	.	PUNCT
easat-4034	294	1	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-4034	294	2	9554	9554	NUM
easat-4034	294	3	edelweiss	edelweiss	PROPN
easat-4034	294	4	applied	apply	VERB
easat-4034	294	5	science	science	NOUN
easat-4034	294	6	and	and	CCONJ
easat-4034	294	7	technology	technology	NOUN
easat-4034	294	8	issn	issn	PROPN
easat-4034	294	9	:	:	PUNCT
easat-4034	294	10	2576	2576	NUM
easat-4034	294	11	-	-	SYM
easat-4034	294	12	8484	8484	NUM
easat-4034	294	13	vol	vol	NOUN
easat-4034	294	14	.	.	PROPN
easat-4034	294	15	8	8	NUM
easat-4034	294	16	,	,	PUNCT
easat-4034	294	17	no	no	INTJ
easat-4034	294	18	.	.	NOUN
easat-4034	295	1	6	6	NUM
easat-4034	295	2	:	:	PUNCT
easat-4034	295	3	9544	9544	NUM
easat-4034	295	4	-	-	SYM
easat-4034	295	5	9554	9554	NUM
easat-4034	295	6	,	,	PUNCT
easat-4034	295	7	2024	2024	NUM
easat-4034	295	8	doi	doi	NOUN
easat-4034	295	9	:	:	PUNCT
easat-4034	295	10	10.55214/25768484.v8i6.4034	10.55214/25768484.v8i6.4034	NUM
easat-4034	295	11	©	©	ADP
easat-4034	295	12	2024	2024	NUM
easat-4034	295	13	by	by	ADP
easat-4034	295	14	the	the	DET
easat-4034	295	15	authors	author	NOUN
easat-4034	295	16	;	;	PUNCT
easat-4034	295	17	licensee	licensee	PROPN
easat-4034	295	18	learning	learning	NOUN
easat-4034	295	19	gate	gate	NOUN
easat-4034	296	1	[	[	X
easat-4034	296	2	4	4	NUM
easat-4034	296	3	]	]	X
easat-4034	296	4	g.	g.	NOUN
easat-4034	296	5	ariyanti	ariyanti	PROPN
easat-4034	296	6	and	and	CCONJ
easat-4034	296	7	a.	a.	PROPN
easat-4034	296	8	e.	e.	PROPN
easat-4034	296	9	r.	r.	PROPN
easat-4034	296	10	m.	m.	PROPN
easat-4034	296	11	sari	sari	PROPN
easat-4034	296	12	,	,	PUNCT
easat-4034	296	13	“	"	PUNCT
easat-4034	296	14	the	the	DET
easat-4034	296	15	discrete	discrete	ADJ
easat-4034	296	16	lyapunov	lyapunov	ADJ
easat-4034	296	17	equation	equation	NOUN
easat-4034	296	18	of	of	ADP
easat-4034	296	19	the	the	DET
easat-4034	296	20	orthogonal	orthogonal	ADJ
easat-4034	296	21	matrix	matrix	NOUN
easat-4034	296	22	in	in	ADP
easat-4034	296	23	semiring	semiring	NOUN
easat-4034	296	24	,	,	PUNCT
easat-4034	296	25	”	"	PUNCT
easat-4034	296	26	eur	eur	PROPN
easat-4034	296	27	.	.	PUNCT
easat-4034	297	1	j.	j.	PROPN
easat-4034	297	2	pure	pure	PROPN
easat-4034	297	3	appl	appl	PROPN
easat-4034	297	4	.	.	PUNCT
easat-4034	297	5	math	math	PROPN
easat-4034	297	6	.	.	PUNCT
easat-4034	298	1	,	,	PUNCT
easat-4034	298	2	vol	vol	NOUN
easat-4034	298	3	.	.	PROPN
easat-4034	299	1	16	16	NUM
easat-4034	299	2	,	,	PUNCT
easat-4034	299	3	no	no	INTJ
easat-4034	299	4	.	.	NOUN
easat-4034	299	5	2	2	NUM
easat-4034	299	6	,	,	PUNCT
easat-4034	299	7	pp	pp	ADJ
easat-4034	299	8	.	.	PUNCT
easat-4034	300	1	784–790	784–790	NUM
easat-4034	300	2	,	,	PUNCT
easat-4034	300	3	2023	2023	NUM
easat-4034	300	4	.	.	PUNCT
easat-4034	301	1	[	[	X
easat-4034	301	2	5	5	X
easat-4034	301	3	]	]	PUNCT
easat-4034	301	4	s.	s.	PROPN
easat-4034	301	5	a.	a.	PROPN
easat-4034	301	6	rosyada	rosyada	PROPN
easat-4034	301	7	,	,	PUNCT
easat-4034	301	8	siswanto	siswanto	ADP
easat-4034	301	9	,	,	PUNCT
easat-4034	301	10	and	and	CCONJ
easat-4034	301	11	v.	v.	ADP
easat-4034	301	12	y.	y.	PROPN
easat-4034	301	13	kurniawan	kurniawan	PROPN
easat-4034	301	14	,	,	PUNCT
easat-4034	301	15	“	"	PUNCT
easat-4034	301	16	bases	basis	NOUN
easat-4034	301	17	in	in	ADP
easat-4034	301	18	min	min	ADJ
easat-4034	301	19	-	-	PUNCT
easat-4034	301	20	plus	plus	ADJ
easat-4034	301	21	algebra	algebra	NOUN
easat-4034	301	22	,	,	PUNCT
easat-4034	301	23	”	"	PUNCT
easat-4034	301	24	proc	proc	NOUN
easat-4034	301	25	.	.	PUNCT
easat-4034	302	1	int	int	NOUN
easat-4034	302	2	.	.	PUNCT
easat-4034	302	3	conf	conf	PROPN
easat-4034	302	4	.	.	PUNCT
easat-4034	303	1	math	math	PROPN
easat-4034	303	2	.	.	PUNCT
easat-4034	304	1	math	math	NOUN
easat-4034	304	2	.	.	PUNCT
easat-4034	305	1	educ	educ	PROPN
easat-4034	305	2	.	.	PUNCT
easat-4034	306	1	(	(	PUNCT
easat-4034	306	2	icmme	icmme	PROPN
easat-4034	306	3	2021	2021	NUM
easat-4034	306	4	)	)	PUNCT
easat-4034	306	5	,	,	PUNCT
easat-4034	306	6	vol	vol	NOUN
easat-4034	306	7	.	.	PROPN
easat-4034	306	8	597	597	NUM
easat-4034	306	9	,	,	PUNCT
easat-4034	306	10	pp	pp	ADJ
easat-4034	306	11	.	.	PUNCT
easat-4034	307	1	313–316	313–316	NUM
easat-4034	307	2	,	,	PUNCT
easat-4034	307	3	2021	2021	NUM
easat-4034	307	4	.	.	PUNCT
easat-4034	308	1	[	[	X
easat-4034	308	2	6	6	NUM
easat-4034	308	3	]	]	X
easat-4034	308	4	g.	g.	PROPN
easat-4034	308	5	ariyanti	ariyanti	PROPN
easat-4034	308	6	,	,	PUNCT
easat-4034	308	7	“	"	PUNCT
easat-4034	308	8	necessary	necessary	ADJ
easat-4034	308	9	and	and	CCONJ
easat-4034	308	10	sufficient	sufficient	ADJ
easat-4034	308	11	conditions	condition	NOUN
easat-4034	308	12	for	for	ADP
easat-4034	308	13	the	the	DET
easat-4034	308	14	solutions	solution	NOUN
easat-4034	308	15	of	of	ADP
easat-4034	308	16	linear	linear	ADJ
easat-4034	308	17	equation	equation	NOUN
easat-4034	308	18	system	system	NOUN
easat-4034	308	19	,	,	PUNCT
easat-4034	308	20	”	"	PUNCT
easat-4034	308	21	j.	j.	PROPN
easat-4034	308	22	mat	mat	PROPN
easat-4034	308	23	.	.	PROPN
easat-4034	308	24	stat	stat	PROPN
easat-4034	308	25	.	.	PUNCT
easat-4034	309	1	dan	dan	PROPN
easat-4034	309	2	komputasi	komputasi	PROPN
easat-4034	309	3	,	,	PUNCT
easat-4034	309	4	vol	vol	NOUN
easat-4034	309	5	.	.	PROPN
easat-4034	309	6	17	17	NUM
easat-4034	309	7	,	,	PUNCT
easat-4034	309	8	no	no	INTJ
easat-4034	309	9	.	.	NOUN
easat-4034	309	10	1	1	NUM
easat-4034	309	11	,	,	PUNCT
easat-4034	309	12	pp	pp	ADJ
easat-4034	309	13	.	.	PUNCT
easat-4034	309	14	82–88	82–88	NUM
easat-4034	309	15	,	,	PUNCT
easat-4034	309	16	2020	2020	NUM
easat-4034	309	17	.	.	PUNCT
easat-4034	310	1	[	[	X
easat-4034	310	2	7	7	X
easat-4034	310	3	]	]	X
easat-4034	310	4	g.	g.	NOUN
easat-4034	310	5	ariyanti	ariyanti	PROPN
easat-4034	310	6	,	,	PUNCT
easat-4034	310	7	“	"	PUNCT
easat-4034	310	8	a	a	DET
easat-4034	310	9	note	note	NOUN
easat-4034	310	10	on	on	ADP
easat-4034	310	11	the	the	DET
easat-4034	310	12	solution	solution	NOUN
easat-4034	310	13	of	of	ADP
easat-4034	310	14	the	the	DET
easat-4034	310	15	characteristic	characteristic	ADJ
easat-4034	310	16	equation	equation	NOUN
easat-4034	310	17	over	over	ADP
easat-4034	310	18	the	the	DET
easat-4034	310	19	symmetrized	symmetrize	VERB
easat-4034	310	20	max	max	PROPN
easat-4034	310	21	-	-	PUNCT
easat-4034	310	22	plus	plus	CCONJ
easat-4034	310	23	algebra	algebra	NOUN
easat-4034	310	24	,	,	PUNCT
easat-4034	310	25	”	"	PUNCT
easat-4034	310	26	barekeng	barekeng	PROPN
easat-4034	310	27	j.	j.	PROPN
easat-4034	310	28	ilmu	ilmu	PROPN
easat-4034	310	29	mat	mat	PROPN
easat-4034	310	30	.	.	PUNCT
easat-4034	311	1	dan	dan	PROPN
easat-4034	311	2	terap	terap	PROPN
easat-4034	311	3	.	.	PUNCT
easat-4034	311	4	,	,	PUNCT
easat-4034	311	5	vol	vol	NOUN
easat-4034	311	6	.	.	PROPN
easat-4034	312	1	16	16	NUM
easat-4034	312	2	,	,	PUNCT
easat-4034	312	3	no	no	INTJ
easat-4034	312	4	.	.	NOUN
easat-4034	312	5	4	4	NUM
easat-4034	312	6	,	,	PUNCT
easat-4034	312	7	pp	pp	ADJ
easat-4034	312	8	.	.	PUNCT
easat-4034	313	1	1347–1354	1347–1354	NUM
easat-4034	313	2	,	,	PUNCT
easat-4034	313	3	2022	2022	NUM
easat-4034	313	4	.	.	PUNCT
easat-4034	314	1	[	[	X
easat-4034	314	2	8	8	NUM
easat-4034	314	3	]	]	PUNCT
easat-4034	314	4	s.	s.	PROPN
easat-4034	314	5	siswanto	siswanto	PROPN
easat-4034	314	6	and	and	CCONJ
easat-4034	314	7	a.	a.	NOUN
easat-4034	314	8	gusmizain	gusmizain	PROPN
easat-4034	314	9	,	,	PUNCT
easat-4034	314	10	“	"	PUNCT
easat-4034	314	11	determining	determine	VERB
easat-4034	314	12	the	the	DET
easat-4034	314	13	inverse	inverse	NOUN
easat-4034	314	14	of	of	ADP
easat-4034	314	15	a	a	DET
easat-4034	314	16	matrix	matrix	NOUN
easat-4034	314	17	over	over	ADP
easat-4034	314	18	min	min	ADJ
easat-4034	314	19	-	-	PUNCT
easat-4034	314	20	plus	plus	ADJ
easat-4034	314	21	algebra	algebra	NOUN
easat-4034	314	22	,	,	PUNCT
easat-4034	314	23	”	"	PUNCT
easat-4034	314	24	jtam	jtam	NOUN
easat-4034	314	25	(	(	PUNCT
easat-4034	314	26	jurnal	jurnal	PROPN
easat-4034	314	27	teor	teor	PROPN
easat-4034	314	28	.	.	PUNCT
easat-4034	315	1	dan	dan	PROPN
easat-4034	315	2	apl	apl	PROPN
easat-4034	315	3	.	.	PROPN
easat-4034	315	4	mat	mat	PROPN
easat-4034	315	5	.	.	PROPN
easat-4034	315	6	,	,	PUNCT
easat-4034	315	7	vol	vol	NOUN
easat-4034	315	8	.	.	PROPN
easat-4034	315	9	8	8	NUM
easat-4034	315	10	,	,	PUNCT
easat-4034	315	11	no	no	INTJ
easat-4034	315	12	.	.	NOUN
easat-4034	315	13	1	1	NUM
easat-4034	315	14	,	,	PUNCT
easat-4034	315	15	p.	p.	NOUN
easat-4034	315	16	244	244	NUM
easat-4034	315	17	,	,	PUNCT
easat-4034	315	18	2024	2024	NUM
easat-4034	315	19	.	.	PUNCT
easat-4034	316	1	[	[	X
easat-4034	316	2	9	9	NUM
easat-4034	316	3	]	]	PUNCT
easat-4034	316	4	z.	z.	PROPN
easat-4034	316	5	n.	n.	PROPN
easat-4034	316	6	r.	r.	PROPN
easat-4034	316	7	putri	putri	PROPN
easat-4034	316	8	,	,	PUNCT
easat-4034	316	9	s.	s.	PROPN
easat-4034	316	10	siswanto	siswanto	PROPN
easat-4034	316	11	,	,	PUNCT
easat-4034	316	12	and	and	CCONJ
easat-4034	316	13	v.	v.	ADP
easat-4034	316	14	y.	y.	PROPN
easat-4034	316	15	kurniawan	kurniawan	PROPN
easat-4034	316	16	,	,	PUNCT
easat-4034	316	17	“	"	PUNCT
easat-4034	316	18	cramer	cramer	PROPN
easat-4034	316	19	’s	’s	PART
easat-4034	316	20	rule	rule	NOUN
easat-4034	316	21	in	in	ADP
easat-4034	316	22	min	min	ADJ
easat-4034	316	23	-	-	ADJ
easat-4034	316	24	plus	plus	ADJ
easat-4034	316	25	algebra	algebra	NOUN
easat-4034	316	26	,	,	PUNCT
easat-4034	316	27	”	"	PUNCT
easat-4034	316	28	barekeng	barekeng	PROPN
easat-4034	316	29	j.	j.	PROPN
easat-4034	316	30	ilmu	ilmu	PROPN
easat-4034	316	31	mat	mat	PROPN
easat-4034	316	32	.	.	PUNCT
easat-4034	316	33	dan	dan	PROPN
easat-4034	316	34	terap	terap	PROPN
easat-4034	316	35	.	.	PUNCT
easat-4034	316	36	,	,	PUNCT
easat-4034	316	37	vol	vol	NOUN
easat-4034	316	38	.	.	PROPN
easat-4034	316	39	18	18	NUM
easat-4034	316	40	,	,	PUNCT
easat-4034	316	41	no	no	INTJ
easat-4034	316	42	.	.	NOUN
easat-4034	316	43	2	2	NUM
easat-4034	316	44	,	,	PUNCT
easat-4034	316	45	pp	pp	ADJ
easat-4034	316	46	.	.	PUNCT
easat-4034	317	1	1147–1154	1147–1154	NOUN
easat-4034	317	2	,	,	PUNCT
easat-4034	317	3	2024	2024	NUM
easat-4034	317	4	.	.	PUNCT
easat-4034	318	1	[	[	X
easat-4034	318	2	10	10	NUM
easat-4034	318	3	]	]	X
easat-4034	318	4	s.	s.	PROPN
easat-4034	318	5	maula	maula	PROPN
easat-4034	318	6	and	and	CCONJ
easat-4034	318	7	a.	a.	NOUN
easat-4034	318	8	maghribi	maghribi	NOUN
easat-4034	318	9	,	,	PUNCT
easat-4034	318	10	“	"	PUNCT
easat-4034	318	11	characteristic	characteristic	ADJ
easat-4034	318	12	min	min	ADJ
easat-4034	318	13	-	-	ADJ
easat-4034	318	14	polynomial	polynomial	ADJ
easat-4034	318	15	and	and	CCONJ
easat-4034	318	16	eigen	eigen	PROPN
easat-4034	318	17	problem	problem	NOUN
easat-4034	318	18	of	of	ADP
easat-4034	318	19	a	a	DET
easat-4034	318	20	matrix	matrix	NOUN
easat-4034	318	21	over	over	ADP
easat-4034	318	22	min	min	ADJ
easat-4034	318	23	-	-	PUNCT
easat-4034	318	24	plus	plus	ADJ
easat-4034	318	25	algebra	algebra	NOUN
easat-4034	318	26	,	,	PUNCT
easat-4034	318	27	”	"	PUNCT
easat-4034	318	28	vol	vol	NOUN
easat-4034	318	29	.	.	PROPN
easat-4034	319	1	7	7	NUM
easat-4034	319	2	,	,	PUNCT
easat-4034	319	3	no	no	INTJ
easat-4034	319	4	.	.	NOUN
easat-4034	319	5	4	4	NUM
easat-4034	319	6	,	,	PUNCT
easat-4034	319	7	pp	pp	ADJ
easat-4034	319	8	.	.	PUNCT
easat-4034	320	1	1108–1117	1108–1117	NUM
easat-4034	320	2	,	,	PUNCT
easat-4034	320	3	2023	2023	NUM
easat-4034	320	4	.	.	PUNCT