id	sid	tid	token	lemma	pos
easat-3854	1	1	edelweiss	edelweiss	PROPN
easat-3854	1	2	applied	apply	VERB
easat-3854	1	3	science	science	NOUN
easat-3854	1	4	and	and	CCONJ
easat-3854	1	5	technology	technology	NOUN
easat-3854	1	6	issn	issn	PROPN
easat-3854	1	7	:	:	PUNCT
easat-3854	1	8	2576	2576	NUM
easat-3854	1	9	-	-	SYM
easat-3854	1	10	8484	8484	NUM
easat-3854	1	11	vol	vol	NOUN
easat-3854	1	12	.	.	PROPN
easat-3854	1	13	8	8	NUM
easat-3854	1	14	,	,	PUNCT
easat-3854	1	15	no	no	INTJ
easat-3854	1	16	.	.	NOUN
easat-3854	1	17	6	6	NUM
easat-3854	1	18	,	,	PUNCT
easat-3854	1	19	8658	8658	NUM
easat-3854	1	20	-	-	SYM
easat-3854	1	21	8666	8666	NUM
easat-3854	1	22	2024	2024	NUM
easat-3854	1	23	publisher	publisher	NOUN
easat-3854	1	24	:	:	PUNCT
easat-3854	1	25	learning	learn	VERB
easat-3854	1	26	gate	gate	NOUN
easat-3854	1	27	doi	doi	PROPN
easat-3854	1	28	:	:	PUNCT
easat-3854	1	29	10.55214/25768484.v8i6.3854	10.55214/25768484.v8i6.3854	NUM
easat-3854	1	30	©	©	ADP
easat-3854	1	31	2024	2024	NUM
easat-3854	1	32	by	by	ADP
easat-3854	1	33	the	the	DET
easat-3854	1	34	authors	author	NOUN
easat-3854	1	35	;	;	PUNCT
easat-3854	1	36	licensee	licensee	PROPN
easat-3854	1	37	learning	learning	NOUN
easat-3854	1	38	gate	gate	NOUN
easat-3854	1	39	©	©	PROPN
easat-3854	1	40	2024	2024	NUM
easat-3854	1	41	by	by	ADP
easat-3854	1	42	the	the	DET
easat-3854	1	43	authors	author	NOUN
easat-3854	1	44	;	;	PUNCT
easat-3854	1	45	licensee	licensee	PROPN
easat-3854	1	46	learning	learn	VERB
easat-3854	1	47	gate	gate	NOUN
easat-3854	1	48	*	*	PUNCT
easat-3854	1	49	correspondence	correspondence	NOUN
easat-3854	1	50	:	:	PUNCT
easat-3854	1	51	fatma.elzhraa6590@gmail.com	fatma.elzhraa6590@gmail.com	X
easat-3854	1	52	steenrod	steenrod	NOUN
easat-3854	1	53	operator	operator	NOUN
easat-3854	1	54	of	of	ADP
easat-3854	1	55	the	the	DET
easat-3854	1	56	dihedral	dihedral	ADJ
easat-3854	1	57	homology	homology	NOUN
easat-3854	1	58	of	of	ADP
easat-3854	1	59	a_∞-algebras	a_∞-algebras	ADP
easat-3854	1	60	alaa	alaa	PROPN
easat-3854	1	61	hassan	hassan	PROPN
easat-3854	1	62	noreldeen1	noreldeen1	PROPN
easat-3854	1	63	,	,	PUNCT
easat-3854	1	64	fatma	fatma	PROPN
easat-3854	1	65	ahmed	ahme	VERB
easat-3854	1	66	mohammed2	mohammed2	PROPN
easat-3854	1	67	*	*	PROPN
easat-3854	1	68	,	,	PUNCT
easat-3854	1	69	faten	faten	ADJ
easat-3854	1	70	ragab	ragab	ADJ
easat-3854	1	71	karar3	karar3	PROPN
easat-3854	1	72	1,2,3department	1,2,3department	NUM
easat-3854	1	73	of	of	ADP
easat-3854	1	74	mathematics	mathematic	NOUN
easat-3854	1	75	,	,	PUNCT
easat-3854	1	76	faculty	faculty	NOUN
easat-3854	1	77	of	of	ADP
easat-3854	1	78	science	science	NOUN
easat-3854	1	79	,	,	PUNCT
easat-3854	1	80	aswan	aswan	PROPN
easat-3854	1	81	university	university	PROPN
easat-3854	1	82	,	,	PUNCT
easat-3854	1	83	aswan	aswan	PROPN
easat-3854	1	84	,	,	PUNCT
easat-3854	1	85	egypt	egypt	PROPN
easat-3854	1	86	;	;	PUNCT
easat-3854	1	87	ala2222000@yahoo.com	ala2222000@yahoo.com	X
easat-3854	1	88	(	(	PUNCT
easat-3854	1	89	a.h.n	a.h.n	ADJ
easat-3854	1	90	.	.	PUNCT
easat-3854	1	91	)	)	PUNCT
easat-3854	2	1	fatma.elzhraa6590@gmail.com	fatma.elzhraa6590@gmail.com	PROPN
easat-3854	2	2	(	(	PUNCT
easat-3854	2	3	f.a.m	f.a.m	ADJ
easat-3854	2	4	.	.	PUNCT
easat-3854	2	5	)	)	PUNCT
easat-3854	2	6	fatenragab2020@yahoo.com	fatenragab2020@yahoo.com	NOUN
easat-3854	3	1	(	(	PUNCT
easat-3854	3	2	f.r.k	f.r.k	ADJ
easat-3854	3	3	.	.	PUNCT
easat-3854	3	4	)	)	PUNCT
easat-3854	3	5	.	.	PUNCT
easat-3854	4	1	abstract	abstract	ADV
easat-3854	4	2	:	:	PUNCT
easat-3854	4	3	in	in	ADP
easat-3854	4	4	this	this	DET
easat-3854	4	5	study	study	NOUN
easat-3854	4	6	,	,	PUNCT
easat-3854	4	7	we	we	PRON
easat-3854	4	8	examine	examine	VERB
easat-3854	4	9	the	the	DET
easat-3854	4	10	interaction	interaction	NOUN
easat-3854	4	11	between	between	ADP
easat-3854	4	12	steenrod	steenrod	NOUN
easat-3854	4	13	operations	operation	NOUN
easat-3854	4	14	and	and	CCONJ
easat-3854	4	15	dihedral	dihedral	ADJ
easat-3854	4	16	homology	homology	NOUN
easat-3854	4	17	within	within	ADP
easat-3854	4	18	the	the	DET
easat-3854	4	19	framework	framework	NOUN
easat-3854	4	20	of	of	ADP
easat-3854	4	21	a	a	DET
easat-3854	4	22	-	-	PUNCT
easat-3854	4	23	infinity	infinity	NOUN
easat-3854	4	24	algebras	algebra	NOUN
easat-3854	4	25	,	,	PUNCT
easat-3854	4	26	aiming	aim	VERB
easat-3854	4	27	to	to	PART
easat-3854	4	28	understand	understand	VERB
easat-3854	4	29	how	how	SCONJ
easat-3854	4	30	these	these	DET
easat-3854	4	31	operations	operation	NOUN
easat-3854	4	32	,	,	PUNCT
easat-3854	4	33	initial	initial	ADJ
easat-3854	4	34	in	in	ADP
easat-3854	4	35	algebraic	algebraic	PROPN
easat-3854	4	36	topology	topology	NOUN
easat-3854	4	37	,	,	PUNCT
easat-3854	4	38	contribute	contribute	VERB
easat-3854	4	39	to	to	ADP
easat-3854	4	40	homological	homological	ADJ
easat-3854	4	41	invariants	invariant	NOUN
easat-3854	4	42	and	and	CCONJ
easat-3854	4	43	influence	influence	NOUN
easat-3854	4	44	dihedral	dihedral	ADJ
easat-3854	4	45	homology	homology	NOUN
easat-3854	4	46	structures	structure	NOUN
easat-3854	4	47	.	.	PUNCT
easat-3854	5	1	we	we	PRON
easat-3854	5	2	begin	begin	VERB
easat-3854	5	3	by	by	ADP
easat-3854	5	4	exploring	explore	VERB
easat-3854	5	5	the	the	DET
easat-3854	5	6	initial	initial	ADJ
easat-3854	5	7	properties	property	NOUN
easat-3854	5	8	of	of	ADP
easat-3854	5	9	a	a	DET
easat-3854	5	10	-	-	PUNCT
easat-3854	5	11	infinity	infinity	NOUN
easat-3854	5	12	algebras	algebra	NOUN
easat-3854	5	13	as	as	ADP
easat-3854	5	14	generalizations	generalization	NOUN
easat-3854	5	15	of	of	ADP
easat-3854	5	16	associative	associative	ADJ
easat-3854	5	17	algebras	algebra	NOUN
easat-3854	5	18	,	,	PUNCT
easat-3854	5	19	focusing	focus	VERB
easat-3854	5	20	on	on	ADP
easat-3854	5	21	their	their	PRON
easat-3854	5	22	volume	volume	NOUN
easat-3854	5	23	to	to	PART
easat-3854	5	24	support	support	VERB
easat-3854	5	25	steenrod	steenrod	NOUN
easat-3854	5	26	operations	operation	NOUN
easat-3854	5	27	.	.	PUNCT
easat-3854	6	1	from	from	ADP
easat-3854	6	2	there	there	ADV
easat-3854	6	3	,	,	PUNCT
easat-3854	6	4	we	we	PRON
easat-3854	6	5	delve	delve	VERB
easat-3854	6	6	into	into	ADP
easat-3854	6	7	the	the	DET
easat-3854	6	8	effects	effect	NOUN
easat-3854	6	9	of	of	ADP
easat-3854	6	10	these	these	DET
easat-3854	6	11	operations	operation	NOUN
easat-3854	6	12	within	within	ADP
easat-3854	6	13	dihedral	dihedral	ADJ
easat-3854	6	14	homology	homology	NOUN
easat-3854	6	15	,	,	PUNCT
easat-3854	6	16	uncovering	uncover	VERB
easat-3854	6	17	their	their	PRON
easat-3854	6	18	role	role	NOUN
easat-3854	6	19	in	in	ADP
easat-3854	6	20	revealing	reveal	VERB
easat-3854	6	21	deeper	deep	ADJ
easat-3854	6	22	algebraic	algebraic	ADJ
easat-3854	6	23	structures	structure	NOUN
easat-3854	6	24	and	and	CCONJ
easat-3854	6	25	improving	improve	VERB
easat-3854	6	26	our	our	PRON
easat-3854	6	27	understanding	understanding	NOUN
easat-3854	6	28	of	of	ADP
easat-3854	6	29	homology	homology	NOUN
easat-3854	6	30	theories	theory	NOUN
easat-3854	6	31	.	.	PUNCT
easat-3854	7	1	we	we	PRON
easat-3854	7	2	also	also	ADV
easat-3854	7	3	analyze	analyze	VERB
easat-3854	7	4	the	the	DET
easat-3854	7	5	connections	connection	NOUN
easat-3854	7	6	between	between	ADP
easat-3854	7	7	steenrod	steenrod	NOUN
easat-3854	7	8	operations	operation	NOUN
easat-3854	7	9	and	and	CCONJ
easat-3854	7	10	projective	projective	ADJ
easat-3854	7	11	varieties	variety	NOUN
easat-3854	7	12	over	over	ADP
easat-3854	7	13	finite	finite	ADJ
easat-3854	7	14	fields	field	NOUN
easat-3854	7	15	,	,	PUNCT
easat-3854	7	16	emphasizing	emphasize	VERB
easat-3854	7	17	their	their	PRON
easat-3854	7	18	actions	action	NOUN
easat-3854	7	19	in	in	ADP
easat-3854	7	20	derived	derived	ADJ
easat-3854	7	21	categories	category	NOUN
easat-3854	7	22	and	and	CCONJ
easat-3854	7	23	their	their	PRON
easat-3854	7	24	significance	significance	NOUN
easat-3854	7	25	in	in	ADP
easat-3854	7	26	the	the	DET
easat-3854	7	27	context	context	NOUN
easat-3854	7	28	of	of	ADP
easat-3854	7	29	α	α	NOUN
easat-3854	7	30	-	-	ADJ
easat-3854	7	31	characteristic	characteristic	ADJ
easat-3854	7	32	fields	field	NOUN
easat-3854	7	33	.	.	PUNCT
easat-3854	8	1	by	by	ADP
easat-3854	8	2	defining	define	VERB
easat-3854	8	3	steenrod	steenrod	NOUN
easat-3854	8	4	operators	operator	NOUN
easat-3854	8	5	within	within	ADP
easat-3854	8	6	dihedral	dihedral	ADJ
easat-3854	8	7	homology	homology	NOUN
easat-3854	8	8	,	,	PUNCT
easat-3854	8	9	we	we	PRON
easat-3854	8	10	explain	explain	VERB
easat-3854	8	11	the	the	DET
easat-3854	8	12	complex	complex	ADJ
easat-3854	8	13	relationships	relationship	NOUN
easat-3854	8	14	between	between	ADP
easat-3854	8	15	these	these	DET
easat-3854	8	16	algebraic	algebraic	ADJ
easat-3854	8	17	structures	structure	NOUN
easat-3854	8	18	and	and	CCONJ
easat-3854	8	19	operations	operation	NOUN
easat-3854	8	20	derived	derive	VERB
easat-3854	8	21	from	from	ADP
easat-3854	8	22	homology	homology	NOUN
easat-3854	8	23	theory	theory	NOUN
easat-3854	8	24	.	.	PUNCT
easat-3854	9	1	through	through	ADP
easat-3854	9	2	specific	specific	ADJ
easat-3854	9	3	examples	example	NOUN
easat-3854	9	4	and	and	CCONJ
easat-3854	9	5	theoretical	theoretical	ADJ
easat-3854	9	6	models	model	NOUN
easat-3854	9	7	,	,	PUNCT
easat-3854	9	8	we	we	PRON
easat-3854	9	9	demonstrate	demonstrate	VERB
easat-3854	9	10	how	how	SCONJ
easat-3854	9	11	these	these	DET
easat-3854	9	12	interactions	interaction	NOUN
easat-3854	9	13	advance	advance	VERB
easat-3854	9	14	our	our	PRON
easat-3854	9	15	understanding	understanding	NOUN
easat-3854	9	16	of	of	ADP
easat-3854	9	17	homological	homological	ADJ
easat-3854	9	18	invariants	invariant	NOUN
easat-3854	9	19	and	and	CCONJ
easat-3854	9	20	provide	provide	VERB
easat-3854	9	21	valuable	valuable	ADJ
easat-3854	9	22	tools	tool	NOUN
easat-3854	9	23	and	and	CCONJ
easat-3854	9	24	perspectives	perspective	NOUN
easat-3854	9	25	for	for	ADP
easat-3854	9	26	the	the	DET
easat-3854	9	27	broader	broad	ADJ
easat-3854	9	28	fields	field	NOUN
easat-3854	9	29	of	of	ADP
easat-3854	9	30	algebraic	algebraic	ADJ
easat-3854	9	31	topology	topology	NOUN
easat-3854	9	32	and	and	CCONJ
easat-3854	9	33	homological	homological	ADJ
easat-3854	9	34	algebra	algebra	NOUN
easat-3854	9	35	.	.	PUNCT
easat-3854	10	1	keywords	keyword	NOUN
easat-3854	10	2	:	:	PUNCT
easat-3854	10	3	a_∞algebras	a_∞algebras	PROPN
easat-3854	10	4	,	,	PUNCT
easat-3854	10	5	adams	adams	PROPN
easat-3854	10	6	operations	operation	NOUN
easat-3854	10	7	,	,	PUNCT
easat-3854	10	8	dihedral	dihedral	NOUN
easat-3854	10	9	,	,	PUNCT
easat-3854	10	10	homology	homology	NOUN
easat-3854	10	11	,	,	PUNCT
easat-3854	10	12	steenrod	steenrod	NOUN
easat-3854	10	13	algebra	algebra	PROPN
easat-3854	10	14	.	.	PUNCT
easat-3854	11	1	jel	jel	PROPN
easat-3854	11	2	classification	classification	NOUN
easat-3854	11	3	:	:	PUNCT
easat-3854	11	4	primary	primary	ADJ
easat-3854	11	5	55n	55n	NOUN
easat-3854	11	6	91	91	NUM
easat-3854	11	7	;	;	PUNCT
easat-3854	11	8	55n20	55n20	NUM
easat-3854	11	9	;	;	PUNCT
easat-3854	11	10	13d03	13d03	NUM
easat-3854	11	11	;	;	PUNCT
easat-3854	11	12	16e40	16e40	NUM
easat-3854	11	13	.	.	PUNCT
easat-3854	12	1	1	1	X
easat-3854	12	2	.	.	X
easat-3854	12	3	introduction	introduction	NOUN
easat-3854	12	4	the	the	DET
easat-3854	12	5	steenrod	steenrod	NOUN
easat-3854	12	6	operations	operation	NOUN
easat-3854	12	7	were	be	AUX
easat-3854	12	8	at	at	ADP
easat-3854	12	9	first	first	ADV
easat-3854	12	10	presented	present	VERB
easat-3854	12	11	in	in	ADP
easat-3854	12	12	the	the	DET
easat-3854	12	13	algebraic	algebraic	ADJ
easat-3854	12	14	topology	topology	NOUN
easat-3854	12	15	over	over	ADP
easat-3854	12	16	the	the	DET
easat-3854	12	17	late	late	ADJ
easat-3854	12	18	1930s	1930	NOUN
easat-3854	12	19	.	.	PUNCT
easat-3854	13	1	this	this	DET
easat-3854	13	2	operation	operation	NOUN
easat-3854	13	3	was	be	AUX
easat-3854	13	4	performed	perform	VERB
easat-3854	13	5	on	on	ADP
easat-3854	13	6	the	the	DET
easat-3854	13	7	topological	topological	ADJ
easat-3854	13	8	spaces	space	NOUN
easat-3854	13	9	'	'	PART
easat-3854	13	10	modulo	modulo	PROPN
easat-3854	13	11	𝛼	𝛼	X
easat-3854	13	12	homology	homology	NOUN
easat-3854	13	13	.	.	PUNCT
easat-3854	14	1	these	these	DET
easat-3854	14	2	operations	operation	NOUN
easat-3854	14	3	have	have	AUX
easat-3854	14	4	used	use	VERB
easat-3854	14	5	to	to	PART
easat-3854	14	6	verify	verify	VERB
easat-3854	14	7	a	a	DET
easat-3854	14	8	number	number	NOUN
easat-3854	14	9	of	of	ADP
easat-3854	14	10	conclusions	conclusion	NOUN
easat-3854	14	11	in	in	ADP
easat-3854	14	12	the	the	DET
easat-3854	14	13	algebraic	algebraic	ADJ
easat-3854	14	14	topology	topology	NOUN
easat-3854	14	15	.	.	PUNCT
easat-3854	15	1	later	later	ADV
easat-3854	15	2	,	,	PUNCT
easat-3854	15	3	they	they	PRON
easat-3854	15	4	were	be	AUX
easat-3854	15	5	employed	employ	VERB
easat-3854	15	6	in	in	ADP
easat-3854	15	7	new	new	ADJ
easat-3854	15	8	ways	way	NOUN
easat-3854	15	9	,	,	PUNCT
easat-3854	15	10	for	for	ADP
easat-3854	15	11	as	as	ADV
easat-3854	15	12	when	when	SCONJ
easat-3854	15	13	discussing	discuss	VERB
easat-3854	15	14	the	the	DET
easat-3854	15	15	sullivan	sullivan	NOUN
easat-3854	15	16	conjecture	conjecture	NOUN
easat-3854	15	17	or	or	CCONJ
easat-3854	15	18	the	the	DET
easat-3854	15	19	adams	adams	PROPN
easat-3854	15	20	spectral	spectral	ADJ
easat-3854	15	21	sequence	sequence	NOUN
easat-3854	15	22	.	.	PUNCT
easat-3854	16	1	the	the	DET
easat-3854	16	2	operations	operation	NOUN
easat-3854	16	3	were	be	AUX
easat-3854	16	4	soon	soon	ADV
easat-3854	16	5	used	use	VERB
easat-3854	16	6	to	to	ADP
easat-3854	16	7	the	the	DET
easat-3854	16	8	study	study	NOUN
easat-3854	16	9	of	of	ADP
easat-3854	16	10	projective	projective	ADJ
easat-3854	16	11	homogeneous	homogeneous	ADJ
easat-3854	16	12	types	type	NOUN
easat-3854	16	13	in	in	ADP
easat-3854	16	14	algebraic	algebraic	ADJ
easat-3854	16	15	geometry	geometry	NOUN
easat-3854	16	16	.	.	PUNCT
easat-3854	17	1	although	although	SCONJ
easat-3854	17	2	steenrod	steenrod	NOUN
easat-3854	17	3	operations	operation	NOUN
easat-3854	17	4	modulo𝛼	modulo𝛼	PROPN
easat-3854	17	5	,	,	PUNCT
easat-3854	17	6	which	which	PRON
easat-3854	17	7	operate	operate	VERB
easat-3854	17	8	over	over	ADP
easat-3854	17	9	fields	field	NOUN
easat-3854	17	10	of	of	ADP
easat-3854	17	11	characteristic	characteristic	ADJ
easat-3854	17	12	𝛼	𝛼	NOUN
easat-3854	17	13	,	,	PUNCT
easat-3854	17	14	do	do	AUX
easat-3854	17	15	not	not	PART
easat-3854	17	16	yet	yet	ADV
easat-3854	17	17	exist	exist	VERB
easat-3854	17	18	,	,	PUNCT
easat-3854	17	19	this	this	PRON
easat-3854	17	20	is	be	AUX
easat-3854	17	21	due	due	ADJ
easat-3854	17	22	to	to	ADP
easat-3854	17	23	a	a	DET
easat-3854	17	24	number	number	NOUN
easat-3854	17	25	of	of	ADP
easat-3854	17	26	factors	factor	NOUN
easat-3854	17	27	.	.	PUNCT
easat-3854	18	1	as	as	ADP
easat-3854	18	2	consequence	consequence	NOUN
easat-3854	18	3	,	,	PUNCT
easat-3854	18	4	given	give	VERB
easat-3854	18	5	specific	specific	ADJ
easat-3854	18	6	values	value	NOUN
easat-3854	18	7	of	of	ADP
easat-3854	18	8	a	a	DET
easat-3854	18	9	characteristic	characteristic	NOUN
easat-3854	18	10	of	of	ADP
easat-3854	18	11	the	the	DET
easat-3854	18	12	basic	basic	ADJ
easat-3854	18	13	field	field	NOUN
easat-3854	18	14	,	,	PUNCT
easat-3854	18	15	a	a	DET
easat-3854	18	16	number	number	NOUN
easat-3854	18	17	of	of	ADP
easat-3854	18	18	significant	significant	ADJ
easat-3854	18	19	concerns	concern	NOUN
easat-3854	18	20	surrounding	surround	VERB
easat-3854	18	21	projective	projective	ADJ
easat-3854	18	22	homogeneous	homogeneous	ADJ
easat-3854	18	23	varieties	variety	NOUN
easat-3854	18	24	remain	remain	VERB
easat-3854	18	25	unanswered	unanswered	ADJ
easat-3854	18	26	.	.	PUNCT
easat-3854	19	1	for	for	ADP
easat-3854	19	2	example	example	NOUN
easat-3854	19	3	,	,	PUNCT
easat-3854	19	4	some	some	PRON
easat-3854	19	5	of	of	ADP
easat-3854	19	6	the	the	DET
easat-3854	19	7	most	most	ADV
easat-3854	19	8	complex	complex	ADJ
easat-3854	19	9	quadratic	quadratic	ADJ
easat-3854	19	10	form	form	NOUN
easat-3854	19	11	theorems	theorem	NOUN
easat-3854	19	12	are	be	AUX
easat-3854	19	13	undefined	undefined	ADJ
easat-3854	19	14	when	when	SCONJ
easat-3854	19	15	the	the	DET
easat-3854	19	16	base	base	NOUN
easat-3854	19	17	domain	domain	NOUN
easat-3854	19	18	of	of	ADP
easat-3854	19	19	characteristic	characteristic	NOUN
easat-3854	19	20	is	be	AUX
easat-3854	19	21	two	two	NUM
easat-3854	19	22	.	.	PUNCT
easat-3854	20	1	the	the	DET
easat-3854	20	2	steenrod	steenrod	NOUN
easat-3854	20	3	operations	operation	NOUN
easat-3854	20	4	constructions	construction	NOUN
easat-3854	20	5	aimed	aim	VERB
easat-3854	20	6	at	at	ADP
easat-3854	20	7	chow	chow	PROPN
easat-3854	20	8	groups	group	NOUN
easat-3854	20	9	modulo	modulo	VERB
easat-3854	20	10	the	the	DET
easat-3854	20	11	major	major	ADJ
easat-3854	20	12	number	number	NOUN
easat-3854	20	13	𝛼	𝛼	NOUN
easat-3854	20	14	designated	designate	VERB
easat-3854	20	15	in	in	ADP
easat-3854	20	16	(	(	PUNCT
easat-3854	20	17	[	[	X
easat-3854	20	18	1	1	NUM
easat-3854	20	19	]	]	PUNCT
easat-3854	20	20	,	,	PUNCT
easat-3854	20	21	[	[	X
easat-3854	20	22	2	2	NUM
easat-3854	20	23	]	]	PUNCT
easat-3854	20	24	)	)	PUNCT
easat-3854	20	25	.	.	PUNCT
easat-3854	21	1	like	like	ADP
easat-3854	21	2	steenrod	steenrod	NOUN
easat-3854	21	3	's	's	PART
easat-3854	21	4	initial	initial	ADJ
easat-3854	21	5	construction	construction	NOUN
easat-3854	21	6	,	,	PUNCT
easat-3854	21	7	they	they	PRON
easat-3854	21	8	all	all	PRON
easat-3854	21	9	include	include	VERB
easat-3854	21	10	taking	take	VERB
easat-3854	21	11	into	into	ADP
easat-3854	21	12	the	the	DET
easat-3854	21	13	action	action	NOUN
easat-3854	21	14	how	how	SCONJ
easat-3854	21	15	a	a	DET
easat-3854	21	16	cyclic	cyclic	ADJ
easat-3854	21	17	group	group	NOUN
easat-3854	21	18	of	of	ADP
easat-3854	21	19	order	order	NOUN
easat-3854	21	20	𝛼	𝛼	PRON
easat-3854	21	21	affects	affect	VERB
easat-3854	21	22	the	the	DET
easat-3854	21	23	result	result	NOUN
easat-3854	21	24	of	of	ADP
easat-3854	21	25	𝛼	𝛼	PRON
easat-3854	21	26	replicas	replica	NOUN
easat-3854	21	27	of	of	ADP
easat-3854	21	28	a	a	DET
easat-3854	21	29	particular	particular	ADJ
easat-3854	21	30	scheme	scheme	NOUN
easat-3854	21	31	.	.	PUNCT
easat-3854	22	1	specifically	specifically	ADV
easat-3854	22	2	on	on	ADP
easat-3854	22	3	a	a	DET
easat-3854	22	4	square	square	NOUN
easat-3854	22	5	of	of	ADP
easat-3854	22	6	the	the	DET
easat-3854	22	7	particular	particular	ADJ
easat-3854	22	8	projective	projective	ADJ
easat-3854	22	9	homogeneous	homogeneous	ADJ
easat-3854	22	10	variation	variation	NOUN
easat-3854	22	11	,	,	PUNCT
easat-3854	22	12	the	the	DET
easat-3854	22	13	steenrod	steenrod	NOUN
easat-3854	22	14	operations	operation	NOUN
easat-3854	22	15	are	be	AUX
easat-3854	22	16	employed	employ	VERB
easat-3854	22	17	as	as	ADP
easat-3854	22	18	a	a	DET
easat-3854	22	19	means	means	NOUN
easat-3854	22	20	of	of	ADP
easat-3854	22	21	generating	generate	VERB
easat-3854	22	22	new	new	ADJ
easat-3854	22	23	algebraic	algebraic	ADJ
easat-3854	22	24	cycles	cycle	NOUN
easat-3854	22	25	and	and	CCONJ
easat-3854	22	26	offering	offer	VERB
easat-3854	22	27	motives	motive	VERB
easat-3854	22	28	decomposition	decomposition	NOUN
easat-3854	22	29	of	of	ADP
easat-3854	22	30	this	this	DET
easat-3854	22	31	variation	variation	NOUN
easat-3854	22	32	.	.	PUNCT
easat-3854	23	1	according	accord	VERB
easat-3854	23	2	to	to	ADP
easat-3854	23	3	the	the	DET
easat-3854	23	4	rost	rost	PROPN
easat-3854	23	5	nilpotence	nilpotence	PROPN
easat-3854	23	6	theorem	theorem	PROPN
easat-3854	23	7	,	,	PUNCT
easat-3854	23	8	the	the	DET
easat-3854	23	9	identical	identical	ADJ
easat-3854	23	10	conclusion	conclusion	NOUN
easat-3854	23	11	obtained	obtain	VERB
easat-3854	23	12	using	use	VERB
easat-3854	23	13	merely	merely	ADV
easat-3854	23	14	reduced	reduce	VERB
easat-3854	23	15	steenrod	steenrod	NOUN
easat-3854	23	16	operations	operation	NOUN
easat-3854	23	17	.	.	PUNCT
easat-3854	24	1	algebraic	algebraic	ADJ
easat-3854	24	2	topology	topology	NOUN
easat-3854	24	3	generally	generally	ADV
easat-3854	24	4	aims	aim	VERB
easat-3854	24	5	to	to	PART
easat-3854	24	6	offer	offer	VERB
easat-3854	24	7	algebraic	algebraic	ADJ
easat-3854	24	8	techniques	technique	NOUN
easat-3854	24	9	to	to	PART
easat-3854	24	10	extricate	extricate	VERB
easat-3854	24	11	topological	topological	ADJ
easat-3854	24	12	spaces	space	NOUN
easat-3854	24	13	.	.	PUNCT
easat-3854	25	1	one	one	NUM
easat-3854	25	2	such	such	ADJ
easat-3854	25	3	diagram	diagram	NOUN
easat-3854	25	4	that	that	PRON
easat-3854	25	5	turns	turn	VERB
easat-3854	25	6	out	out	ADP
easat-3854	25	7	to	to	PART
easat-3854	25	8	be	be	AUX
easat-3854	25	9	very	very	ADV
easat-3854	25	10	exciting	exciting	ADJ
easat-3854	25	11	is	be	AUX
easat-3854	25	12	the	the	DET
easat-3854	25	13	infinity	infinity	NOUN
easat-3854	25	14	homology	homology	NOUN
easat-3854	25	15	𝐻•(𝑉,𝒜	𝐻•(𝑉,𝒜	PROPN
easat-3854	25	16	)	)	PUNCT
easat-3854	25	17	for	for	ADP
easat-3854	25	18	a	a	DET
easat-3854	25	19	space	space	NOUN
easat-3854	25	20	𝑉.	𝑉.	NOUN
easat-3854	25	21	in	in	ADP
easat-3854	25	22	this	this	DET
easat-3854	25	23	case	case	NOUN
easat-3854	25	24	,	,	PUNCT
easat-3854	25	25	we	we	PRON
easat-3854	25	26	obtain	obtain	VERB
easat-3854	25	27	an	an	DET
easat-3854	25	28	additional	additional	ADJ
easat-3854	25	29	infinity	infinity	NOUN
easat-3854	25	30	algebras	algebras	NOUN
easat-3854	25	31	structure	structure	NOUN
easat-3854	25	32	not	not	PART
easat-3854	25	33	immediately	immediately	ADV
easat-3854	25	34	offered	offer	VERB
easat-3854	25	35	by	by	ADP
easat-3854	25	36	homology	homology	NOUN
easat-3854	25	37	groups	group	NOUN
easat-3854	25	38	,	,	PUNCT
easat-3854	25	39	and	and	CCONJ
easat-3854	25	40	we	we	PRON
easat-3854	25	41	are	be	AUX
easat-3854	25	42	able	able	ADJ
easat-3854	25	43	to	to	PART
easat-3854	25	44	compute	compute	VERB
easat-3854	25	45	this	this	DET
easat-3854	25	46	algebra	algebra	NOUN
easat-3854	25	47	more	more	ADV
easat-3854	25	48	easily	easily	ADV
easat-3854	25	49	than	than	SCONJ
easat-3854	25	50	we	we	PRON
easat-3854	25	51	can	can	AUX
easat-3854	25	52	with	with	ADP
easat-3854	25	53	homotopy	homotopy	NOUN
easat-3854	25	54	groups	group	NOUN
easat-3854	25	55	.	.	PUNCT
easat-3854	26	1	8659	8659	NUM
easat-3854	26	2	edelweiss	edelweiss	PROPN
easat-3854	26	3	applied	apply	VERB
easat-3854	26	4	science	science	NOUN
easat-3854	26	5	and	and	CCONJ
easat-3854	26	6	technology	technology	NOUN
easat-3854	26	7	issn	issn	PROPN
easat-3854	26	8	:	:	PUNCT
easat-3854	26	9	2576	2576	NUM
easat-3854	26	10	-	-	SYM
easat-3854	26	11	8484	8484	NUM
easat-3854	26	12	vol	vol	NOUN
easat-3854	26	13	.	.	PROPN
easat-3854	26	14	8	8	NUM
easat-3854	26	15	,	,	PUNCT
easat-3854	26	16	no	no	INTJ
easat-3854	26	17	.	.	NOUN
easat-3854	27	1	6	6	NUM
easat-3854	27	2	:	:	SYM
easat-3854	27	3	8658	8658	NUM
easat-3854	27	4	-	-	SYM
easat-3854	27	5	8666	8666	NUM
easat-3854	27	6	,	,	PUNCT
easat-3854	27	7	2024	2024	NUM
easat-3854	27	8	doi	doi	NOUN
easat-3854	27	9	:	:	PUNCT
easat-3854	27	10	10.55214/25768484.v8i6.3854	10.55214/25768484.v8i6.3854	NUM
easat-3854	27	11	©	©	ADP
easat-3854	27	12	2024	2024	NUM
easat-3854	27	13	by	by	ADP
easat-3854	27	14	the	the	DET
easat-3854	27	15	authors	author	NOUN
easat-3854	27	16	;	;	PUNCT
easat-3854	27	17	licensee	licensee	PROPN
easat-3854	27	18	learning	learning	NOUN
easat-3854	27	19	gate	gate	NOUN
easat-3854	27	20	furthermore	furthermore	ADV
easat-3854	27	21	,	,	PUNCT
easat-3854	27	22	it	it	PRON
easat-3854	27	23	goes	go	VERB
easat-3854	27	24	out	out	ADP
easat-3854	27	25	that	that	SCONJ
easat-3854	27	26	if	if	SCONJ
easat-3854	27	27	we	we	PRON
easat-3854	27	28	select	select	VERB
easat-3854	27	29	the	the	DET
easat-3854	27	30	constant	constant	ADJ
easat-3854	27	31	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-3854	27	32	ℳ	ℳ	NOUN
easat-3854	27	33	to	to	PART
easat-3854	27	34	be	be	AUX
easat-3854	27	35	one	one	NUM
easat-3854	27	36	of	of	ADP
easat-3854	27	37	infinite	infinite	ADJ
easat-3854	27	38	fields	field	NOUN
easat-3854	27	39	with	with	ADP
easat-3854	27	40	the	the	DET
easat-3854	27	41	form	form	NOUN
easat-3854	27	42	𝐹𝛼	𝐹𝛼	PROPN
easat-3854	27	43	,	,	PUNCT
easat-3854	27	44	for	for	ADP
easat-3854	27	45	a	a	DET
easat-3854	27	46	prime	prime	ADJ
easat-3854	27	47	𝛼	𝛼	NOUN
easat-3854	27	48	,	,	PUNCT
easat-3854	27	49	we	we	PRON
easat-3854	27	50	have	have	VERB
easat-3854	27	51	even	even	ADV
easat-3854	27	52	additional	additional	ADJ
easat-3854	27	53	structure	structure	NOUN
easat-3854	27	54	.	.	PUNCT
easat-3854	28	1	in	in	ADP
easat-3854	28	2	this	this	DET
easat-3854	28	3	instance	instance	NOUN
easat-3854	28	4	,	,	PUNCT
easat-3854	28	5	steenrod	steenrod	NOUN
easat-3854	28	6	presented	present	VERB
easat-3854	28	7	the	the	DET
easat-3854	28	8	stable	stable	ADJ
easat-3854	28	9	homology	homology	NOUN
easat-3854	28	10	operations	operation	NOUN
easat-3854	28	11	,	,	PUNCT
easat-3854	28	12	which	which	PRON
easat-3854	28	13	are	be	AUX
easat-3854	28	14	natural	natural	ADJ
easat-3854	28	15	transformations	transformation	NOUN
easat-3854	28	16	𝜓:𝐻𝑛(−	𝜓:𝐻𝑛(−	NOUN
easat-3854	28	17	,	,	PUNCT
easat-3854	28	18	𝐹𝛼	𝐹𝛼	NOUN
easat-3854	28	19	)	)	PUNCT
easat-3854	28	20	→	→	NOUN
easat-3854	28	21	𝐻𝑚(−	𝐻𝑚(−	NOUN
easat-3854	28	22	,	,	PUNCT
easat-3854	28	23	𝐹𝛼	𝐹𝛼	PROPN
easat-3854	28	24	)	)	PUNCT
easat-3854	28	25	with	with	ADP
easat-3854	28	26	specific	specific	ADJ
easat-3854	28	27	properties	property	NOUN
easat-3854	28	28	that	that	PRON
easat-3854	28	29	it	it	PRON
easat-3854	28	30	turns	turn	VERB
easat-3854	28	31	out	out	ADP
easat-3854	28	32	combine	combine	NOUN
easat-3854	28	33	to	to	PART
easat-3854	28	34	produce	produce	VERB
easat-3854	28	35	an	an	DET
easat-3854	28	36	infinity	infinity	NOUN
easat-3854	28	37	algebra	algebra	NOUN
easat-3854	28	38	.	.	PUNCT
easat-3854	29	1	without	without	ADP
easat-3854	29	2	ever	ever	ADV
easat-3854	29	3	taking	take	VERB
easat-3854	29	4	into	into	ADP
easat-3854	29	5	account	account	NOUN
easat-3854	29	6	topological	topological	ADJ
easat-3854	29	7	spaces	space	NOUN
easat-3854	29	8	,	,	PUNCT
easat-3854	29	9	it	it	PRON
easat-3854	29	10	is	be	AUX
easat-3854	29	11	possible	possible	ADJ
easat-3854	29	12	to	to	PART
easat-3854	29	13	study	study	VERB
easat-3854	29	14	the	the	DET
easat-3854	29	15	steenrod	steenrod	NOUN
easat-3854	29	16	algebra	algebra	NOUN
easat-3854	29	17	,	,	PUNCT
easat-3854	29	18	as	as	SCONJ
easat-3854	29	19	it	it	PRON
easat-3854	29	20	is	be	AUX
easat-3854	29	21	known	know	VERB
easat-3854	29	22	.	.	PUNCT
easat-3854	30	1	a	a	DET
easat-3854	30	2	further	further	ADJ
easat-3854	30	3	restriction	restriction	NOUN
easat-3854	30	4	on	on	ADP
easat-3854	30	5	the	the	DET
easat-3854	30	6	existence	existence	NOUN
easat-3854	30	7	and	and	CCONJ
easat-3854	30	8	behavior	behavior	NOUN
easat-3854	30	9	of	of	ADP
easat-3854	30	10	such	such	ADJ
easat-3854	30	11	spaces	space	NOUN
easat-3854	30	12	imposed	impose	VERB
easat-3854	30	13	by	by	ADP
easat-3854	30	14	the	the	DET
easat-3854	30	15	fact	fact	NOUN
easat-3854	30	16	that	that	SCONJ
easat-3854	30	17	homology	homology	NOUN
easat-3854	30	18	operations	operation	NOUN
easat-3854	30	19	with	with	ADP
easat-3854	30	20	these	these	DET
easat-3854	30	21	abilities	ability	NOUN
easat-3854	30	22	can	can	AUX
easat-3854	30	23	be	be	AUX
easat-3854	30	24	explicitly	explicitly	ADV
easat-3854	30	25	create	create	VERB
easat-3854	30	26	for	for	ADP
easat-3854	30	27	any	any	DET
easat-3854	30	28	topological	topological	ADJ
easat-3854	30	29	space	space	NOUN
easat-3854	30	30	.	.	PUNCT
easat-3854	31	1	the	the	DET
easat-3854	31	2	aim	aim	NOUN
easat-3854	31	3	of	of	ADP
easat-3854	31	4	this	this	DET
easat-3854	31	5	search	search	NOUN
easat-3854	31	6	:	:	PUNCT
easat-3854	31	7	lapin	lapin	NOUN
easat-3854	31	8	in	in	ADP
easat-3854	31	9	[	[	X
easat-3854	31	10	2	2	NUM
easat-3854	31	11	]	]	PUNCT
easat-3854	31	12	has	have	AUX
easat-3854	31	13	examined	examine	VERB
easat-3854	31	14	how	how	SCONJ
easat-3854	31	15	generalized	generalize	VERB
easat-3854	31	16	the	the	DET
easat-3854	31	17	steenrod	steenrod	NOUN
easat-3854	31	18	operations	operation	NOUN
easat-3854	31	19	can	can	AUX
easat-3854	31	20	be	be	AUX
easat-3854	31	21	constructed	construct	VERB
easat-3854	31	22	in	in	ADP
easat-3854	31	23	relations	relation	NOUN
easat-3854	31	24	of	of	ADP
easat-3854	31	25	multiplicative	multiplicative	ADJ
easat-3854	31	26	spectral	spectral	ADJ
easat-3854	31	27	sequences	sequence	NOUN
easat-3854	31	28	.	.	PUNCT
easat-3854	32	1	j.m	j.m	PROPN
easat-3854	32	2	.	.	PROPN
easat-3854	32	3	lopez	lopez	PROPN
easat-3854	33	1	[	[	X
easat-3854	33	2	3	3	NUM
easat-3854	33	3	]	]	PUNCT
easat-3854	33	4	.	.	PUNCT
easat-3854	34	1	burghelea	burghelea	PROPN
easat-3854	34	2	[	[	X
easat-3854	34	3	4	4	X
easat-3854	34	4	]	]	PUNCT
easat-3854	34	5	has	have	AUX
easat-3854	34	6	been	be	AUX
easat-3854	34	7	use	use	NOUN
easat-3854	34	8	to	to	PART
easat-3854	34	9	study	study	VERB
easat-3854	34	10	adams	adams	PROPN
easat-3854	34	11	operations	operation	NOUN
easat-3854	34	12	in	in	ADP
easat-3854	34	13	the	the	DET
easat-3854	34	14	hochschild	hochschild	ADJ
easat-3854	34	15	and	and	CCONJ
easat-3854	34	16	cyclic	cyclic	ADJ
easat-3854	34	17	homologies	homology	NOUN
easat-3854	34	18	of	of	ADP
easat-3854	34	19	the	the	DET
easat-3854	34	20	de	de	PROPN
easat-3854	34	21	-	-	NOUN
easat-3854	34	22	rham	rham	PROPN
easat-3854	34	23	algebras	algebras	PROPN
easat-3854	34	24	and	and	CCONJ
easat-3854	34	25	allowed	allow	VERB
easat-3854	34	26	loop	loop	NOUN
easat-3854	34	27	spaces	space	NOUN
easat-3854	34	28	.	.	PUNCT
easat-3854	35	1	we	we	PRON
easat-3854	35	2	study	study	VERB
easat-3854	35	3	adams	adam	NOUN
easat-3854	35	4	operations	operation	NOUN
easat-3854	35	5	on	on	ADP
easat-3854	35	6	the	the	DET
easat-3854	35	7	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-3854	35	8	by	by	ADP
easat-3854	35	9	the	the	DET
easat-3854	35	10	rational	rational	ADJ
easat-3854	35	11	coefficients	coefficient	NOUN
easat-3854	35	12	are	be	AUX
easat-3854	35	13	developed	develop	VERB
easat-3854	35	14	and	and	CCONJ
easat-3854	35	15	proved	prove	VERB
easat-3854	35	16	to	to	PART
easat-3854	35	17	descend	descend	VERB
easat-3854	35	18	to	to	ADP
easat-3854	35	19	the	the	DET
easat-3854	35	20	universal	universal	ADJ
easat-3854	35	21	relating	relate	VERB
easat-3854	35	22	to	to	ADP
easat-3854	35	23	their	their	PRON
easat-3854	35	24	group	group	NOUN
easat-3854	35	25	law	law	NOUN
easat-3854	35	26	oriented	orient	VERB
easat-3854	35	27	cohomology	cohomology	NOUN
easat-3854	35	28	theories	theory	NOUN
easat-3854	35	29	.	.	PUNCT
easat-3854	36	1	we	we	PRON
easat-3854	36	2	introduce	introduce	VERB
easat-3854	36	3	and	and	CCONJ
easat-3854	36	4	study	study	VERB
easat-3854	36	5	some	some	DET
easat-3854	36	6	basic	basic	ADJ
easat-3854	36	7	statement	statement	NOUN
easat-3854	36	8	of	of	ADP
easat-3854	36	9	the	the	DET
easat-3854	36	10	dihedral	dihedral	ADJ
easat-3854	36	11	homology	homology	NOUN
easat-3854	36	12	theory	theory	NOUN
easat-3854	36	13	of	of	ADP
easat-3854	36	14	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-3854	36	15	and	and	CCONJ
easat-3854	36	16	we	we	PRON
easat-3854	36	17	define	define	VERB
easat-3854	36	18	adams	adam	NOUN
easat-3854	36	19	and	and	CCONJ
easat-3854	36	20	steenrod	steenrod	NOUN
easat-3854	36	21	operators	operator	NOUN
easat-3854	36	22	in	in	ADP
easat-3854	36	23	algebras	algebras	PROPN
easat-3854	36	24	.	.	PUNCT
easat-3854	37	1	the	the	DET
easat-3854	37	2	main	main	ADJ
easat-3854	37	3	study	study	NOUN
easat-3854	37	4	of	of	ADP
easat-3854	37	5	this	this	DET
easat-3854	37	6	paper	paper	NOUN
easat-3854	37	7	is	be	AUX
easat-3854	37	8	the	the	DET
easat-3854	37	9	form	form	NOUN
easat-3854	37	10	of	of	ADP
easat-3854	37	11	the	the	DET
easat-3854	37	12	adam`s	adam`s	PROPN
easat-3854	37	13	and	and	CCONJ
easat-3854	37	14	steenrod`s	steenrod`s	NOUN
easat-3854	37	15	of	of	ADP
easat-3854	37	16	the	the	DET
easat-3854	37	17	dihedral	dihedral	ADJ
easat-3854	37	18	homology	homology	NOUN
easat-3854	37	19	of	of	ADP
easat-3854	37	20	an	an	DET
easat-3854	37	21	𝒜∞-algebras	𝒜∞-algebras	ADJ
easat-3854	37	22	.	.	PUNCT
easat-3854	38	1	we	we	PRON
easat-3854	38	2	introduce	introduce	VERB
easat-3854	38	3	the	the	DET
easat-3854	38	4	steenrod	steenrod	NOUN
easat-3854	38	5	operator	operator	NOUN
easat-3854	38	6	in	in	ADP
easat-3854	38	7	dihedral	dihedral	ADJ
easat-3854	38	8	homology	homology	NOUN
easat-3854	38	9	on	on	ADP
easat-3854	38	10	𝒜∞-algebras	𝒜∞-algebra	NOUN
easat-3854	38	11	.	.	PUNCT
easat-3854	39	1	2	2	NUM
easat-3854	39	2	.	.	X
easat-3854	39	3	the	the	DET
easat-3854	39	4	dihedral	dihedral	ADJ
easat-3854	39	5	homology	homology	NOUN
easat-3854	39	6	of	of	ADP
easat-3854	39	7	𝓐∞-algebras	𝓐∞-algebras	PUNCT
easat-3854	39	8	𝒜∞-algebras	𝒜∞-algebras	PRON
easat-3854	39	9	is	be	AUX
easat-3854	39	10	one	one	NUM
easat-3854	39	11	of	of	ADP
easat-3854	39	12	several	several	ADJ
easat-3854	39	13	branches	branch	NOUN
easat-3854	39	14	of	of	ADP
easat-3854	39	15	algebras	algebra	NOUN
easat-3854	39	16	,	,	PUNCT
easat-3854	39	17	and	and	CCONJ
easat-3854	39	18	it	it	PRON
easat-3854	39	19	has	have	VERB
easat-3854	39	20	certain	certain	ADJ
easat-3854	39	21	unique	unique	ADJ
easat-3854	39	22	properties	property	NOUN
easat-3854	39	23	.	.	PUNCT
easat-3854	40	1	it	it	PRON
easat-3854	40	2	is	be	AUX
easat-3854	40	3	described	describe	VERB
easat-3854	40	4	as	as	ADP
easat-3854	40	5	a	a	DET
easat-3854	40	6	graded	grade	VERB
easat-3854	40	7	algebra	algebra	NOUN
easat-3854	40	8	with	with	ADP
easat-3854	40	9	graded	grade	VERB
easat-3854	40	10	maps	map	NOUN
easat-3854	40	11	,	,	PUNCT
easat-3854	40	12	which	which	PRON
easat-3854	40	13	satisfies	satisfy	VERB
easat-3854	40	14	some	some	DET
easat-3854	40	15	conditions	condition	NOUN
easat-3854	40	16	.	.	PUNCT
easat-3854	41	1	stasheff	stasheff	PROPN
easat-3854	41	2	introduced	introduce	VERB
easat-3854	41	3	infinity	infinity	NOUN
easat-3854	41	4	algebras	algebra	NOUN
easat-3854	41	5	in	in	ADP
easat-3854	41	6	the	the	DET
easat-3854	41	7	1960	1960	NUM
easat-3854	41	8	's	's	PART
easat-3854	41	9	and	and	CCONJ
easat-3854	41	10	provides	provide	VERB
easat-3854	41	11	the	the	DET
easat-3854	41	12	properties	property	NOUN
easat-3854	41	13	of	of	ADP
easat-3854	41	14	topological	topological	ADJ
easat-3854	41	15	algebras	algebra	NOUN
easat-3854	41	16	.	.	PUNCT
easat-3854	42	1	its	its	PRON
easat-3854	42	2	homological	homological	ADJ
easat-3854	42	3	theory	theory	NOUN
easat-3854	42	4	was	be	AUX
easat-3854	42	5	also	also	ADV
easat-3854	42	6	studied	study	VERB
easat-3854	42	7	in	in	ADP
easat-3854	42	8	(	(	PUNCT
easat-3854	42	9	2013	2013	NUM
easat-3854	42	10	)	)	PUNCT
easat-3854	42	11	by	by	ADP
easat-3854	42	12	alaa	alaa	PROPN
easat-3854	42	13	.	.	PUNCT
easat-3854	43	1	h.	h.	PROPN
easat-3854	43	2	,	,	PUNCT
easat-3854	43	3	y.	y.	PROPN
easat-3854	43	4	gouda	gouda	NOUN
easat-3854	43	5	.	.	PUNCT
easat-3854	44	1	in	in	ADP
easat-3854	44	2	view	view	NOUN
easat-3854	44	3	of	of	ADP
easat-3854	44	4	this	this	PRON
easat-3854	44	5	,	,	PUNCT
easat-3854	44	6	we	we	PRON
easat-3854	44	7	will	will	AUX
easat-3854	44	8	show	show	VERB
easat-3854	44	9	some	some	DET
easat-3854	44	10	previous	previous	ADJ
easat-3854	44	11	studies	study	NOUN
easat-3854	44	12	of	of	ADP
easat-3854	44	13	some	some	DET
easat-3854	44	14	definitions	definition	NOUN
easat-3854	44	15	,	,	PUNCT
easat-3854	44	16	theorems	theorem	NOUN
easat-3854	44	17	and	and	CCONJ
easat-3854	44	18	algebraic	algebraic	ADJ
easat-3854	44	19	properties	property	NOUN
easat-3854	44	20	of	of	ADP
easat-3854	44	21	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-3854	44	22	and	and	CCONJ
easat-3854	44	23	its	its	PRON
easat-3854	44	24	homological	homological	ADJ
easat-3854	44	25	properties	property	NOUN
easat-3854	44	26	,	,	PUNCT
easat-3854	44	27	by	by	ADP
easat-3854	44	28	using	use	VERB
easat-3854	44	29	the	the	DET
easat-3854	44	30	references	reference	NOUN
easat-3854	44	31	:	:	PUNCT
easat-3854	45	1	[	[	X
easat-3854	45	2	5	5	NUM
easat-3854	45	3	]	]	PUNCT
easat-3854	45	4	,	,	PUNCT
easat-3854	45	5	[	[	X
easat-3854	45	6	6	6	NUM
easat-3854	45	7	]	]	PUNCT
easat-3854	45	8	and	and	CCONJ
easat-3854	45	9	[	[	X
easat-3854	45	10	7	7	NUM
easat-3854	45	11	]	]	PUNCT
easat-3854	45	12	.	.	PUNCT
easat-3854	46	1	2.1	2.1	NUM
easat-3854	46	2	definition	definition	NOUN
easat-3854	46	3	[	[	X
easat-3854	46	4	8	8	NUM
easat-3854	46	5	]	]	PUNCT
easat-3854	46	6	by	by	ADP
easat-3854	46	7	considering	consider	VERB
easat-3854	46	8	a	a	DET
easat-3854	46	9	differential	differential	ADJ
easat-3854	46	10	module	module	NOUN
easat-3854	46	11	(	(	PUNCT
easat-3854	46	12	𝒞	𝒞	PROPN
easat-3854	46	13	,	,	PUNCT
easat-3854	46	14	𝛿	𝛿	ADJ
easat-3854	46	15	)	)	PUNCT
easat-3854	46	16	such	such	ADJ
easat-3854	46	17	as	as	ADP
easat-3854	46	18	𝛿	𝛿	ADJ
easat-3854	46	19	:	:	PUNCT
easat-3854	46	20	𝒞𝓅	𝒞𝓅	PROPN
easat-3854	46	21	→	→	SYM
easat-3854	46	22	𝒞𝓅−1	𝒞𝓅−1	PROPN
easat-3854	46	23	,	,	PUNCT
easat-3854	46	24	then	then	ADV
easat-3854	46	25	we	we	PRON
easat-3854	46	26	can	can	AUX
easat-3854	46	27	define	define	VERB
easat-3854	46	28	a	a	DET
easat-3854	46	29	simplicial	simplicial	ADJ
easat-3854	46	30	faces	face	NOUN
easat-3854	46	31	as	as	ADP
easat-3854	46	32	𝜕𝚤	𝜕𝚤	NOUN
easat-3854	46	33	:	:	PUNCT
easat-3854	46	34	𝒞𝓅	𝒞𝓅	PROPN
easat-3854	46	35	→	→	SYM
easat-3854	46	36	𝒞𝓅−1	𝒞𝓅−1	PROPN
easat-3854	46	37	,	,	PUNCT
easat-3854	46	38	0	0	NUM
easat-3854	46	39	≤	≤	NUM
easat-3854	46	40	𝚤	𝚤	PRON
easat-3854	46	41	≤	≤	NUM
easat-3854	46	42	𝑛	𝑛	NOUN
easat-3854	46	43	,	,	PUNCT
easat-3854	46	44	where	where	SCONJ
easat-3854	46	45	𝜕𝚤𝜕𝑗	𝜕𝚤𝜕𝑗	NOUN
easat-3854	46	46	=	=	SYM
easat-3854	46	47	𝜕𝑗−1𝜕𝚤	𝜕𝑗−1𝜕𝚤	PROPN
easat-3854	46	48	,	,	PUNCT
easat-3854	46	49	𝚤	𝚤	X
easat-3854	46	50	<	<	X
easat-3854	46	51	𝑗	𝑗	INTJ
easat-3854	46	52	,	,	PUNCT
easat-3854	46	53	additionally	additionally	ADV
easat-3854	46	54	𝜕𝚤	𝜕𝚤	NOUN
easat-3854	46	55	refers	refer	VERB
easat-3854	46	56	to	to	PART
easat-3854	46	57	be	be	AUX
easat-3854	46	58	the	the	DET
easat-3854	46	59	(	(	PUNCT
easat-3854	46	60	𝒞	𝒞	PROPN
easat-3854	46	61	,	,	PUNCT
easat-3854	46	62	𝛿)simplicial	𝛿)simplicial	ADJ
easat-3854	46	63	faces	face	VERB
easat-3854	46	64	.	.	PUNCT
easat-3854	47	1	let	let	VERB
easat-3854	47	2	the	the	DET
easat-3854	47	3	permutation	permutation	NOUN
easat-3854	47	4	𝜎	𝜎	PROPN
easat-3854	47	5	of	of	ADP
easat-3854	47	6	a	a	DET
easat-3854	47	7	symmetrical	symmetrical	ADJ
easat-3854	47	8	group	group	NOUN
easat-3854	47	9	𝛴𝓆	𝛴𝓆	PROPN
easat-3854	47	10	of	of	ADP
easat-3854	47	11	𝓆-elements	𝓆-element	NOUN
easat-3854	47	12	of	of	ADP
easat-3854	47	13	permutations	permutation	NOUN
easat-3854	47	14	,	,	PUNCT
easat-3854	47	15	in	in	ADP
easat-3854	47	16	which	which	PRON
easat-3854	47	17	its	its	PRON
easat-3854	47	18	components	component	NOUN
easat-3854	47	19	are	be	AUX
easat-3854	47	20	(	(	PUNCT
easat-3854	47	21	𝜎	𝜎	X
easat-3854	47	22	(	(	PUNCT
easat-3854	47	23	𝚤1	𝚤1	PROPN
easat-3854	47	24	)	)	PUNCT
easat-3854	47	25	,	,	PUNCT
easat-3854	47	26	…	…	PUNCT
easat-3854	47	27	,	,	PUNCT
easat-3854	47	28	𝜎	𝜎	PROPN
easat-3854	47	29	(	(	PUNCT
easat-3854	47	30	𝚤𝓆	𝚤𝓆	NOUN
easat-3854	47	31	)	)	PUNCT
easat-3854	47	32	)	)	PUNCT
easat-3854	47	33	that	that	PRON
easat-3854	47	34	operates	operate	VERB
easat-3854	47	35	on	on	ADP
easat-3854	47	36	(	(	PUNCT
easat-3854	47	37	𝚤1	𝚤1	PROPN
easat-3854	47	38	,	,	PUNCT
easat-3854	47	39	…	…	PUNCT
easat-3854	47	40	,	,	PUNCT
easat-3854	47	41	𝚤𝓆	𝚤𝓆	NOUN
easat-3854	47	42	)	)	PUNCT
easat-3854	47	43	where	where	SCONJ
easat-3854	47	44	𝚤1	𝚤1	PROPN
easat-3854	47	45	<	<	X
easat-3854	47	46	⋯	⋯	X
easat-3854	47	47	<	<	X
easat-3854	47	48	𝚤𝓆	𝚤𝓆	NOUN
easat-3854	47	49	,	,	PUNCT
easat-3854	47	50	then	then	ADV
easat-3854	47	51	(	(	PUNCT
easat-3854	47	52	𝜎	𝜎	PROPN
easat-3854	47	53	(	(	PUNCT
easat-3854	47	54	𝚤1)̂	𝚤1)̂	PROPN
easat-3854	47	55	,	,	PUNCT
easat-3854	47	56	…	…	PUNCT
easat-3854	47	57	,	,	PUNCT
easat-3854	47	58	𝜎	𝜎	PROPN
easat-3854	47	59	(	(	PUNCT
easat-3854	47	60	𝚤𝓆	𝚤𝓆	NOUN
easat-3854	47	61	)	)	PUNCT
easat-3854	48	1	̂	̂	VERB
easat-3854	48	2	)	)	PUNCT
easat-3854	49	1	write	write	VERB
easat-3854	49	2	as	as	ADP
easat-3854	49	3	:	:	PUNCT
easat-3854	49	4	𝜎	𝜎	X
easat-3854	49	5	(	(	PUNCT
easat-3854	49	6	𝚤𝓀)̂	𝚤𝓀)̂	PROPN
easat-3854	49	7	=	=	SYM
easat-3854	49	8	𝜎	𝜎	PROPN
easat-3854	49	9	(	(	PUNCT
easat-3854	49	10	𝚤𝓀	𝚤𝓀	PROPN
easat-3854	49	11	)	)	PUNCT
easat-3854	49	12	−	−	PROPN
easat-3854	49	13	𝛾(𝜎	𝛾(𝜎	NOUN
easat-3854	49	14	(	(	PUNCT
easat-3854	49	15	𝚤𝓀	𝚤𝓀	PROPN
easat-3854	49	16	)	)	PUNCT
easat-3854	49	17	)	)	PUNCT
easat-3854	49	18	,	,	PUNCT
easat-3854	49	19	1	1	NUM
easat-3854	49	20	≤	≤	NUM
easat-3854	49	21	𝓀	𝓀	PRON
easat-3854	49	22	≤	≤	PROPN
easat-3854	49	23	𝓆	𝓆	NOUN
easat-3854	49	24	,	,	PUNCT
easat-3854	49	25	while	while	SCONJ
easat-3854	49	26	,	,	PUNCT
easat-3854	49	27	𝛾(𝜎	𝛾(𝜎	NOUN
easat-3854	49	28	(	(	PUNCT
easat-3854	49	29	𝚤𝓀	𝚤𝓀	PROPN
easat-3854	49	30	)	)	PUNCT
easat-3854	49	31	)	)	PUNCT
easat-3854	49	32	is	be	AUX
easat-3854	49	33	a	a	DET
easat-3854	49	34	number	number	NOUN
easat-3854	49	35	of	of	ADP
easat-3854	49	36	(	(	PUNCT
easat-3854	49	37	𝜎	𝜎	PROPN
easat-3854	49	38	(	(	PUNCT
easat-3854	49	39	𝚤1	𝚤1	PROPN
easat-3854	49	40	)	)	PUNCT
easat-3854	49	41	,	,	PUNCT
easat-3854	49	42	…	…	PUNCT
easat-3854	49	43	,	,	PUNCT
easat-3854	49	44	𝜎	𝜎	PROPN
easat-3854	49	45	(	(	PUNCT
easat-3854	49	46	𝚤𝓀	𝚤𝓀	PROPN
easat-3854	49	47	)	)	PUNCT
easat-3854	49	48	,	,	PUNCT
easat-3854	49	49	…	…	PUNCT
easat-3854	49	50	,	,	PUNCT
easat-3854	49	51	𝜎	𝜎	PROPN
easat-3854	49	52	(	(	PUNCT
easat-3854	49	53	𝚤𝓆	𝚤𝓆	NOUN
easat-3854	49	54	)	)	PUNCT
easat-3854	49	55	)	)	PUNCT
easat-3854	49	56	.	.	PUNCT
easat-3854	50	1	as	as	ADP
easat-3854	50	2	of	of	ADP
easat-3854	50	3	right	right	ADV
easat-3854	50	4	now	now	ADV
easat-3854	50	5	,	,	PUNCT
easat-3854	50	6	the	the	DET
easat-3854	50	7	differential	differential	NOUN
easat-3854	50	8	module	module	NOUN
easat-3854	50	9	(	(	PUNCT
easat-3854	50	10	𝒞	𝒞	PROPN
easat-3854	50	11	,	,	PUNCT
easat-3854	50	12	𝛿	𝛿	NOUN
easat-3854	50	13	)	)	PUNCT
easat-3854	50	14	with	with	ADP
easat-3854	50	15	the	the	DET
easat-3854	50	16	family	family	NOUN
easat-3854	50	17	map	map	NOUN
easat-3854	50	18	:	:	PUNCT
easat-3854	50	19	�	�	PROPN
easat-3854	50	20	̃	̃	PROPN
easat-3854	50	21	�	�	NOUN
easat-3854	50	22	=	=	SYM
easat-3854	50	23	𝜕(𝚤1,	𝜕(𝚤1,	NOUN
easat-3854	50	24	…	…	SYM
easat-3854	50	25	,𝚤𝓆	,𝚤𝓆	NUM
easat-3854	50	26	):	):	PUNCT
easat-3854	50	27	𝒞𝓅	𝒞𝓅	PROPN
easat-3854	50	28	→	→	SYM
easat-3854	50	29	𝒞𝓅−𝓆	𝒞𝓅−𝓆	NOUN
easat-3854	50	30	,	,	PUNCT
easat-3854	50	31	𝚤	𝚤	PROPN
easat-3854	50	32	≤	≤	NOUN
easat-3854	50	33	𝓆	𝓆	DET
easat-3854	50	34	≤	≤	NUM
easat-3854	50	35	𝓅	𝓅	NOUN
easat-3854	50	36	,	,	PUNCT
easat-3854	50	37	0	0	NUM
easat-3854	50	38	≤	≤	NUM
easat-3854	50	39	𝚤1	𝚤1	PROPN
easat-3854	50	40	<	<	X
easat-3854	50	41	⋯	⋯	X
easat-3854	50	42	<	<	X
easat-3854	50	43	𝚤𝓆	𝚤𝓆	PROPN
easat-3854	50	44	≤	≤	PROPN
easat-3854	50	45	𝓅	𝓅	PROPN
easat-3854	50	46	,	,	PUNCT
easat-3854	50	47	𝚤1	𝚤1	PROPN
easat-3854	50	48	,	,	PUNCT
easat-3854	50	49	…	…	PUNCT
easat-3854	50	50	,	,	PUNCT
easat-3854	50	51	𝚤𝓆	𝚤𝓆	PROPN
easat-3854	50	52	∈	∈	PROPN
easat-3854	50	53	ℤ	ℤ	PROPN
easat-3854	50	54	,	,	PUNCT
easat-3854	50	55	can	can	AUX
easat-3854	50	56	be	be	AUX
easat-3854	50	57	used	use	VERB
easat-3854	50	58	to	to	PART
easat-3854	50	59	define	define	VERB
easat-3854	50	60	the	the	DET
easat-3854	50	61	ℱ∞-module	ℱ∞-module	PROPN
easat-3854	50	62	(	(	PUNCT
easat-3854	50	63	𝒞	𝒞	PROPN
easat-3854	50	64	,	,	PUNCT
easat-3854	50	65	𝛿	𝛿	ADJ
easat-3854	50	66	,	,	PUNCT
easat-3854	50	67	�	�	PROPN
easat-3854	50	68	̃	̃	PROPN
easat-3854	50	69	�	�	PROPN
easat-3854	50	70	)	)	PUNCT
easat-3854	50	71	,	,	PUNCT
easat-3854	50	72	which	which	PRON
easat-3854	50	73	satisfy	satisfy	VERB
easat-3854	50	74	that	that	PRON
easat-3854	50	75	:	:	PUNCT
easat-3854	50	76	𝛿	𝛿	ADJ
easat-3854	50	77	(	(	PUNCT
easat-3854	50	78	𝜕(𝚤1,	𝜕(𝚤1,	NOUN
easat-3854	50	79	…	…	SYM
easat-3854	50	80	,𝚤𝓆	,𝚤𝓆	NUM
easat-3854	50	81	)	)	PUNCT
easat-3854	50	82	)	)	PUNCT
easat-3854	51	1	=	=	PUNCT
easat-3854	51	2	∑	∑	PUNCT
easat-3854	51	3	∑	∑	PUNCT
easat-3854	51	4	(	(	PUNCT
easat-3854	51	5	−1)1+𝑠𝑖𝑔𝑛(𝜎)𝜕(𝜎	−1)1+𝑠𝑖𝑔𝑛(𝜎)𝜕(𝜎	PROPN
easat-3854	51	6	(	(	PUNCT
easat-3854	51	7	𝚤1)̂	𝚤1)̂	PROPN
easat-3854	51	8	,	,	PUNCT
easat-3854	51	9	…	…	PUNCT
easat-3854	51	10	,	,	PUNCT
easat-3854	51	11	𝜎	𝜎	PROPN
easat-3854	51	12	(	(	PUNCT
easat-3854	51	13	𝚤ℓ)̂	𝚤ℓ)̂	PROPN
easat-3854	51	14	)	)	PUNCT
easat-3854	51	15	𝜕(𝜎	𝜕(𝜎	NOUN
easat-3854	51	16	(	(	PUNCT
easat-3854	51	17	𝚤ℓ+1)̂	𝚤ℓ+1)̂	NOUN
easat-3854	51	18	,	,	PUNCT
easat-3854	51	19	…	…	PUNCT
easat-3854	51	20	,	,	PUNCT
easat-3854	51	21	𝜎	𝜎	PROPN
easat-3854	51	22	(	(	PUNCT
easat-3854	51	23	𝚤𝓆	𝚤𝓆	NOUN
easat-3854	51	24	)	)	PUNCT
easat-3854	51	25	̂	̂	NUM
easat-3854	51	26	)	)	PUNCT
easat-3854	51	27	𝐼𝜎𝜎∈𝛴𝓆	𝐼𝜎𝜎∈𝛴𝓆	NOUN
easat-3854	51	28	.	.	PUNCT
easat-3854	52	1	(	(	PUNCT
easat-3854	52	2	1	1	X
easat-3854	52	3	)	)	PUNCT
easat-3854	52	4	since	since	SCONJ
easat-3854	52	5	iσ	iσ	NOUN
easat-3854	52	6	indicates	indicate	VERB
easat-3854	52	7	the	the	DET
easat-3854	52	8	permutations	permutation	NOUN
easat-3854	52	9	of	of	ADP
easat-3854	52	10	(	(	PUNCT
easat-3854	52	11	𝜎	𝜎	PROPN
easat-3854	52	12	(	(	PUNCT
easat-3854	52	13	𝚤1)̂	𝚤1)̂	PROPN
easat-3854	52	14	,	,	PUNCT
easat-3854	52	15	…	…	PUNCT
easat-3854	52	16	,	,	PUNCT
easat-3854	52	17	𝜎	𝜎	PROPN
easat-3854	52	18	(	(	PUNCT
easat-3854	52	19	𝚤𝓆	𝚤𝓆	NOUN
easat-3854	52	20	)	)	PUNCT
easat-3854	52	21	̂	̂	VERB
easat-3854	52	22	)	)	PUNCT
easat-3854	52	23	such	such	ADJ
easat-3854	52	24	as	as	ADP
easat-3854	52	25	:	:	PUNCT
easat-3854	52	26	𝜎	𝜎	PROPN
easat-3854	52	27	(	(	PUNCT
easat-3854	52	28	𝚤1)̂	𝚤1)̂	VERB
easat-3854	52	29	<	<	X
easat-3854	52	30	⋯	⋯	X
easat-3854	52	31	<	<	X
easat-3854	52	32	𝜎	𝜎	PROPN
easat-3854	52	33	(	(	PUNCT
easat-3854	52	34	𝚤ℓ)̂	𝚤ℓ)̂	PROPN
easat-3854	52	35	<	<	X
easat-3854	52	36	𝜎	𝜎	PROPN
easat-3854	52	37	(	(	PUNCT
easat-3854	52	38	𝚤ℓ+1)̂	𝚤ℓ+1)̂	X
easat-3854	52	39	<	<	X
easat-3854	52	40	⋯	⋯	X
easat-3854	52	41	<	<	X
easat-3854	52	42	𝜎	𝜎	PROPN
easat-3854	52	43	(	(	PUNCT
easat-3854	52	44	𝚤𝓆	𝚤𝓆	NOUN
easat-3854	52	45	)	)	PUNCT
easat-3854	52	46	̂	̂	PUNCT
easat-3854	52	47	.	.	PUNCT
easat-3854	53	1	then	then	ADV
easat-3854	53	2	,	,	PUNCT
easat-3854	53	3	�	�	PROPN
easat-3854	53	4	̃	̃	PROPN
easat-3854	53	5	�	�	NOUN
easat-3854	53	6	=	=	SYM
easat-3854	53	7	𝜕(𝚤1,	𝜕(𝚤1,	NOUN
easat-3854	53	8	…	…	SYM
easat-3854	53	9	,𝚤𝓆	,𝚤𝓆	NUM
easat-3854	53	10	)	)	PUNCT
easat-3854	53	11	is	be	AUX
easat-3854	53	12	the	the	DET
easat-3854	53	13	ℱ∞-differential	ℱ∞-differential	PROPN
easat-3854	53	14	of	of	ADP
easat-3854	53	15	(	(	PUNCT
easat-3854	53	16	𝒞	𝒞	PROPN
easat-3854	53	17	,	,	PUNCT
easat-3854	53	18	𝛿	𝛿	ADJ
easat-3854	53	19	)	)	PUNCT
easat-3854	53	20	is	be	AUX
easat-3854	53	21	the	the	DET
easat-3854	53	22	∞-simplicial	∞-simplicial	NOUN
easat-3854	53	23	of	of	ADP
easat-3854	53	24	faces	face	NOUN
easat-3854	53	25	of	of	ADP
easat-3854	53	26	ℱ∞-module	ℱ∞-module	PROPN
easat-3854	53	27	.	.	PUNCT
easat-3854	54	1	therefore	therefore	ADV
easat-3854	54	2	,	,	PUNCT
easat-3854	54	3	let	let	VERB
easat-3854	54	4	𝓆	𝓆	X
easat-3854	54	5	=	=	SYM
easat-3854	54	6	1	1	NUM
easat-3854	54	7	,	,	PUNCT
easat-3854	54	8	then	then	ADV
easat-3854	54	9	𝛿(𝜕(𝚤1	𝛿(𝜕(𝚤1	VERB
easat-3854	54	10	)	)	PUNCT
easat-3854	54	11	)	)	PUNCT
easat-3854	55	1	=	=	SYM
easat-3854	55	2	0	0	NUM
easat-3854	55	3	,	,	PUNCT
easat-3854	55	4	𝚤1	𝚤1	PROPN
easat-3854	55	5	≥	≥	NUM
easat-3854	55	6	0	0	NUM
easat-3854	55	7	,	,	PUNCT
easat-3854	55	8	let	let	VERB
easat-3854	55	9	𝓆	𝓆	X
easat-3854	55	10	=	=	SYM
easat-3854	55	11	2	2	NUM
easat-3854	55	12	,	,	PUNCT
easat-3854	55	13	then	then	ADV
easat-3854	55	14	:	:	PUNCT
easat-3854	55	15	𝛿(𝜕(𝚤1,𝚤2	𝛿(𝜕(𝚤1,𝚤2	NOUN
easat-3854	55	16	)	)	PUNCT
easat-3854	55	17	)	)	PUNCT
easat-3854	56	1	=	=	SYM
easat-3854	56	2	𝜕(𝚤2−1)𝜕(𝚤1	𝜕(𝚤2−1)𝜕(𝚤1	PROPN
easat-3854	56	3	)	)	PUNCT
easat-3854	56	4	−	−	PROPN
easat-3854	56	5	𝜕(𝚤1)𝜕(𝚤2	𝜕(𝚤1)𝜕(𝚤2	NUM
easat-3854	56	6	)	)	PUNCT
easat-3854	56	7	,	,	PUNCT
easat-3854	56	8	𝚤1	𝚤1	VERB
easat-3854	56	9	<	<	X
easat-3854	56	10	𝚤2	𝚤2	PROPN
easat-3854	56	11	,	,	PUNCT
easat-3854	56	12	8660	8660	NUM
easat-3854	56	13	edelweiss	edelweiss	PROPN
easat-3854	56	14	applied	apply	VERB
easat-3854	56	15	science	science	NOUN
easat-3854	56	16	and	and	CCONJ
easat-3854	56	17	technology	technology	NOUN
easat-3854	56	18	issn	issn	PROPN
easat-3854	56	19	:	:	PUNCT
easat-3854	56	20	2576	2576	NUM
easat-3854	56	21	-	-	SYM
easat-3854	56	22	8484	8484	NUM
easat-3854	56	23	vol	vol	NOUN
easat-3854	56	24	.	.	PROPN
easat-3854	56	25	8	8	NUM
easat-3854	56	26	,	,	PUNCT
easat-3854	56	27	no	no	INTJ
easat-3854	56	28	.	.	NOUN
easat-3854	57	1	6	6	NUM
easat-3854	57	2	:	:	SYM
easat-3854	57	3	8658	8658	NUM
easat-3854	57	4	-	-	SYM
easat-3854	57	5	8666	8666	NUM
easat-3854	57	6	,	,	PUNCT
easat-3854	57	7	2024	2024	NUM
easat-3854	57	8	doi	doi	NOUN
easat-3854	57	9	:	:	PUNCT
easat-3854	57	10	10.55214/25768484.v8i6.3854	10.55214/25768484.v8i6.3854	NUM
easat-3854	57	11	©	©	ADP
easat-3854	57	12	2024	2024	NUM
easat-3854	57	13	by	by	ADP
easat-3854	57	14	the	the	DET
easat-3854	57	15	authors	author	NOUN
easat-3854	57	16	;	;	PUNCT
easat-3854	57	17	licensee	licensee	PROPN
easat-3854	57	18	learning	learn	VERB
easat-3854	57	19	gate	gate	NOUN
easat-3854	57	20	let	let	VERB
easat-3854	57	21	𝓆	𝓆	NOUN
easat-3854	57	22	=	=	SYM
easat-3854	57	23	3	3	NUM
easat-3854	57	24	,	,	PUNCT
easat-3854	57	25	then	then	ADV
easat-3854	57	26	:	:	PUNCT
easat-3854	57	27	𝛿(𝜕(𝚤1,𝚤2,𝚤3	𝛿(𝜕(𝚤1,𝚤2,𝚤3	X
easat-3854	57	28	)	)	PUNCT
easat-3854	57	29	)	)	PUNCT
easat-3854	57	30	=	=	PUNCT
easat-3854	58	1	−𝜕(𝚤1)𝜕(𝚤2,𝚤3	−𝜕(𝚤1)𝜕(𝚤2,𝚤3	NUM
easat-3854	58	2	)	)	PUNCT
easat-3854	58	3	−	−	PROPN
easat-3854	58	4	𝜕(𝚤1,𝚤2)𝜕(𝚤3	𝜕(𝚤1,𝚤2)𝜕(𝚤3	PROPN
easat-3854	58	5	)	)	PUNCT
easat-3854	58	6	−	−	PROPN
easat-3854	58	7	𝜕(𝚤3−2)𝜕(𝚤1,𝚤2	𝜕(𝚤3−2)𝜕(𝚤1,𝚤2	PROPN
easat-3854	58	8	)	)	PUNCT
easat-3854	58	9	−	−	PROPN
easat-3854	58	10	𝜕(𝚤2−1,𝚤3−1)𝜕(𝚤1	𝜕(𝚤2−1,𝚤3−1)𝜕(𝚤1	VERB
easat-3854	58	11	)	)	PUNCT
easat-3854	58	12	+	+	CCONJ
easat-3854	58	13	𝜕(𝚤2−1)𝜕(𝚤1,𝚤3	𝜕(𝚤2−1)𝜕(𝚤1,𝚤3	NOUN
easat-3854	58	14	)	)	PUNCT
easat-3854	58	15	+	+	NUM
easat-3854	58	16	𝜕(𝚤1,𝚤3−1)𝜕(𝚤2	𝜕(𝚤1,𝚤3−1)𝜕(𝚤2	NOUN
easat-3854	58	17	)	)	PUNCT
easat-3854	58	18	,	,	PUNCT
easat-3854	58	19	𝚤1	𝚤1	VERB
easat-3854	58	20	<	<	X
easat-3854	58	21	𝚤2	𝚤2	NOUN
easat-3854	58	22	<	<	X
easat-3854	58	23	𝚤3	𝚤3	NOUN
easat-3854	58	24	,	,	PUNCT
easat-3854	58	25	we	we	PRON
easat-3854	58	26	can	can	AUX
easat-3854	58	27	define	define	VERB
easat-3854	58	28	a	a	DET
easat-3854	58	29	module	module	NOUN
easat-3854	58	30	of	of	ADP
easat-3854	58	31	cyclic	cyclic	ADJ
easat-3854	58	32	differential	differential	NOUN
easat-3854	58	33	(	(	PUNCT
easat-3854	58	34	𝒞	𝒞	PROPN
easat-3854	58	35	,	,	PUNCT
easat-3854	58	36	𝛿	𝛿	ADJ
easat-3854	58	37	,	,	PUNCT
easat-3854	58	38	𝓉	𝓉	PROPN
easat-3854	58	39	)	)	PUNCT
easat-3854	58	40	by	by	ADP
easat-3854	58	41	using	use	VERB
easat-3854	58	42	define	define	NOUN
easat-3854	58	43	of	of	ADP
easat-3854	58	44	differential	differential	NOUN
easat-3854	58	45	module	module	NOUN
easat-3854	58	46	(	(	PUNCT
easat-3854	58	47	𝒞	𝒞	PROPN
easat-3854	58	48	,	,	PUNCT
easat-3854	58	49	𝛿	𝛿	ADJ
easat-3854	58	50	)	)	PUNCT
easat-3854	58	51	and	and	CCONJ
easat-3854	58	52	the	the	DET
easat-3854	58	53	map	map	NOUN
easat-3854	58	54	𝓉	𝓉	PROPN
easat-3854	58	55	=	=	PRON
easat-3854	58	56	{	{	PUNCT
easat-3854	58	57	𝓉𝓅	𝓉𝓅	NOUN
easat-3854	58	58	:	:	PUNCT
easat-3854	58	59	𝒞𝓅	𝒞𝓅	PROPN
easat-3854	58	60	→	→	SYM
easat-3854	58	61	𝒞𝓅	𝒞𝓅	PROPN
easat-3854	58	62	}	}	PUNCT
easat-3854	58	63	,	,	PUNCT
easat-3854	58	64	∀𝓅	∀𝓅	VERB
easat-3854	58	65	≥	≥	NOUN
easat-3854	58	66	0	0	NUM
easat-3854	58	67	,	,	PUNCT
easat-3854	58	68	𝓉𝓅	𝓉𝓅	NOUN
easat-3854	58	69	𝓅+1	𝓅+1	NOUN
easat-3854	58	70	=	=	PUNCT
easat-3854	58	71	𝐼𝒞𝓅	𝐼𝒞𝓅	ADJ
easat-3854	58	72	,	,	PUNCT
easat-3854	58	73	𝛿𝓉𝓅	𝛿𝓉𝓅	NOUN
easat-3854	58	74	=	=	SYM
easat-3854	58	75	𝓉𝓅𝛿.	𝓉𝓅𝛿.	NOUN
easat-3854	58	76	similarly	similarly	ADV
easat-3854	58	77	,	,	PUNCT
easat-3854	58	78	we	we	PRON
easat-3854	58	79	can	can	AUX
easat-3854	58	80	get	get	VERB
easat-3854	58	81	a	a	DET
easat-3854	58	82	module	module	NOUN
easat-3854	58	83	of	of	ADP
easat-3854	58	84	dihedral	dihedral	ADJ
easat-3854	58	85	differential	differential	NOUN
easat-3854	58	86	(	(	PUNCT
easat-3854	58	87	𝒞	𝒞	PROPN
easat-3854	58	88	,	,	PUNCT
easat-3854	58	89	𝛿	𝛿	ADJ
easat-3854	58	90	,	,	PUNCT
easat-3854	58	91	𝓉	𝓉	PROPN
easat-3854	58	92	,	,	PUNCT
easat-3854	58	93	𝓇	𝓇	NOUN
easat-3854	58	94	)	)	PUNCT
easat-3854	58	95	by	by	ADP
easat-3854	58	96	define	define	VERB
easat-3854	58	97	the	the	DET
easat-3854	58	98	map	map	NOUN
easat-3854	58	99	:	:	PUNCT
easat-3854	59	1	𝓇	𝓇	X
easat-3854	59	2	=	=	SYM
easat-3854	59	3	{	{	PUNCT
easat-3854	59	4	𝓇𝓅	𝓇𝓅	NOUN
easat-3854	59	5	:	:	PUNCT
easat-3854	59	6	𝒞𝓅	𝒞𝓅	PROPN
easat-3854	59	7	→	→	SYM
easat-3854	59	8	𝒞𝓅	𝒞𝓅	PROPN
easat-3854	59	9	}	}	PUNCT
easat-3854	59	10	,	,	PUNCT
easat-3854	59	11	∀𝓅	∀𝓅	VERB
easat-3854	59	12	≥	≥	NOUN
easat-3854	59	13	0	0	NUM
easat-3854	59	14	,	,	PUNCT
easat-3854	59	15	𝓇𝓅	𝓇𝓅	PROPN
easat-3854	59	16	2	2	NUM
easat-3854	59	17	=	=	NOUN
easat-3854	59	18	𝐼𝒞𝓅	𝐼𝒞𝓅	NOUN
easat-3854	59	19	,	,	PUNCT
easat-3854	59	20	then	then	ADV
easat-3854	59	21	we	we	PRON
easat-3854	59	22	have	have	VERB
easat-3854	59	23	:	:	PUNCT
easat-3854	59	24	𝓇𝓅𝓉𝓅	𝓇𝓅𝓉𝓅	PROPN
easat-3854	59	25	=	=	PUNCT
easat-3854	59	26	𝓉𝓅	𝓉𝓅	PROPN
easat-3854	59	27	−1𝓇𝓅	−1𝓇𝓅	PROPN
easat-3854	59	28	,	,	PUNCT
easat-3854	59	29	𝛿𝓇𝓅	𝛿𝓇𝓅	NOUN
easat-3854	59	30	=	=	NOUN
easat-3854	59	31	𝓇𝓅𝛿.	𝓇𝓅𝛿.	NOUN
easat-3854	59	32	for	for	ADP
easat-3854	59	33	the	the	DET
easat-3854	59	34	cyclic	cyclic	NOUN
easat-3854	59	35	and	and	CCONJ
easat-3854	59	36	the	the	DET
easat-3854	59	37	dihedral	dihedral	ADJ
easat-3854	59	38	modules	module	NOUN
easat-3854	59	39	,	,	PUNCT
easat-3854	59	40	we	we	PRON
easat-3854	59	41	have	have	VERB
easat-3854	59	42	:	:	PUNCT
easat-3854	59	43	𝜕𝚤𝓉𝓅	𝜕𝚤𝓉𝓅	NOUN
easat-3854	59	44	=	=	SYM
easat-3854	59	45	𝓉𝓅−1𝜕𝚤−1	𝓉𝓅−1𝜕𝚤−1	NOUN
easat-3854	59	46	,	,	PUNCT
easat-3854	59	47	0	0	PUNCT
easat-3854	59	48	<	<	X
easat-3854	59	49	𝚤	𝚤	X
easat-3854	59	50	≤	≤	NOUN
easat-3854	59	51	𝓅	𝓅	NOUN
easat-3854	59	52	𝜕0𝓉𝓅	𝜕0𝓉𝓅	PROPN
easat-3854	59	53	=	=	SYM
easat-3854	59	54	𝜕𝓅	𝜕𝓅	PROPN
easat-3854	59	55	,	,	PUNCT
easat-3854	59	56	𝜕𝚤𝓇𝓅	𝜕𝚤𝓇𝓅	NOUN
easat-3854	59	57	=	=	SYM
easat-3854	59	58	𝓇𝓅−1𝜕𝚤−1	𝓇𝓅−1𝜕𝚤−1	NOUN
easat-3854	59	59	,	,	PUNCT
easat-3854	59	60	0	0	NUM
easat-3854	59	61	≤	≤	NUM
easat-3854	59	62	𝚤	𝚤	DET
easat-3854	59	63	≤	≤	ADJ
easat-3854	59	64	𝓅.	𝓅.	NOUN
easat-3854	59	65	then	then	ADV
easat-3854	59	66	,	,	PUNCT
easat-3854	59	67	the	the	DET
easat-3854	59	68	𝓓𝓕∞-module	𝓓𝓕∞-module	NOUN
easat-3854	59	69	(	(	PUNCT
easat-3854	59	70	𝓒	𝓒	PROPN
easat-3854	59	71	,	,	PUNCT
easat-3854	59	72	𝜹	𝜹	X
easat-3854	59	73	,	,	PUNCT
easat-3854	59	74	𝓽	𝓽	X
easat-3854	59	75	,	,	PUNCT
easat-3854	59	76	𝓻	𝓻	NOUN
easat-3854	59	77	,	,	PUNCT
easat-3854	59	78	�	�	PROPN
easat-3854	59	79	̃	̃	PROPN
easat-3854	59	80	�	�	PROPN
easat-3854	59	81	)	)	PUNCT
easat-3854	59	82	can	can	AUX
easat-3854	59	83	be	be	AUX
easat-3854	59	84	identified	identify	VERB
easat-3854	59	85	as	as	ADP
easat-3854	59	86	the	the	DET
easat-3854	59	87	dihedral	dihedral	ADJ
easat-3854	59	88	module	module	NOUN
easat-3854	59	89	,	,	PUNCT
easat-3854	59	90	such	such	ADJ
easat-3854	59	91	that	that	SCONJ
easat-3854	59	92	it	it	PRON
easat-3854	59	93	is	be	AUX
easat-3854	59	94	through	through	ADP
easat-3854	59	95	∞-simplicial	∞-simplicial	NOUN
easat-3854	59	96	of	of	ADP
easat-3854	59	97	faces	face	NOUN
easat-3854	59	98	,	,	PUNCT
easat-3854	59	99	subsequently	subsequently	ADV
easat-3854	59	100	(	(	PUNCT
easat-3854	59	101	𝓒	𝓒	PROPN
easat-3854	59	102	,	,	PUNCT
easat-3854	59	103	𝜹	𝜹	X
easat-3854	59	104	,	,	PUNCT
easat-3854	59	105	𝓽	𝓽	X
easat-3854	59	106	,	,	PUNCT
easat-3854	59	107	𝓻	𝓻	NOUN
easat-3854	59	108	)	)	PUNCT
easat-3854	59	109	denotes	denote	VERB
easat-3854	59	110	a	a	DET
easat-3854	59	111	module	module	NOUN
easat-3854	59	112	of	of	ADP
easat-3854	59	113	dihedral	dihedral	ADJ
easat-3854	59	114	differential	differential	NOUN
easat-3854	59	115	and	and	CCONJ
easat-3854	59	116	fulfils	fulfil	VERB
easat-3854	59	117	that	that	SCONJ
easat-3854	59	118	:	:	PUNCT
easat-3854	59	119	𝝏(𝚤𝟏,	𝝏(𝚤𝟏,	NOUN
easat-3854	59	120	…	…	PUNCT
easat-3854	59	121	,𝚤𝓺)𝓽𝓹	,𝚤𝓺)𝓽𝓹	NOUN
easat-3854	59	122	=	=	SYM
easat-3854	59	123	{	{	PUNCT
easat-3854	59	124	𝓽𝓹−𝓺𝝏(𝚤𝟏−𝟏,	𝓽𝓹−𝓺𝝏(𝚤𝟏−𝟏,	NOUN
easat-3854	59	125	…	…	PUNCT
easat-3854	59	126	,𝚤𝓺−𝟏	,𝚤𝓺−𝟏	NUM
easat-3854	59	127	)	)	PUNCT
easat-3854	59	128	,	,	PUNCT
easat-3854	59	129	𝚤𝟏	𝚤𝟏	NOUN
easat-3854	59	130	>	>	SYM
easat-3854	59	131	𝟎	𝟎	PROPN
easat-3854	59	132	(	(	PUNCT
easat-3854	59	133	−𝟏)𝓺−𝟏𝝏(𝚤𝟐−𝟏,	−𝟏)𝓺−𝟏𝝏(𝚤𝟐−𝟏,	NOUN
easat-3854	59	134	…	…	PUNCT
easat-3854	59	135	,𝚤𝓺−𝟏,𝓹	,𝚤𝓺−𝟏,𝓹	PUNCT
easat-3854	59	136	)	)	PUNCT
easat-3854	59	137	,	,	PUNCT
easat-3854	59	138	𝚤𝟏	𝚤𝟏	NOUN
easat-3854	59	139	=	=	SYM
easat-3854	59	140	𝟎	𝟎	X
easat-3854	59	141	,	,	PUNCT
easat-3854	59	142	(	(	PUNCT
easat-3854	59	143	2	2	NUM
easat-3854	59	144	)	)	PUNCT
easat-3854	59	145	𝜕(𝚤1,	𝜕(𝚤1,	NOUN
easat-3854	59	146	…	…	SYM
easat-3854	59	147	,𝚤𝓆)𝓇𝓅	,𝚤𝓆)𝓇𝓅	NOUN
easat-3854	59	148	=	=	PUNCT
easat-3854	59	149	(	(	PUNCT
easat-3854	59	150	−1	−1	NOUN
easat-3854	59	151	)	)	PUNCT
easat-3854	59	152	𝓆(𝓆−1	𝓆(𝓆−1	X
easat-3854	59	153	)	)	PUNCT
easat-3854	59	154	2	2	NUM
easat-3854	59	155	𝓇𝓅−𝓆𝜕(𝓅−𝚤𝓆	𝓇𝓅−𝓆𝜕(𝓅−𝚤𝓆	NOUN
easat-3854	59	156	,	,	PUNCT
easat-3854	59	157	…	…	PUNCT
easat-3854	59	158	,	,	PUNCT
easat-3854	59	159	𝓅−𝚤1	𝓅−𝚤1	X
easat-3854	59	160	)	)	PUNCT
easat-3854	59	161	,	,	PUNCT
easat-3854	59	162	𝚤1	𝚤1	PROPN
easat-3854	59	163	=	=	SYM
easat-3854	59	164	0	0	PROPN
easat-3854	59	165	.	.	PUNCT
easat-3854	60	1	(	(	PUNCT
easat-3854	60	2	3	3	X
easat-3854	60	3	)	)	PUNCT
easat-3854	60	4	2.2	2.2	NUM
easat-3854	60	5	definition	definition	NOUN
easat-3854	60	6	[	[	X
easat-3854	60	7	6	6	NUM
easat-3854	60	8	]	]	PUNCT
easat-3854	60	9	assume	assume	VERB
easat-3854	60	10	that	that	SCONJ
easat-3854	60	11	ℳ	ℳ	PROPN
easat-3854	60	12	=	=	SYM
easat-3854	60	13	{	{	PUNCT
easat-3854	60	14	ℳ𝓅	ℳ𝓅	PROPN
easat-3854	60	15	}	}	PUNCT
easat-3854	60	16	,	,	PUNCT
easat-3854	60	17	∀𝓅	∀𝓅	PRON
easat-3854	60	18	∈	∈	PROPN
easat-3854	60	19	ℤ	ℤ	PROPN
easat-3854	60	20	,	,	PUNCT
easat-3854	60	21	𝓅	𝓅	PROPN
easat-3854	60	22	>	>	X
easat-3854	60	23	0	0	NUM
easat-3854	60	24	,	,	PUNCT
easat-3854	60	25	is	be	AUX
easat-3854	60	26	a	a	DET
easat-3854	60	27	unital	unital	ADJ
easat-3854	60	28	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-3854	60	29	where	where	SCONJ
easat-3854	60	30	the	the	DET
easat-3854	60	31	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-3854	60	32	(	(	PUNCT
easat-3854	60	33	ℳ	ℳ	PROPN
easat-3854	60	34	,	,	PUNCT
easat-3854	60	35	𝛿	𝛿	ADJ
easat-3854	60	36	,	,	PUNCT
easat-3854	60	37	𝜑𝓅	𝜑𝓅	NOUN
easat-3854	60	38	)	)	PUNCT
easat-3854	60	39	be	be	VERB
easat-3854	60	40	any	any	DET
easat-3854	60	41	differential	differential	ADJ
easat-3854	60	42	module	module	NOUN
easat-3854	60	43	(	(	PUNCT
easat-3854	60	44	ℳ	ℳ	PROPN
easat-3854	60	45	,	,	PUNCT
easat-3854	60	46	𝛿	𝛿	ADJ
easat-3854	60	47	)	)	PUNCT
easat-3854	60	48	,	,	PUNCT
easat-3854	60	49	as	as	SCONJ
easat-3854	60	50	𝛿:ℳ⋆	𝛿:ℳ⋆	PROPN
easat-3854	60	51	→ℳ⋆−1	→ℳ⋆−1	VERB
easat-3854	60	52	,	,	PUNCT
easat-3854	60	53	prepared	prepare	VERB
easat-3854	60	54	through	through	ADP
easat-3854	60	55	a	a	DET
easat-3854	60	56	family	family	NOUN
easat-3854	60	57	of	of	ADP
easat-3854	60	58	functions	function	NOUN
easat-3854	60	59	{	{	PUNCT
easat-3854	60	60	𝜑𝓅	𝜑𝓅	NOUN
easat-3854	60	61	:	:	PUNCT
easat-3854	60	62	(	(	PUNCT
easat-3854	60	63	ℳ	ℳ	PROPN
easat-3854	60	64	⊗(𝓅+2))⋆	⊗(𝓅+2))⋆	PRON
easat-3854	60	65	→ℳ⋆+𝓅	→ℳ⋆+𝓅	PROPN
easat-3854	60	66	}	}	PUNCT
easat-3854	60	67	,	,	PUNCT
easat-3854	60	68	fulfilling	fulfil	VERB
easat-3854	60	69	the	the	DET
easat-3854	60	70	preceding	precede	VERB
easat-3854	60	71	relations	relation	NOUN
easat-3854	60	72	for	for	ADP
easat-3854	60	73	each	each	DET
easat-3854	60	74	integer	integer	NOUN
easat-3854	60	75	𝓅	𝓅	PROPN
easat-3854	60	76	>	>	X
easat-3854	60	77	1	1	NUM
easat-3854	60	78	,	,	PUNCT
easat-3854	60	79	since	since	SCONJ
easat-3854	60	80	𝛿(𝜑𝓅−1	𝛿(𝜑𝓅−1	NOUN
easat-3854	60	81	)	)	PUNCT
easat-3854	60	82	=	=	VERB
easat-3854	61	1	𝛿𝜑𝓅−1	𝛿𝜑𝓅−1	PROPN
easat-3854	61	2	+	+	CCONJ
easat-3854	61	3	(	(	PUNCT
easat-3854	61	4	−1	−1	NOUN
easat-3854	61	5	)	)	PUNCT
easat-3854	61	6	𝓅𝜑𝓅−1𝛿	𝓅𝜑𝓅−1𝛿	ADV
easat-3854	61	7	,	,	PUNCT
easat-3854	61	8	𝛿(𝜑𝓅−1	𝛿(𝜑𝓅−1	NOUN
easat-3854	61	9	)	)	PUNCT
easat-3854	61	10	=	=	SYM
easat-3854	61	11	∑	∑	PUNCT
easat-3854	61	12	∑	∑	PUNCT
easat-3854	61	13	(	(	PUNCT
easat-3854	61	14	−1)𝑠(𝓅−𝓆)+𝓅+1𝜋𝓆−1	−1)𝑠(𝓅−𝓆)+𝓅+1𝜋𝓆−1	PROPN
easat-3854	61	15	(	(	PUNCT
easat-3854	61	16	1⊗	1⊗	NUM
easat-3854	61	17	…	…	SYM
easat-3854	61	18	⊗1⏟	⊗1⏟	NOUN
easat-3854	61	19	𝑠−1	𝑠−1	PROPN
easat-3854	61	20	⊗𝜑𝓅−𝓆−1⊗1⊗	⊗𝜑𝓅−𝓆−1⊗1⊗	PROPN
easat-3854	61	21	…	…	SYM
easat-3854	61	22	⊗1⏟	⊗1⏟	NOUN
easat-3854	61	23	𝓆−𝑠−1	𝓆−𝑠−1	PROPN
easat-3854	61	24	)	)	PUNCT
easat-3854	61	25	𝓆+1	𝓆+1	X
easat-3854	61	26	𝑠=1	𝑠=1	PUNCT
easat-3854	61	27	𝓅−1	𝓅−1	PROPN
easat-3854	61	28	𝓆=1	𝓆=1	PROPN
easat-3854	61	29	,	,	PUNCT
easat-3854	61	30	(	(	PUNCT
easat-3854	61	31	4	4	X
easat-3854	61	32	)	)	PUNCT
easat-3854	61	33	for	for	ADP
easat-3854	61	34	example	example	NOUN
easat-3854	61	35	,	,	PUNCT
easat-3854	61	36	the	the	DET
easat-3854	61	37	relations	relation	NOUN
easat-3854	61	38	(	(	PUNCT
easat-3854	61	39	4	4	X
easat-3854	61	40	)	)	PUNCT
easat-3854	61	41	have	have	VERB
easat-3854	61	42	the	the	DET
easat-3854	61	43	following	follow	VERB
easat-3854	61	44	forms	form	NOUN
easat-3854	61	45	:	:	PUNCT
easat-3854	61	46	•	•	NOUN
easat-3854	61	47	for	for	ADP
easat-3854	61	48	𝓅	𝓅	NOUN
easat-3854	61	49	=	=	SYM
easat-3854	61	50	1	1	NUM
easat-3854	61	51	:	:	PUNCT
easat-3854	61	52	then	then	ADV
easat-3854	61	53	𝛿(𝜑0	𝛿(𝜑0	ADV
easat-3854	61	54	)	)	PUNCT
easat-3854	62	1	=	=	SYM
easat-3854	62	2	0	0	NUM
easat-3854	62	3	,	,	PUNCT
easat-3854	62	4	•	•	NOUN
easat-3854	62	5	for	for	ADP
easat-3854	62	6	𝓅	𝓅	NOUN
easat-3854	62	7	=	=	SYM
easat-3854	62	8	2	2	NUM
easat-3854	62	9	:	:	PUNCT
easat-3854	62	10	𝛿(𝜑1	𝛿(𝜑1	NOUN
easat-3854	62	11	)	)	PUNCT
easat-3854	62	12	=	=	SYM
easat-3854	62	13	𝜑0(𝜑0⊗1	𝜑0(𝜑0⊗1	PROPN
easat-3854	62	14	)	)	PUNCT
easat-3854	62	15	−	−	ADP
easat-3854	62	16	𝜑0(1⊗𝜑0	𝜑0(1⊗𝜑0	ADV
easat-3854	62	17	)	)	PUNCT
easat-3854	62	18	,	,	PUNCT
easat-3854	62	19	•	•	NOUN
easat-3854	62	20	for	for	ADP
easat-3854	62	21	𝓅	𝓅	NOUN
easat-3854	62	22	=	=	SYM
easat-3854	62	23	3	3	NUM
easat-3854	62	24	:	:	PUNCT
easat-3854	62	25	𝛿(𝜑2	𝛿(𝜑2	NOUN
easat-3854	62	26	)	)	PUNCT
easat-3854	63	1	=	=	SYM
easat-3854	63	2	𝜑0(𝜑1⊗1	𝜑0(𝜑1⊗1	PROPN
easat-3854	63	3	+	+	NOUN
easat-3854	63	4	1⊗𝜑1	1⊗𝜑1	NUM
easat-3854	63	5	)	)	PUNCT
easat-3854	63	6	−	−	PROPN
easat-3854	63	7	𝜑1(𝜑0⊗1⊗2	𝜑1(𝜑0⊗1⊗2	PROPN
easat-3854	63	8	−	−	PROPN
easat-3854	63	9	1⊗𝜑0⊗1	1⊗𝜑0⊗1	NUM
easat-3854	63	10	+	+	NOUN
easat-3854	63	11	1⊗2⊗𝜑0	1⊗2⊗𝜑0	NUM
easat-3854	63	12	)	)	PUNCT
easat-3854	63	13	.	.	PUNCT
easat-3854	64	1	since	since	SCONJ
easat-3854	64	2	(	(	PUNCT
easat-3854	64	3	ℳ	ℳ	PROPN
easat-3854	64	4	,	,	PUNCT
easat-3854	64	5	δ	δ	PROPN
easat-3854	64	6	,	,	PUNCT
easat-3854	64	7	φ𝓅	φ𝓅	ADP
easat-3854	64	8	)	)	PUNCT
easat-3854	64	9	is	be	AUX
easat-3854	64	10	the	the	DET
easat-3854	64	11	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-3854	64	12	and	and	CCONJ
easat-3854	64	13	given	give	VERB
easat-3854	64	14	by	by	ADP
easat-3854	64	15	auto	auto	NOUN
easat-3854	64	16	-	-	PUNCT
easat-3854	64	17	morphism	morphism	NOUN
easat-3854	64	18	⋆∶	⋆∶	NOUN
easat-3854	64	19	ℳ𝓅	ℳ𝓅	PROPN
easat-3854	64	20	→ℳ𝓅	→ℳ𝓅	PUNCT
easat-3854	64	21	,	,	PUNCT
easat-3854	64	22	the	the	DET
easat-3854	64	23	involutive	involutive	ADJ
easat-3854	64	24	𝒜∞algebra	𝒜∞algebra	PUNCT
easat-3854	64	25	also	also	ADV
easat-3854	64	26	can	can	AUX
easat-3854	64	27	be	be	AUX
easat-3854	64	28	identifying	identify	VERB
easat-3854	64	29	as	as	ADP
easat-3854	64	30	the	the	DET
easat-3854	64	31	complex	complex	ADJ
easat-3854	64	32	(	(	PUNCT
easat-3854	64	33	ℳ	ℳ	PROPN
easat-3854	64	34	,	,	PUNCT
easat-3854	64	35	𝛿	𝛿	ADJ
easat-3854	64	36	,	,	PUNCT
easat-3854	64	37	𝜑𝓅	𝜑𝓅	NOUN
easat-3854	64	38	,	,	PUNCT
easat-3854	64	39	⋆):ℳ𝓅	⋆):ℳ𝓅	PROPN
easat-3854	64	40	→ℳ𝓅	→ℳ𝓅	INTJ
easat-3854	64	41	such	such	ADJ
easat-3854	64	42	as	as	ADP
easat-3854	64	43	∀𝓂	∀𝓂	NOUN
easat-3854	64	44	∈	∈	PROPN
easat-3854	64	45	ℳ	ℳ	PROPN
easat-3854	64	46	,	,	PUNCT
easat-3854	64	47	⋆	⋆	X
easat-3854	64	48	(	(	PUNCT
easat-3854	64	49	𝓂	𝓂	NOUN
easat-3854	64	50	)	)	PUNCT
easat-3854	64	51	=	=	SYM
easat-3854	64	52	𝓂⋆	𝓂⋆	NOUN
easat-3854	64	53	and	and	CCONJ
easat-3854	64	54	the	the	DET
easat-3854	64	55	conditions	condition	NOUN
easat-3854	64	56	are	be	AUX
easat-3854	64	57	fulfilled	fulfil	VERB
easat-3854	64	58	as	as	SCONJ
easat-3854	64	59	follows	follow	VERB
easat-3854	64	60	:	:	PUNCT
easat-3854	64	61	(	(	PUNCT
easat-3854	64	62	𝓂⋆)⋆	𝓂⋆)⋆	PROPN
easat-3854	64	63	=	=	SYM
easat-3854	64	64	𝓂	𝓂	NOUN
easat-3854	64	65	,	,	PUNCT
easat-3854	64	66	𝛿(𝓂⋆	𝛿(𝓂⋆	NOUN
easat-3854	64	67	)	)	PUNCT
easat-3854	64	68	=	=	PUNCT
easat-3854	65	1	𝛿(𝓂)∗	𝛿(𝓂)∗	ADJ
easat-3854	65	2	,	,	PUNCT
easat-3854	65	3	𝜑𝑛(𝓂0⊗𝓂1⊗	𝜑𝑛(𝓂0⊗𝓂1⊗	NOUN
easat-3854	65	4	…	…	SYM
easat-3854	65	5	⊗𝓂𝓅⊗𝓂𝓅+1	⊗𝓂𝓅⊗𝓂𝓅+1	ADV
easat-3854	65	6	)	)	PUNCT
easat-3854	65	7	⋆	⋆	PUNCT
easat-3854	66	1	=	=	PUNCT
easat-3854	66	2	(	(	PUNCT
easat-3854	66	3	−1)𝜉𝜑𝓅(𝓂𝓅+1	−1)𝜉𝜑𝓅(𝓂𝓅+1	NOUN
easat-3854	66	4	∗	∗	NOUN
easat-3854	66	5	⊗𝓂𝓅	⊗𝓂𝓅	NUM
easat-3854	66	6	∗	∗	PROPN
easat-3854	66	7	⊗	⊗	NUM
easat-3854	66	8	…	…	SYM
easat-3854	66	9	⊗𝓂1	⊗𝓂1	PROPN
easat-3854	66	10	∗	∗	NOUN
easat-3854	66	11	⊗𝓂0	⊗𝓂0	NOUN
easat-3854	66	12	∗	∗	NOUN
easat-3854	66	13	)	)	PUNCT
easat-3854	66	14	.	.	PUNCT
easat-3854	67	1	8661	8661	NUM
easat-3854	67	2	edelweiss	edelweiss	PROPN
easat-3854	67	3	applied	apply	VERB
easat-3854	67	4	science	science	NOUN
easat-3854	67	5	and	and	CCONJ
easat-3854	67	6	technology	technology	NOUN
easat-3854	67	7	issn	issn	PROPN
easat-3854	67	8	:	:	PUNCT
easat-3854	67	9	2576	2576	NUM
easat-3854	67	10	-	-	SYM
easat-3854	67	11	8484	8484	NUM
easat-3854	67	12	vol	vol	NOUN
easat-3854	67	13	.	.	PROPN
easat-3854	67	14	8	8	NUM
easat-3854	67	15	,	,	PUNCT
easat-3854	67	16	no	no	INTJ
easat-3854	67	17	.	.	NOUN
easat-3854	68	1	6	6	NUM
easat-3854	68	2	:	:	SYM
easat-3854	68	3	8658	8658	NUM
easat-3854	68	4	-	-	SYM
easat-3854	68	5	8666	8666	NUM
easat-3854	68	6	,	,	PUNCT
easat-3854	68	7	2024	2024	NUM
easat-3854	68	8	doi	doi	NOUN
easat-3854	68	9	:	:	PUNCT
easat-3854	68	10	10.55214/25768484.v8i6.3854	10.55214/25768484.v8i6.3854	NUM
easat-3854	68	11	©	©	ADP
easat-3854	68	12	2024	2024	NUM
easat-3854	68	13	by	by	ADP
easat-3854	68	14	the	the	DET
easat-3854	68	15	authors	author	NOUN
easat-3854	68	16	;	;	PUNCT
easat-3854	68	17	licensee	licensee	PROPN
easat-3854	68	18	learning	learning	NOUN
easat-3854	68	19	gate	gate	VERB
easat-3854	68	20	such	such	ADJ
easat-3854	68	21	that	that	PRON
easat-3854	68	22	𝜉	𝜉	NOUN
easat-3854	68	23	=	=	PUNCT
easat-3854	68	24	𝓅(𝓅−1	𝓅(𝓅−1	PROPN
easat-3854	68	25	)	)	PUNCT
easat-3854	68	26	2	2	NUM
easat-3854	68	27	+	+	NOUN
easat-3854	68	28	∑	∑	PROPN
easat-3854	68	29	|𝓂𝚤||𝓂𝑗|0≤𝚤<𝑗≤𝓅	|𝓂𝚤||𝓂𝑗|0≤𝚤<𝑗≤𝓅	X
easat-3854	68	30	,	,	PUNCT
easat-3854	68	31	𝓅	𝓅	X
easat-3854	68	32	≥	≥	NUM
easat-3854	68	33	0	0	NUM
easat-3854	68	34	.	.	PUNCT
easat-3854	69	1	therefore	therefore	ADV
easat-3854	69	2	,	,	PUNCT
easat-3854	69	3	a	a	DET
easat-3854	69	4	module	module	NOUN
easat-3854	69	5	of	of	ADP
easat-3854	69	6	a	a	DET
easat-3854	69	7	dihedral	dihedral	ADJ
easat-3854	69	8	differential	differential	NOUN
easat-3854	69	9	remains	remain	VERB
easat-3854	69	10	the	the	DET
easat-3854	69	11	complex	complex	ADJ
easat-3854	69	12	(	(	PUNCT
easat-3854	69	13	ℳ	ℳ	NOUN
easat-3854	69	14	𝜚	𝜚	NOUN
easat-3854	69	15	(	(	PUNCT
easat-3854	69	16	ℳ	ℳ	PROPN
easat-3854	69	17	)	)	PUNCT
easat-3854	69	18	,	,	PUNCT
easat-3854	69	19	𝓉	𝓉	PROPN
easat-3854	69	20	,	,	PUNCT
easat-3854	69	21	𝓇	𝓇	PROPN
easat-3854	69	22	,	,	PUNCT
easat-3854	69	23	𝛿	𝛿	ADJ
easat-3854	69	24	)	)	PUNCT
easat-3854	69	25	,	,	PUNCT
easat-3854	69	26	such	such	ADJ
easat-3854	69	27	as	as	ADP
easat-3854	69	28	𝜚	𝜚	NOUN
easat-3854	69	29	=	=	SYM
easat-3854	69	30	±1	±1	VERB
easat-3854	69	31	,	,	PUNCT
easat-3854	69	32	also	also	ADV
easat-3854	69	33	𝓉𝓅(𝓂0⊗	𝓉𝓅(𝓂0⊗	NOUN
easat-3854	69	34	…	…	X
easat-3854	69	35	⊗𝓂𝓅	⊗𝓂𝓅	X
easat-3854	69	36	)	)	PUNCT
easat-3854	69	37	=	=	SYM
easat-3854	69	38	(	(	PUNCT
easat-3854	69	39	−1	−1	NOUN
easat-3854	69	40	)	)	PUNCT
easat-3854	70	1	𝛽𝓂𝓅⊗𝓂0⊗𝓂1⊗	𝛽𝓂𝓅⊗𝓂0⊗𝓂1⊗	NOUN
easat-3854	70	2	…	…	SYM
easat-3854	70	3	⊗𝓂𝓅−1	⊗𝓂𝓅−1	NOUN
easat-3854	70	4	,	,	PUNCT
easat-3854	70	5	𝓇𝓅(𝓂0⊗	𝓇𝓅(𝓂0⊗	NUM
easat-3854	70	6	…	…	SYM
easat-3854	70	7	⊗𝓂𝓅	⊗𝓂𝓅	X
easat-3854	70	8	)	)	PUNCT
easat-3854	70	9	=	=	SYM
easat-3854	70	10	𝜚(−1	𝜚(−1	NOUN
easat-3854	70	11	)	)	PUNCT
easat-3854	70	12	𝛾𝓂0	𝛾𝓂0	ADP
easat-3854	70	13	∗	∗	NOUN
easat-3854	70	14	⊗𝓂𝓅	⊗𝓂𝓅	X
easat-3854	70	15	∗	∗	NOUN
easat-3854	70	16	⊗𝓂𝓅−1	⊗𝓂𝓅−1	PROPN
easat-3854	70	17	∗	∗	PROPN
easat-3854	70	18	⊗	⊗	NUM
easat-3854	70	19	…	…	SYM
easat-3854	70	20	⊗𝓂1	⊗𝓂1	NUM
easat-3854	70	21	∗	∗	NOUN
easat-3854	70	22	,	,	PUNCT
easat-3854	70	23	𝛿(𝓂0⊗	𝛿(𝓂0⊗	NUM
easat-3854	70	24	…	…	X
easat-3854	70	25	⊗𝓂𝓅	⊗𝓂𝓅	X
easat-3854	70	26	)	)	PUNCT
easat-3854	70	27	=	=	SYM
easat-3854	70	28	∑	∑	PUNCT
easat-3854	70	29	(	(	PUNCT
easat-3854	70	30	−1)𝜇𝓂0⊗	−1)𝜇𝓂0⊗	X
easat-3854	70	31	…	…	PUNCT
easat-3854	70	32	⊗𝓂𝚤−1⊗𝛿𝓂𝚤⊗	⊗𝓂𝚤−1⊗𝛿𝓂𝚤⊗	NUM
easat-3854	70	33	…	…	PUNCT
easat-3854	70	34	⊗𝓂𝓅	⊗𝓂𝓅	X
easat-3854	70	35	𝓅	𝓅	X
easat-3854	70	36	𝚤=0	𝚤=0	NUM
easat-3854	70	37	.	.	PUNCT
easat-3854	71	1	2.3	2.3	NUM
easat-3854	71	2	.	.	PUNCT
easat-3854	72	1	theorem	theorem	VERB
easat-3854	72	2	[	[	X
easat-3854	72	3	5	5	NUM
easat-3854	72	4	]	]	PUNCT
easat-3854	72	5	the	the	DET
easat-3854	72	6	dihedral	dihedral	ADJ
easat-3854	72	7	module	module	NOUN
easat-3854	72	8	(	(	PUNCT
easat-3854	72	9	𝒟ℱ∞-module	𝒟ℱ∞-module	ADJ
easat-3854	72	10	)	)	PUNCT
easat-3854	72	11	is	be	AUX
easat-3854	72	12	defined	define	VERB
easat-3854	72	13	as	as	ADP
easat-3854	72	14	(	(	PUNCT
easat-3854	72	15	ℳ	ℳ	PROPN
easat-3854	72	16	𝜚	𝜚	NOUN
easat-3854	72	17	(	(	PUNCT
easat-3854	72	18	ℳ	ℳ	PROPN
easat-3854	72	19	)	)	PUNCT
easat-3854	72	20	,	,	PUNCT
easat-3854	72	21	𝓉	𝓉	PROPN
easat-3854	72	22	,	,	PUNCT
easat-3854	72	23	𝓇	𝓇	PROPN
easat-3854	72	24	,	,	PUNCT
easat-3854	72	25	𝛿	𝛿	ADJ
easat-3854	72	26	)	)	PUNCT
easat-3854	72	27	,	,	PUNCT
easat-3854	72	28	if	if	SCONJ
easat-3854	72	29	(	(	PUNCT
easat-3854	72	30	ℳ	ℳ	PROPN
easat-3854	72	31	,	,	PUNCT
easat-3854	72	32	𝛿	𝛿	ADJ
easat-3854	72	33	,	,	PUNCT
easat-3854	72	34	𝜑𝓅,⋆	𝜑𝓅,⋆	NOUN
easat-3854	72	35	)	)	PUNCT
easat-3854	72	36	seems	seem	VERB
easat-3854	72	37	to	to	PART
easat-3854	72	38	be	be	AUX
easat-3854	72	39	the	the	DET
easat-3854	72	40	involutive	involutive	ADJ
easat-3854	72	41	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-3854	72	42	.	.	PUNCT
easat-3854	73	1	2.4	2.4	NUM
easat-3854	73	2	.	.	PUNCT
easat-3854	73	3	definition	definition	NOUN
easat-3854	73	4	[	[	X
easat-3854	73	5	6	6	NUM
easat-3854	73	6	]	]	PUNCT
easat-3854	73	7	for	for	ADP
easat-3854	73	8	the	the	DET
easat-3854	73	9	field	field	NOUN
easat-3854	73	10	𝒦	𝒦	PROPN
easat-3854	73	11	of	of	ADP
easat-3854	73	12	a	a	DET
easat-3854	73	13	characteristic	characteristic	ADJ
easat-3854	73	14	zero	zero	NUM
easat-3854	73	15	,	,	PUNCT
easat-3854	73	16	the	the	DET
easat-3854	73	17	one	one	NUM
easat-3854	73	18	-	-	PUNCT
easat-3854	73	19	dimensional	dimensional	ADJ
easat-3854	73	20	vector	vector	NOUN
easat-3854	73	21	spaces	space	NOUN
easat-3854	73	22	of	of	ADP
easat-3854	73	23	degrees	degree	NOUN
easat-3854	73	24	−1	−1	NOUN
easat-3854	73	25	and	and	CCONJ
easat-3854	73	26	1	1	NUM
easat-3854	73	27	with	with	ADP
easat-3854	73	28	0	0	NUM
easat-3854	73	29	-	-	PUNCT
easat-3854	73	30	differential	differential	ADJ
easat-3854	73	31	,	,	PUNCT
easat-3854	73	32	respectively	respectively	ADV
easat-3854	73	33	,	,	PUNCT
easat-3854	73	34	are	be	AUX
easat-3854	73	35	denoted	denote	VERB
easat-3854	73	36	by	by	ADP
easat-3854	73	37	the	the	DET
easat-3854	73	38	notations	notation	NOUN
easat-3854	73	39	𝛴𝒦	𝛴𝒦	NOUN
easat-3854	73	40	and	and	CCONJ
easat-3854	73	41	𝛴−1𝒦.	𝛴−1𝒦.	ADP
easat-3854	73	42	the	the	DET
easat-3854	73	43	free	free	ADJ
easat-3854	73	44	formalized	formalize	VERB
easat-3854	73	45	augment	augment	NOUN
easat-3854	73	46	differential	differential	NOUN
easat-3854	73	47	of	of	ADP
easat-3854	73	48	the	the	DET
easat-3854	73	49	graded	grade	VERB
easat-3854	73	50	associative	associative	ADJ
easat-3854	73	51	algebra	algebra	NOUN
easat-3854	73	52	denoted	denote	VERB
easat-3854	73	53	by	by	ADP
easat-3854	73	54	�	�	PROPN
easat-3854	73	55	̂	̂	NOUN
easat-3854	73	56	�	�	NOUN
easat-3854	73	57	𝒱	𝒱	PROPN
easat-3854	73	58	,	,	PUNCT
easat-3854	73	59	which	which	PRON
easat-3854	73	60	is	be	AUX
easat-3854	73	61	produced	produce	VERB
easat-3854	73	62	by	by	ADP
easat-3854	73	63	𝒱	𝒱	PROPN
easat-3854	73	64	and	and	CCONJ
easat-3854	73	65	given	give	VERB
easat-3854	73	66	by	by	ADP
easat-3854	73	67	:	:	PUNCT
easat-3854	73	68	�	�	PROPN
easat-3854	73	69	̂	̂	VERB
easat-3854	73	70	�	�	NOUN
easat-3854	73	71	𝒱	𝒱	NOUN
easat-3854	73	72	=	=	PROPN
easat-3854	73	73	∏𝒱⊗𝓅	∏𝒱⊗𝓅	ADV
easat-3854	73	74	∞	∞	NUM
easat-3854	73	75	𝓅=0	𝓅=0	NUM
easat-3854	74	1	=	=	SYM
easat-3854	75	1	𝒦	𝒦	ADP
easat-3854	75	2	×	×	PROPN
easat-3854	75	3	𝒱	𝒱	PROPN
easat-3854	75	4	×	×	NOUN
easat-3854	75	5	(	(	PUNCT
easat-3854	75	6	𝒱	𝒱	PROPN
easat-3854	75	7	⊗𝒱	⊗𝒱	NOUN
easat-3854	75	8	)	)	PUNCT
easat-3854	75	9	×	×	NOUN
easat-3854	75	10	…	…	PUNCT
easat-3854	75	11	.	.	PUNCT
easat-3854	76	1	over	over	ADP
easat-3854	76	2	�	�	PROPN
easat-3854	76	3	̂	̂	NUM
easat-3854	76	4	�	�	NOUN
easat-3854	76	5	≥𝚤𝒱	≥𝚤𝒱	NUM
easat-3854	76	6	,	,	PUNCT
easat-3854	76	7	we	we	PRON
easat-3854	76	8	refer	refer	VERB
easat-3854	76	9	to	to	ADP
easat-3854	76	10	the	the	DET
easat-3854	76	11	sub	sub	NOUN
easat-3854	76	12	-	-	NOUN
easat-3854	76	13	algebra	algebra	NOUN
easat-3854	76	14	with	with	ADP
easat-3854	76	15	element	element	NOUN
easat-3854	76	16	orders	order	NOUN
easat-3854	76	17	equal	equal	ADJ
easat-3854	76	18	to	to	ADP
easat-3854	76	19	or	or	CCONJ
easat-3854	76	20	greater	great	ADJ
easat-3854	76	21	than	than	ADP
easat-3854	76	22	𝚤.	𝚤.	NOUN
easat-3854	76	23	2.5	2.5	NUM
easat-3854	76	24	.	.	PUNCT
easat-3854	77	1	definition	definition	NOUN
easat-3854	77	2	[	[	X
easat-3854	77	3	7	7	X
easat-3854	77	4	]	]	PUNCT
easat-3854	77	5	suppose	suppose	VERB
easat-3854	77	6	that	that	SCONJ
easat-3854	77	7	(	(	PUNCT
easat-3854	77	8	ℳ,𝒬	ℳ,𝒬	NOUN
easat-3854	77	9	)	)	PUNCT
easat-3854	77	10	and	and	CCONJ
easat-3854	77	11	(	(	PUNCT
easat-3854	77	12	𝒩	𝒩	PROPN
easat-3854	77	13	,	,	PUNCT
easat-3854	77	14	𝒬′	𝒬′	NUM
easat-3854	77	15	)	)	PUNCT
easat-3854	77	16	are	be	AUX
easat-3854	77	17	𝒜∞-algebras	𝒜∞-algebras	PRON
easat-3854	77	18	.	.	PUNCT
easat-3854	78	1	then	then	ADV
easat-3854	78	2	the	the	DET
easat-3854	78	3	𝒜∞-morphism	𝒜∞-morphism	PROPN
easat-3854	78	4	of	of	ADP
easat-3854	78	5	𝒜∞–algebras	𝒜∞–algebra	NOUN
easat-3854	78	6	are	be	AUX
easat-3854	78	7	a	a	DET
easat-3854	78	8	map	map	NOUN
easat-3854	79	1	𝑓	𝑓	PRON
easat-3854	79	2	of	of	ADP
easat-3854	79	3	associative	associative	ADJ
easat-3854	79	4	algebras	algebra	NOUN
easat-3854	79	5	:	:	PUNCT
easat-3854	79	6	𝑓	𝑓	X
easat-3854	79	7	:	:	PUNCT
easat-3854	79	8	�	�	PROPN
easat-3854	79	9	̂	̂	VERB
easat-3854	79	10	�	�	PROPN
easat-3854	79	11	≥1∑	≥1∑	PROPN
easat-3854	79	12	−1𝒩∗	−1𝒩∗	PROPN
easat-3854	79	13	→	→	SYM
easat-3854	79	14	�	�	PROPN
easat-3854	79	15	̂	̂	VERB
easat-3854	79	16	�	�	PROPN
easat-3854	79	17	≥1∑	≥1∑	ADP
easat-3854	79	18	−1ℳ⋆	−1ℳ⋆	PROPN
easat-3854	79	19	,	,	PUNCT
easat-3854	79	20	such	such	ADJ
easat-3854	79	21	that	that	SCONJ
easat-3854	79	22	𝒬	𝒬	PROPN
easat-3854	79	23	∘	∘	NOUN
easat-3854	79	24	𝑓	𝑓	PROPN
easat-3854	79	25	=	=	X
easat-3854	79	26	𝑓	𝑓	DET
easat-3854	79	27	∘	∘	PROPN
easat-3854	79	28	𝒬′	𝒬′	X
easat-3854	79	29	and	and	CCONJ
easat-3854	79	30	f	f	PROPN
easat-3854	79	31	preserves	preserve	VERB
easat-3854	79	32	the	the	DET
easat-3854	79	33	involution	involution	NOUN
easat-3854	79	34	:	:	PUNCT
easat-3854	79	35	𝑓(𝓂⋆	𝑓(𝓂⋆	NOUN
easat-3854	79	36	)	)	PUNCT
easat-3854	79	37	=	=	PUNCT
easat-3854	79	38	𝑓(𝓂)⋆.	𝑓(𝓂)⋆.	VERB
easat-3854	79	39	2.6	2.6	NUM
easat-3854	79	40	.	.	PUNCT
easat-3854	80	1	definition	definition	NOUN
easat-3854	80	2	[	[	X
easat-3854	80	3	8	8	X
easat-3854	80	4	]	]	X
easat-3854	80	5	if	if	SCONJ
easat-3854	80	6	the	the	DET
easat-3854	80	7	space	space	NOUN
easat-3854	80	8	of	of	ADP
easat-3854	80	9	derivations	derivation	NOUN
easat-3854	80	10	𝐷𝑒𝑟(	𝐷𝑒𝑟(	PROPN
easat-3854	80	11	�	�	PROPN
easat-3854	80	12	̂	̂	NUM
easat-3854	80	13	�	�	PROPN
easat-3854	80	14	≥1∑	≥1∑	NOUN
easat-3854	80	15	−1ℳ∗	−1ℳ∗	NUM
easat-3854	80	16	)	)	PUNCT
easat-3854	80	17	for	for	ADP
easat-3854	80	18	the	the	DET
easat-3854	80	19	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-3854	80	20	(	(	PUNCT
easat-3854	80	21	ℳ,𝒬	ℳ,𝒬	NOUN
easat-3854	80	22	)	)	PUNCT
easat-3854	80	23	,	,	PUNCT
easat-3854	80	24	then	then	ADV
easat-3854	80	25	the	the	DET
easat-3854	80	26	differential	differential	NOUN
easat-3854	80	27	graded	grade	VERB
easat-3854	80	28	vector	vector	NOUN
easat-3854	80	29	space	space	NOUN
easat-3854	80	30	ℳ	ℳ	PROPN
easat-3854	80	31	is	be	AUX
easat-3854	80	32	the	the	DET
easat-3854	80	33	hochschild	hochschild	ADJ
easat-3854	80	34	homology	homology	NOUN
easat-3854	80	35	complex	complex	ADJ
easat-3854	80	36	ℋℋ∎(ℳ,ℳ	ℋℋ∎(ℳ,ℳ	NOUN
easat-3854	80	37	)	)	PUNCT
easat-3854	80	38	of	of	ADP
easat-3854	80	39	ℳ	ℳ	PROPN
easat-3854	80	40	with	with	ADP
easat-3854	80	41	coefficients	coefficient	NOUN
easat-3854	80	42	in	in	ADP
easat-3854	80	43	itself	itself	PRON
easat-3854	80	44	:	:	PUNCT
easat-3854	80	45	𝒞ℋ∎(ℳ,ℳ	𝒞ℋ∎(ℳ,ℳ	X
easat-3854	80	46	)	)	PUNCT
easat-3854	80	47	=	=	SYM
easat-3854	80	48	∑	∑	PUNCT
easat-3854	80	49	−1𝐷𝑒𝑟(	−1𝐷𝑒𝑟(	PROPN
easat-3854	80	50	�	�	PROPN
easat-3854	80	51	̂	̂	VERB
easat-3854	80	52	�	�	PROPN
easat-3854	80	53	≥1∑	≥1∑	VERB
easat-3854	80	54	−1ℳ⋆	−1ℳ⋆	NOUN
easat-3854	80	55	)	)	PUNCT
easat-3854	80	56	.	.	PUNCT
easat-3854	81	1	(	(	PUNCT
easat-3854	81	2	5	5	X
easat-3854	81	3	)	)	PUNCT
easat-3854	81	4	2.7	2.7	NUM
easat-3854	81	5	.	.	PUNCT
easat-3854	82	1	definition	definition	NOUN
easat-3854	82	2	[	[	X
easat-3854	82	3	5	5	NUM
easat-3854	82	4	]	]	X
easat-3854	82	5	let	let	ADJ
easat-3854	82	6	(	(	PUNCT
easat-3854	82	7	ℳ,𝒬	ℳ,𝒬	NOUN
easat-3854	82	8	)	)	PUNCT
easat-3854	82	9	be	be	AUX
easat-3854	82	10	the	the	DET
easat-3854	82	11	involutive	involutive	ADJ
easat-3854	82	12	𝒜∞-algebras	𝒜∞-algebras	ADJ
easat-3854	82	13	.	.	PUNCT
easat-3854	83	1	so	so	ADV
easat-3854	83	2	the	the	DET
easat-3854	83	3	cyclic	cyclic	ADJ
easat-3854	83	4	homology	homology	NOUN
easat-3854	83	5	of	of	ADP
easat-3854	83	6	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-3854	83	7	ℋ𝒞∎(ℳ	ℋ𝒞∎(ℳ	X
easat-3854	83	8	)	)	PUNCT
easat-3854	83	9	is	be	AUX
easat-3854	83	10	a	a	DET
easat-3854	83	11	differential	differential	ADJ
easat-3854	83	12	graded	grade	VERB
easat-3854	83	13	vector	vector	NOUN
easat-3854	83	14	space	space	NOUN
easat-3854	83	15	𝒞𝒞∎(ℳ	𝒞𝒞∎(ℳ	PROPN
easat-3854	83	16	)	)	PUNCT
easat-3854	83	17	,	,	PUNCT
easat-3854	83	18	which	which	PRON
easat-3854	83	19	is	be	AUX
easat-3854	83	20	represented	represent	VERB
easat-3854	83	21	by	by	ADP
easat-3854	83	22	:	:	PUNCT
easat-3854	83	23	𝒞𝒞∎(ℳ	𝒞𝒞∎(ℳ	PROPN
easat-3854	83	24	)	)	PUNCT
easat-3854	84	1	=	=	SYM
easat-3854	84	2	∑∏	∑∏	NOUN
easat-3854	84	3	∞	∞	PROPN
easat-3854	84	4	𝚤=1	𝚤=1	PROPN
easat-3854	85	1	[	[	X
easat-3854	85	2	(	(	PUNCT
easat-3854	85	3	∑	∑	INTJ
easat-3854	85	4	−1ℳ⋆)⊗𝚤]𝒵𝚤	−1ℳ⋆)⊗𝚤]𝒵𝚤	PROPN
easat-3854	85	5	,	,	PUNCT
easat-3854	85	6	(	(	PUNCT
easat-3854	85	7	6	6	NUM
easat-3854	85	8	)	)	PUNCT
easat-3854	85	9	such	such	ADJ
easat-3854	85	10	as	as	SCONJ
easat-3854	85	11	,	,	PUNCT
easat-3854	85	12	𝒵𝚤	𝒵𝚤	PROPN
easat-3854	85	13	is	be	AUX
easat-3854	85	14	the	the	DET
easat-3854	85	15	cyclic	cyclic	ADJ
easat-3854	85	16	group	group	NOUN
easat-3854	85	17	of	of	ADP
easat-3854	85	18	order	order	NOUN
easat-3854	85	19	𝚤.	𝚤.	ADJ
easat-3854	85	20	2.8	2.8	NUM
easat-3854	85	21	.	.	PUNCT
easat-3854	86	1	definition	definition	NOUN
easat-3854	86	2	[	[	X
easat-3854	86	3	9	9	NUM
easat-3854	86	4	]	]	PUNCT
easat-3854	86	5	by	by	ADP
easat-3854	86	6	considering	consider	VERB
easat-3854	86	7	ℳ	ℳ	PROPN
easat-3854	86	8	is	be	AUX
easat-3854	86	9	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-3854	86	10	(	(	PUNCT
easat-3854	86	11	ℳ	ℳ	PROPN
easat-3854	86	12	,	,	PUNCT
easat-3854	86	13	𝛿	𝛿	ADJ
easat-3854	86	14	,	,	PUNCT
easat-3854	86	15	𝜑𝑛	𝜑𝑛	NOUN
easat-3854	86	16	)	)	PUNCT
easat-3854	86	17	,	,	PUNCT
easat-3854	86	18	then	then	ADV
easat-3854	86	19	the	the	DET
easat-3854	86	20	cyclic	cyclic	ADJ
easat-3854	86	21	differential	differential	NOUN
easat-3854	86	22	module	module	NOUN
easat-3854	86	23	(	(	PUNCT
easat-3854	86	24	𝒞(ℳ	𝒞(ℳ	PROPN
easat-3854	86	25	)	)	PUNCT
easat-3854	86	26	,	,	PUNCT
easat-3854	86	27	𝓉	𝓉	PROPN
easat-3854	86	28	,	,	PUNCT
easat-3854	86	29	𝛿	𝛿	ADJ
easat-3854	86	30	)	)	PUNCT
easat-3854	86	31	is	be	AUX
easat-3854	86	32	denoted	denote	VERB
easat-3854	86	33	by	by	ADP
easat-3854	86	34	:	:	PUNCT
easat-3854	86	35	𝒞(ℳ	𝒞(ℳ	X
easat-3854	86	36	)	)	PUNCT
easat-3854	86	37	=	=	PUNCT
easat-3854	86	38	{	{	PUNCT
easat-3854	86	39	𝒞(ℳ)𝑠,𝑛	𝒞(ℳ)𝑠,𝑛	PROPN
easat-3854	86	40	}	}	PUNCT
easat-3854	86	41	,	,	PUNCT
easat-3854	86	42	𝑠𝑖𝑛𝑐𝑒	𝑠𝑖𝑛𝑐𝑒	VERB
easat-3854	86	43	𝒞(ℳ)𝑠,𝑛	𝒞(ℳ)𝑠,𝑛	NUM
easat-3854	86	44	=	=	SYM
easat-3854	86	45	(	(	PUNCT
easat-3854	86	46	ℳ	ℳ	PROPN
easat-3854	86	47	⊗(𝑛+2	⊗(𝑛+2	NOUN
easat-3854	86	48	)	)	PUNCT
easat-3854	86	49	)	)	PUNCT
easat-3854	87	1	𝑠	𝑠	PROPN
easat-3854	87	2	,	,	PUNCT
easat-3854	87	3	∀𝑛	∀𝑛	PROPN
easat-3854	87	4	,	,	PUNCT
easat-3854	87	5	𝑠	𝑠	X
easat-3854	87	6	≥	≥	NOUN
easat-3854	87	7	0	0	NUM
easat-3854	87	8	,	,	PUNCT
easat-3854	87	9	𝓉𝑛(𝓂0⊗	𝓉𝑛(𝓂0⊗	NUM
easat-3854	87	10	…	…	X
easat-3854	87	11	⊗𝓂𝑠	⊗𝓂𝑠	NOUN
easat-3854	87	12	)	)	PUNCT
easat-3854	87	13	=	=	SYM
easat-3854	87	14	(	(	PUNCT
easat-3854	87	15	−1	−1	NOUN
easat-3854	87	16	)	)	PUNCT
easat-3854	87	17	|𝓂𝑛|(|𝓂0	|𝓂𝑛|(|𝓂0	NOUN
easat-3854	87	18	|+⋯+|𝓂𝑠−1|	|+⋯+|𝓂𝑠−1|	NUM
easat-3854	87	19	𝓂𝑛⊗𝓂0	𝓂𝑛⊗𝓂0	PRON
easat-3854	87	20	⊗	⊗	NUM
easat-3854	87	21	…	…	SYM
easat-3854	87	22	⊗𝓂𝑠−1	⊗𝓂𝑠−1	ADJ
easat-3854	87	23	,	,	PUNCT
easat-3854	87	24	𝛿𝑛(𝓂0⊗	𝛿𝑛(𝓂0⊗	NUM
easat-3854	87	25	…	…	SYM
easat-3854	87	26	⊗𝓂𝑠	⊗𝓂𝑠	NOUN
easat-3854	87	27	)	)	PUNCT
easat-3854	87	28	=	=	PUNCT
easat-3854	88	1	∑(−1)|𝓂0	∑(−1)|𝓂0	PROPN
easat-3854	88	2	|+⋯+|𝓂𝑘−1|	|+⋯+|𝓂𝑘−1|	PROPN
easat-3854	88	3	𝓂0	𝓂0	PROPN
easat-3854	88	4	⊗	⊗	PROPN
easat-3854	88	5	…	…	PROPN
easat-3854	88	6	⊗𝓂𝑘−1	⊗𝓂𝑘−1	PROPN
easat-3854	88	7	𝑛	𝑛	PROPN
easat-3854	88	8	𝑘=0	𝑘=0	PRON
easat-3854	88	9	⊗𝛿𝓂𝑘⊗𝓂𝑘+1⊗𝓂𝑛	⊗𝛿𝓂𝑘⊗𝓂𝑘+1⊗𝓂𝑛	NOUN
easat-3854	88	10	,	,	PUNCT
easat-3854	88	11	8662	8662	NUM
easat-3854	88	12	edelweiss	edelweiss	PROPN
easat-3854	88	13	applied	apply	VERB
easat-3854	88	14	science	science	NOUN
easat-3854	88	15	and	and	CCONJ
easat-3854	88	16	technology	technology	NOUN
easat-3854	88	17	issn	issn	PROPN
easat-3854	88	18	:	:	PUNCT
easat-3854	88	19	2576	2576	NUM
easat-3854	88	20	-	-	SYM
easat-3854	88	21	8484	8484	NUM
easat-3854	88	22	vol	vol	NOUN
easat-3854	88	23	.	.	PROPN
easat-3854	88	24	8	8	NUM
easat-3854	88	25	,	,	PUNCT
easat-3854	88	26	no	no	INTJ
easat-3854	88	27	.	.	NOUN
easat-3854	89	1	6	6	NUM
easat-3854	89	2	:	:	SYM
easat-3854	89	3	8658	8658	NUM
easat-3854	89	4	-	-	SYM
easat-3854	89	5	8666	8666	NUM
easat-3854	89	6	,	,	PUNCT
easat-3854	89	7	2024	2024	NUM
easat-3854	89	8	doi	doi	NOUN
easat-3854	89	9	:	:	PUNCT
easat-3854	89	10	10.55214/25768484.v8i6.3854	10.55214/25768484.v8i6.3854	NUM
easat-3854	89	11	©	©	ADP
easat-3854	89	12	2024	2024	NUM
easat-3854	89	13	by	by	ADP
easat-3854	89	14	the	the	DET
easat-3854	89	15	authors	author	NOUN
easat-3854	89	16	;	;	PUNCT
easat-3854	89	17	licensee	licensee	PROPN
easat-3854	89	18	learning	learning	NOUN
easat-3854	89	19	gate	gate	VERB
easat-3854	89	20	such	such	ADJ
easat-3854	89	21	|𝓂|	|𝓂|	PROPN
easat-3854	89	22	=	=	NOUN
easat-3854	89	23	∗	∗	NOUN
easat-3854	89	24	means	mean	VERB
easat-3854	89	25	that	that	SCONJ
easat-3854	89	26	,	,	PUNCT
easat-3854	89	27	𝓂	𝓂	PROPN
easat-3854	89	28	∈ℳ∗.	∈ℳ∗.	PROPN
easat-3854	89	29	also	also	ADV
easat-3854	89	30	,	,	PUNCT
easat-3854	89	31	suppose	suppose	VERB
easat-3854	89	32	that	that	SCONJ
easat-3854	89	33	the	the	DET
easat-3854	89	34	family	family	NOUN
easat-3854	89	35	of	of	ADP
easat-3854	89	36	maps	map	NOUN
easat-3854	89	37	:	:	PUNCT
easat-3854	89	38	𝒷′	𝒷′	PROPN
easat-3854	89	39	=	=	SYM
easat-3854	89	40	{	{	PUNCT
easat-3854	89	41	𝒷(𝑘1,	𝒷(𝑘1,	PROPN
easat-3854	89	42	…	…	X
easat-3854	89	43	,𝑘𝑠	,𝑘𝑠	NUM
easat-3854	89	44	):	):	PUNCT
easat-3854	89	45	𝒞(ℳ)𝑛,𝑝	𝒞(ℳ)𝑛,𝑝	PROPN
easat-3854	89	46	→	→	SYM
easat-3854	89	47	𝒞(ℳ)𝑛−𝑠,𝑝+𝑠−1	𝒞(ℳ)𝑛−𝑠,𝑝+𝑠−1	PROPN
easat-3854	89	48	}	}	PUNCT
easat-3854	89	49	,	,	PUNCT
easat-3854	89	50	0	0	NUM
easat-3854	89	51	≤	≤	NOUN
easat-3854	89	52	𝑘1	𝑘1	PROPN
easat-3854	89	53	<	<	X
easat-3854	89	54	⋯	⋯	X
easat-3854	89	55	<	<	X
easat-3854	89	56	𝑘𝑠	𝑘𝑠	X
easat-3854	89	57	<	<	X
easat-3854	89	58	𝑛	𝑛	PROPN
easat-3854	89	59	,	,	PUNCT
easat-3854	89	60	𝑛	𝑛	PROPN
easat-3854	89	61	,	,	PUNCT
easat-3854	89	62	𝑝	𝑝	PRON
easat-3854	89	63	≥	≥	NOUN
easat-3854	89	64	0	0	NUM
easat-3854	89	65	,	,	PUNCT
easat-3854	89	66	denoted	denote	VERB
easat-3854	89	67	by	by	ADP
easat-3854	89	68	:	:	PUNCT
easat-3854	89	69	𝒷(𝑘1,	𝒷(𝑘1,	PROPN
easat-3854	89	70	…	…	X
easat-3854	89	71	,𝑘𝑠	,𝑘𝑠	PUNCT
easat-3854	89	72	)	)	PUNCT
easat-3854	89	73	=	=	PRON
easat-3854	89	74	{	{	PUNCT
easat-3854	89	75	(	(	PUNCT
easat-3854	89	76	−1)𝑠(𝑝−1)1⊗𝑗⊗𝜑𝑠−1⊗1⊗(𝑛−𝑠−𝑗	−1)𝑠(𝑝−1)1⊗𝑗⊗𝜑𝑠−1⊗1⊗(𝑛−𝑠−𝑗	NUM
easat-3854	89	77	)	)	PUNCT
easat-3854	89	78	,	,	PUNCT
easat-3854	89	79	𝑖𝑓	𝑖𝑓	ADP
easat-3854	89	80	0	0	NUM
easat-3854	89	81	≤	≤	NUM
easat-3854	90	1	𝑗	𝑗	PRON
easat-3854	90	2	≤	≤	NUM
easat-3854	90	3	𝑛	𝑛	DET
easat-3854	90	4	−	−	PROPN
easat-3854	90	5	𝑠	𝑠	PROPN
easat-3854	90	6	,	,	PUNCT
easat-3854	90	7	(	(	PUNCT
easat-3854	90	8	𝑘1	𝑘1	PROPN
easat-3854	90	9	,	,	PUNCT
easat-3854	90	10	…	…	PUNCT
easat-3854	90	11	,	,	PUNCT
easat-3854	90	12	𝑘𝑠	𝑘𝑠	NOUN
easat-3854	90	13	)	)	PUNCT
easat-3854	90	14	=	=	SYM
easat-3854	90	15	(	(	PUNCT
easat-3854	90	16	𝑗	𝑗	PROPN
easat-3854	90	17	,	,	PUNCT
easat-3854	90	18	𝑗	𝑗	PROPN
easat-3854	90	19	+	+	ADJ
easat-3854	90	20	1	1	NUM
easat-3854	90	21	,	,	PUNCT
easat-3854	90	22	…	…	PUNCT
easat-3854	90	23	,	,	PUNCT
easat-3854	91	1	𝑗	𝑗	X
easat-3854	91	2	+	+	NUM
easat-3854	91	3	𝑠	𝑠	INTJ
easat-3854	91	4	−	−	NOUN
easat-3854	91	5	1	1	NUM
easat-3854	91	6	)	)	PUNCT
easat-3854	91	7	(	(	PUNCT
easat-3854	91	8	−1)𝑞(𝑠−1)𝒷(0,1,	−1)𝑞(𝑠−1)𝒷(0,1,	NOUN
easat-3854	91	9	…	…	SYM
easat-3854	91	10	,𝑠−1)𝓉𝑛	,𝑠−1)𝓉𝑛	PUNCT
easat-3854	91	11	𝑞	𝑞	X
easat-3854	91	12	,	,	PUNCT
easat-3854	91	13	𝑖𝑓	𝑖𝑓	ADP
easat-3854	91	14	1	1	NUM
easat-3854	91	15	≤	≤	NOUN
easat-3854	91	16	𝑞	𝑞	X
easat-3854	91	17	≤	≤	PROPN
easat-3854	91	18	𝑠	𝑠	PROPN
easat-3854	91	19	,	,	PUNCT
easat-3854	91	20	𝑎𝑛𝑑	𝑎𝑛𝑑	X
easat-3854	91	21	(	(	PUNCT
easat-3854	91	22	𝑘1	𝑘1	PROPN
easat-3854	91	23	,	,	PUNCT
easat-3854	91	24	…	…	PUNCT
easat-3854	91	25	,	,	PUNCT
easat-3854	91	26	𝑘𝑠	𝑘𝑠	NOUN
easat-3854	91	27	)	)	PUNCT
easat-3854	91	28	=	=	SYM
easat-3854	91	29	(	(	PUNCT
easat-3854	91	30	0,1	0,1	NUM
easat-3854	91	31	,	,	PUNCT
easat-3854	91	32	…	…	PUNCT
easat-3854	91	33	,	,	PUNCT
easat-3854	91	34	𝑠	𝑠	INTJ
easat-3854	91	35	−	−	NOUN
easat-3854	91	36	𝑞	𝑞	X
easat-3854	91	37	−	−	PROPN
easat-3854	91	38	1	1	NUM
easat-3854	91	39	,	,	PUNCT
easat-3854	91	40	𝑛	𝑛	PRON
easat-3854	91	41	−	−	NOUN
easat-3854	91	42	𝑞	𝑞	X
easat-3854	91	43	+	+	PROPN
easat-3854	91	44	1	1	NUM
easat-3854	91	45	,	,	PUNCT
easat-3854	91	46	𝑛	𝑛	PRON
easat-3854	91	47	−	−	NOUN
easat-3854	91	48	𝑞	𝑞	X
easat-3854	91	49	+	+	PROPN
easat-3854	91	50	2	2	NUM
easat-3854	91	51	,	,	PUNCT
easat-3854	91	52	…	…	PUNCT
easat-3854	91	53	,	,	PUNCT
easat-3854	91	54	𝑛	𝑛	NOUN
easat-3854	91	55	)	)	PUNCT
easat-3854	91	56	0	0	NUM
easat-3854	91	57	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.	PROPN
easat-3854	92	1	the	the	DET
easat-3854	92	2	quadruple	quadruple	NOUN
easat-3854	92	3	(	(	PUNCT
easat-3854	92	4	𝒞(ℳ	𝒞(ℳ	PROPN
easat-3854	92	5	)	)	PUNCT
easat-3854	92	6	,	,	PUNCT
easat-3854	92	7	𝓉	𝓉	PROPN
easat-3854	92	8	,	,	PUNCT
easat-3854	92	9	𝜕	𝜕	PROPN
easat-3854	92	10	,	,	PUNCT
easat-3854	92	11	𝒷′	𝒷′	X
easat-3854	92	12	)	)	PUNCT
easat-3854	92	13	is	be	AUX
easat-3854	92	14	the	the	DET
easat-3854	92	15	cyclic	cyclic	ADJ
easat-3854	92	16	modules	module	NOUN
easat-3854	92	17	of	of	ADP
easat-3854	92	18	∞-simplicial	∞-simplicial	NOUN
easat-3854	92	19	sides	side	NOUN
easat-3854	92	20	,	,	PUNCT
easat-3854	92	21	for	for	ADP
easat-3854	92	22	each	each	PRON
easat-3854	92	23	𝒜∞algebras	𝒜∞algebras	NUM
easat-3854	92	24	(	(	PUNCT
easat-3854	92	25	ℳ	ℳ	PROPN
easat-3854	92	26	,	,	PUNCT
easat-3854	92	27	𝛿	𝛿	ADJ
easat-3854	92	28	,	,	PUNCT
easat-3854	92	29	𝜑𝑛	𝜑𝑛	NOUN
easat-3854	92	30	)	)	PUNCT
easat-3854	92	31	.	.	PUNCT
easat-3854	93	1	also	also	ADV
easat-3854	93	2	,	,	PUNCT
easat-3854	93	3	we	we	PRON
easat-3854	93	4	can	can	AUX
easat-3854	93	5	define	define	VERB
easat-3854	93	6	a	a	DET
easat-3854	93	7	cyclic	cyclic	ADJ
easat-3854	93	8	homology	homology	NOUN
easat-3854	93	9	ℋ𝒞(ℳ	ℋ𝒞(ℳ	NOUN
easat-3854	93	10	)	)	PUNCT
easat-3854	93	11	of	of	ADP
easat-3854	93	12	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-3854	93	13	by	by	ADP
easat-3854	93	14	the	the	DET
easat-3854	93	15	cyclic	cyclic	ADJ
easat-3854	93	16	homology	homology	NOUN
easat-3854	93	17	ℋ𝒞(𝒞(ℳ	ℋ𝒞(𝒞(ℳ	PROPN
easat-3854	93	18	)	)	PUNCT
easat-3854	93	19	)	)	PUNCT
easat-3854	93	20	of	of	ADP
easat-3854	93	21	the	the	DET
easat-3854	93	22	cyclic	cyclic	ADJ
easat-3854	93	23	modules	module	NOUN
easat-3854	93	24	of	of	ADP
easat-3854	93	25	∞-simplicial	∞-simplicial	NOUN
easat-3854	93	26	sides	side	NOUN
easat-3854	93	27	(	(	PUNCT
easat-3854	93	28	𝒞(ℳ	𝒞(ℳ	X
easat-3854	93	29	)	)	PUNCT
easat-3854	93	30	,	,	PUNCT
easat-3854	93	31	𝓉	𝓉	PROPN
easat-3854	93	32	,	,	PUNCT
easat-3854	93	33	∂	∂	NUM
easat-3854	93	34	,	,	PUNCT
easat-3854	93	35	𝒷′	𝒷′	NUM
easat-3854	93	36	)	)	PUNCT
easat-3854	93	37	.	.	PUNCT
easat-3854	94	1	ℋ𝒞(ℳ	ℋ𝒞(ℳ	NOUN
easat-3854	94	2	)	)	PUNCT
easat-3854	95	1	=	=	PUNCT
easat-3854	95	2	(	(	PUNCT
easat-3854	95	3	𝑇𝑜𝑡𝒞(𝒞(ℳ))̅̅	𝑇𝑜𝑡𝒞(𝒞(ℳ))̅̅	PROPN
easat-3854	95	4	̅̅	̅̅	PROPN
easat-3854	95	5	̅̅	̅̅	PROPN
easat-3854	95	6	̅̅	̅̅	PROPN
easat-3854	95	7	̅̅	̅̅	PROPN
easat-3854	95	8	̅	̅	PROPN
easat-3854	95	9	,	,	PUNCT
easat-3854	95	10	𝐷	𝐷	NOUN
easat-3854	95	11	)	)	PUNCT
easat-3854	95	12	,	,	PUNCT
easat-3854	95	13	𝐷	𝐷	NOUN
easat-3854	95	14	=	=	PUNCT
easat-3854	95	15	𝐷1	𝐷1	PROPN
easat-3854	95	16	+	+	NUM
easat-3854	95	17	𝐷2	𝐷2	NOUN
easat-3854	95	18	,	,	PUNCT
easat-3854	95	19	(	(	PUNCT
easat-3854	95	20	7	7	X
easat-3854	95	21	)	)	PUNCT
easat-3854	95	22	consequently	consequently	ADV
easat-3854	95	23	,	,	PUNCT
easat-3854	95	24	the	the	DET
easat-3854	95	25	cyclic	cyclic	ADJ
easat-3854	95	26	homology	homology	NOUN
easat-3854	95	27	ℋ𝒞(ℳ	ℋ𝒞(ℳ	NOUN
easat-3854	95	28	)	)	PUNCT
easat-3854	95	29	of	of	ADP
easat-3854	95	30	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-3854	95	31	is	be	AUX
easat-3854	95	32	the	the	DET
easat-3854	95	33	homology	homology	NOUN
easat-3854	95	34	of	of	ADP
easat-3854	95	35	the	the	DET
easat-3854	95	36	chain	chain	NOUN
easat-3854	95	37	complex	complex	ADJ
easat-3854	95	38	ℋ𝒞(ℳ	ℋ𝒞(ℳ	NOUN
easat-3854	95	39	)	)	PUNCT
easat-3854	95	40	=	=	PUNCT
easat-3854	96	1	(	(	PUNCT
easat-3854	96	2	𝑇𝑜𝑡𝒞(𝒞(ℳ))̅̅	𝑇𝑜𝑡𝒞(𝒞(ℳ))̅̅	PROPN
easat-3854	96	3	̅̅	̅̅	PROPN
easat-3854	96	4	̅̅	̅̅	PROPN
easat-3854	96	5	̅̅	̅̅	PROPN
easat-3854	96	6	̅̅	̅̅	PROPN
easat-3854	96	7	̅	̅	PROPN
easat-3854	96	8	,	,	PUNCT
easat-3854	96	9	𝐷	𝐷	NOUN
easat-3854	96	10	)	)	PUNCT
easat-3854	96	11	,	,	PUNCT
easat-3854	96	12	𝐷	𝐷	NOUN
easat-3854	96	13	=	=	PUNCT
easat-3854	96	14	𝐷1	𝐷1	PROPN
easat-3854	96	15	+	+	NUM
easat-3854	96	16	𝐷2	𝐷2	NOUN
easat-3854	96	17	,	,	PUNCT
easat-3854	96	18	associated	associate	VERB
easat-3854	96	19	with	with	ADP
easat-3854	96	20	the	the	DET
easat-3854	96	21	chain	chain	NOUN
easat-3854	96	22	bicomplex	bicomplex	NOUN
easat-3854	96	23	(	(	PUNCT
easat-3854	96	24	𝒞(𝒞(ℳ)̅̅	𝒞(𝒞(ℳ)̅̅	PROPN
easat-3854	96	25	̅̅	̅̅	PROPN
easat-3854	96	26	̅̅	̅̅	PROPN
easat-3854	96	27	̅̅	̅̅	PROPN
easat-3854	96	28	)	)	PUNCT
easat-3854	96	29	,	,	PUNCT
easat-3854	96	30	𝐷1	𝐷1	PROPN
easat-3854	96	31	,	,	PUNCT
easat-3854	96	32	𝐷2	𝐷2	PROPN
easat-3854	96	33	)	)	PUNCT
easat-3854	96	34	.	.	PUNCT
easat-3854	97	1	note	note	VERB
easat-3854	97	2	that	that	SCONJ
easat-3854	97	3	if	if	SCONJ
easat-3854	97	4	an	an	DET
easat-3854	97	5	𝒜∞-algebra	𝒜∞-algebra	PROPN
easat-3854	97	6	is	be	AUX
easat-3854	97	7	a	a	DET
easat-3854	97	8	differential	differential	ADJ
easat-3854	97	9	associative	associative	NOUN
easat-3854	97	10	algebra	algebra	NOUN
easat-3854	97	11	(	(	PUNCT
easat-3854	97	12	ℳ	ℳ	PROPN
easat-3854	97	13	,	,	PUNCT
easat-3854	97	14	𝛿	𝛿	ADJ
easat-3854	97	15	,	,	PUNCT
easat-3854	97	16	𝜑𝑛	𝜑𝑛	NOUN
easat-3854	97	17	)	)	PUNCT
easat-3854	97	18	,	,	PUNCT
easat-3854	97	19	where	where	SCONJ
easat-3854	97	20	𝜑0	𝜑0	NOUN
easat-3854	97	21	=	=	SYM
easat-3854	97	22	𝜑	𝜑	PROPN
easat-3854	97	23	and	and	CCONJ
easat-3854	97	24	𝜑𝑛	𝜑𝑛	ADP
easat-3854	97	25	=	=	NOUN
easat-3854	97	26	0	0	PROPN
easat-3854	97	27	,	,	PUNCT
easat-3854	97	28	𝑛	𝑛	PROPN
easat-3854	97	29	>	>	X
easat-3854	97	30	0	0	NUM
easat-3854	97	31	,	,	PUNCT
easat-3854	97	32	then	then	ADV
easat-3854	97	33	the	the	DET
easat-3854	97	34	chain	chain	NOUN
easat-3854	97	35	bicomplex	bicomplex	NOUN
easat-3854	97	36	(	(	PUNCT
easat-3854	97	37	𝒞(𝒞(ℳ)̅̅	𝒞(𝒞(ℳ)̅̅	PROPN
easat-3854	97	38	̅̅	̅̅	PROPN
easat-3854	97	39	̅̅	̅̅	PROPN
easat-3854	97	40	̅̅	̅̅	PROPN
easat-3854	97	41	)	)	PUNCT
easat-3854	97	42	,	,	PUNCT
easat-3854	97	43	𝐷1	𝐷1	PROPN
easat-3854	97	44	,	,	PUNCT
easat-3854	97	45	𝐷2	𝐷2	PROPN
easat-3854	97	46	)	)	PUNCT
easat-3854	97	47	coincides	coincide	VERB
easat-3854	97	48	with	with	ADP
easat-3854	97	49	the	the	DET
easat-3854	97	50	tsygan	tsygan	ADJ
easat-3854	97	51	chain	chain	NOUN
easat-3854	97	52	bicomplex	bicomplex	NOUN
easat-3854	97	53	for	for	ADP
easat-3854	97	54	the	the	DET
easat-3854	97	55	differential	differential	ADJ
easat-3854	97	56	associative	associative	NOUN
easat-3854	97	57	algebra	algebra	NOUN
easat-3854	97	58	(	(	PUNCT
easat-3854	97	59	ℳ	ℳ	PROPN
easat-3854	97	60	,	,	PUNCT
easat-3854	97	61	𝛿	𝛿	ADJ
easat-3854	97	62	,	,	PUNCT
easat-3854	97	63	𝜑𝑛	𝜑𝑛	NOUN
easat-3854	97	64	)	)	PUNCT
easat-3854	97	65	.	.	PUNCT
easat-3854	98	1	2.9	2.9	NUM
easat-3854	98	2	.	.	PUNCT
easat-3854	98	3	definition	definition	NOUN
easat-3854	98	4	[	[	X
easat-3854	98	5	10	10	NUM
easat-3854	98	6	]	]	PUNCT
easat-3854	98	7	for	for	ADP
easat-3854	98	8	a	a	DET
easat-3854	98	9	graded	grade	VERB
easat-3854	98	10	vector	vector	NOUN
easat-3854	98	11	space	space	NOUN
easat-3854	98	12	ℳ	ℳ	NOUN
easat-3854	98	13	be	be	VERB
easat-3854	98	14	with	with	ADP
easat-3854	98	15	an	an	DET
easat-3854	98	16	involution	involution	NOUN
easat-3854	98	17	.	.	PUNCT
easat-3854	99	1	then	then	ADV
easat-3854	99	2	the	the	DET
easat-3854	99	3	dihedral	dihedral	ADJ
easat-3854	99	4	group	group	NOUN
easat-3854	99	5	of	of	ADP
easat-3854	99	6	order	order	NOUN
easat-3854	99	7	2𝓅	2𝓅	NOUN
easat-3854	99	8	symbolized	symbolize	VERB
easat-3854	99	9	by	by	ADP
easat-3854	99	10	𝒟𝓅	𝒟𝓅	PROPN
easat-3854	99	11	,	,	PUNCT
easat-3854	99	12	since	since	SCONJ
easat-3854	99	13	𝒟𝓅	𝒟𝓅	PROPN
easat-3854	99	14	=	=	PUNCT
easat-3854	100	1	[	[	X
easat-3854	100	2	𝓇	𝓇	X
easat-3854	100	3	,	,	PUNCT
easat-3854	100	4	𝓈|𝓇	𝓈|𝓇	INTJ
easat-3854	100	5	𝓅	𝓅	NOUN
easat-3854	100	6	=	=	SYM
easat-3854	100	7	𝓈2	𝓈2	PROPN
easat-3854	100	8	=	=	SYM
easat-3854	100	9	1	1	NUM
easat-3854	100	10	,	,	PUNCT
easat-3854	100	11	𝓈𝓇𝓈−1	𝓈𝓇𝓈−1	NOUN
easat-3854	100	12	=	=	SYM
easat-3854	100	13	𝓇−1	𝓇−1	ADV
easat-3854	100	14	]	]	PUNCT
easat-3854	100	15	.	.	PUNCT
easat-3854	101	1	then	then	ADV
easat-3854	101	2	there	there	PRON
easat-3854	101	3	are	be	VERB
easat-3854	101	4	the	the	DET
easat-3854	101	5	following	follow	VERB
easat-3854	101	6	two	two	NUM
easat-3854	101	7	actions	action	NOUN
easat-3854	101	8	of	of	ADP
easat-3854	101	9	𝒟𝓅	𝒟𝓅	PROPN
easat-3854	101	10	on	on	ADP
easat-3854	101	11	ℳ⊗𝓅	ℳ⊗𝓅	PRON
easat-3854	101	12	,	,	PUNCT
easat-3854	102	1	∀𝓂𝚤	∀𝓂𝚤	PROPN
easat-3854	102	2	∈	∈	PROPN
easat-3854	102	3	ℳ.	ℳ.	NOUN
easat-3854	102	4	1the	1the	PRON
easat-3854	102	5	dihedral	dihedral	ADJ
easat-3854	102	6	action	action	NOUN
easat-3854	102	7	can	can	AUX
easat-3854	102	8	be	be	AUX
easat-3854	102	9	described	describe	VERB
easat-3854	102	10	as	as	ADP
easat-3854	102	11	:	:	PUNCT
easat-3854	102	12	𝑟(𝓂1⊗𝓂2⊗	𝑟(𝓂1⊗𝓂2⊗	NOUN
easat-3854	102	13	…	…	PUNCT
easat-3854	102	14	⊗𝓂𝓅	⊗𝓂𝓅	X
easat-3854	102	15	)	)	PUNCT
easat-3854	102	16	=	=	SYM
easat-3854	102	17	(	(	PUNCT
easat-3854	102	18	−1	−1	NOUN
easat-3854	102	19	)	)	PUNCT
easat-3854	102	20	𝜀𝓂𝓅⊗𝓂1⊗⋯⊗𝓂𝓅−1	𝜀𝓂𝓅⊗𝓂1⊗⋯⊗𝓂𝓅−1	NOUN
easat-3854	102	21	,	,	PUNCT
easat-3854	102	22	𝑠(𝓂1⊗𝓂2⊗	𝑠(𝓂1⊗𝓂2⊗	NOUN
easat-3854	102	23	…	…	SYM
easat-3854	102	24	⊗𝓂𝓅	⊗𝓂𝓅	X
easat-3854	102	25	)	)	PUNCT
easat-3854	103	1	=	=	SYM
easat-3854	103	2	(	(	PUNCT
easat-3854	103	3	𝓂1⊗𝓂2⊗	𝓂1⊗𝓂2⊗	NOUN
easat-3854	103	4	…	…	SYM
easat-3854	103	5	⊗𝓂𝓅	⊗𝓂𝓅	X
easat-3854	103	6	)	)	PUNCT
easat-3854	103	7	⋆	⋆	NOUN
easat-3854	103	8	.	.	PUNCT
easat-3854	104	1	2the	2the	DET
easat-3854	104	2	skew	skew	ADJ
easat-3854	104	3	-	-	PUNCT
easat-3854	104	4	dihedral	dihedral	ADJ
easat-3854	104	5	action	action	NOUN
easat-3854	104	6	can	can	AUX
easat-3854	104	7	be	be	AUX
easat-3854	104	8	described	describe	VERB
easat-3854	104	9	as	as	ADP
easat-3854	104	10	:	:	PUNCT
easat-3854	104	11	𝑟(𝓂1⊗𝓂2⊗	𝑟(𝓂1⊗𝓂2⊗	NOUN
easat-3854	104	12	…	…	PUNCT
easat-3854	104	13	⊗𝓂𝓅	⊗𝓂𝓅	X
easat-3854	104	14	)	)	PUNCT
easat-3854	104	15	=	=	SYM
easat-3854	104	16	(	(	PUNCT
easat-3854	104	17	−1	−1	NOUN
easat-3854	104	18	)	)	PUNCT
easat-3854	104	19	𝜀𝓂𝓅⊗𝓂1⊗⋯⊗𝓂𝓅−1	𝜀𝓂𝓅⊗𝓂1⊗⋯⊗𝓂𝓅−1	NOUN
easat-3854	104	20	,	,	PUNCT
easat-3854	104	21	𝑠(𝓂1⊗𝓂2⊗	𝑠(𝓂1⊗𝓂2⊗	NOUN
easat-3854	104	22	…	…	SYM
easat-3854	104	23	⊗𝓂𝓅	⊗𝓂𝓅	X
easat-3854	104	24	)	)	PUNCT
easat-3854	104	25	=	=	SYM
easat-3854	104	26	−(𝓂1⊗𝓂2⊗	−(𝓂1⊗𝓂2⊗	NOUN
easat-3854	104	27	…	…	SYM
easat-3854	104	28	⊗𝓂𝓅	⊗𝓂𝓅	X
easat-3854	104	29	)	)	PUNCT
easat-3854	104	30	⋆	⋆	VERB
easat-3854	104	31	.	.	PUNCT
easat-3854	105	1	3	3	X
easat-3854	105	2	.	.	X
easat-3854	105	3	steenrod	steenrod	NOUN
easat-3854	105	4	operator	operator	NOUN
easat-3854	105	5	through	through	ADP
easat-3854	105	6	this	this	DET
easat-3854	105	7	section	section	NOUN
easat-3854	105	8	,	,	PUNCT
easat-3854	105	9	we	we	PRON
easat-3854	105	10	discuss	discuss	VERB
easat-3854	105	11	and	and	CCONJ
easat-3854	105	12	study	study	VERB
easat-3854	105	13	the	the	DET
easat-3854	105	14	steenrod	steenrod	NOUN
easat-3854	105	15	’s	’s	PART
easat-3854	105	16	operator	operator	NOUN
easat-3854	105	17	for	for	ADP
easat-3854	105	18	the	the	DET
easat-3854	105	19	dihedral	dihedral	ADJ
easat-3854	105	20	homology	homology	NOUN
easat-3854	105	21	of	of	ADP
easat-3854	105	22	𝓐∞-algebras	𝓐∞-algebras	PROPN
easat-3854	105	23	.	.	PUNCT
easat-3854	105	24	by	by	ADP
easat-3854	105	25	using	use	VERB
easat-3854	105	26	[	[	X
easat-3854	105	27	11	11	NUM
easat-3854	105	28	,	,	PUNCT
easat-3854	105	29	12	12	NUM
easat-3854	105	30	]	]	PUNCT
easat-3854	105	31	,	,	PUNCT
easat-3854	105	32	let	let	VERB
easat-3854	105	33	us	we	PRON
easat-3854	105	34	assume	assume	VERB
easat-3854	105	35	that	that	SCONJ
easat-3854	105	36	𝓚	𝓚	PROPN
easat-3854	105	37	be	be	VERB
easat-3854	105	38	the	the	DET
easat-3854	105	39	field	field	NOUN
easat-3854	105	40	with	with	ADP
easat-3854	105	41	characteristic	characteristic	ADJ
easat-3854	105	42	zero	zero	NUM
easat-3854	105	43	,	,	PUNCT
easat-3854	105	44	where	where	SCONJ
easat-3854	105	45	𝓜	𝓜	PROPN
easat-3854	105	46	is	be	AUX
easat-3854	105	47	the	the	DET
easat-3854	105	48	commutative	commutative	ADJ
easat-3854	105	49	𝓚-infinity	𝓚-infinity	PROPN
easat-3854	105	50	algebras	algebra	NOUN
easat-3854	105	51	.	.	PUNCT
easat-3854	105	52	suppose	suppose	VERB
easat-3854	105	53	that	that	SCONJ
easat-3854	105	54	[	[	X
easat-3854	105	55	𝓓𝒑	𝓓𝒑	X
easat-3854	105	56	]	]	X
easat-3854	105	57	is	be	AUX
easat-3854	105	58	a	a	DET
easat-3854	105	59	dihedral	dihedral	ADJ
easat-3854	105	60	category	category	NOUN
easat-3854	105	61	,	,	PUNCT
easat-3854	105	62	then	then	ADV
easat-3854	105	63	𝓚[𝓓𝒑	𝓚[𝓓𝒑	ADJ
easat-3854	105	64	]	]	X
easat-3854	105	65	is	be	AUX
easat-3854	105	66	an	an	DET
easat-3854	105	67	𝓐∞algebras	𝓐∞algebra	NOUN
easat-3854	105	68	related	relate	VERB
easat-3854	105	69	with	with	ADP
easat-3854	105	70	[	[	X
easat-3854	105	71	𝓓𝒑	𝓓𝒑	X
easat-3854	105	72	]	]	X
easat-3854	105	73	over	over	ADP
easat-3854	105	74	𝓚	𝓚	PROPN
easat-3854	105	75	(	(	PUNCT
easat-3854	105	76	see	see	VERB
easat-3854	105	77	[	[	X
easat-3854	105	78	1	1	NUM
easat-3854	105	79	]	]	PUNCT
easat-3854	105	80	,	,	PUNCT
easat-3854	105	81	[	[	X
easat-3854	105	82	4	4	NUM
easat-3854	105	83	]	]	PUNCT
easat-3854	105	84	,	,	PUNCT
easat-3854	106	1	[	[	X
easat-3854	106	2	8	8	NUM
easat-3854	106	3	]	]	NUM
easat-3854	106	4	)	)	PUNCT
easat-3854	106	5	.	.	PUNCT
easat-3854	107	1	also	also	ADV
easat-3854	107	2	,	,	PUNCT
easat-3854	107	3	𝓐𝓓	𝓐𝓓	PROPN
easat-3854	107	4	𝜺	𝜺	NOUN
easat-3854	107	5	is	be	AUX
easat-3854	107	6	describing	describe	VERB
easat-3854	107	7	on	on	ADP
easat-3854	107	8	the	the	DET
easat-3854	107	9	𝓚[𝓓𝒑]-module	𝓚[𝓓𝒑]-module	NOUN
easat-3854	107	10	,	,	PUNCT
easat-3854	107	11	the	the	DET
easat-3854	107	12	construction	construction	NOUN
easat-3854	107	13	of	of	ADP
easat-3854	107	14	the	the	DET
easat-3854	107	15	commutative	commutative	ADJ
easat-3854	107	16	𝓚[𝓓𝒑]-algebra	𝓚[𝓓𝒑]-algebra	NOUN
easat-3854	107	17	as	as	SCONJ
easat-3854	107	18	determined	determine	VERB
easat-3854	107	19	by	by	ADP
easat-3854	107	20	:	:	PUNCT
easat-3854	107	21	(	(	PUNCT
easat-3854	107	22	ℳ⊗𝑛)𝐷	ℳ⊗𝑛)𝐷	NOUN
easat-3854	107	23	𝜀	𝜀	NOUN
easat-3854	107	24	∆	∆	X
easat-3854	107	25	→	→	SYM
easat-3854	107	26	(	(	PUNCT
easat-3854	107	27	ℳ⊗𝑛⊗ℳ	ℳ⊗𝑛⊗ℳ	PROPN
easat-3854	107	28	)	)	PUNCT
easat-3854	107	29	𝜀	𝜀	X
easat-3854	107	30	𝑓	𝑓	PROPN
easat-3854	107	31	→	→	SYM
easat-3854	107	32	(	(	PUNCT
easat-3854	107	33	ℳ⊗𝑛)𝐷	ℳ⊗𝑛)𝐷	PROPN
easat-3854	107	34	𝜀	𝜀	PROPN
easat-3854	107	35	⊗	⊗	PROPN
easat-3854	107	36	(	(	PUNCT
easat-3854	107	37	ℳ)𝐷	ℳ)𝐷	NOUN
easat-3854	107	38	𝜀	𝜀	X
easat-3854	107	39	(	(	PUNCT
easat-3854	107	40	8)	8)	NUM
easat-3854	107	41	where	where	SCONJ
easat-3854	107	42	∆	∆	PROPN
easat-3854	107	43	is	be	AUX
easat-3854	107	44	the	the	DET
easat-3854	107	45	homomorphism	homomorphism	NOUN
easat-3854	107	46	of	of	ADP
easat-3854	107	47	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	107	48	]	]	PUNCT
easat-3854	107	49	,	,	PUNCT
easat-3854	107	50	and	and	CCONJ
easat-3854	107	51	𝑓	𝑓	PRON
easat-3854	107	52	is	be	AUX
easat-3854	107	53	defined	define	VERB
easat-3854	107	54	by	by	ADP
easat-3854	107	55	:	:	SYM
easat-3854	107	56	8663	8663	NUM
easat-3854	107	57	edelweiss	edelweiss	PROPN
easat-3854	107	58	applied	apply	VERB
easat-3854	107	59	science	science	NOUN
easat-3854	107	60	and	and	CCONJ
easat-3854	107	61	technology	technology	NOUN
easat-3854	107	62	issn	issn	PROPN
easat-3854	107	63	:	:	PUNCT
easat-3854	107	64	2576	2576	NUM
easat-3854	107	65	-	-	SYM
easat-3854	107	66	8484	8484	NUM
easat-3854	107	67	vol	vol	NOUN
easat-3854	107	68	.	.	PROPN
easat-3854	108	1	8	8	NUM
easat-3854	108	2	,	,	PUNCT
easat-3854	108	3	no	no	INTJ
easat-3854	108	4	.	.	NOUN
easat-3854	109	1	6	6	NUM
easat-3854	109	2	:	:	SYM
easat-3854	109	3	8658	8658	NUM
easat-3854	109	4	-	-	SYM
easat-3854	109	5	8666	8666	NUM
easat-3854	109	6	,	,	PUNCT
easat-3854	109	7	2024	2024	NUM
easat-3854	109	8	doi	doi	NOUN
easat-3854	109	9	:	:	PUNCT
easat-3854	109	10	10.55214/25768484.v8i6.3854	10.55214/25768484.v8i6.3854	NUM
easat-3854	109	11	©	©	ADP
easat-3854	109	12	2024	2024	NUM
easat-3854	109	13	by	by	ADP
easat-3854	109	14	the	the	DET
easat-3854	109	15	authors	author	NOUN
easat-3854	109	16	;	;	PUNCT
easat-3854	109	17	licensee	licensee	PROPN
easat-3854	109	18	learning	learning	NOUN
easat-3854	109	19	gate	gate	VERB
easat-3854	109	20	𝑓((𝓂0⊗𝓃0)⊗	𝑓((𝓂0⊗𝓃0)⊗	PROPN
easat-3854	109	21	(	(	PUNCT
easat-3854	109	22	𝓂1⊗𝓃1)⊗	𝓂1⊗𝓃1)⊗	PROPN
easat-3854	109	23	…	…	PUNCT
easat-3854	109	24	⊗	⊗	PROPN
easat-3854	109	25	(	(	PUNCT
easat-3854	109	26	𝓂𝑠⊗𝓃𝑠	𝓂𝑠⊗𝓃𝑠	NOUN
easat-3854	109	27	)	)	PUNCT
easat-3854	109	28	)	)	PUNCT
easat-3854	110	1	=	=	SYM
easat-3854	110	2	(	(	PUNCT
easat-3854	110	3	𝓂0⊗𝓂1⊗	𝓂0⊗𝓂1⊗	NOUN
easat-3854	110	4	…	…	SYM
easat-3854	110	5	⊗𝓂𝑠	⊗𝓂𝑠	NOUN
easat-3854	110	6	)	)	PUNCT
easat-3854	110	7	⊗	⊗	PROPN
easat-3854	110	8	(	(	PUNCT
easat-3854	110	9	𝓃0⊗𝓃1⊗	𝓃0⊗𝓃1⊗	NOUN
easat-3854	110	10	…	…	SYM
easat-3854	110	11	⊗𝓃𝑠	⊗𝓃𝑠	NOUN
easat-3854	110	12	)	)	PUNCT
easat-3854	110	13	.	.	PUNCT
easat-3854	111	1	assume	assume	VERB
easat-3854	111	2	that	that	SCONJ
easat-3854	111	3	𝑓	𝑓	DET
easat-3854	111	4	∘	∘	PROPN
easat-3854	111	5	∆=	∆=	PROPN
easat-3854	111	6	∆𝒟	∆𝒟	ADJ
easat-3854	111	7	𝜀	𝜀	PROPN
easat-3854	111	8	provides	provide	VERB
easat-3854	111	9	the	the	DET
easat-3854	111	10	commutative	commutative	ADJ
easat-3854	111	11	multiplication	multiplication	NOUN
easat-3854	111	12	in	in	ADP
easat-3854	111	13	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	111	14	𝜀	𝜀	NOUN
easat-3854	111	15	.	.	PUNCT
easat-3854	112	1	we	we	PRON
easat-3854	112	2	indicate	indicate	VERB
easat-3854	112	3	that	that	SCONJ
easat-3854	112	4	∆𝒟	∆𝒟	ADJ
easat-3854	112	5	𝜀	𝜀	PROPN
easat-3854	112	6	is	be	AUX
easat-3854	112	7	the	the	DET
easat-3854	112	8	𝒦[𝒟𝑝]-homomorphism	𝒦[𝒟𝑝]-homomorphism	NOUN
easat-3854	112	9	known	know	VERB
easat-3854	112	10	on	on	ADP
easat-3854	112	11	the	the	DET
easat-3854	112	12	𝒜∞-algebras𝒦[𝒟𝑝	𝒜∞-algebras𝒦[𝒟𝑝	NOUN
easat-3854	112	13	]	]	PUNCT
easat-3854	112	14	,	,	PUNCT
easat-3854	112	15	the	the	DET
easat-3854	112	16	multiplication	multiplication	NOUN
easat-3854	112	17	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	112	18	]	]	PUNCT
easat-3854	112	19	→	→	SYM
easat-3854	112	20	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	112	21	]	]	X
easat-3854	112	22	⊗𝓀	⊗𝓀	PROPN
easat-3854	112	23	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	112	24	]	]	PUNCT
easat-3854	112	25	,	,	PUNCT
easat-3854	112	26	like	like	ADP
easat-3854	112	27	that	that	PRON
easat-3854	112	28	𝜅	𝜅	X
easat-3854	112	29	→	→	X
easat-3854	112	30	𝜅	𝜅	PROPN
easat-3854	112	31	⊗	⊗	PROPN
easat-3854	112	32	𝜅	𝜅	PROPN
easat-3854	112	33	,	,	PUNCT
easat-3854	112	34	𝜅	𝜅	PROPN
easat-3854	112	35	∈	∈	PROPN
easat-3854	112	36	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	112	37	]	]	PUNCT
easat-3854	112	38	.	.	PUNCT
easat-3854	113	1	as	as	ADP
easat-3854	113	2	(	(	PUNCT
easat-3854	113	3	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	113	4	𝜀	𝜀	PROPN
easat-3854	113	5	⊗𝓀	⊗𝓀	NOUN
easat-3854	113	6	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	113	7	𝜀	𝜀	NOUN
easat-3854	113	8	)	)	PUNCT
easat-3854	113	9	is	be	AUX
easat-3854	113	10	(	(	PUNCT
easat-3854	113	11	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	113	12	]	]	X
easat-3854	113	13	⊗𝑘	⊗𝑘	PROPN
easat-3854	113	14	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	113	15	]	]	PUNCT
easat-3854	113	16	)	)	PUNCT
easat-3854	113	17	module	module	NOUN
easat-3854	113	18	,	,	PUNCT
easat-3854	113	19	then	then	ADV
easat-3854	113	20	through	through	ADP
easat-3854	113	21	the	the	DET
easat-3854	113	22	multiplication	multiplication	NOUN
easat-3854	113	23	on	on	ADP
easat-3854	113	24	(	(	PUNCT
easat-3854	113	25	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	113	26	𝜀	𝜀	PROPN
easat-3854	113	27	⊗	⊗	PROPN
easat-3854	113	28	𝓀	𝓀	PROPN
easat-3854	113	29	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	113	30	𝜀	𝜀	PROPN
easat-3854	113	31	)	)	PUNCT
easat-3854	113	32	one	one	PRON
easat-3854	113	33	can	can	AUX
easat-3854	113	34	describe	describe	VERB
easat-3854	113	35	𝒦[𝒟𝑝]-module	𝒦[𝒟𝑝]-module	ADP
easat-3854	113	36	construction	construction	NOUN
easat-3854	113	37	and	and	CCONJ
easat-3854	113	38	𝒦[𝒟𝑝]-module	𝒦[𝒟𝑝]-module	VERB
easat-3854	113	39	homomorphism	homomorphism	PROPN
easat-3854	113	40	𝑓	𝑓	X
easat-3854	113	41	,	,	PUNCT
easat-3854	113	42	subsequently	subsequently	ADV
easat-3854	113	43	:	:	PUNCT
easat-3854	114	1	𝑓	𝑓	DET
easat-3854	114	2	(	(	PUNCT
easat-3854	114	3	𝜅((𝓂0⊗𝓃0	𝜅((𝓂0⊗𝓃0	NOUN
easat-3854	114	4	)	)	PUNCT
easat-3854	114	5	⊗	⊗	PROPN
easat-3854	114	6	(	(	PUNCT
easat-3854	114	7	𝓂1⊗𝓃1)⊗	𝓂1⊗𝓃1)⊗	PROPN
easat-3854	114	8	…	…	PUNCT
easat-3854	114	9	⊗	⊗	PROPN
easat-3854	114	10	(	(	PUNCT
easat-3854	114	11	𝓂𝑠⊗𝓃𝑠	𝓂𝑠⊗𝓃𝑠	NOUN
easat-3854	114	12	)	)	PUNCT
easat-3854	114	13	)	)	PUNCT
easat-3854	114	14	)	)	PUNCT
easat-3854	115	1	=	=	PUNCT
easat-3854	115	2	𝜅(𝓂0⊗𝓂1⊗	𝜅(𝓂0⊗𝓂1⊗	NOUN
easat-3854	115	3	…	…	SYM
easat-3854	115	4	⊗𝓂𝑠	⊗𝓂𝑠	NOUN
easat-3854	115	5	)	)	PUNCT
easat-3854	115	6	⊗	⊗	NUM
easat-3854	115	7	𝜅(𝓃0⊗𝓃1⊗	𝜅(𝓃0⊗𝓃1⊗	X
easat-3854	115	8	…	…	SYM
easat-3854	115	9	⊗𝓃𝑠	⊗𝓃𝑠	NOUN
easat-3854	115	10	)	)	PUNCT
easat-3854	115	11	=	=	SYM
easat-3854	115	12	𝜅((𝓂0⊗𝓂1⊗	𝜅((𝓂0⊗𝓂1⊗	NOUN
easat-3854	115	13	…	…	SYM
easat-3854	115	14	⊗𝓂𝑠	⊗𝓂𝑠	NOUN
easat-3854	115	15	)	)	PUNCT
easat-3854	115	16	⊗	⊗	PROPN
easat-3854	115	17	(	(	PUNCT
easat-3854	115	18	𝓃0⊗𝓃1⊗	𝓃0⊗𝓃1⊗	NOUN
easat-3854	115	19	…	…	SYM
easat-3854	115	20	⊗𝓃𝑠	⊗𝓃𝑠	NOUN
easat-3854	115	21	)	)	PUNCT
easat-3854	115	22	)	)	PUNCT
easat-3854	116	1	=	=	SYM
easat-3854	116	2	𝜅𝑓((𝓂0⊗𝓃0	𝜅𝑓((𝓂0⊗𝓃0	NOUN
easat-3854	116	3	)	)	PUNCT
easat-3854	117	1	⊗	⊗	PROPN
easat-3854	117	2	(	(	PUNCT
easat-3854	117	3	𝓂1⊗𝓃1)⊗	𝓂1⊗𝓃1)⊗	PROPN
easat-3854	117	4	…	…	PUNCT
easat-3854	117	5	⊗	⊗	PROPN
easat-3854	117	6	(	(	PUNCT
easat-3854	117	7	𝓂𝑠⊗𝓃𝑠	𝓂𝑠⊗𝓃𝑠	PROPN
easat-3854	117	8	)	)	PUNCT
easat-3854	117	9	)	)	PUNCT
easat-3854	117	10	,	,	PUNCT
easat-3854	117	11	,	,	PUNCT
easat-3854	117	12	𝜅	𝜅	PROPN
easat-3854	117	13	∈	∈	PROPN
easat-3854	117	14	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	117	15	]	]	PUNCT
easat-3854	117	16	(	(	PUNCT
easat-3854	117	17	9	9	NUM
easat-3854	117	18	)	)	PUNCT
easat-3854	117	19	therefore	therefore	ADV
easat-3854	117	20	,	,	PUNCT
easat-3854	117	21	the	the	DET
easat-3854	117	22	morphism	morphism	NOUN
easat-3854	117	23	∆𝒟	∆𝒟	PROPN
easat-3854	117	24	𝜀	𝜀	PROPN
easat-3854	117	25	is	be	AUX
easat-3854	117	26	the	the	DET
easat-3854	117	27	𝒦[𝒟𝑝]-module	𝒦[𝒟𝑝]-module	PROPN
easat-3854	117	28	homomorphism	homomorphism	NOUN
easat-3854	117	29	.	.	PUNCT
easat-3854	118	1	then	then	ADV
easat-3854	118	2	the	the	DET
easat-3854	118	3	dihedral	dihedral	ADJ
easat-3854	118	4	homology	homology	NOUN
easat-3854	118	5	𝐸𝑥𝑡𝒦[𝒟𝑝	𝐸𝑥𝑡𝒦[𝒟𝑝	PROPN
easat-3854	118	6	]	]	X
easat-3854	118	7	𝑚	𝑚	X
easat-3854	118	8	(	(	PUNCT
easat-3854	118	9	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	118	10	𝜀	𝜀	PROPN
easat-3854	118	11	,	,	PUNCT
easat-3854	118	12	(	(	PUNCT
easat-3854	118	13	𝒦𝒟)⋆	𝒦𝒟)⋆	X
easat-3854	118	14	)	)	PUNCT
easat-3854	118	15	can	can	AUX
easat-3854	118	16	be	be	AUX
easat-3854	118	17	determined	determine	VERB
easat-3854	118	18	by	by	ADP
easat-3854	118	19	applying	apply	VERB
easat-3854	118	20	the	the	DET
easat-3854	118	21	normalized	normalize	VERB
easat-3854	118	22	bar	bar	NOUN
easat-3854	118	23	construction	construction	NOUN
easat-3854	118	24	𝛽(ℒ	𝛽(ℒ	NOUN
easat-3854	118	25	)	)	PUNCT
easat-3854	118	26	(	(	PUNCT
easat-3854	118	27	see	see	VERB
easat-3854	118	28	[	[	X
easat-3854	118	29	5	5	NUM
easat-3854	118	30	]	]	PUNCT
easat-3854	118	31	)	)	PUNCT
easat-3854	118	32	.	.	PUNCT
easat-3854	119	1	by	by	ADP
easat-3854	119	2	assuming	assume	VERB
easat-3854	119	3	that	that	SCONJ
easat-3854	119	4	ℒ	ℒ	PROPN
easat-3854	119	5	be	be	VERB
easat-3854	119	6	the	the	DET
easat-3854	119	7	triples	triple	NOUN
easat-3854	119	8	(	(	PUNCT
easat-3854	119	9	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	119	10	𝜀	𝜀	PROPN
easat-3854	119	11	,	,	PUNCT
easat-3854	119	12	𝒦[𝒟𝑝],𝒦𝒟	𝒦[𝒟𝑝],𝒦𝒟	PROPN
easat-3854	119	13	)	)	PUNCT
easat-3854	119	14	,	,	PUNCT
easat-3854	119	15	(	(	PUNCT
easat-3854	119	16	𝒦[𝒟𝑝],𝒦[𝒟𝑝],𝒦𝒟	𝒦[𝒟𝑝],𝒦[𝒟𝑝],𝒦𝒟	X
easat-3854	119	17	)	)	PUNCT
easat-3854	119	18	,	,	PUNCT
easat-3854	119	19	and	and	CCONJ
easat-3854	119	20	also	also	ADV
easat-3854	119	21	let	let	VERB
easat-3854	119	22	𝒥𝒦[𝒟𝑝	𝒥𝒦[𝒟𝑝	PRON
easat-3854	119	23	]	]	PUNCT
easat-3854	119	24	be	be	VERB
easat-3854	119	25	the	the	DET
easat-3854	119	26	kernel	kernel	PROPN
easat-3854	119	27	identity	identity	NOUN
easat-3854	119	28	𝓀	𝓀	PROPN
easat-3854	119	29	→	→	SYM
easat-3854	119	30	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	119	31	]	]	PUNCT
easat-3854	119	32	.	.	PUNCT
easat-3854	120	1	we	we	PRON
easat-3854	120	2	establish	establish	VERB
easat-3854	120	3	the	the	DET
easat-3854	120	4	identity	identity	NOUN
easat-3854	120	5	of	of	ADP
easat-3854	120	6	the	the	DET
easat-3854	120	7	normalized	normalize	VERB
easat-3854	120	8	bar	bar	NOUN
easat-3854	120	9	structure	structure	NOUN
easat-3854	120	10	𝛽(ℒ	𝛽(ℒ	PROPN
easat-3854	120	11	)	)	PUNCT
easat-3854	120	12	with	with	ADP
easat-3854	120	13	the	the	DET
easat-3854	120	14	𝓀-module	𝓀-module	NOUN
easat-3854	120	15	:	:	PUNCT
easat-3854	120	16	𝛽(ℒ	𝛽(ℒ	NOUN
easat-3854	120	17	)	)	PUNCT
easat-3854	120	18	=	=	SYM
easat-3854	120	19	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	120	20	𝜀	𝜀	NOUN
easat-3854	120	21	⊗𝒦[𝒟𝑝	⊗𝒦[𝒟𝑝	NOUN
easat-3854	120	22	]	]	PUNCT
easat-3854	120	23	𝒯(𝒥𝒦[𝒟𝑝	𝒯(𝒥𝒦[𝒟𝑝	PROPN
easat-3854	120	24	]	]	PUNCT
easat-3854	120	25	)	)	PUNCT
easat-3854	120	26	⊗𝒦[𝒟𝑝	⊗𝒦[𝒟𝑝	NOUN
easat-3854	120	27	]	]	X
easat-3854	120	28	𝒦𝒟	𝒦𝒟	PROPN
easat-3854	120	29	,	,	PUNCT
easat-3854	120	30	where	where	SCONJ
easat-3854	120	31	𝒯(𝒥𝒦[𝒟𝑝	𝒯(𝒥𝒦[𝒟𝑝	PROPN
easat-3854	120	32	]	]	PUNCT
easat-3854	120	33	)	)	PUNCT
easat-3854	120	34	be	be	AUX
easat-3854	120	35	the	the	DET
easat-3854	120	36	algebraic	algebraic	ADJ
easat-3854	120	37	tensor	tensor	NOUN
easat-3854	120	38	of	of	ADP
easat-3854	120	39	𝒥𝒦[𝒟𝑝	𝒥𝒦[𝒟𝑝	PROPN
easat-3854	120	40	]	]	PUNCT
easat-3854	120	41	.	.	PUNCT
easat-3854	121	1	obviously	obviously	ADV
easat-3854	121	2	the	the	DET
easat-3854	121	3	𝒦-module	𝒦-module	PROPN
easat-3854	121	4	𝛽(ℒ	𝛽(ℒ	PROPN
easat-3854	121	5	)	)	PUNCT
easat-3854	121	6	can	can	AUX
easat-3854	121	7	be	be	AUX
easat-3854	121	8	graded	grade	VERB
easat-3854	121	9	.	.	PUNCT
easat-3854	122	1	then	then	ADV
easat-3854	122	2	the	the	DET
easat-3854	122	3	elements	element	NOUN
easat-3854	122	4	of	of	ADP
easat-3854	122	5	𝒦-module	𝒦-module	PROPN
easat-3854	122	6	𝛽(ℒ	𝛽(ℒ	PROPN
easat-3854	122	7	)	)	PUNCT
easat-3854	122	8	is	be	AUX
easat-3854	122	9	possible	possible	ADJ
easat-3854	122	10	to	to	PART
easat-3854	122	11	write	write	VERB
easat-3854	122	12	:	:	PUNCT
easat-3854	122	13	𝓂[𝜐1	𝓂[𝜐1	ADJ
easat-3854	122	14	,	,	PUNCT
easat-3854	122	15	𝜐2	𝜐2	ADJ
easat-3854	122	16	,	,	PUNCT
easat-3854	122	17	…	…	PUNCT
easat-3854	122	18	,	,	PUNCT
easat-3854	122	19	𝜐𝑠]𝓀	𝜐𝑠]𝓀	PROPN
easat-3854	122	20	∈	∈	PROPN
easat-3854	122	21	𝛽(ℒ)𝑠	𝛽(ℒ)𝑠	NOUN
easat-3854	122	22	,	,	PUNCT
easat-3854	122	23	𝓂	𝓂	PROPN
easat-3854	122	24	∈	∈	PROPN
easat-3854	122	25	ℳ	ℳ	PROPN
easat-3854	122	26	𝜀	𝜀	NOUN
easat-3854	122	27	,	,	PUNCT
easat-3854	122	28	𝜐𝑖	𝜐𝑖	NOUN
easat-3854	122	29	∈	∈	PROPN
easat-3854	122	30	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	122	31	]	]	PUNCT
easat-3854	122	32	,	,	PUNCT
easat-3854	122	33	𝓀	𝓀	PROPN
easat-3854	122	34	∈	∈	PROPN
easat-3854	122	35	𝒦𝒟.	𝒦𝒟.	VERB
easat-3854	123	1	the	the	DET
easat-3854	123	2	differential	differential	NOUN
easat-3854	123	3	𝛿	𝛿	ADJ
easat-3854	123	4	:	:	PUNCT
easat-3854	123	5	𝛽(ℒ)𝑠	𝛽(ℒ)𝑠	NUM
easat-3854	123	6	→	→	SYM
easat-3854	123	7	𝛽(ℒ)𝑠−1	𝛽(ℒ)𝑠−1	NOUN
easat-3854	123	8	and	and	CCONJ
easat-3854	123	9	the	the	DET
easat-3854	123	10	argument	argument	NOUN
easat-3854	123	11	𝑓	𝑓	X
easat-3854	123	12	:	:	PUNCT
easat-3854	123	13	𝛽(ℒ	𝛽(ℒ	PROPN
easat-3854	123	14	)	)	PUNCT
easat-3854	123	15	→	→	SYM
easat-3854	123	16	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	123	17	𝜀	𝜀	PROPN
easat-3854	123	18	⊗𝒦[𝒟𝑝	⊗𝒦[𝒟𝑝	NOUN
easat-3854	123	19	]	]	X
easat-3854	123	20	𝒦𝒟	𝒦𝒟	PROPN
easat-3854	123	21	written	write	VERB
easat-3854	123	22	as	as	ADP
easat-3854	123	23	:	:	PUNCT
easat-3854	123	24	𝛿[𝓂[𝜐1|𝜐2	𝛿[𝓂[𝜐1|𝜐2	X
easat-3854	123	25	…	…	SYM
easat-3854	123	26	|𝜐𝑠]𝓀	|𝜐𝑠]𝓀	NOUN
easat-3854	123	27	]	]	X
easat-3854	124	1	=	=	PUNCT
easat-3854	124	2	𝓂𝜐1[𝜐2|𝜐3|	𝓂𝜐1[𝜐2|𝜐3|	NUM
easat-3854	124	3	…	…	PUNCT
easat-3854	124	4	|𝓀	|𝓀	NOUN
easat-3854	124	5	+	+	ADJ
easat-3854	124	6	∑(−1)𝑖𝓂[𝜐1|	∑(−1)𝑖𝓂[𝜐1|	ADJ
easat-3854	124	7	…	…	SYM
easat-3854	124	8	|𝜐𝑖−1|𝜐𝑖𝜐𝑖+1|𝜐𝑖+2|	|𝜐𝑖−1|𝜐𝑖𝜐𝑖+1|𝜐𝑖+2|	NOUN
easat-3854	124	9	…	…	PUNCT
easat-3854	124	10	|𝜐𝑠]𝓀	|𝜐𝑠]𝓀	NOUN
easat-3854	124	11	+	+	CCONJ
easat-3854	124	12	(	(	PUNCT
easat-3854	124	13	−1	−1	NOUN
easat-3854	124	14	)	)	PUNCT
easat-3854	124	15	𝑠𝓂[𝜐1|	𝑠𝓂[𝜐1|	PROPN
easat-3854	124	16	…	…	PUNCT
easat-3854	124	17	|𝜐𝑠−1𝜐𝑠]𝓀.	|𝜐𝑠−1𝜐𝑠]𝓀.	NOUN
easat-3854	124	18	𝑠−1	𝑠−1	PROPN
easat-3854	124	19	𝑖=1	𝑖=1	PROPN
easat-3854	124	20	(	(	PUNCT
easat-3854	124	21	10	10	NUM
easat-3854	124	22	)	)	PUNCT
easat-3854	124	23	,	,	PUNCT
easat-3854	124	24	and	and	CCONJ
easat-3854	124	25	𝑓[𝜐1|	𝑓[𝜐1|	PROPN
easat-3854	124	26	…	…	PUNCT
easat-3854	124	27	|𝜐𝑠]𝓀	|𝜐𝑠]𝓀	NOUN
easat-3854	124	28	=	=	SYM
easat-3854	124	29	0	0	NUM
easat-3854	124	30	,	,	PUNCT
easat-3854	124	31	𝑓(𝓂[]𝓀	𝑓(𝓂[]𝓀	PROPN
easat-3854	124	32	)	)	PUNCT
easat-3854	124	33	=	=	SYM
easat-3854	125	1	0	0	X
easat-3854	125	2	.	.	PUNCT
easat-3854	126	1	also	also	ADV
easat-3854	126	2	,	,	PUNCT
easat-3854	126	3	the	the	DET
easat-3854	126	4	maps	map	NOUN
easat-3854	126	5	𝛿	𝛿	NOUN
easat-3854	126	6	and	and	CCONJ
easat-3854	126	7	𝑓	𝑓	PRON
easat-3854	126	8	can	can	AUX
easat-3854	126	9	be	be	AUX
easat-3854	126	10	defined	define	VERB
easat-3854	126	11	for	for	ADP
easat-3854	126	12	𝜁	𝜁	PROPN
easat-3854	126	13	in	in	ADP
easat-3854	126	14	the	the	DET
easat-3854	126	15	similar	similar	ADJ
easat-3854	126	16	way	way	NOUN
easat-3854	126	17	.	.	PUNCT
easat-3854	127	1	as	as	ADP
easat-3854	127	2	a	a	DET
easat-3854	127	3	reminder	reminder	NOUN
easat-3854	127	4	,	,	PUNCT
easat-3854	127	5	the	the	DET
easat-3854	127	6	differential	differential	NOUN
easat-3854	127	7	𝛿	𝛿	NOUN
easat-3854	127	8	for	for	ADP
easat-3854	127	9	ℒ	ℒ	PROPN
easat-3854	127	10	seems	seem	VERB
easat-3854	127	11	to	to	PART
easat-3854	127	12	be	be	AUX
easat-3854	127	13	the	the	DET
easat-3854	127	14	left	left	NOUN
easat-3854	127	15	𝒦[𝒟𝑝]-module	𝒦[𝒟𝑝]-module	ADP
easat-3854	127	16	homomorphism	homomorphism	NOUN
easat-3854	127	17	,	,	PUNCT
easat-3854	127	18	with	with	ADP
easat-3854	127	19	𝛿𝑆	𝛿𝑆	PROPN
easat-3854	127	20	+	+	CCONJ
easat-3854	128	1	𝑆𝛿	𝑆𝛿	PROPN
easat-3854	128	2	=	=	SYM
easat-3854	128	3	1	1	NUM
easat-3854	128	4	−	−	PROPN
easat-3854	128	5	𝜎𝑓	𝜎𝑓	PROPN
easat-3854	128	6	,	,	PUNCT
easat-3854	128	7	where	where	SCONJ
easat-3854	128	8	𝜎	𝜎	PROPN
easat-3854	128	9	is	be	AUX
easat-3854	128	10	the	the	DET
easat-3854	128	11	homomorphism	homomorphism	NOUN
easat-3854	128	12	as	as	ADP
easat-3854	128	13	:	:	PUNCT
easat-3854	128	14	𝜎:𝒦𝒟	𝜎:𝒦𝒟	PROPN
easat-3854	128	15	→	→	SYM
easat-3854	128	16	𝛽(𝜁	𝛽(𝜁	ADJ
easat-3854	128	17	)	)	PUNCT
easat-3854	128	18	,	,	PUNCT
easat-3854	128	19	and	and	CCONJ
easat-3854	128	20	𝑆	𝑆	PROPN
easat-3854	128	21	:	:	PUNCT
easat-3854	128	22	𝛽(𝜁)𝑠	𝛽(𝜁)𝑠	PROPN
easat-3854	128	23	→	→	SYM
easat-3854	128	24	𝛽(𝜁)𝑠+1	𝛽(𝜁)𝑠+1	PROPN
easat-3854	128	25	,	,	PUNCT
easat-3854	128	26	is	be	AUX
easat-3854	128	27	assumed	assume	VERB
easat-3854	128	28	by	by	ADP
easat-3854	128	29	the	the	DET
easat-3854	128	30	forms	form	NOUN
easat-3854	128	31	:	:	PUNCT
easat-3854	128	32	𝜎(𝓀	𝜎(𝓀	X
easat-3854	128	33	)	)	PUNCT
easat-3854	128	34	=	=	PUNCT
easat-3854	129	1	[	[	X
easat-3854	129	2	]	]	X
easat-3854	129	3	𝓀	𝓀	X
easat-3854	129	4	⊗	⊗	PROPN
easat-3854	129	5	[	[	X
easat-3854	129	6	]	]	X
easat-3854	129	7	,	,	PUNCT
easat-3854	129	8	𝑆(𝜐[𝜐1|	𝑆(𝜐[𝜐1|	ADJ
easat-3854	129	9	…	…	SYM
easat-3854	129	10	|𝜐𝑠]𝓀	|𝜐𝑠]𝓀	NOUN
easat-3854	129	11	)	)	PUNCT
easat-3854	129	12	=	=	SYM
easat-3854	130	1	𝜐[𝜐1|	𝜐[𝜐1|	PROPN
easat-3854	130	2	…	…	PUNCT
easat-3854	130	3	|𝜐𝑠]𝓀.	|𝜐𝑠]𝓀.	NOUN
easat-3854	130	4	obviously	obviously	ADV
easat-3854	130	5	,	,	PUNCT
easat-3854	130	6	in	in	ADP
easat-3854	130	7	the	the	DET
easat-3854	130	8	complex	complex	ADJ
easat-3854	130	9	𝛽(ℓ)𝑠	𝛽(ℓ)𝑠	NOUN
easat-3854	130	10	→	→	SYM
easat-3854	130	11	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	130	12	𝜀	𝜀	PROPN
easat-3854	130	13	⊗	⊗	PROPN
easat-3854	130	14	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	130	15	]	]	PUNCT
easat-3854	130	16	𝛽(𝜁	𝛽(𝜁	NOUN
easat-3854	130	17	)	)	PUNCT
easat-3854	130	18	,	,	PUNCT
easat-3854	130	19	there	there	PRON
easat-3854	130	20	is	be	VERB
easat-3854	130	21	the	the	DET
easat-3854	130	22	differential	differential	ADJ
easat-3854	130	23	𝛿	𝛿	ADJ
easat-3854	130	24	=	=	SYM
easat-3854	130	25	1⊗	1⊗	NUM
easat-3854	130	26	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	130	27	]	]	X
easat-3854	130	28	𝛿	𝛿	X
easat-3854	130	29	.	.	PUNCT
easat-3854	131	1	from	from	ADP
easat-3854	131	2	[	[	X
easat-3854	131	3	11	11	NUM
easat-3854	131	4	]	]	PUNCT
easat-3854	131	5	,	,	PUNCT
easat-3854	131	6	we	we	PRON
easat-3854	131	7	have	have	VERB
easat-3854	131	8	the	the	DET
easat-3854	131	9	next	next	ADJ
easat-3854	131	10	:	:	PUNCT
easat-3854	131	11	𝐻𝑜𝑚𝒦[𝒟𝑝](𝛽(𝜁	𝐻𝑜𝑚𝒦[𝒟𝑝](𝛽(𝜁	PROPN
easat-3854	131	12	)	)	PUNCT
easat-3854	131	13	,	,	PUNCT
easat-3854	131	14	(	(	PUNCT
easat-3854	131	15	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	131	16	𝜀	𝜀	PROPN
easat-3854	131	17	)	)	PUNCT
easat-3854	131	18	)	)	PUNCT
easat-3854	131	19	⋆	⋆	X
easat-3854	132	1	=	=	SYM
easat-3854	132	2	(	(	PUNCT
easat-3854	132	3	𝛽(𝜁))⋆	𝛽(𝜁))⋆	PROPN
easat-3854	132	4	=	=	SYM
easat-3854	132	5	𝐻𝑜𝑚𝒦[𝒟𝑝](𝛽	𝐻𝑜𝑚𝒦[𝒟𝑝](𝛽	PROPN
easat-3854	132	6	(	(	PUNCT
easat-3854	132	7	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	132	8	𝜀	𝜀	PROPN
easat-3854	132	9	)	)	PUNCT
easat-3854	132	10	,	,	PUNCT
easat-3854	132	11	𝒦[𝒟𝑝	𝒦[𝒟𝑝	PROPN
easat-3854	132	12	]	]	PUNCT
easat-3854	132	13	,	,	PUNCT
easat-3854	132	14	(	(	PUNCT
easat-3854	132	15	𝒦𝒟)⋆	𝒦𝒟)⋆	X
easat-3854	132	16	)	)	PUNCT
easat-3854	132	17	then	then	ADV
easat-3854	132	18	,	,	PUNCT
easat-3854	132	19	ℋ𝒟𝑚(𝒜	ℋ𝒟𝑚(𝒜	NUM
easat-3854	132	20	)	)	PUNCT
easat-3854	132	21	𝜀	𝜀	NOUN
easat-3854	132	22	=	=	SYM
easat-3854	132	23	𝐸𝑥𝑡𝒦[𝒟𝑝	𝐸𝑥𝑡𝒦[𝒟𝑝	PROPN
easat-3854	132	24	]	]	X
easat-3854	132	25	𝑚	𝑚	X
easat-3854	132	26	(	(	PUNCT
easat-3854	132	27	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	132	28	𝜀	𝜀	PROPN
easat-3854	132	29	,	,	PUNCT
easat-3854	132	30	(	(	PUNCT
easat-3854	132	31	𝒦𝒟)⋆	𝒦𝒟)⋆	X
easat-3854	132	32	)	)	PUNCT
easat-3854	132	33	=	=	PUNCT
easat-3854	132	34	ℋ𝑚𝛽(ℒ)⋆	ℋ𝑚𝛽(ℒ)⋆	NOUN
easat-3854	132	35	)	)	PUNCT
easat-3854	132	36	.	.	PUNCT
easat-3854	133	1	(	(	PUNCT
easat-3854	133	2	11	11	NUM
easat-3854	133	3	)	)	PUNCT
easat-3854	133	4	8664	8664	NUM
easat-3854	133	5	edelweiss	edelweiss	PROPN
easat-3854	133	6	applied	apply	VERB
easat-3854	133	7	science	science	NOUN
easat-3854	133	8	and	and	CCONJ
easat-3854	133	9	technology	technology	NOUN
easat-3854	133	10	issn	issn	PROPN
easat-3854	133	11	:	:	PUNCT
easat-3854	133	12	2576	2576	NUM
easat-3854	133	13	-	-	SYM
easat-3854	133	14	8484	8484	NUM
easat-3854	133	15	vol	vol	NOUN
easat-3854	133	16	.	.	PROPN
easat-3854	133	17	8	8	NUM
easat-3854	133	18	,	,	PUNCT
easat-3854	133	19	no	no	INTJ
easat-3854	133	20	.	.	NOUN
easat-3854	134	1	6	6	NUM
easat-3854	134	2	:	:	SYM
easat-3854	134	3	8658	8658	NUM
easat-3854	134	4	-	-	SYM
easat-3854	134	5	8666	8666	NUM
easat-3854	134	6	,	,	PUNCT
easat-3854	134	7	2024	2024	NUM
easat-3854	134	8	doi	doi	NOUN
easat-3854	134	9	:	:	PUNCT
easat-3854	134	10	10.55214/25768484.v8i6.3854	10.55214/25768484.v8i6.3854	NUM
easat-3854	134	11	©	©	ADP
easat-3854	134	12	2024	2024	NUM
easat-3854	134	13	by	by	ADP
easat-3854	134	14	the	the	DET
easat-3854	134	15	authors	author	NOUN
easat-3854	134	16	;	;	PUNCT
easat-3854	134	17	licensee	licensee	PROPN
easat-3854	134	18	learning	learn	VERB
easat-3854	134	19	gate	gate	PROPN
easat-3854	134	20	assume	assume	VERB
easat-3854	134	21	the	the	DET
easat-3854	134	22	triples	triple	NOUN
easat-3854	134	23	ℒ	ℒ	NOUN
easat-3854	134	24	=	=	SYM
easat-3854	134	25	(	(	PUNCT
easat-3854	134	26	(	(	PUNCT
easat-3854	134	27	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	134	28	𝜀	𝜀	PROPN
easat-3854	134	29	)	)	PUNCT
easat-3854	134	30	,	,	PUNCT
easat-3854	134	31	𝒦[𝒟𝑝],𝒦𝒟)and	𝒦[𝒟𝑝],𝒦𝒟)and	NOUN
easat-3854	134	32	𝜁	𝜁	X
easat-3854	134	33	=	=	PUNCT
easat-3854	134	34	(	(	PUNCT
easat-3854	134	35	(	(	PUNCT
easat-3854	134	36	ℳ̂𝒟	ℳ̂𝒟	NUM
easat-3854	134	37	𝜀	𝜀	NOUN
easat-3854	134	38	)	)	PUNCT
easat-3854	134	39	,	,	PUNCT
easat-3854	134	40	�	�	PROPN
easat-3854	134	41	̂	̂	NOUN
easat-3854	134	42	�	�	NOUN
easat-3854	134	43	[𝒟𝑝	[𝒟𝑝	X
easat-3854	134	44	]	]	X
easat-3854	134	45	,	,	PUNCT
easat-3854	134	46	�	�	PROPN
easat-3854	134	47	̂	̂	SYM
easat-3854	134	48	�	�	NOUN
easat-3854	134	49	𝒟	𝒟	NOUN
easat-3854	134	50	)	)	PUNCT
easat-3854	134	51	and	and	CCONJ
easat-3854	134	52	ruminate	ruminate	VERB
easat-3854	134	53	the	the	DET
easat-3854	134	54	product	product	NOUN
easat-3854	134	55	𝛤	𝛤	PROPN
easat-3854	134	56	:	:	PUNCT
easat-3854	134	57	𝛽(ℒ	𝛽(ℒ	NUM
easat-3854	134	58	⊗	⊗	PROPN
easat-3854	134	59	𝜁	𝜁	NOUN
easat-3854	134	60	)	)	PUNCT
easat-3854	134	61	→	→	SYM
easat-3854	134	62	𝛽(ℒ)⊗	𝛽(ℒ)⊗	NOUN
easat-3854	134	63	𝛽(𝜁	𝛽(𝜁	ADJ
easat-3854	134	64	)	)	PUNCT
easat-3854	134	65	.	.	PUNCT
easat-3854	135	1	describe	describe	VERB
easat-3854	135	2	on	on	ADP
easat-3854	135	3	𝛽(ℒ	𝛽(ℒ	NOUN
easat-3854	135	4	)	)	PUNCT
easat-3854	135	5	such	such	ADJ
easat-3854	135	6	that	that	SCONJ
easat-3854	135	7	a	a	DET
easat-3854	135	8	construction	construction	NOUN
easat-3854	135	9	of	of	ADP
easat-3854	135	10	associative	associative	ADJ
easat-3854	135	11	algebra	algebra	NOUN
easat-3854	135	12	through	through	ADP
easat-3854	135	13	multiplication	multiplication	NOUN
easat-3854	135	14	∆̃=	∆̃=	ADP
easat-3854	135	15	𝛤𝛽	𝛤𝛽	PROPN
easat-3854	135	16	(	(	PUNCT
easat-3854	135	17	∆𝒟	∆𝒟	PROPN
easat-3854	135	18	𝜀	𝜀	PROPN
easat-3854	135	19	,	,	PUNCT
easat-3854	135	20	∆𝒦[𝒟𝑝	∆𝒦[𝒟𝑝	PROPN
easat-3854	135	21	]	]	PUNCT
easat-3854	135	22	,	,	PUNCT
easat-3854	135	23	∆𝒦𝒟	∆𝒦𝒟	PROPN
easat-3854	135	24	)	)	PUNCT
easat-3854	135	25	:	:	PUNCT
easat-3854	135	26	𝛽(ℒ	𝛽(ℒ	NOUN
easat-3854	135	27	)	)	PUNCT
easat-3854	135	28	→	→	SYM
easat-3854	135	29	𝛽(ℒ	𝛽(ℒ	NOUN
easat-3854	135	30	)	)	PUNCT
easat-3854	135	31	⊗	⊗	PROPN
easat-3854	135	32	𝛽(ℒ	𝛽(ℒ	PROPN
easat-3854	135	33	)	)	PUNCT
easat-3854	135	34	and	and	CCONJ
easat-3854	135	35	on	on	ADP
easat-3854	135	36	a	a	DET
easat-3854	135	37	complex	complex	NOUN
easat-3854	135	38	(	(	PUNCT
easat-3854	135	39	𝛽(ℒ))⋆	𝛽(ℒ))⋆	ADP
easat-3854	135	40	the	the	DET
easat-3854	135	41	next	next	ADJ
easat-3854	135	42	multiplication	multiplication	NOUN
easat-3854	135	43	:	:	PUNCT
easat-3854	135	44	(	(	PUNCT
easat-3854	135	45	𝛽(ℒ))⋆⊗	𝛽(ℒ))⋆⊗	PROPN
easat-3854	135	46	(	(	PUNCT
easat-3854	135	47	𝛽(ℒ))⋆	𝛽(ℒ))⋆	PROPN
easat-3854	135	48	→	→	SYM
easat-3854	135	49	(	(	PUNCT
easat-3854	135	50	𝛽(ℒ)⊗	𝛽(ℒ)⊗	NOUN
easat-3854	135	51	𝛽(ℒ	𝛽(ℒ	NOUN
easat-3854	135	52	)	)	PUNCT
easat-3854	135	53	)	)	PUNCT
easat-3854	135	54	⋆	⋆	X
easat-3854	135	55	(	(	PUNCT
easat-3854	135	56	∆̃)⋆	∆̃)⋆	NUM
easat-3854	135	57	→	→	SYM
easat-3854	135	58	(	(	PUNCT
easat-3854	135	59	𝛽(ℒ))⋆.	𝛽(ℒ))⋆.	VERB
easat-3854	135	60	(	(	PUNCT
easat-3854	135	61	12	12	NUM
easat-3854	135	62	)	)	PUNCT
easat-3854	135	63	the	the	DET
easat-3854	135	64	following	follow	VERB
easat-3854	135	65	lemma	lemma	PROPN
easat-3854	135	66	is	be	AUX
easat-3854	135	67	simply	simply	ADV
easat-3854	135	68	confirmed	confirm	VERB
easat-3854	135	69	by	by	ADP
easat-3854	135	70	applying	apply	VERB
easat-3854	135	71	the	the	DET
easat-3854	135	72	standard	standard	ADJ
easat-3854	135	73	methods	method	NOUN
easat-3854	135	74	of	of	ADP
easat-3854	135	75	the	the	DET
easat-3854	135	76	homological	homological	ADJ
easat-3854	135	77	𝒜∞algebras	𝒜∞algebras	NUM
easat-3854	135	78	.	.	PUNCT
easat-3854	136	1	3.1	3.1	NUM
easat-3854	136	2	.	.	PUNCT
easat-3854	137	1	lemma	lemma	PROPN
easat-3854	137	2	by	by	ADP
easat-3854	137	3	assuming	assume	VERB
easat-3854	137	4	that	that	SCONJ
easat-3854	137	5	𝜂	𝜂	NOUN
easat-3854	137	6	is	be	AUX
easat-3854	137	7	a	a	DET
easat-3854	137	8	subgroup	subgroup	NOUN
easat-3854	137	9	of	of	ADP
easat-3854	137	10	a	a	DET
easat-3854	137	11	symmetrical	symmetrical	ADJ
easat-3854	137	12	group	group	NOUN
easat-3854	137	13	𝜉𝓇	𝜉𝓇	NOUN
easat-3854	137	14	and	and	CCONJ
easat-3854	137	15	ℰ	ℰ	PRON
easat-3854	137	16	seems	seem	VERB
easat-3854	137	17	to	to	PART
easat-3854	137	18	be	be	AUX
easat-3854	137	19	the	the	DET
easat-3854	137	20	𝒦[𝜂]-free	𝒦[𝜂]-free	ADJ
easat-3854	137	21	resolution	resolution	NOUN
easat-3854	137	22	𝒦[𝜂]-module	𝒦[𝜂]-module	VERB
easat-3854	138	1	𝒦	𝒦	ADP
easat-3854	138	2	such	such	ADJ
easat-3854	138	3	ℰ0	ℰ0	NOUN
easat-3854	138	4	=	=	SYM
easat-3854	138	5	𝒦[𝜂	𝒦[𝜂	X
easat-3854	138	6	]	]	PUNCT
easat-3854	138	7	through	through	ADP
easat-3854	138	8	𝓋0	𝓋0	NOUN
easat-3854	138	9	the	the	DET
easat-3854	138	10	generator	generator	NOUN
easat-3854	138	11	of	of	ADP
easat-3854	138	12	𝒦[𝑚	𝒦[𝑚	PROPN
easat-3854	138	13	]	]	PUNCT
easat-3854	138	14	,	,	PUNCT
easat-3854	138	15	since	since	SCONJ
easat-3854	138	16	[	[	X
easat-3854	138	17	ℰ	ℰ	PROPN
easat-3854	138	18	⊗	⊗	PROPN
easat-3854	138	19	𝛽(ℒ)]𝑠	𝛽(ℒ)]𝑠	PROPN
easat-3854	138	20	=	=	PUNCT
easat-3854	138	21	∑	∑	PUNCT
easat-3854	138	22	ℰ𝑖⊗𝛽𝑗(ℒ)𝑖+𝑗=𝑠	ℰ𝑖⊗𝛽𝑗(ℒ)𝑖+𝑗=𝑠	ADJ
easat-3854	138	23	,	,	PUNCT
easat-3854	138	24	the	the	DET
easat-3854	138	25	module	module	NOUN
easat-3854	138	26	ℰ	ℰ	PROPN
easat-3854	138	27	⊗	⊗	PROPN
easat-3854	138	28	𝛽(ℒ	𝛽(ℒ	PROPN
easat-3854	138	29	)	)	PUNCT
easat-3854	138	30	is	be	AUX
easat-3854	138	31	graded	grade	VERB
easat-3854	138	32	.	.	PUNCT
easat-3854	139	1	then	then	ADV
easat-3854	139	2	the	the	DET
easat-3854	139	3	graded	grade	VERB
easat-3854	139	4	𝒦[𝑚]complexes	𝒦[𝑚]complexes	PROPN
easat-3854	139	5	exist	exist	VERB
easat-3854	139	6	with	with	ADP
easat-3854	139	7	the	the	DET
easat-3854	139	8	next	next	ADJ
easat-3854	139	9	conditions	condition	NOUN
easat-3854	139	10	of	of	ADP
easat-3854	139	11	the	the	DET
easat-3854	139	12	homomorphism	homomorphism	PROPN
easat-3854	139	13	λ	λ	NOUN
easat-3854	139	14	:	:	PUNCT
easat-3854	139	15	ℰ	ℰ	PROPN
easat-3854	139	16	⊗	⊗	PROPN
easat-3854	139	17	𝛽(ℒ	𝛽(ℒ	PROPN
easat-3854	139	18	)	)	PUNCT
easat-3854	139	19	→	→	SYM
easat-3854	139	20	𝛽(ℒ)⊗𝓇	𝛽(ℒ)⊗𝓇	X
easat-3854	139	21	such	such	ADJ
easat-3854	139	22	as	as	ADP
easat-3854	139	23	:	:	PUNCT
easat-3854	139	24	(	(	PUNCT
easat-3854	139	25	i	i	NOUN
easat-3854	139	26	)	)	PUNCT
easat-3854	140	1	𝛬(ℯ	𝛬(ℯ	PUNCT
easat-3854	140	2	⊗	⊗	PROPN
easat-3854	140	3	𝒷	𝒷	PROPN
easat-3854	140	4	)	)	PUNCT
easat-3854	140	5	=	=	SYM
easat-3854	140	6	0	0	NUM
easat-3854	140	7	,	,	PUNCT
easat-3854	140	8	𝒷	𝒷	PROPN
easat-3854	140	9	∈	∈	PROPN
easat-3854	140	10	𝛽(ℒ)0	𝛽(ℒ)0	PROPN
easat-3854	140	11	and	and	CCONJ
easat-3854	140	12	ℯ	ℯ	PROPN
easat-3854	140	13	∈	∈	PROPN
easat-3854	140	14	ℰ𝑖	ℰ𝑖	PROPN
easat-3854	140	15	,	,	PUNCT
easat-3854	140	16	𝑖	𝑖	X
easat-3854	140	17	>	>	X
easat-3854	140	18	0	0	NUM
easat-3854	140	19	.	.	PUNCT
easat-3854	141	1	(	(	PUNCT
easat-3854	141	2	ii	ii	NOUN
easat-3854	141	3	)	)	PUNCT
easat-3854	141	4	𝛬(𝓋0⊗𝒷	𝛬(𝓋0⊗𝒷	PROPN
easat-3854	141	5	)	)	PUNCT
easat-3854	141	6	=	=	SYM
easat-3854	141	7	∆̃⊗𝓇(𝒷	∆̃⊗𝓇(𝒷	PROPN
easat-3854	141	8	)	)	PUNCT
easat-3854	141	9	,	,	PUNCT
easat-3854	141	10	𝑖𝑓	𝑖𝑓	ADP
easat-3854	141	11	𝒷	𝒷	DET
easat-3854	141	12	∈	∈	PROPN
easat-3854	141	13	𝛽(ℒ	𝛽(ℒ	PROPN
easat-3854	141	14	)	)	PUNCT
easat-3854	141	15	,	,	PUNCT
easat-3854	141	16	∆̃⊗𝓇	∆̃⊗𝓇	PROPN
easat-3854	141	17	:	:	PUNCT
easat-3854	141	18	𝛽(ℒ	𝛽(ℒ	NOUN
easat-3854	141	19	)	)	PUNCT
easat-3854	141	20	→	→	SYM
easat-3854	141	21	𝛽(ℒ)⊗𝓇.	𝛽(ℒ)⊗𝓇.	X
easat-3854	141	22	(	(	PUNCT
easat-3854	141	23	iii	iii	NOUN
easat-3854	141	24	)	)	PUNCT
easat-3854	141	25	the	the	DET
easat-3854	141	26	map	map	NOUN
easat-3854	141	27	𝛬	𝛬	NOUN
easat-3854	141	28	for	for	ADP
easat-3854	141	29	𝛽(ℒ	𝛽(ℒ	NOUN
easat-3854	141	30	)	)	PUNCT
easat-3854	141	31	is	be	AUX
easat-3854	141	32	the	the	DET
easat-3854	141	33	homomorphism	homomorphism	NOUN
easat-3854	141	34	of	of	ADP
easat-3854	141	35	the	the	DET
easat-3854	141	36	left	left	ADJ
easat-3854	141	37	𝒦[𝒟𝑝]-module	𝒦[𝒟𝑝]-module	NOUN
easat-3854	141	38	,	,	PUNCT
easat-3854	141	39	as	as	SCONJ
easat-3854	141	40	𝒦[𝒟𝑝	𝒦[𝒟𝑝	NOUN
easat-3854	141	41	]	]	PUNCT
easat-3854	141	42	works	work	VERB
easat-3854	141	43	on	on	ADP
easat-3854	141	44	ℰ	ℰ	PROPN
easat-3854	141	45	⊗	⊗	PROPN
easat-3854	141	46	𝛽(ℒ	𝛽(ℒ	PROPN
easat-3854	141	47	)	)	PUNCT
easat-3854	141	48	through	through	ADP
easat-3854	141	49	the	the	DET
easat-3854	141	50	relation	relation	NOUN
easat-3854	141	51	𝒦(ℯ	𝒦(ℯ	PUNCT
easat-3854	141	52	⊗	⊗	PROPN
easat-3854	141	53	𝒷)ℯ	𝒷)ℯ	PUNCT
easat-3854	142	1	⊗	⊗	PROPN
easat-3854	142	2	𝓀𝒷.	𝓀𝒷.	X
easat-3854	142	3	(	(	PUNCT
easat-3854	142	4	iv	iv	X
easat-3854	142	5	)	)	PUNCT
easat-3854	142	6	𝛬(ℯ𝑖⊗𝛽(ℒ)𝑠	𝛬(ℯ𝑖⊗𝛽(ℒ)𝑠	PROPN
easat-3854	142	7	)	)	PUNCT
easat-3854	142	8	=	=	NOUN
easat-3854	143	1	0	0	PUNCT
easat-3854	143	2	when	when	SCONJ
easat-3854	143	3	𝑖	𝑖	X
easat-3854	143	4	>	>	X
easat-3854	143	5	(	(	PUNCT
easat-3854	143	6	𝓇	𝓇	X
easat-3854	143	7	−	−	PROPN
easat-3854	143	8	1	1	NUM
easat-3854	143	9	)	)	PUNCT
easat-3854	143	10	.	.	PUNCT
easat-3854	144	1	additionally	additionally	ADV
easat-3854	144	2	,	,	PUNCT
easat-3854	144	3	each	each	DET
easat-3854	144	4	pair	pair	NOUN
easat-3854	144	5	of	of	ADP
easat-3854	144	6	homomorphisms	homomorphism	NOUN
easat-3854	144	7	with	with	ADP
easat-3854	144	8	similar	similar	ADJ
easat-3854	144	9	properties	property	NOUN
easat-3854	144	10	has	have	VERB
easat-3854	144	11	𝒦[𝜂]-homotopy	𝒦[𝜂]-homotopy	NOUN
easat-3854	144	12	.	.	PUNCT
easat-3854	145	1	now	now	ADV
easat-3854	145	2	,	,	PUNCT
easat-3854	145	3	give	give	VERB
easat-3854	145	4	the	the	DET
easat-3854	145	5	𝒦[𝒟𝑝]-homomorphism	𝒦[𝒟𝑝]-homomorphism	NOUN
easat-3854	145	6	ω	ω	NOUN
easat-3854	145	7	the	the	DET
easat-3854	145	8	following	follow	VERB
easat-3854	145	9	definition	definition	NOUN
easat-3854	145	10	:	:	PUNCT
easat-3854	145	11	ω	ω	NUM
easat-3854	145	12	:	:	PUNCT
easat-3854	145	13	ℰ	ℰ	PROPN
easat-3854	145	14	⊗	⊗	PROPN
easat-3854	145	15	(	(	PUNCT
easat-3854	145	16	𝛽(ℒ)⋆	𝛽(ℒ)⋆	NOUN
easat-3854	145	17	)	)	PUNCT
easat-3854	145	18	⊗𝓇	⊗𝓇	NOUN
easat-3854	145	19	→	→	SYM
easat-3854	145	20	𝛽(ℒ)⋆	𝛽(ℒ)⋆	NOUN
easat-3854	145	21	,	,	PUNCT
easat-3854	145	22	since	since	SCONJ
easat-3854	145	23	;	;	PUNCT
easat-3854	145	24	ω(ℯ	ω(ℯ	PROPN
easat-3854	145	25	⊗	⊗	PROPN
easat-3854	145	26	𝓍)(ℓ	𝓍)(ℓ	PUNCT
easat-3854	145	27	)	)	PUNCT
easat-3854	146	1	=	=	SYM
easat-3854	146	2	𝔅(𝓍)𝛬(ℯ	𝔅(𝓍)𝛬(ℯ	ADJ
easat-3854	146	3	⊗	⊗	PROPN
easat-3854	146	4	ℓ	ℓ	PROPN
easat-3854	146	5	)	)	PUNCT
easat-3854	146	6	,	,	PUNCT
easat-3854	146	7	ℯ	ℯ	PROPN
easat-3854	146	8	∈	∈	PROPN
easat-3854	146	9	ℰ	ℰ	PROPN
easat-3854	146	10	,	,	PUNCT
easat-3854	146	11	𝓍	𝓍	X
easat-3854	146	12	∈	∈	PROPN
easat-3854	146	13	(	(	PUNCT
easat-3854	146	14	𝛽(ℒ)⋆	𝛽(ℒ)⋆	NOUN
easat-3854	146	15	)	)	PUNCT
easat-3854	146	16	⊗𝓇𝑎𝑛𝑑	⊗𝓇𝑎𝑛𝑑	NOUN
easat-3854	146	17	ℓ	ℓ	PROPN
easat-3854	146	18	∈	∈	PROPN
easat-3854	146	19	𝛽(ℒ	𝛽(ℒ	PROPN
easat-3854	146	20	)	)	PUNCT
easat-3854	146	21	.	.	PUNCT
easat-3854	147	1	𝔅	𝔅	NOUN
easat-3854	147	2	:	:	PUNCT
easat-3854	147	3	(	(	PUNCT
easat-3854	147	4	𝛽(ℒ)⋆	𝛽(ℒ)⋆	NOUN
easat-3854	147	5	)	)	PUNCT
easat-3854	147	6	⊗𝓇	⊗𝓇	NOUN
easat-3854	147	7	→	→	SYM
easat-3854	147	8	(	(	PUNCT
easat-3854	147	9	𝛽(ℒ)⊗𝓇)⋆	𝛽(ℒ)⊗𝓇)⋆	NOUN
easat-3854	147	10	,	,	PUNCT
easat-3854	147	11	is	be	AUX
easat-3854	147	12	a	a	DET
easat-3854	147	13	homomorphism	homomorphism	NOUN
easat-3854	147	14	that	that	PRON
easat-3854	147	15	is	be	AUX
easat-3854	147	16	trivial	trivial	ADJ
easat-3854	147	17	.	.	PUNCT
easat-3854	148	1	proof	proof	NOUN
easat-3854	148	2	:	:	PUNCT
easat-3854	148	3	now	now	ADV
easat-3854	148	4	,	,	PUNCT
easat-3854	148	5	the	the	DET
easat-3854	148	6	operator	operator	NOUN
easat-3854	148	7	in	in	ADP
easat-3854	148	8	ℋ(𝛽(ℒ)⋆	ℋ(𝛽(ℒ)⋆	NOUN
easat-3854	148	9	)	)	PUNCT
easat-3854	148	10	will	will	AUX
easat-3854	148	11	be	be	AUX
easat-3854	148	12	defined	define	VERB
easat-3854	148	13	.	.	PUNCT
easat-3854	149	1	in	in	ADP
easat-3854	149	2	lemma	lemma	PROPN
easat-3854	149	3	(	(	PUNCT
easat-3854	149	4	3.1	3.1	NUM
easat-3854	149	5	)	)	PUNCT
easat-3854	149	6	,	,	PUNCT
easat-3854	149	7	considering	consider	VERB
easat-3854	149	8	𝒦	𝒦	PROPN
easat-3854	149	9	=	=	SYM
easat-3854	149	10	𝒵/𝒫.	𝒵/𝒫.	PROPN
easat-3854	149	11	assume	assume	VERB
easat-3854	149	12	that	that	SCONJ
easat-3854	149	13	ℰ	ℰ	PRON
easat-3854	149	14	has	have	VERB
easat-3854	149	15	the	the	DET
easat-3854	149	16	normal	normal	ADJ
easat-3854	149	17	𝒦(𝒵/𝒫)-free	𝒦(𝒵/𝒫)-free	NUM
easat-3854	149	18	resolution	resolution	NOUN
easat-3854	149	19	.	.	PUNCT
easat-3854	150	1	here	here	ADV
easat-3854	150	2	,	,	PUNCT
easat-3854	150	3	the	the	DET
easat-3854	150	4	free	free	ADJ
easat-3854	150	5	𝒦(𝒵/𝒫	𝒦(𝒵/𝒫	NOUN
easat-3854	150	6	)	)	PUNCT
easat-3854	150	7	-module	-module	NOUN
easat-3854	150	8	using	use	VERB
easat-3854	150	9	the	the	DET
easat-3854	150	10	generator	generator	NOUN
easat-3854	150	11	𝓋𝑖	𝓋𝑖	NOUN
easat-3854	150	12	,	,	PUNCT
easat-3854	150	13	denoted	denote	VERB
easat-3854	150	14	as	as	ADP
easat-3854	150	15	ℰ𝑖	ℰ𝑖	PROPN
easat-3854	150	16	for	for	ADP
easat-3854	150	17	𝑖	𝑖	PRON
easat-3854	150	18	≥	≥	NOUN
easat-3854	150	19	0	0	NUM
easat-3854	150	20	.	.	PUNCT
easat-3854	151	1	assuming	assume	VERB
easat-3854	151	2	that	that	SCONJ
easat-3854	151	3	the	the	DET
easat-3854	151	4	graded	grade	VERB
easat-3854	151	5	ℰ𝑖	ℰ𝑖	VERB
easat-3854	151	6	=	=	PUNCT
easat-3854	151	7	ℰ	ℰ	PROPN
easat-3854	151	8	−𝑖	−𝑖	ADJ
easat-3854	151	9	remains	remain	VERB
easat-3854	151	10	the	the	DET
easat-3854	151	11	free	free	ADJ
easat-3854	151	12	𝒦(𝒵/𝒫)-module	𝒦(𝒵/𝒫)-module	NOUN
easat-3854	151	13	using	use	VERB
easat-3854	151	14	the	the	DET
easat-3854	151	15	generator	generator	NOUN
easat-3854	151	16	𝓋−𝑖.	𝓋−𝑖.	VERB
easat-3854	151	17	letting	let	VERB
easat-3854	151	18	𝒶	𝒶	PRON
easat-3854	151	19	∈	∈	PROPN
easat-3854	151	20	ℋ𝓆(𝛽(ℒ)⋆	ℋ𝓆(𝛽(ℒ)⋆	X
easat-3854	151	21	and	and	CCONJ
easat-3854	151	22	define	define	VERB
easat-3854	151	23	the	the	DET
easat-3854	151	24	homomorphism	homomorphism	NOUN
easat-3854	151	25	:	:	PUNCT
easat-3854	151	26	𝔑𝑖:ℋ𝓆(𝛽(ℒ)⋆	𝔑𝑖:ℋ𝓆(𝛽(ℒ)⋆	NOUN
easat-3854	151	27	)	)	PUNCT
easat-3854	151	28	→	→	SYM
easat-3854	151	29	ℋ𝓅𝓆−𝑖(𝛽(ℒ)⋆	ℋ𝓅𝓆−𝑖(𝛽(ℒ)⋆	NOUN
easat-3854	151	30	as	as	ADP
easat-3854	151	31	𝔑𝑖(𝒶	𝔑𝑖(𝒶	NOUN
easat-3854	151	32	)	)	PUNCT
easat-3854	151	33	=	=	PUNCT
easat-3854	151	34	ω⋆(𝓋	ω⋆(𝓋	PROPN
easat-3854	151	35	−𝑖⊗𝒶𝓅	−𝑖⊗𝒶𝓅	NOUN
easat-3854	151	36	)	)	PUNCT
easat-3854	151	37	,	,	PUNCT
easat-3854	151	38	𝑖	𝑖	X
easat-3854	151	39	≥	≥	NOUN
easat-3854	151	40	0	0	NUM
easat-3854	151	41	.	.	PUNCT
easat-3854	152	1	now	now	ADV
easat-3854	152	2	,	,	PUNCT
easat-3854	152	3	the	the	DET
easat-3854	152	4	steenrod	steenrod	NOUN
easat-3854	152	5	operator	operator	NOUN
easat-3854	152	6	𝒫𝑖	𝒫𝑖	PROPN
easat-3854	152	7	defined	define	VERB
easat-3854	152	8	with	with	ADP
easat-3854	152	9	the	the	DET
easat-3854	152	10	operator	operator	NOUN
easat-3854	152	11	ℛ𝑖	ℛ𝑖	PROPN
easat-3854	152	12	,	,	PUNCT
easat-3854	152	13	as	as	SCONJ
easat-3854	152	14	follows	follow	VERB
easat-3854	152	15	:	:	PUNCT
easat-3854	152	16	1	1	X
easat-3854	152	17	)	)	PUNCT
easat-3854	152	18	if	if	SCONJ
easat-3854	152	19	𝓅	𝓅	NOUN
easat-3854	152	20	=	=	SYM
easat-3854	152	21	2	2	NUM
easat-3854	152	22	then	then	ADV
easat-3854	152	23	,	,	PUNCT
easat-3854	152	24	𝓅𝑠(𝒶	𝓅𝑠(𝒶	X
easat-3854	152	25	)	)	PUNCT
easat-3854	152	26	=	=	PUNCT
easat-3854	152	27	𝔑𝑞−𝑠(𝓍	𝔑𝑞−𝑠(𝓍	NOUN
easat-3854	152	28	)	)	PUNCT
easat-3854	152	29	∈	∈	PROPN
easat-3854	152	30	ℋ𝓆+𝑠(𝛽(ℒ)⋆	ℋ𝓆+𝑠(𝛽(ℒ)⋆	NOUN
easat-3854	152	31	)	)	PUNCT
easat-3854	152	32	,	,	PUNCT
easat-3854	152	33	since	since	SCONJ
easat-3854	152	34	𝔑𝑖	𝔑𝑖	PROPN
easat-3854	152	35	=	=	PUNCT
easat-3854	152	36	0	0	PUNCT
easat-3854	153	1	if	if	SCONJ
easat-3854	153	2	𝑖	𝑖	X
easat-3854	153	3	<	<	X
easat-3854	153	4	0	0	NUM
easat-3854	153	5	.	.	NOUN
easat-3854	153	6	2	2	NUM
easat-3854	153	7	)	)	PUNCT
easat-3854	153	8	if	if	SCONJ
easat-3854	153	9	𝓅	𝓅	PROPN
easat-3854	153	10	>	>	X
easat-3854	153	11	2	2	NUM
easat-3854	153	12	then	then	ADV
easat-3854	153	13	,	,	PUNCT
easat-3854	153	14	𝓅𝑠(𝒶	𝓅𝑠(𝒶	PROPN
easat-3854	153	15	)	)	PUNCT
easat-3854	153	16	=	=	SYM
easat-3854	153	17	(	(	PUNCT
easat-3854	153	18	−1	−1	NOUN
easat-3854	153	19	)	)	PUNCT
easat-3854	153	20	𝑠𝛾(−𝓆)𝔑(𝓆−2𝑠)(𝓅−1)(𝒶	𝑠𝛾(−𝓆)𝔑(𝓆−2𝑠)(𝓅−1)(𝒶	PROPN
easat-3854	153	21	)	)	PUNCT
easat-3854	153	22	∈	∈	PROPN
easat-3854	153	23	ℋ𝓆+2𝑠(𝓅−1)(𝛽(ℒ)⋆	ℋ𝓆+2𝑠(𝓅−1)(𝛽(ℒ)⋆	NOUN
easat-3854	153	24	)	)	PUNCT
easat-3854	153	25	,	,	PUNCT
easat-3854	153	26	𝔅𝓅𝑠(𝒶	𝔅𝓅𝑠(𝒶	ADJ
easat-3854	153	27	)	)	PUNCT
easat-3854	153	28	=	=	SYM
easat-3854	153	29	(	(	PUNCT
easat-3854	153	30	−1	−1	NOUN
easat-3854	153	31	)	)	PUNCT
easat-3854	153	32	𝑠𝛾(−𝓆)𝔑(𝓆−2𝑠)(𝓅−1)−1(𝒶	𝑠𝛾(−𝓆)𝔑(𝓆−2𝑠)(𝓅−1)−1(𝒶	PROPN
easat-3854	153	33	)	)	PUNCT
easat-3854	153	34	∈	∈	PROPN
easat-3854	153	35	ℋ𝓆+2𝑠(𝓅−1)+1(𝛽(ℒ)⋆	ℋ𝓆+2𝑠(𝓅−1)+1(𝛽(ℒ)⋆	NOUN
easat-3854	153	36	)	)	PUNCT
easat-3854	153	37	,	,	PUNCT
easat-3854	153	38	where	where	SCONJ
easat-3854	153	39	𝔑𝑖	𝔑𝑖	PROPN
easat-3854	153	40	=	=	PROPN
easat-3854	153	41	0	0	NUM
easat-3854	153	42	and	and	CCONJ
easat-3854	153	43	ℓ	ℓ	X
easat-3854	153	44	=	=	SYM
easat-3854	153	45	0	0	NUM
easat-3854	153	46	𝑜𝑟1	𝑜𝑟1	NOUN
easat-3854	153	47	and	and	CCONJ
easat-3854	153	48	𝛾(−𝓆	𝛾(−𝓆	NOUN
easat-3854	153	49	)	)	PUNCT
easat-3854	153	50	=	=	SYM
easat-3854	154	1	(	(	PUNCT
easat-3854	154	2	−1)𝒿(𝓂𝐼)ℒ,𝓂	−1)𝒿(𝓂𝐼)ℒ,𝓂	PROPN
easat-3854	154	3	=	=	PUNCT
easat-3854	154	4	𝓅−1	𝓅−1	PROPN
easat-3854	154	5	2	2	NUM
easat-3854	154	6	if	if	SCONJ
easat-3854	154	7	𝓆	𝓆	X
easat-3854	154	8	=	=	NOUN
easat-3854	154	9	2ℐ	2ℐ	NOUN
easat-3854	154	10	−	−	PROPN
easat-3854	154	11	ℓ	ℓ	PROPN
easat-3854	154	12	,	,	PUNCT
easat-3854	154	13	𝑖	𝑖	X
easat-3854	154	14	<	<	X
easat-3854	154	15	0	0	NUM
easat-3854	154	16	.	.	PROPN
easat-3854	154	17	3.2	3.2	NUM
easat-3854	154	18	.	.	PUNCT
easat-3854	155	1	theorem	theorem	NOUN
easat-3854	155	2	given	give	VERB
easat-3854	155	3	that	that	SCONJ
easat-3854	155	4	𝒦	𝒦	PROPN
easat-3854	155	5	=	=	SYM
easat-3854	155	6	𝒵/𝒫	𝒵/𝒫	NOUN
easat-3854	155	7	and	and	CCONJ
easat-3854	155	8	ℳ	ℳ	PROPN
easat-3854	155	9	is	be	AUX
easat-3854	155	10	the	the	DET
easat-3854	155	11	commutative	commutative	ADJ
easat-3854	155	12	𝒦-infinity	𝒦-infinity	PROPN
easat-3854	155	13	algebras	algebra	NOUN
easat-3854	155	14	,	,	PUNCT
easat-3854	155	15	so	so	SCONJ
easat-3854	155	16	the	the	DET
easat-3854	155	17	next	next	ADJ
easat-3854	155	18	homomorphisms	homomorphism	NOUN
easat-3854	155	19	"	"	PUNCT
easat-3854	155	20	steenrod	steenrod	NOUN
easat-3854	155	21	map	map	NOUN
easat-3854	155	22	"	"	PUNCT
easat-3854	155	23	are	be	AUX
easat-3854	155	24	defined	define	VERB
easat-3854	155	25	for	for	ADP
easat-3854	155	26	the	the	DET
easat-3854	155	27	dihedral	dihedral	ADJ
easat-3854	155	28	homology	homology	NOUN
easat-3854	155	29	group	group	NOUN
easat-3854	155	30	ℋ𝒟(ℳ)𝜀	ℋ𝒟(ℳ)𝜀	NUM
easat-3854	155	31	as	as	ADP
easat-3854	155	32	:	:	PUNCT
easat-3854	155	33	8665	8665	NUM
easat-3854	155	34	edelweiss	edelweiss	PROPN
easat-3854	155	35	applied	apply	VERB
easat-3854	155	36	science	science	NOUN
easat-3854	155	37	and	and	CCONJ
easat-3854	155	38	technology	technology	NOUN
easat-3854	155	39	issn	issn	PROPN
easat-3854	155	40	:	:	PUNCT
easat-3854	155	41	2576	2576	NUM
easat-3854	155	42	-	-	SYM
easat-3854	155	43	8484	8484	NUM
easat-3854	155	44	vol	vol	NOUN
easat-3854	155	45	.	.	PROPN
easat-3854	156	1	8	8	NUM
easat-3854	156	2	,	,	PUNCT
easat-3854	156	3	no	no	INTJ
easat-3854	156	4	.	.	NOUN
easat-3854	157	1	6	6	NUM
easat-3854	157	2	:	:	SYM
easat-3854	157	3	8658	8658	NUM
easat-3854	157	4	-	-	SYM
easat-3854	157	5	8666	8666	NUM
easat-3854	157	6	,	,	PUNCT
easat-3854	157	7	2024	2024	NUM
easat-3854	157	8	doi	doi	NOUN
easat-3854	157	9	:	:	PUNCT
easat-3854	157	10	10.55214/25768484.v8i6.3854	10.55214/25768484.v8i6.3854	NUM
easat-3854	157	11	©	©	ADP
easat-3854	157	12	2024	2024	NUM
easat-3854	157	13	by	by	ADP
easat-3854	157	14	the	the	DET
easat-3854	157	15	authors	author	NOUN
easat-3854	157	16	;	;	PUNCT
easat-3854	157	17	licensee	licensee	PROPN
easat-3854	157	18	learning	learning	NOUN
easat-3854	157	19	gate	gate	NOUN
easat-3854	157	20	(	(	PUNCT
easat-3854	157	21	i	i	NOUN
easat-3854	157	22	)	)	PUNCT
easat-3854	157	23	𝒫𝑖	𝒫𝑖	NOUN
easat-3854	157	24	:	:	PUNCT
easat-3854	157	25	ℋ𝒟𝑠(ℳ)𝜀	ℋ𝒟𝑠(ℳ)𝜀	NUM
easat-3854	157	26	→	→	SYM
easat-3854	157	27	ℋ𝒟𝑠+𝑖(ℳ)𝜀	ℋ𝒟𝑠+𝑖(ℳ)𝜀	NOUN
easat-3854	157	28	,	,	PUNCT
easat-3854	157	29	if	if	SCONJ
easat-3854	157	30	𝓅	𝓅	PROPN
easat-3854	157	31	=	=	SYM
easat-3854	157	32	2	2	NUM
easat-3854	157	33	,	,	PUNCT
easat-3854	157	34	(	(	PUNCT
easat-3854	157	35	ii	ii	NOUN
easat-3854	157	36	)	)	PUNCT
easat-3854	157	37	𝒫𝑖	𝒫𝑖	PROPN
easat-3854	157	38	:	:	PUNCT
easat-3854	157	39	ℋ𝒟𝑠(ℳ)𝜀	ℋ𝒟𝑠(ℳ)𝜀	NUM
easat-3854	157	40	→	→	SYM
easat-3854	157	41	ℋ𝒟𝑠+2𝑖(𝓅−1)(ℳ)𝜀	ℋ𝒟𝑠+2𝑖(𝓅−1)(ℳ)𝜀	NOUN
easat-3854	157	42	,	,	PUNCT
easat-3854	157	43	and	and	CCONJ
easat-3854	157	44	𝔅𝒫𝑖	𝔅𝒫𝑖	NOUN
easat-3854	157	45	:	:	PUNCT
easat-3854	157	46	ℋ𝒟𝑠(ℳ)𝜀	ℋ𝒟𝑠(ℳ)𝜀	NUM
easat-3854	157	47	→	→	SYM
easat-3854	157	48	ℋ𝒟𝑠+𝑖+2𝑖(𝓅−1)(ℳ)𝜀	ℋ𝒟𝑠+𝑖+2𝑖(𝓅−1)(ℳ)𝜀	NOUN
easat-3854	157	49	if	if	SCONJ
easat-3854	157	50	𝓅	𝓅	PROPN
easat-3854	157	51	>	>	X
easat-3854	157	52	2	2	NUM
easat-3854	157	53	.	.	PUNCT
easat-3854	158	1	the	the	DET
easat-3854	158	2	following	follow	VERB
easat-3854	158	3	characteristics	characteristic	NOUN
easat-3854	158	4	apply	apply	VERB
easat-3854	158	5	to	to	ADP
easat-3854	158	6	the	the	DET
easat-3854	158	7	operators	operator	NOUN
easat-3854	158	8	𝒫𝑖	𝒫𝑖	PROPN
easat-3854	158	9	and	and	CCONJ
easat-3854	158	10	𝔅𝒫𝑖	𝔅𝒫𝑖	NOUN
easat-3854	158	11	:	:	PUNCT
easat-3854	159	1	1	1	X
easat-3854	159	2	)	)	PUNCT
easat-3854	159	3	{	{	PUNCT
easat-3854	159	4	𝒫𝑖|	𝒫𝑖|	PROPN
easat-3854	159	5	ℋ𝒟𝑠(ℳ	ℋ𝒟𝑠(ℳ	NUM
easat-3854	159	6	)	)	PUNCT
easat-3854	159	7	=	=	SYM
easat-3854	159	8	0	0	NUM
easat-3854	159	9	,	,	PUNCT
easat-3854	159	10	𝑖𝑓	𝑖𝑓	ADP
easat-3854	159	11	𝓅	𝓅	NOUN
easat-3854	159	12	=	=	SYM
easat-3854	159	13	2	2	NUM
easat-3854	159	14	,	,	PUNCT
easat-3854	159	15	𝑖	𝑖	PROPN
easat-3854	159	16	>	>	X
easat-3854	159	17	𝑠	𝑠	PROPN
easat-3854	159	18	,	,	PUNCT
easat-3854	159	19	𝒫𝑖|	𝒫𝑖|	PROPN
easat-3854	159	20	ℋ𝒟𝑠(ℳ	ℋ𝒟𝑠(ℳ	X
easat-3854	159	21	)	)	PUNCT
easat-3854	159	22	=	=	SYM
easat-3854	160	1	0	0	NUM
easat-3854	160	2	,	,	PUNCT
easat-3854	160	3	𝑖𝑓	𝑖𝑓	ADP
easat-3854	160	4	𝓅	𝓅	X
easat-3854	160	5	>	>	X
easat-3854	160	6	2	2	NUM
easat-3854	160	7	,	,	PUNCT
easat-3854	160	8	2𝑖	2𝑖	NOUN
easat-3854	160	9	>	>	X
easat-3854	160	10	𝑠	𝑠	PROPN
easat-3854	160	11	,	,	PUNCT
easat-3854	160	12	𝔅𝒫𝑖|	𝔅𝒫𝑖|	PROPN
easat-3854	160	13	ℋ𝒟𝑠(ℳ	ℋ𝒟𝑠(ℳ	NUM
easat-3854	160	14	)	)	PUNCT
easat-3854	161	1	=	=	SYM
easat-3854	161	2	0	0	NUM
easat-3854	161	3	,	,	PUNCT
easat-3854	161	4	𝑖𝑓	𝑖𝑓	ADP
easat-3854	161	5	𝓅	𝓅	X
easat-3854	161	6	>	>	X
easat-3854	161	7	2	2	NUM
easat-3854	161	8	,	,	PUNCT
easat-3854	161	9	2𝑖	2𝑖	NUM
easat-3854	161	10	≥	≥	X
easat-3854	161	11	𝑠	𝑠	PROPN
easat-3854	161	12	,	,	PUNCT
easat-3854	161	13	2	2	NUM
easat-3854	161	14	)	)	PUNCT
easat-3854	161	15	𝒫𝑖(𝒶	𝒫𝑖(𝒶	NOUN
easat-3854	161	16	)	)	PUNCT
easat-3854	161	17	=	=	SYM
easat-3854	162	1	𝒶𝓅	𝒶𝓅	VERB
easat-3854	162	2	,	,	PUNCT
easat-3854	162	3	𝑖𝑓	𝑖𝑓	AUX
easat-3854	162	4	𝓅	𝓅	NOUN
easat-3854	162	5	=	=	SYM
easat-3854	162	6	2	2	NUM
easat-3854	162	7	,	,	PUNCT
easat-3854	162	8	𝑖	𝑖	PROPN
easat-3854	162	9	=	=	SYM
easat-3854	162	10	𝑠	𝑠	PROPN
easat-3854	162	11	,	,	PUNCT
easat-3854	162	12	𝑜𝑟	𝑜𝑟	VERB
easat-3854	162	13	𝓅	𝓅	NOUN
easat-3854	162	14	>	>	X
easat-3854	162	15	2	2	NUM
easat-3854	162	16	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-3854	162	17	2𝑖	2𝑖	NOUN
easat-3854	162	18	=	=	SYM
easat-3854	162	19	𝑠	𝑠	PROPN
easat-3854	162	20	,	,	PUNCT
easat-3854	162	21	3	3	X
easat-3854	162	22	)	)	PUNCT
easat-3854	162	23	𝒫𝒿	𝒫𝒿	PROPN
easat-3854	162	24	=	=	SYM
easat-3854	162	25	∑𝒫𝑖⊗𝒫𝒿−𝑖	∑𝒫𝑖⊗𝒫𝒿−𝑖	NUM
easat-3854	162	26	and	and	CCONJ
easat-3854	162	27	𝔅𝒫𝒿	𝔅𝒫𝒿	NOUN
easat-3854	162	28	=	=	NOUN
easat-3854	162	29	∑𝔅𝒫𝒿−𝑖	∑𝔅𝒫𝒿−𝑖	NOUN
easat-3854	163	1	+	+	PUNCT
easat-3854	163	2	𝒫𝑖⊗𝒫𝒿−𝑖	𝒫𝑖⊗𝒫𝒿−𝑖	NOUN
easat-3854	163	3	.	.	PUNCT
easat-3854	164	1	4	4	X
easat-3854	164	2	)	)	PUNCT
easat-3854	164	3	the	the	DET
easat-3854	164	4	next	next	ADJ
easat-3854	164	5	relations	relation	NOUN
easat-3854	164	6	of	of	ADP
easat-3854	164	7	adam	adam	PROPN
easat-3854	164	8	are	be	AUX
easat-3854	164	9	satisfied	satisfied	ADJ
easat-3854	164	10	by	by	ADP
easat-3854	164	11	the	the	DET
easat-3854	164	12	operators	operator	NOUN
easat-3854	164	13	𝒫𝑖	𝒫𝑖	PROPN
easat-3854	164	14	and	and	CCONJ
easat-3854	164	15	𝔅𝒫𝑖	𝔅𝒫𝑖	NOUN
easat-3854	164	16	:	:	PUNCT
easat-3854	164	17	(	(	PUNCT
easat-3854	164	18	a	a	X
easat-3854	164	19	)	)	PUNCT
easat-3854	164	20	if	if	SCONJ
easat-3854	164	21	𝓎	𝓎	X
easat-3854	164	22	<	<	X
easat-3854	164	23	𝓅𝒷	𝓅𝒷	NOUN
easat-3854	164	24	and	and	CCONJ
easat-3854	164	25	𝓅	𝓅	X
easat-3854	164	26	≥	≥	NUM
easat-3854	164	27	2	2	NUM
easat-3854	164	28	,	,	PUNCT
easat-3854	164	29	we	we	PRON
easat-3854	164	30	have	have	VERB
easat-3854	164	31	:	:	PUNCT
easat-3854	164	32	𝔅𝛾𝒫𝓎𝒫𝒷∑(−1)𝓎+𝑖(𝓎	𝔅𝛾𝒫𝓎𝒫𝒷∑(−1)𝓎+𝑖(𝓎	PROPN
easat-3854	164	33	−	−	PROPN
easat-3854	164	34	𝓅𝑖	𝓅𝑖	ADP
easat-3854	164	35	,	,	PUNCT
easat-3854	164	36	(	(	PUNCT
easat-3854	164	37	𝓅	𝓅	PROPN
easat-3854	164	38	−	−	PROPN
easat-3854	164	39	1)𝒷	1)𝒷	NUM
easat-3854	164	40	−	−	PROPN
easat-3854	164	41	𝓎	𝓎	PROPN
easat-3854	164	42	+	+	NOUN
easat-3854	164	43	𝑖	𝑖	SYM
easat-3854	164	44	−	−	NOUN
easat-3854	164	45	1).𝔅𝛾𝒫𝓎+𝒷−𝑖𝒫𝑖	1).𝔅𝛾𝒫𝓎+𝒷−𝑖𝒫𝑖	NUM
easat-3854	164	46	𝑖	𝑖	NOUN
easat-3854	164	47	since	since	SCONJ
easat-3854	164	48	𝛾	𝛾	PROPN
easat-3854	164	49	=	=	SYM
easat-3854	164	50	0	0	NUM
easat-3854	164	51	𝑜𝑟	𝑜𝑟	ADP
easat-3854	164	52	1	1	NUM
easat-3854	164	53	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
easat-3854	164	54	𝓅	𝓅	X
easat-3854	164	55	=	=	SYM
easat-3854	164	56	2	2	NUM
easat-3854	164	57	,	,	PUNCT
easat-3854	164	58	also	also	ADV
easat-3854	164	59	𝛾	𝛾	ADP
easat-3854	164	60	=	=	SYM
easat-3854	164	61	1	1	NUM
easat-3854	164	62	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
easat-3854	164	63	𝓅	𝓅	X
easat-3854	164	64	>	>	X
easat-3854	164	65	2	2	NUM
easat-3854	164	66	and	and	CCONJ
easat-3854	164	67	for	for	ADP
easat-3854	164	68	any	any	DET
easat-3854	164	69	two	two	NUM
easat-3854	164	70	integers	integer	NOUN
easat-3854	164	71	𝑖	𝑖	ADP
easat-3854	164	72	and	and	CCONJ
easat-3854	164	73	𝒿	𝒿	X
easat-3854	164	74	,	,	PUNCT
easat-3854	164	75	there	there	PRON
easat-3854	164	76	exist	exist	VERB
easat-3854	164	77	:	:	PUNCT
easat-3854	164	78	(	(	PUNCT
easat-3854	164	79	𝑖	𝑖	SYM
easat-3854	164	80	,	,	PUNCT
easat-3854	164	81	𝒿	𝒿	NOUN
easat-3854	164	82	)	)	PUNCT
easat-3854	165	1	=	=	PUNCT
easat-3854	166	1	[	[	PUNCT
easat-3854	166	2	(	(	PUNCT
easat-3854	166	3	𝑖	𝑖	SYM
easat-3854	166	4	,	,	PUNCT
easat-3854	166	5	𝒿	𝒿	NOUN
easat-3854	166	6	)	)	PUNCT
easat-3854	166	7	!	!	PUNCT
easat-3854	167	1	𝑖	𝑖	X
easat-3854	167	2	!	!	NOUN
easat-3854	167	3	,	,	PUNCT
easat-3854	167	4	𝒿	𝒿	PROPN
easat-3854	167	5	!	!	PUNCT
easat-3854	167	6	,	,	PUNCT
easat-3854	167	7	𝑖𝑓	𝑖𝑓	ADP
easat-3854	167	8	𝑖	𝑖	PRON
easat-3854	167	9	≥	≥	NOUN
easat-3854	167	10	0	0	NUM
easat-3854	167	11	,	,	PUNCT
easat-3854	167	12	𝒿	𝒿	X
easat-3854	167	13	≥	≥	NOUN
easat-3854	167	14	0	0	NUM
easat-3854	167	15	,	,	PUNCT
easat-3854	167	16	0	0	NUM
easat-3854	168	1	𝑖𝑓	𝑖𝑓	ADP
easat-3854	168	2	𝑖	𝑖	X
easat-3854	168	3	<	<	X
easat-3854	168	4	0	0	PROPN
easat-3854	168	5	,	,	PUNCT
easat-3854	168	6	𝒿	𝒿	X
easat-3854	168	7	<	<	X
easat-3854	168	8	0	0	NUM
easat-3854	168	9	,	,	PUNCT
easat-3854	168	10	(	(	PUNCT
easat-3854	168	11	b	b	X
easat-3854	168	12	)	)	PUNCT
easat-3854	168	13	if	if	SCONJ
easat-3854	168	14	𝓎	𝓎	X
easat-3854	168	15	<	<	X
easat-3854	168	16	𝓅𝒷,𝓅	𝓅𝒷,𝓅	X
easat-3854	168	17	=	=	SYM
easat-3854	168	18	2	2	NUM
easat-3854	168	19	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
easat-3854	168	20	𝛾	𝛾	NOUN
easat-3854	168	21	=	=	SYM
easat-3854	168	22	0	0	NUM
easat-3854	168	23	𝑜𝑟	𝑜𝑟	ADP
easat-3854	168	24	1	1	NUM
easat-3854	168	25	,	,	PUNCT
easat-3854	168	26	then	then	ADV
easat-3854	168	27	:	:	PUNCT
easat-3854	168	28	𝔅𝛾𝒫𝓎𝒫𝒷	𝔅𝛾𝒫𝓎𝒫𝒷	NOUN
easat-3854	168	29	=	=	SYM
easat-3854	168	30	(	(	PUNCT
easat-3854	168	31	1	1	NUM
easat-3854	168	32	−	−	NOUN
easat-3854	168	33	𝛾)∑(−1)𝓎+𝑖(𝓎	𝛾)∑(−1)𝓎+𝑖(𝓎	NOUN
easat-3854	168	34	−	−	NOUN
easat-3854	168	35	𝓅𝑖	𝓅𝑖	ADP
easat-3854	168	36	,	,	PUNCT
easat-3854	168	37	(	(	PUNCT
easat-3854	168	38	𝓅	𝓅	PROPN
easat-3854	168	39	−	−	PROPN
easat-3854	168	40	1)𝒷	1)𝒷	NUM
easat-3854	168	41	−𝓎	−𝓎	NOUN
easat-3854	169	1	+	+	CCONJ
easat-3854	170	1	𝑖	𝑖	ADP
easat-3854	170	2	−	−	NOUN
easat-3854	170	3	1).𝔅𝒫𝓎+𝒷−𝑖𝒫𝑖	1).𝔅𝒫𝓎+𝒷−𝑖𝒫𝑖	NUM
easat-3854	170	4	𝑖	𝑖	ADP
easat-3854	170	5	−∑(−1)𝓎+𝑖(𝓎	−∑(−1)𝓎+𝑖(𝓎	NOUN
easat-3854	170	6	−	−	NOUN
easat-3854	170	7	𝓅𝑖	𝓅𝑖	ADP
easat-3854	170	8	−	−	PROPN
easat-3854	170	9	1	1	NUM
easat-3854	170	10	,	,	PUNCT
easat-3854	170	11	(	(	PUNCT
easat-3854	170	12	𝓅	𝓅	PROPN
easat-3854	170	13	−	−	PROPN
easat-3854	170	14	1)𝒷	1)𝒷	NUM
easat-3854	170	15	−𝓎	−𝓎	NOUN
easat-3854	170	16	+	+	CCONJ
easat-3854	170	17	𝑖).𝔅𝛾𝒫𝓎+𝒷−𝑖𝔅𝒫𝑖	𝑖).𝔅𝛾𝒫𝓎+𝒷−𝑖𝔅𝒫𝑖	NOUN
easat-3854	170	18	𝑖	𝑖	X
easat-3854	170	19	.	.	PUNCT
easat-3854	171	1	by	by	ADP
easat-3854	171	2	noted	note	VERB
easat-3854	171	3	that	that	SCONJ
easat-3854	171	4	,	,	PUNCT
easat-3854	171	5	the	the	DET
easat-3854	171	6	operators	operator	NOUN
easat-3854	171	7	𝔅0𝒫𝑠	𝔅0𝒫𝑠	VERB
easat-3854	171	8	and	and	CCONJ
easat-3854	171	9	𝔅1𝒫𝑠	𝔅1𝒫𝑠	PROPN
easat-3854	171	10	represent	represent	NOUN
easat-3854	171	11	,	,	PUNCT
easat-3854	171	12	respectively	respectively	ADV
easat-3854	171	13	,	,	PUNCT
easat-3854	171	14	𝒫𝑠	𝒫𝑠	PROPN
easat-3854	171	15	and	and	CCONJ
easat-3854	171	16	𝔅𝒫𝑠.	𝔅𝒫𝑠.	PROPN
easat-3854	171	17	proof	proof	NOUN
easat-3854	171	18	:	:	PUNCT
easat-3854	171	19	by	by	ADP
easat-3854	171	20	assuming	assume	VERB
easat-3854	171	21	that	that	SCONJ
easat-3854	171	22	the	the	DET
easat-3854	171	23	triple	triple	ADJ
easat-3854	171	24	ℭ	ℭ	NOUN
easat-3854	171	25	=	=	PUNCT
easat-3854	171	26	(	(	PUNCT
easat-3854	171	27	ℛ,ℳ,𝒯	ℛ,ℳ,𝒯	ADJ
easat-3854	171	28	)	)	PUNCT
easat-3854	171	29	,	,	PUNCT
easat-3854	171	30	when	when	SCONJ
easat-3854	171	31	ℳ	ℳ	PROPN
easat-3854	171	32	is	be	AUX
easat-3854	171	33	the	the	DET
easat-3854	171	34	commutative	commutative	ADJ
easat-3854	171	35	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-3854	171	36	for	for	ADP
easat-3854	171	37	𝒦	𝒦	PROPN
easat-3854	171	38	=	=	PUNCT
easat-3854	171	39	𝒵/𝒫	𝒵/𝒫	NOUN
easat-3854	171	40	,	,	PUNCT
easat-3854	171	41	such	such	ADJ
easat-3854	171	42	𝒯	𝒯	PROPN
easat-3854	171	43	and	and	CCONJ
easat-3854	171	44	ℛ	ℛ	PROPN
easat-3854	171	45	are	be	AUX
easat-3854	171	46	the	the	DET
easat-3854	171	47	left	left	NOUN
easat-3854	171	48	and	and	CCONJ
easat-3854	171	49	the	the	DET
easat-3854	171	50	right	right	ADJ
easat-3854	171	51	commutative	commutative	ADJ
easat-3854	171	52	ℳ-algebras	ℳ-algebras	PROPN
easat-3854	171	53	,	,	PUNCT
easat-3854	171	54	respectively	respectively	ADV
easat-3854	171	55	.	.	PUNCT
easat-3854	172	1	because	because	SCONJ
easat-3854	172	2	of	of	ADP
easat-3854	172	3	the	the	DET
easat-3854	172	4	explanation	explanation	NOUN
easat-3854	172	5	above	above	ADV
easat-3854	172	6	and	and	CCONJ
easat-3854	172	7	taking	take	VERB
easat-3854	172	8	into	into	ADP
easat-3854	172	9	consideration	consideration	NOUN
easat-3854	172	10	the	the	DET
easat-3854	172	11	triple	triple	ADJ
easat-3854	172	12	ℒ	ℒ	NOUN
easat-3854	172	13	=	=	SYM
easat-3854	172	14	(	(	PUNCT
easat-3854	172	15	(	(	PUNCT
easat-3854	172	16	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	172	17	𝜀	𝜀	PROPN
easat-3854	172	18	)	)	PUNCT
easat-3854	172	19	,	,	PUNCT
easat-3854	172	20	𝒦[𝒟𝑝],𝒦𝒟	𝒦[𝒟𝑝],𝒦𝒟	PROPN
easat-3854	172	21	)	)	PUNCT
easat-3854	172	22	,	,	PUNCT
easat-3854	172	23	we	we	PRON
easat-3854	172	24	get	get	VERB
easat-3854	172	25	that	that	DET
easat-3854	172	26	𝒦[𝒟𝑝	𝒦[𝒟𝑝	NOUN
easat-3854	172	27	]	]	PUNCT
easat-3854	172	28	is	be	AUX
easat-3854	172	29	the	the	DET
easat-3854	172	30	commutative	commutative	ADJ
easat-3854	172	31	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-3854	172	32	over	over	ADP
easat-3854	172	33	𝒦	𝒦	PROPN
easat-3854	172	34	=	=	PUNCT
easat-3854	172	35	𝒵/𝒫	𝒵/𝒫	NOUN
easat-3854	172	36	,	,	PUNCT
easat-3854	172	37	ℳ𝒟	ℳ𝒟	PROPN
easat-3854	172	38	𝜀	𝜀	PROPN
easat-3854	172	39	and	and	CCONJ
easat-3854	172	40	𝒦𝒟	𝒦𝒟	PROPN
easat-3854	172	41	,	,	PUNCT
easat-3854	172	42	respectively	respectively	ADV
easat-3854	172	43	,	,	PUNCT
easat-3854	172	44	the	the	DET
easat-3854	172	45	left	left	ADJ
easat-3854	172	46	and	and	CCONJ
easat-3854	172	47	right	right	ADJ
easat-3854	172	48	commutative	commutative	ADJ
easat-3854	172	49	𝒦[𝒟𝑝]-algebras	𝒦[𝒟𝑝]-algebras	NOUN
easat-3854	172	50	and	and	CCONJ
easat-3854	172	51	hence	hence	ADV
easat-3854	172	52	ℋ((𝛽(ℒ)⋆	ℋ((𝛽(ℒ)⋆	NOUN
easat-3854	172	53	)	)	PUNCT
easat-3854	173	1	=	=	SYM
easat-3854	173	2	ℋ𝒟(ℳ)𝜀	ℋ𝒟(ℳ)𝜀	NUM
easat-3854	173	3	.	.	PUNCT
easat-3854	174	1	3.1	3.1	NUM
easat-3854	174	2	.	.	X
easat-3854	174	3	concluding	conclude	VERB
easat-3854	174	4	remarks	remark	NOUN
easat-3854	174	5	in	in	ADP
easat-3854	174	6	our	our	PRON
easat-3854	174	7	paper	paper	NOUN
easat-3854	174	8	,	,	PUNCT
easat-3854	174	9	we	we	PRON
easat-3854	174	10	introduced	introduce	VERB
easat-3854	174	11	and	and	CCONJ
easat-3854	174	12	studied	study	VERB
easat-3854	174	13	another	another	DET
easat-3854	174	14	definition	definition	NOUN
easat-3854	174	15	for	for	ADP
easat-3854	174	16	the	the	DET
easat-3854	174	17	steenrod	steenrod	NOUN
easat-3854	174	18	operator	operator	NOUN
easat-3854	174	19	for	for	ADP
easat-3854	174	20	dihedral	dihedral	ADJ
easat-3854	174	21	homology	homology	NOUN
easat-3854	174	22	of	of	ADP
easat-3854	174	23	𝒜∞-algebras	𝒜∞-algebras	PROPN
easat-3854	174	24	.	.	PUNCT
easat-3854	175	1	by	by	ADP
easat-3854	175	2	employing	employ	VERB
easat-3854	175	3	the	the	DET
easat-3854	175	4	tensor	tensor	NOUN
easat-3854	175	5	product	product	NOUN
easat-3854	175	6	of	of	ADP
easat-3854	175	7	the	the	DET
easat-3854	175	8	symmetry	symmetry	NOUN
easat-3854	175	9	group	group	NOUN
easat-3854	175	10	’s	’s	PART
easat-3854	175	11	free	free	ADJ
easat-3854	175	12	resolution	resolution	NOUN
easat-3854	175	13	beside	beside	ADP
easat-3854	175	14	the	the	DET
easat-3854	175	15	standard	standard	ADJ
easat-3854	175	16	𝒜∞-algebras	𝒜∞-algebras	PUNCT
easat-3854	175	17	resolution	resolution	NOUN
easat-3854	175	18	influenced	influence	VERB
easat-3854	175	19	by	by	ADP
easat-3854	175	20	the	the	DET
easat-3854	175	21	dihedral	dihedral	ADJ
easat-3854	175	22	group	group	NOUN
easat-3854	175	23	,	,	PUNCT
easat-3854	175	24	we	we	PRON
easat-3854	175	25	established	establish	VERB
easat-3854	175	26	a	a	DET
easat-3854	175	27	framework	framework	NOUN
easat-3854	175	28	for	for	ADP
easat-3854	175	29	conducting	conduct	VERB
easat-3854	175	30	sporadic	sporadic	ADJ
easat-3854	175	31	steenrod	steenrod	NOUN
easat-3854	175	32	operation	operation	NOUN
easat-3854	175	33	computations	computation	NOUN
easat-3854	175	34	.	.	PUNCT
easat-3854	176	1	this	this	DET
easat-3854	176	2	approach	approach	NOUN
easat-3854	176	3	provided	provide	VERB
easat-3854	176	4	approximate	approximate	ADJ
easat-3854	176	5	results	result	NOUN
easat-3854	176	6	that	that	PRON
easat-3854	176	7	enriched	enrich	VERB
easat-3854	176	8	our	our	PRON
easat-3854	176	9	understanding	understanding	NOUN
easat-3854	176	10	of	of	ADP
easat-3854	176	11	these	these	DET
easat-3854	176	12	intricate	intricate	ADJ
easat-3854	176	13	algebraic	algebraic	ADJ
easat-3854	176	14	structures	structure	NOUN
easat-3854	176	15	.	.	PUNCT
easat-3854	177	1	copyright	copyright	NOUN
easat-3854	177	2	:	:	PUNCT
easat-3854	177	3	©	©	PROPN
easat-3854	177	4	2024	2024	NUM
easat-3854	177	5	by	by	ADP
easat-3854	177	6	the	the	DET
easat-3854	177	7	authors	author	NOUN
easat-3854	177	8	.	.	PUNCT
easat-3854	178	1	this	this	DET
easat-3854	178	2	article	article	NOUN
easat-3854	178	3	is	be	AUX
easat-3854	178	4	an	an	DET
easat-3854	178	5	open	open	ADJ
easat-3854	178	6	access	access	NOUN
easat-3854	178	7	article	article	NOUN
easat-3854	178	8	distributed	distribute	VERB
easat-3854	178	9	under	under	ADP
easat-3854	178	10	the	the	DET
easat-3854	178	11	terms	term	NOUN
easat-3854	178	12	and	and	CCONJ
easat-3854	178	13	conditions	condition	NOUN
easat-3854	178	14	of	of	ADP
easat-3854	178	15	the	the	DET
easat-3854	178	16	creative	creative	ADJ
easat-3854	178	17	commons	common	NOUN
easat-3854	178	18	attribution	attribution	NOUN
easat-3854	178	19	(	(	PUNCT
easat-3854	178	20	cc	cc	NOUN
easat-3854	178	21	by	by	ADP
easat-3854	178	22	)	)	PUNCT
easat-3854	178	23	license	license	NOUN
easat-3854	178	24	(	(	PUNCT
easat-3854	178	25	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-3854	178	26	)	)	PUNCT
easat-3854	178	27	.	.	PUNCT
easat-3854	179	1	references	reference	NOUN
easat-3854	179	2	[	[	X
easat-3854	179	3	1	1	NUM
easat-3854	179	4	]	]	PUNCT
easat-3854	179	5	r.	r.	PROPN
easat-3854	179	6	e.	e.	PROPN
easat-3854	179	7	mosher	mosher	PROPN
easat-3854	179	8	and	and	CCONJ
easat-3854	179	9	m.	m.	PROPN
easat-3854	179	10	c.	c.	PROPN
easat-3854	179	11	tangora	tangora	PROPN
easat-3854	179	12	,	,	PUNCT
easat-3854	179	13	"	"	PUNCT
easat-3854	179	14	cohomology	cohomology	NOUN
easat-3854	179	15	operations	operation	NOUN
easat-3854	179	16	and	and	CCONJ
easat-3854	179	17	applications	application	NOUN
easat-3854	179	18	in	in	ADP
easat-3854	179	19	homotopytheory	homotopytheory	NOUN
easat-3854	179	20	,	,	PUNCT
easat-3854	179	21	"	"	PUNCT
easat-3854	179	22	handbook	handbook	NOUN
easat-3854	179	23	of	of	ADP
easat-3854	179	24	mathematic	mathematic	ADJ
easat-3854	179	25	,	,	PUNCT
easat-3854	179	26	harper	harper	NOUN
easat-3854	179	27	and	and	CCONJ
easat-3854	179	28	row	row	NOUN
easat-3854	179	29	,	,	PUNCT
easat-3854	179	30	publishers	publisher	VERB
easat-3854	179	31	new	new	PROPN
easat-3854	179	32	york	york	PROPN
easat-3854	179	33	,	,	PUNCT
easat-3854	179	34	evanston	evanston	PROPN
easat-3854	179	35	,	,	PUNCT
easat-3854	179	36	and	and	CCONJ
easat-3854	179	37	london	london	PROPN
easat-3854	179	38	,	,	PUNCT
easat-3854	179	39	dc	dc	PROPN
easat-3854	179	40	:	:	PUNCT
easat-3854	179	41	1968	1968	NUM
easat-3854	179	42	,	,	PUNCT
easat-3854	179	43	pp	pp	ADP
easat-3854	179	44	.	.	PUNCT
easat-3854	180	1	45	45	NUM
easat-3854	180	2	-	-	SYM
easat-3854	180	3	85	85	NUM
easat-3854	180	4	.	.	PUNCT
easat-3854	180	5	https://www.maths.ed.ac.uk/~v1ranick/papers/moshtang.pdf	https://www.maths.ed.ac.uk/~v1ranick/papers/moshtang.pdf	X
easat-3854	181	1	[	[	X
easat-3854	181	2	2	2	NUM
easat-3854	181	3	]	]	PUNCT
easat-3854	181	4	s.	s.	PROPN
easat-3854	181	5	v.	v.	PROPN
easat-3854	181	6	lapin	lapin	PROPN
easat-3854	181	7	,	,	PUNCT
easat-3854	181	8	"	"	PUNCT
easat-3854	181	9	𝐷-differential	𝐷-differential	PROPN
easat-3854	181	10	𝐸-algebra	𝐸-algebra	PROPN
easat-3854	181	11	and	and	CCONJ
easat-3854	181	12	steenrod	steenrod	NOUN
easat-3854	181	13	operation	operation	NOUN
easat-3854	181	14	spectral	spectral	ADJ
easat-3854	181	15	sequences	sequence	NOUN
easat-3854	181	16	,	,	PUNCT
easat-3854	181	17	"	"	PUNCT
easat-3854	181	18	j.	j.	PROPN
easat-3854	181	19	math	math	PROPN
easat-3854	181	20	.	.	PUNCT
easat-3854	182	1	science	science	NOUN
easat-3854	182	2	,	,	PUNCT
easat-3854	182	3	vol	vol	NOUN
easat-3854	182	4	.	.	PROPN
easat-3854	182	5	152	152	NUM
easat-3854	182	6	,	,	PUNCT
easat-3854	182	7	no	no	INTJ
easat-3854	182	8	.	.	NOUN
easat-3854	182	9	3	3	NUM
easat-3854	182	10	,	,	PUNCT
easat-3854	182	11	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-3854	182	12	https://www.maths.ed.ac.uk/~v1ranick/papers/moshtang.pdf	https://www.maths.ed.ac.uk/~v1ranick/papers/moshtang.pdf	PROPN
easat-3854	182	13	8666	8666	NUM
easat-3854	182	14	edelweiss	edelweiss	PROPN
easat-3854	182	15	applied	apply	VERB
easat-3854	182	16	science	science	NOUN
easat-3854	182	17	and	and	CCONJ
easat-3854	182	18	technology	technology	NOUN
easat-3854	182	19	issn	issn	PROPN
easat-3854	182	20	:	:	PUNCT
easat-3854	182	21	2576	2576	NUM
easat-3854	182	22	-	-	SYM
easat-3854	182	23	8484	8484	NUM
easat-3854	182	24	vol	vol	NOUN
easat-3854	182	25	.	.	PROPN
easat-3854	182	26	8	8	NUM
easat-3854	182	27	,	,	PUNCT
easat-3854	182	28	no	no	INTJ
easat-3854	182	29	.	.	NOUN
easat-3854	183	1	6	6	NUM
easat-3854	183	2	:	:	SYM
easat-3854	183	3	8658	8658	NUM
easat-3854	183	4	-	-	SYM
easat-3854	183	5	8666	8666	NUM
easat-3854	183	6	,	,	PUNCT
easat-3854	183	7	2024	2024	NUM
easat-3854	183	8	doi	doi	NOUN
easat-3854	183	9	:	:	PUNCT
easat-3854	183	10	10.55214/25768484.v8i6.3854	10.55214/25768484.v8i6.3854	NUM
easat-3854	183	11	©	©	ADP
easat-3854	183	12	2024	2024	NUM
easat-3854	183	13	by	by	ADP
easat-3854	183	14	the	the	DET
easat-3854	183	15	authors	author	NOUN
easat-3854	183	16	;	;	PUNCT
easat-3854	183	17	licensee	licensee	PROPN
easat-3854	183	18	learning	learn	VERB
easat-3854	183	19	gate	gate	NOUN
easat-3854	183	20	pp	pp	PROPN
easat-3854	183	21	.	.	PUNCT
easat-3854	184	1	372	372	NUM
easat-3854	184	2	-	-	SYM
easat-3854	184	3	410	410	NUM
easat-3854	184	4	,	,	PUNCT
easat-3854	184	5	2008	2008	NUM
easat-3854	184	6	.	.	PUNCT
easat-3854	185	1	https://doi.org/10.1007/s10958-008-9070-y	https://doi.org/10.1007/s10958-008-9070-y	PROPN
easat-3854	186	1	[	[	X
easat-3854	186	2	3	3	NUM
easat-3854	186	3	]	]	PUNCT
easat-3854	186	4	m.	m.	NOUN
easat-3854	186	5	lopez	lopez	NOUN
easat-3854	186	6	,	,	PUNCT
easat-3854	186	7	"	"	PUNCT
easat-3854	186	8	adams	adam	NOUN
easat-3854	186	9	operations	operation	NOUN
easat-3854	186	10	and	and	CCONJ
easat-3854	186	11	𝜆-operations	𝜆-operation	NOUN
easat-3854	186	12	on	on	ADP
easat-3854	186	13	classifying	classify	VERB
easat-3854	186	14	oriented	oriented	ADJ
easat-3854	186	15	cohomology	cohomology	NOUN
easat-3854	186	16	theories	theory	NOUN
easat-3854	186	17	,	,	PUNCT
easat-3854	186	18	"	"	PUNCT
easat-3854	186	19	journal	journal	NOUN
easat-3854	186	20	of	of	ADP
easat-3854	186	21	pure	pure	ADJ
easat-3854	186	22	and	and	CCONJ
easat-3854	186	23	applied	applied	ADJ
easat-3854	186	24	algebra	algebra	NOUN
easat-3854	186	25	,	,	PUNCT
easat-3854	186	26	vol	vol	NOUN
easat-3854	186	27	.	.	NOUN
easat-3854	186	28	213	213	NUM
easat-3854	186	29	,	,	PUNCT
easat-3854	186	30	no	no	NOUN
easat-3854	186	31	.	.	NOUN
easat-3854	186	32	4	4	NUM
easat-3854	186	33	,	,	PUNCT
easat-3854	186	34	pp	pp	ADJ
easat-3854	186	35	.	.	PUNCT
easat-3854	187	1	409	409	NUM
easat-3854	187	2	-	-	SYM
easat-3854	187	3	420	420	NUM
easat-3854	187	4	,	,	PUNCT
easat-3854	187	5	2009	2009	NUM
easat-3854	187	6	.	.	PUNCT
easat-3854	188	1	https://doi.org/10.1016/j.jpaa.2008.07.017	https://doi.org/10.1016/j.jpaa.2008.07.017	PRON
easat-3854	188	2	[	[	X
easat-3854	188	3	4	4	X
easat-3854	188	4	]	]	PUNCT
easat-3854	188	5	d.	d.	NOUN
easat-3854	188	6	burghelea	burghelea	PROPN
easat-3854	189	1	,	,	PUNCT
easat-3854	189	2	z.	z.	PROPN
easat-3854	189	3	fiedorwicz	fiedorwicz	PROPN
easat-3854	189	4	and	and	CCONJ
easat-3854	189	5	w.	w.	PROPN
easat-3854	189	6	gojda	gojda	PROPN
easat-3854	189	7	,	,	PUNCT
easat-3854	189	8	"	"	PUNCT
easat-3854	189	9	adams	adam	NOUN
easat-3854	189	10	operations	operation	NOUN
easat-3854	189	11	in	in	ADP
easat-3854	189	12	hochschild	hochschild	ADJ
easat-3854	189	13	and	and	CCONJ
easat-3854	189	14	cyclic	cyclic	ADJ
easat-3854	189	15	homology	homology	NOUN
easat-3854	189	16	of	of	ADP
easat-3854	189	17	de	de	X
easat-3854	189	18	rham	rham	PROPN
easat-3854	189	19	algebra	algebra	PROPN
easat-3854	189	20	and	and	CCONJ
easat-3854	189	21	free	free	ADJ
easat-3854	189	22	loop	loop	NOUN
easat-3854	189	23	spaces	space	NOUN
easat-3854	189	24	,	,	PUNCT
easat-3854	189	25	"	"	PUNCT
easat-3854	189	26	k	k	NOUN
easat-3854	189	27	-	-	NOUN
easat-3854	189	28	theory	theory	NOUN
easat-3854	189	29	,	,	PUNCT
easat-3854	189	30	vol	vol	NOUN
easat-3854	189	31	.	.	PROPN
easat-3854	189	32	4	4	NUM
easat-3854	189	33	,	,	PUNCT
easat-3854	189	34	no	no	INTJ
easat-3854	189	35	.	.	NOUN
easat-3854	189	36	3	3	NUM
easat-3854	189	37	,	,	PUNCT
easat-3854	189	38	pp	pp	ADJ
easat-3854	189	39	.	.	PUNCT
easat-3854	190	1	269	269	NUM
easat-3854	190	2	-	-	SYM
easat-3854	190	3	287	287	NUM
easat-3854	190	4	,	,	PUNCT
easat-3854	190	5	1991	1991	NUM
easat-3854	190	6	.	.	PUNCT
easat-3854	191	1	doi:10.1007	doi:10.1007	PROPN
easat-3854	191	2	/	/	SYM
easat-3854	191	3	bf00569450	bf00569450	PROPN
easat-3854	192	1	[	[	X
easat-3854	192	2	5	5	NUM
easat-3854	192	3	]	]	PUNCT
easat-3854	192	4	a.	a.	NOUN
easat-3854	192	5	h.	h.	PROPN
easat-3854	192	6	noreldeen	noreldeen	PROPN
easat-3854	192	7	.	.	PUNCT
easat-3854	193	1	"	"	PUNCT
easat-3854	193	2	on	on	ADP
easat-3854	193	3	the	the	DET
easat-3854	193	4	hochschild	hochschild	ADJ
easat-3854	193	5	cohomology	cohomology	NOUN
easat-3854	193	6	theory	theory	NOUN
easat-3854	193	7	of	of	ADP
easat-3854	193	8	𝐴∞-algebra	𝐴∞-algebra	PROPN
easat-3854	193	9	"	"	PUNCT
easat-3854	193	10	.	.	PUNCT
easat-3854	194	1	scientific	scientific	ADJ
easat-3854	194	2	african	african	ADJ
easat-3854	194	3	,	,	PUNCT
easat-3854	194	4	vol	vol	NOUN
easat-3854	194	5	.	.	PROPN
easat-3854	194	6	5	5	NUM
easat-3854	194	7	,	,	PUNCT
easat-3854	194	8	e00115	e00115	PROPN
easat-3854	194	9	,	,	PUNCT
easat-3854	194	10	2019	2019	NUM
easat-3854	194	11	.	.	PUNCT
easat-3854	194	12	https://doi.org/10.1016/j.sciaf.2019.e00115	https://doi.org/10.1016/j.sciaf.2019.e00115	PUNCT
easat-3854	195	1	[	[	X
easat-3854	195	2	6	6	NUM
easat-3854	195	3	]	]	PUNCT
easat-3854	195	4	a.	a.	NOUN
easat-3854	195	5	h.	h.	PROPN
easat-3854	195	6	noreldeen	noreldeen	PROPN
easat-3854	195	7	,	,	PUNCT
easat-3854	195	8	y.	y.	PROPN
easat-3854	195	9	gh	gh	PROPN
easat-3854	195	10	.	.	PUNCT
easat-3854	196	1	gouda	gouda	NOUN
easat-3854	196	2	and	and	CCONJ
easat-3854	196	3	m.	m.	NOUN
easat-3854	196	4	saad	saad	NOUN
easat-3854	196	5	,	,	PUNCT
easat-3854	196	6	"	"	PUNCT
easat-3854	196	7	reflexive	reflexive	ADJ
easat-3854	196	8	and	and	CCONJ
easat-3854	196	9	dihedral	dihedral	ADJ
easat-3854	196	10	(	(	PUNCT
easat-3854	196	11	co)homology	co)homology	NOUN
easat-3854	196	12	of	of	ADP
easat-3854	196	13	𝑍/2	𝑍/2	VERB
easat-3854	196	14	graded	grade	VERB
easat-3854	196	15	algebras	algebra	NOUN
easat-3854	196	16	,	,	PUNCT
easat-3854	196	17	"	"	PUNCT
easat-3854	196	18	international	international	ADJ
easat-3854	196	19	journal	journal	NOUN
easat-3854	196	20	of	of	ADP
easat-3854	196	21	mathematics	mathematics	PROPN
easat-3854	196	22	and	and	CCONJ
easat-3854	196	23	statistics	statistic	NOUN
easat-3854	196	24	invention	invention	NOUN
easat-3854	196	25	(	(	PUNCT
easat-3854	196	26	ijmsi	ijmsi	NOUN
easat-3854	196	27	)	)	PUNCT
easat-3854	196	28	,	,	PUNCT
easat-3854	196	29	vol	vol	NOUN
easat-3854	196	30	.	.	PROPN
easat-3854	196	31	5	5	NUM
easat-3854	196	32	,	,	PUNCT
easat-3854	196	33	no	no	INTJ
easat-3854	196	34	.	.	NOUN
easat-3854	196	35	1	1	NUM
easat-3854	196	36	,	,	PUNCT
easat-3854	196	37	pp	pp	ADJ
easat-3854	196	38	.	.	PUNCT
easat-3854	197	1	23	23	NUM
easat-3854	197	2	-	-	SYM
easat-3854	197	3	31	31	NUM
easat-3854	197	4	,	,	PUNCT
easat-3854	197	5	2017	2017	NUM
easat-3854	197	6	.	.	PUNCT
easat-3854	198	1	https://www.ijmsi.org/papers/volume.5.issue.1/c05012331.pdf	https://www.ijmsi.org/papers/volume.5.issue.1/c05012331.pdf	PUNCT
easat-3854	199	1	[	[	X
easat-3854	199	2	7	7	X
easat-3854	199	3	]	]	X
easat-3854	199	4	s.	s.	PROPN
easat-3854	199	5	v.	v.	PROPN
easat-3854	199	6	lapin	lapin	PROPN
easat-3854	199	7	,	,	PUNCT
easat-3854	199	8	"	"	PUNCT
easat-3854	199	9	dihedral	dihedral	ADJ
easat-3854	199	10	and	and	CCONJ
easat-3854	199	11	reflexive	reflexive	ADJ
easat-3854	199	12	modules	module	NOUN
easat-3854	199	13	with	with	ADP
easat-3854	199	14	∞-simplicial	∞-simplicial	NOUN
easat-3854	199	15	faces	face	NOUN
easat-3854	199	16	and	and	CCONJ
easat-3854	199	17	dihedral	dihedral	ADJ
easat-3854	199	18	and	and	CCONJ
easat-3854	199	19	reflexive	reflexive	ADJ
easat-3854	199	20	homology	homology	NOUN
easat-3854	199	21	of	of	ADP
easat-3854	199	22	involutive	involutive	ADJ
easat-3854	199	23	𝐴∞-algebras	𝐴∞-algebras	PROPN
easat-3854	199	24	over	over	ADP
easat-3854	199	25	unital	unital	ADJ
easat-3854	199	26	commutative	commutative	ADJ
easat-3854	199	27	rings	ring	NOUN
easat-3854	199	28	,	,	PUNCT
easat-3854	199	29	"	"	PUNCT
easat-3854	199	30	math	math	NOUN
easat-3854	199	31	at	at	ADP
easat-3854	199	32	,	,	PUNCT
easat-3854	199	33	vol	vol	NOUN
easat-3854	199	34	.	.	PROPN
easat-3854	199	35	1	1	NUM
easat-3854	199	36	,	,	PUNCT
easat-3854	199	37	2018	2018	NUM
easat-3854	199	38	.	.	PUNCT
easat-3854	200	1	https://doi.org/10.48550/arxiv.1809.07510	https://doi.org/10.48550/arxiv.1809.07510	NOUN
easat-3854	200	2	.	.	PUNCT
easat-3854	201	1	[	[	X
easat-3854	201	2	8	8	NUM
easat-3854	201	3	]	]	X
easat-3854	201	4	s.a	s.a	PROPN
easat-3854	201	5	.	.	PROPN
easat-3854	201	6	quota	quota	PROPN
easat-3854	201	7	,	,	PUNCT
easat-3854	201	8	f.	f.	PROPN
easat-3854	201	9	r.	r.	PROPN
easat-3854	201	10	karar	karar	PROPN
easat-3854	201	11	,	,	PUNCT
easat-3854	201	12	o.	o.	PROPN
easat-3854	201	13	h.	h.	PROPN
easat-3854	201	14	fathy	fathy	PROPN
easat-3854	201	15	and	and	CCONJ
easat-3854	201	16	w.	w.	PROPN
easat-3854	201	17	m.	m.	PROPN
easat-3854	201	18	mahmoud	mahmoud	PROPN
easat-3854	201	19	,	,	PUNCT
easat-3854	201	20	"	"	PUNCT
easat-3854	201	21	some	some	DET
easat-3854	201	22	properties	property	NOUN
easat-3854	201	23	of	of	ADP
easat-3854	201	24	cyclic	cyclic	ADJ
easat-3854	201	25	and	and	CCONJ
easat-3854	201	26	dihedral	dihedral	ADJ
easat-3854	201	27	homology	homology	NOUN
easat-3854	201	28	for	for	ADP
easat-3854	201	29	schemes	scheme	NOUN
easat-3854	201	30	,	,	PUNCT
easat-3854	201	31	"	"	PUNCT
easat-3854	201	32	mathematics	mathematic	NOUN
easat-3854	201	33	and	and	CCONJ
easat-3854	201	34	statistics	statistic	NOUN
easat-3854	201	35	,	,	PUNCT
easat-3854	201	36	vol.11	vol.11	PROPN
easat-3854	201	37	,	,	PUNCT
easat-3854	201	38	no	no	INTJ
easat-3854	201	39	.	.	NOUN
easat-3854	201	40	6	6	NUM
easat-3854	201	41	,	,	PUNCT
easat-3854	201	42	pp	pp	ADJ
easat-3854	201	43	.	.	PUNCT
easat-3854	201	44	874	874	NUM
easat-3854	201	45	-	-	SYM
easat-3854	201	46	882	882	NUM
easat-3854	201	47	,	,	PUNCT
easat-3854	201	48	2023	2023	NUM
easat-3854	201	49	.	.	PUNCT
easat-3854	202	1	doi	doi	NOUN
easat-3854	202	2	:	:	PUNCT
easat-3854	202	3	10.13189	10.13189	NUM
easat-3854	202	4	/	/	SYM
easat-3854	202	5	ms.2023.110602	ms.2023.110602	PROPN
easat-3854	202	6	.	.	PUNCT
easat-3854	203	1	https://www.hrpub.org/download/20231030/ms2-13433629.pdf	https://www.hrpub.org/download/20231030/ms2-13433629.pdf	X
easat-3854	204	1	[	[	X
easat-3854	204	2	9	9	NUM
easat-3854	204	3	]	]	SYM
easat-3854	204	4	yu	yu	PROPN
easat-3854	204	5	.	.	PUNCT
easat-3854	205	1	p.	p.	NOUN
easat-3854	205	2	solovev	solovev	PROPN
easat-3854	205	3	,	,	PUNCT
easat-3854	205	4	r.	r.	PROPN
easat-3854	205	5	l.	l.	PROPN
easat-3854	205	6	krasauskas	krasauskas	PROPN
easat-3854	205	7	,	,	PUNCT
easat-3854	205	8	and	and	CCONJ
easat-3854	205	9	s.v.lapin	s.v.lapin	NOUN
easat-3854	205	10	.	.	PUNCT
easat-3854	206	1	"	"	PUNCT
easat-3854	206	2	dihedral	dihedral	ADJ
easat-3854	206	3	homology	homology	NOUN
easat-3854	206	4	and	and	CCONJ
easat-3854	206	5	cohomology	cohomology	NOUN
easat-3854	206	6	,	,	PUNCT
easat-3854	206	7	basic	basic	ADJ
easat-3854	206	8	notions	notion	NOUN
easat-3854	206	9	and	and	CCONJ
easat-3854	206	10	constructions	construction	NOUN
easat-3854	206	11	"	"	PUNCT
easat-3854	206	12	.	.	PUNCT
easat-3854	207	1	math	math	NOUN
easat-3854	207	2	.	.	PUNCT
easat-3854	208	1	ussr	ussr	PROPN
easat-3854	208	2	spornic	spornic	PROPN
easat-3854	208	3	,	,	PUNCT
easat-3854	208	4	vol	vol	NOUN
easat-3854	208	5	.	.	PROPN
easat-3854	208	6	61	61	NUM
easat-3854	208	7	,	,	PUNCT
easat-3854	208	8	no	no	INTJ
easat-3854	208	9	.	.	NOUN
easat-3854	208	10	1	1	NUM
easat-3854	208	11	,	,	PUNCT
easat-3854	208	12	pp	pp	ADJ
easat-3854	208	13	.	.	PUNCT
easat-3854	209	1	23	23	NUM
easat-3854	209	2	-	-	SYM
easat-3854	209	3	47	47	NUM
easat-3854	209	4	1988	1988	NUM
easat-3854	209	5	.	.	PUNCT
easat-3854	210	1	doi	doi	NOUN
easat-3854	210	2	:	:	PUNCT
easat-3854	210	3	10.1070	10.1070	NUM
easat-3854	210	4	/	/	SYM
easat-3854	210	5	sm1988v061n01abeh003190	sm1988v061n01abeh003190	NOUN
easat-3854	210	6	[	[	X
easat-3854	210	7	10	10	NUM
easat-3854	210	8	]	]	X
easat-3854	210	9	w.	w.	PROPN
easat-3854	210	10	m.	m.	PROPN
easat-3854	210	11	mahmoud	mahmoud	PROPN
easat-3854	210	12	,	,	PUNCT
easat-3854	210	13	a.	a.	PROPN
easat-3854	210	14	h.	h.	PROPN
easat-3854	210	15	noreldeen	noreldeen	PROPN
easat-3854	210	16	,	,	PUNCT
easat-3854	210	17	o.	o.	PROPN
easat-3854	210	18	h.	h.	PROPN
easat-3854	210	19	fathy	fathy	PROPN
easat-3854	210	20	and	and	CCONJ
easat-3854	210	21	s.	s.	PROPN
easat-3854	210	22	a.	a.	PROPN
easat-3854	210	23	a.	a.	PROPN
easat-3854	210	24	quota	quota	PROPN
easat-3854	210	25	,	,	PUNCT
easat-3854	210	26	"	"	PUNCT
easat-3854	210	27	on	on	ADP
easat-3854	210	28	the	the	DET
easat-3854	210	29	cyclic	cyclic	ADJ
easat-3854	210	30	homology	homology	NOUN
easat-3854	210	31	theory	theory	NOUN
easat-3854	210	32	of	of	ADP
easat-3854	210	33	algebras	algebra	NOUN
easat-3854	210	34	,	,	PUNCT
easat-3854	210	35	”	"	PUNCT
easat-3854	210	36	scientific	scientific	ADJ
easat-3854	210	37	african	african	ADJ
easat-3854	210	38	,	,	PUNCT
easat-3854	210	39	vol	vol	NOUN
easat-3854	210	40	.	.	PROPN
easat-3854	210	41	25	25	NUM
easat-3854	210	42	,	,	PUNCT
easat-3854	210	43	e02288	e02288	PROPN
easat-3854	210	44	,	,	PUNCT
easat-3854	210	45	september	september	PROPN
easat-3854	210	46	2024	2024	NUM
easat-3854	210	47	.	.	PUNCT
easat-3854	211	1	https://doi.org/10.1016/j.sciaf.2024.e02288	https://doi.org/10.1016/j.sciaf.2024.e02288	ADJ
easat-3854	211	2	.	.	PUNCT
easat-3854	212	1	[	[	X
easat-3854	212	2	11	11	NUM
easat-3854	212	3	]	]	X
easat-3854	212	4	y.	y.	PROPN
easat-3854	212	5	gh	gh	PROPN
easat-3854	212	6	.	.	PUNCT
easat-3854	212	7	gouda	gouda	NOUN
easat-3854	212	8	and	and	CCONJ
easat-3854	212	9	m.	m.	NOUN
easat-3854	212	10	elhamdadi	elhamdadi	NOUN
easat-3854	212	11	,	,	PUNCT
easat-3854	212	12	"	"	PUNCT
easat-3854	212	13	on	on	ADP
easat-3854	212	14	the	the	DET
easat-3854	212	15	steenrod	steenrod	NOUN
easat-3854	212	16	operations	operation	NOUN
easat-3854	212	17	on	on	ADP
easat-3854	212	18	cyclic	cyclic	ADJ
easat-3854	212	19	homology	homology	NOUN
easat-3854	212	20	,	,	PUNCT
easat-3854	212	21	"	"	PUNCT
easat-3854	212	22	int	int	NOUN
easat-3854	212	23	.	.	PUNCT
easat-3854	213	1	jornal	jornal	ADJ
easat-3854	213	2	of	of	ADP
easat-3854	213	3	math	math	NOUN
easat-3854	213	4	.	.	PUNCT
easat-3854	214	1	vol	vol	NOUN
easat-3854	214	2	.	.	PROPN
easat-3854	215	1	72	72	NUM
easat-3854	215	2	,	,	PUNCT
easat-3854	215	3	pp	pp	ADJ
easat-3854	215	4	.	.	PUNCT
easat-3854	216	1	4539	4539	NUM
easat-3854	216	2	-	-	SYM
easat-3854	216	3	4545	4545	NUM
easat-3854	216	4	,	,	PUNCT
easat-3854	216	5	2003	2003	NUM
easat-3854	216	6	.	.	PUNCT
easat-3854	217	1	doi:10.1155	doi:10.1155	PROPN
easat-3854	217	2	/	/	SYM
easat-3854	217	3	s016117120320908x	s016117120320908x	AUX
easat-3854	217	4	https://www.emis.de/journals/hoa/ijmms/volume2003_72/502342.pdf	https://www.emis.de/journals/hoa/ijmms/volume2003_72/502342.pdf	NOUN
easat-3854	217	5	[	[	X
easat-3854	217	6	12	12	NUM
easat-3854	217	7	]	]	X
easat-3854	217	8	y.	y.	NOUN
easat-3854	217	9	gouda	gouda	NOUN
easat-3854	217	10	,	,	PUNCT
easat-3854	217	11	"	"	PUNCT
easat-3854	217	12	adams	adam	NOUN
easat-3854	217	13	and	and	CCONJ
easat-3854	217	14	steenrod	steenrod	NOUN
easat-3854	217	15	operators	operator	NOUN
easat-3854	217	16	in	in	ADP
easat-3854	217	17	dihedral	dihedral	ADJ
easat-3854	217	18	homology	homology	NOUN
easat-3854	217	19	"	"	PUNCT
easat-3854	217	20	,	,	PUNCT
easat-3854	217	21	international	international	ADJ
easat-3854	217	22	journal	journal	NOUN
easat-3854	217	23	of	of	ADP
easat-3854	217	24	mathematics	mathematics	PROPN
easat-3854	217	25	and	and	CCONJ
easat-3854	217	26	mathematical	mathematical	ADJ
easat-3854	217	27	sciences	science	NOUN
easat-3854	217	28	,	,	PUNCT
easat-3854	217	29	vol	vol	NOUN
easat-3854	217	30	.	.	PROPN
easat-3854	218	1	21	21	NUM
easat-3854	218	2	,	,	PUNCT
easat-3854	218	3	no	no	INTJ
easat-3854	218	4	.	.	NOUN
easat-3854	218	5	2	2	NUM
easat-3854	218	6	,	,	PUNCT
easat-3854	218	7	pp	pp	ADJ
easat-3854	218	8	.	.	PUNCT
easat-3854	219	1	341	341	NUM
easat-3854	219	2	-	-	SYM
easat-3854	219	3	346	346	NUM
easat-3854	219	4	,	,	PUNCT
easat-3854	219	5	1998	1998	NUM
easat-3854	219	6	.	.	PUNCT
easat-3854	220	1	https://www.emis.de/journals/hoa/ijmms/volume21_2/857542.pdf	https://www.emis.de/journals/hoa/ijmms/volume21_2/857542.pdf	PROPN
easat-3854	220	2	https://doi.org/10.1007/s10958-008-9070-y	https://doi.org/10.1007/s10958-008-9070-y	VERB
easat-3854	220	3	https://doi.org/10.1016/j.jpaa.2008.07.017	https://doi.org/10.1016/j.jpaa.2008.07.017	PRON
easat-3854	220	4	https://doi.org/10.1007/bf00569450	https://doi.org/10.1007/bf00569450	PRON
easat-3854	220	5	https://doi.org/10.1016/j.sciaf.2019.e00115	https://doi.org/10.1016/j.sciaf.2019.e00115	NOUN
easat-3854	220	6	https://www.ijmsi.org/papers/volume.5.issue.1/c05012331.pdf	https://www.ijmsi.org/papers/volume.5.issue.1/c05012331.pdf	PROPN
easat-3854	220	7	https://doi.org/10.48550/arxiv.1809.07510	https://doi.org/10.48550/arxiv.1809.07510	PROPN
easat-3854	220	8	https://www.hrpub.org/download/20231030/ms2-13433629.pdf	https://www.hrpub.org/download/20231030/ms2-13433629.pdf	PROPN
easat-3854	221	1	https://ui.adsabs.harvard.edu/link_gateway/1988sbmat..61...23k/doi:10.1070/sm1988v061n01abeh003190	https://ui.adsabs.harvard.edu/link_gateway/1988sbmat..61...23k/doi:10.1070/sm1988v061n01abeh003190	PROPN
easat-3854	221	2	https://doi.org/10.1016/j.sciaf.2024.e02288	https://doi.org/10.1016/j.sciaf.2024.e02288	ADJ
easat-3854	221	3	http://dx.doi.org/10.1155/s016117120320908x	http://dx.doi.org/10.1155/s016117120320908x	PUNCT
easat-3854	221	4	https://www.emis.de/journals/hoa/ijmms/volume2003_72/502342.pdf	https://www.emis.de/journals/hoa/ijmms/volume2003_72/502342.pdf	NOUN
easat-3854	221	5	https://www.emis.de/journals/hoa/ijmms/volume21_2/857542.pdf	https://www.emis.de/journals/hoa/ijmms/volume21_2/857542.pdf	PROPN