id	sid	tid	token	lemma	pos
ap-1815	1	1	acta	acta	PROPN
ap-1815	1	2	polytechnica	polytechnica	PROPN
ap-1815	1	3	acta	acta	PROPN
ap-1815	1	4	polytechnica	polytechnica	PROPN
ap-1815	1	5	53(3):308–313	53(3):308–313	NUM
ap-1815	1	6	,	,	PUNCT
ap-1815	1	7	2013	2013	NUM
ap-1815	1	8	©	©	PROPN
ap-1815	1	9	czech	czech	PROPN
ap-1815	1	10	technical	technical	PROPN
ap-1815	1	11	university	university	PROPN
ap-1815	1	12	in	in	ADP
ap-1815	1	13	prague	prague	PROPN
ap-1815	1	14	,	,	PUNCT
ap-1815	1	15	2013	2013	NUM
ap-1815	1	16	available	available	ADJ
ap-1815	1	17	online	online	ADV
ap-1815	1	18	at	at	ADP
ap-1815	1	19	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	ADJ
ap-1815	1	20	inherited	inherit	VERB
ap-1815	1	21	properties	property	NOUN
ap-1815	1	22	of	of	ADP
ap-1815	1	23	effect	effect	NOUN
ap-1815	1	24	algebras	algebra	NOUN
ap-1815	1	25	preserved	preserve	VERB
ap-1815	1	26	by	by	ADP
ap-1815	1	27	isomorphisms	isomorphisms	PROPN
ap-1815	1	28	jan	jan	PROPN
ap-1815	1	29	pasekaa,∗	pasekaa,∗	PROPN
ap-1815	1	30	,	,	PUNCT
ap-1815	1	31	zdenka	zdenka	PROPN
ap-1815	1	32	riečanováb	riečanováb	PROPN
ap-1815	1	33	a	a	DET
ap-1815	1	34	department	department	NOUN
ap-1815	1	35	of	of	ADP
ap-1815	1	36	mathematics	mathematic	NOUN
ap-1815	1	37	and	and	CCONJ
ap-1815	1	38	statistics	statistic	NOUN
ap-1815	1	39	,	,	PUNCT
ap-1815	1	40	faculty	faculty	NOUN
ap-1815	1	41	of	of	ADP
ap-1815	1	42	science	science	NOUN
ap-1815	1	43	,	,	PUNCT
ap-1815	1	44	masaryk	masaryk	PROPN
ap-1815	1	45	university	university	NOUN
ap-1815	1	46	,	,	PUNCT
ap-1815	1	47	kotlářská	kotlářská	NOUN
ap-1815	1	48	2	2	NUM
ap-1815	1	49	,	,	PUNCT
ap-1815	1	50	cz-611	cz-611	VERB
ap-1815	1	51	37	37	NUM
ap-1815	1	52	brno	brno	NOUN
ap-1815	1	53	,	,	PUNCT
ap-1815	1	54	czech	czech	PROPN
ap-1815	1	55	republic	republic	PROPN
ap-1815	1	56	b	b	PROPN
ap-1815	1	57	department	department	PROPN
ap-1815	1	58	of	of	ADP
ap-1815	1	59	mathematics	mathematic	NOUN
ap-1815	1	60	,	,	PUNCT
ap-1815	1	61	faculty	faculty	NOUN
ap-1815	1	62	of	of	ADP
ap-1815	1	63	electrical	electrical	ADJ
ap-1815	1	64	engineering	engineering	NOUN
ap-1815	1	65	and	and	CCONJ
ap-1815	1	66	information	information	NOUN
ap-1815	1	67	technology	technology	NOUN
ap-1815	1	68	,	,	PUNCT
ap-1815	1	69	slovak	slovak	ADJ
ap-1815	1	70	university	university	NOUN
ap-1815	1	71	of	of	ADP
ap-1815	1	72	technology	technology	NOUN
ap-1815	1	73	,	,	PUNCT
ap-1815	1	74	ilkovičova	ilkovičova	VERB
ap-1815	1	75	3	3	NUM
ap-1815	1	76	,	,	PUNCT
ap-1815	1	77	sk-812	sk-812	PROPN
ap-1815	1	78	19	19	NUM
ap-1815	1	79	bratislava	bratislava	NOUN
ap-1815	1	80	,	,	PUNCT
ap-1815	1	81	slovak	slovak	ADJ
ap-1815	1	82	republic	republic	NOUN
ap-1815	1	83	∗	∗	NOUN
ap-1815	1	84	corresponding	correspond	VERB
ap-1815	1	85	author	author	NOUN
ap-1815	1	86	:	:	PUNCT
ap-1815	1	87	paseka@math.muni.cz	paseka@math.muni.cz	NOUN
ap-1815	1	88	abstract	abstract	ADJ
ap-1815	1	89	.	.	PUNCT
ap-1815	2	1	we	we	PRON
ap-1815	2	2	show	show	VERB
ap-1815	2	3	that	that	SCONJ
ap-1815	2	4	isomorphism	isomorphism	NOUN
ap-1815	2	5	of	of	ADP
ap-1815	2	6	effect	effect	NOUN
ap-1815	2	7	algebras	algebra	NOUN
ap-1815	2	8	preserves	preserve	VERB
ap-1815	2	9	properties	property	NOUN
ap-1815	2	10	of	of	ADP
ap-1815	2	11	effect	effect	NOUN
ap-1815	2	12	algebras	algebra	NOUN
ap-1815	2	13	derived	derive	VERB
ap-1815	2	14	from	from	ADP
ap-1815	2	15	effect	effect	NOUN
ap-1815	2	16	algebraic	algebraic	ADJ
ap-1815	2	17	sum	sum	NOUN
ap-1815	2	18	⊕	⊕	PROPN
ap-1815	2	19	of	of	ADP
ap-1815	2	20	elements	element	NOUN
ap-1815	2	21	.	.	PUNCT
ap-1815	3	1	these	these	PRON
ap-1815	3	2	are	be	AUX
ap-1815	3	3	partial	partial	ADJ
ap-1815	3	4	order	order	NOUN
ap-1815	3	5	,	,	PUNCT
ap-1815	3	6	order	order	NOUN
ap-1815	3	7	convergence	convergence	NOUN
ap-1815	3	8	,	,	PUNCT
ap-1815	3	9	order	order	NOUN
ap-1815	3	10	topology	topology	NOUN
ap-1815	3	11	,	,	PUNCT
ap-1815	3	12	existence	existence	NOUN
ap-1815	3	13	of	of	ADP
ap-1815	3	14	states	state	NOUN
ap-1815	3	15	and	and	CCONJ
ap-1815	3	16	other	other	ADJ
ap-1815	3	17	important	important	ADJ
ap-1815	3	18	properties	property	NOUN
ap-1815	3	19	.	.	PUNCT
ap-1815	4	1	however	however	ADV
ap-1815	4	2	,	,	PUNCT
ap-1815	4	3	there	there	PRON
ap-1815	4	4	are	be	VERB
ap-1815	4	5	properties	property	NOUN
ap-1815	4	6	of	of	ADP
ap-1815	4	7	effect	effect	NOUN
ap-1815	4	8	algebras	algebra	NOUN
ap-1815	4	9	for	for	ADP
ap-1815	4	10	which	which	PRON
ap-1815	4	11	the	the	DET
ap-1815	4	12	preservation	preservation	NOUN
ap-1815	4	13	of	of	ADP
ap-1815	4	14	the	the	DET
ap-1815	4	15	⊕-operation	⊕-operation	NOUN
ap-1815	4	16	is	be	AUX
ap-1815	4	17	not	not	PART
ap-1815	4	18	substantial	substantial	ADJ
ap-1815	4	19	and	and	CCONJ
ap-1815	4	20	they	they	PRON
ap-1815	4	21	need	need	AUX
ap-1815	4	22	not	not	PART
ap-1815	4	23	be	be	AUX
ap-1815	4	24	preserved	preserve	VERB
ap-1815	4	25	.	.	PUNCT
ap-1815	5	1	keywords	keyword	NOUN
ap-1815	5	2	:	:	PUNCT
ap-1815	5	3	effect	effect	NOUN
ap-1815	5	4	algebras	algebra	NOUN
ap-1815	5	5	;	;	PUNCT
ap-1815	5	6	operator	operator	NOUN
ap-1815	5	7	effect	effect	NOUN
ap-1815	5	8	algebras	algebra	NOUN
ap-1815	5	9	;	;	PUNCT
ap-1815	5	10	isomorphisms	isomorphism	NOUN
ap-1815	5	11	;	;	PUNCT
ap-1815	5	12	operator	operator	NOUN
ap-1815	5	13	representations	representation	NOUN
ap-1815	5	14	of	of	ADP
ap-1815	5	15	effect	effect	NOUN
ap-1815	5	16	algebras	algebra	NOUN
ap-1815	5	17	..	..	PUNCT
ap-1815	5	18	ams	am	NOUN
ap-1815	5	19	mathematics	mathematics	PROPN
ap-1815	5	20	subject	subject	ADJ
ap-1815	5	21	classification	classification	NOUN
ap-1815	5	22	:	:	PUNCT
ap-1815	5	23	06c15	06c15	NOUN
ap-1815	5	24	,	,	PUNCT
ap-1815	5	25	(	(	PUNCT
ap-1815	5	26	03g12	03g12	NUM
ap-1815	5	27	,	,	PUNCT
ap-1815	5	28	81p10	81p10	NUM
ap-1815	5	29	)	)	PUNCT
ap-1815	5	30	.	.	PUNCT
ap-1815	6	1	1	1	X
ap-1815	6	2	.	.	X
ap-1815	6	3	introduction	introduction	NOUN
ap-1815	6	4	in	in	ADP
ap-1815	6	5	the	the	DET
ap-1815	6	6	quantum	quantum	ADJ
ap-1815	6	7	-	-	ADJ
ap-1815	6	8	mechanical	mechanical	ADJ
ap-1815	6	9	framework	framework	NOUN
ap-1815	6	10	,	,	PUNCT
ap-1815	6	11	the	the	DET
ap-1815	6	12	elements	element	NOUN
ap-1815	6	13	of	of	ADP
ap-1815	6	14	an	an	DET
ap-1815	6	15	effect	effect	NOUN
ap-1815	6	16	algebra	algebra	NOUN
ap-1815	6	17	represent	represent	VERB
ap-1815	6	18	quantum	quantum	ADJ
ap-1815	6	19	effects	effect	NOUN
ap-1815	6	20	,	,	PUNCT
ap-1815	6	21	meaning	mean	VERB
ap-1815	6	22	elementary	elementary	ADJ
ap-1815	6	23	yes	yes	NOUN
ap-1815	6	24	-	-	PUNCT
ap-1815	6	25	no	no	PRON
ap-1815	6	26	measurements	measurement	NOUN
ap-1815	6	27	that	that	PRON
ap-1815	6	28	may	may	AUX
ap-1815	6	29	be	be	AUX
ap-1815	6	30	unsharp	unsharp	ADJ
ap-1815	6	31	.	.	PUNCT
ap-1815	7	1	the	the	DET
ap-1815	7	2	standard	standard	ADJ
ap-1815	7	3	hilbert	hilbert	PROPN
ap-1815	7	4	space	space	NOUN
ap-1815	7	5	effect	effect	NOUN
ap-1815	7	6	algebra	algebra	VERB
ap-1815	7	7	e(h	e(h	PROPN
ap-1815	7	8	)	)	PUNCT
ap-1815	7	9	on	on	ADP
ap-1815	7	10	a	a	DET
ap-1815	7	11	complex	complex	ADJ
ap-1815	7	12	hilbert	hilbert	NOUN
ap-1815	7	13	space	space	NOUN
ap-1815	7	14	h	h	NOUN
ap-1815	7	15	is	be	AUX
ap-1815	7	16	the	the	DET
ap-1815	7	17	set	set	ADJ
ap-1815	7	18	e(h	e(h	PROPN
ap-1815	7	19	)	)	PUNCT
ap-1815	7	20	of	of	ADP
ap-1815	7	21	all	all	DET
ap-1815	7	22	positive	positive	ADJ
ap-1815	7	23	operators	operator	NOUN
ap-1815	7	24	dominated	dominate	VERB
ap-1815	7	25	by	by	ADP
ap-1815	7	26	the	the	DET
ap-1815	7	27	identity	identity	NOUN
ap-1815	7	28	operator	operator	NOUN
ap-1815	7	29	i	i	PRON
ap-1815	7	30	on	on	ADP
ap-1815	7	31	h.	h.	PROPN
ap-1815	7	32	so	so	ADV
ap-1815	7	33	called	call	VERB
ap-1815	7	34	interval	interval	NOUN
ap-1815	7	35	effect	effect	NOUN
ap-1815	7	36	algebras	algebra	NOUN
ap-1815	7	37	form	form	VERB
ap-1815	7	38	a	a	DET
ap-1815	7	39	further	further	ADJ
ap-1815	7	40	important	important	ADJ
ap-1815	7	41	class	class	NOUN
ap-1815	7	42	of	of	ADP
ap-1815	7	43	effect	effect	NOUN
ap-1815	7	44	algebras	algebra	VERB
ap-1815	7	45	.	.	PUNCT
ap-1815	8	1	these	these	PRON
ap-1815	8	2	are	be	AUX
ap-1815	8	3	effect	effect	NOUN
ap-1815	8	4	algebras	algebra	NOUN
ap-1815	8	5	possessing	possess	VERB
ap-1815	8	6	an	an	DET
ap-1815	8	7	ordering	ordering	NOUN
ap-1815	8	8	set	set	NOUN
ap-1815	8	9	of	of	ADP
ap-1815	8	10	states	state	NOUN
ap-1815	8	11	,	,	PUNCT
ap-1815	8	12	which	which	PRON
ap-1815	8	13	is	be	AUX
ap-1815	8	14	equivalent	equivalent	ADJ
ap-1815	8	15	to	to	ADP
ap-1815	8	16	the	the	DET
ap-1815	8	17	condition	condition	NOUN
ap-1815	8	18	that	that	SCONJ
ap-1815	8	19	these	these	DET
ap-1815	8	20	effect	effect	NOUN
ap-1815	8	21	algebras	algebra	NOUN
ap-1815	8	22	can	can	AUX
ap-1815	8	23	be	be	AUX
ap-1815	8	24	represented	represent	VERB
ap-1815	8	25	by	by	ADP
ap-1815	8	26	positive	positive	ADJ
ap-1815	8	27	linear	linear	PROPN
ap-1815	8	28	operators	operator	NOUN
ap-1815	8	29	densely	densely	ADV
ap-1815	8	30	defined	define	VERB
ap-1815	8	31	in	in	ADP
ap-1815	8	32	an	an	DET
ap-1815	8	33	infinite	infinite	ADJ
ap-1815	8	34	-	-	PUNCT
ap-1815	8	35	dimensional	dimensional	ADJ
ap-1815	8	36	complex	complex	ADJ
ap-1815	8	37	hilbert	hilbert	NOUN
ap-1815	8	38	space	space	NOUN
ap-1815	8	39	h	h	NOUN
ap-1815	8	40	(	(	PUNCT
ap-1815	8	41	see	see	VERB
ap-1815	8	42	[	[	X
ap-1815	8	43	18	18	NUM
ap-1815	8	44	]	]	NUM
ap-1815	8	45	)	)	PUNCT
ap-1815	8	46	.	.	PUNCT
ap-1815	9	1	here	here	ADV
ap-1815	9	2	,	,	PUNCT
ap-1815	9	3	by	by	ADP
ap-1815	9	4	the	the	DET
ap-1815	9	5	operator	operator	NOUN
ap-1815	9	6	representation	representation	NOUN
ap-1815	9	7	of	of	ADP
ap-1815	9	8	effect	effect	NOUN
ap-1815	9	9	algebras	algebra	NOUN
ap-1815	9	10	(	(	PUNCT
ap-1815	9	11	initiated	initiate	VERB
ap-1815	9	12	by	by	ADP
ap-1815	9	13	questions	question	NOUN
ap-1815	9	14	of	of	ADP
ap-1815	9	15	m.	m.	NOUN
ap-1815	9	16	znojil	znojil	NOUN
ap-1815	9	17	at	at	ADP
ap-1815	9	18	the	the	DET
ap-1815	9	19	9th	9th	ADJ
ap-1815	9	20	phhqp	phhqp	ADJ
ap-1815	9	21	workshop	workshop	NOUN
ap-1815	9	22	in	in	ADP
ap-1815	9	23	hangzhou	hangzhou	PROPN
ap-1815	9	24	,	,	PUNCT
ap-1815	9	25	china	china	PROPN
ap-1815	9	26	)	)	PUNCT
ap-1815	9	27	we	we	PRON
ap-1815	9	28	mean	mean	VERB
ap-1815	9	29	their	their	PRON
ap-1815	9	30	isomorphism	isomorphism	NOUN
ap-1815	9	31	with	with	ADP
ap-1815	9	32	sub	sub	ADJ
ap-1815	9	33	-	-	ADJ
ap-1815	9	34	effect	effect	ADJ
ap-1815	9	35	algebras	algebra	NOUN
ap-1815	9	36	of	of	ADP
ap-1815	9	37	the	the	DET
ap-1815	9	38	standard	standard	ADJ
ap-1815	9	39	hilbert	hilbert	NOUN
ap-1815	9	40	space	space	NOUN
ap-1815	9	41	effect	effect	NOUN
ap-1815	9	42	algebra	algebra	VERB
ap-1815	9	43	e(h	e(h	PROPN
ap-1815	9	44	)	)	PUNCT
ap-1815	9	45	on	on	ADP
ap-1815	9	46	the	the	DET
ap-1815	9	47	complex	complex	ADJ
ap-1815	9	48	hilbert	hilbert	PROPN
ap-1815	9	49	space	space	NOUN
ap-1815	9	50	h.	h.	PROPN
ap-1815	9	51	in	in	ADP
ap-1815	9	52	this	this	DET
ap-1815	9	53	paper	paper	NOUN
ap-1815	9	54	we	we	PRON
ap-1815	9	55	show	show	VERB
ap-1815	9	56	that	that	SCONJ
ap-1815	9	57	isomorphisms	isomorphism	NOUN
ap-1815	9	58	of	of	ADP
ap-1815	9	59	effect	effect	NOUN
ap-1815	9	60	algebras	algebra	NOUN
ap-1815	9	61	inherit	inherit	VERB
ap-1815	9	62	the	the	DET
ap-1815	9	63	partial	partial	ADJ
ap-1815	9	64	order	order	NOUN
ap-1815	9	65	on	on	ADP
ap-1815	9	66	them	they	PRON
ap-1815	9	67	,	,	PUNCT
ap-1815	9	68	and	and	CCONJ
ap-1815	9	69	consequently	consequently	ADV
ap-1815	9	70	also	also	ADV
ap-1815	9	71	the	the	DET
ap-1815	9	72	order	order	NOUN
ap-1815	9	73	convergence	convergence	NOUN
ap-1815	9	74	on	on	ADP
ap-1815	9	75	them	they	PRON
ap-1815	9	76	and	and	CCONJ
ap-1815	9	77	other	other	ADJ
ap-1815	9	78	important	important	ADJ
ap-1815	9	79	properties	property	NOUN
ap-1815	9	80	.	.	PUNCT
ap-1815	10	1	however	however	ADV
ap-1815	10	2	,	,	PUNCT
ap-1815	10	3	we	we	PRON
ap-1815	10	4	also	also	ADV
ap-1815	10	5	show	show	VERB
ap-1815	10	6	examples	example	NOUN
ap-1815	10	7	of	of	ADP
ap-1815	10	8	properties	property	NOUN
ap-1815	10	9	that	that	PRON
ap-1815	10	10	need	need	AUX
ap-1815	10	11	not	not	PART
ap-1815	10	12	be	be	AUX
ap-1815	10	13	inherited	inherit	VERB
ap-1815	10	14	by	by	ADP
ap-1815	10	15	isomorphisms	isomorphism	NOUN
ap-1815	10	16	of	of	ADP
ap-1815	10	17	effect	effect	NOUN
ap-1815	10	18	algebras	algebra	NOUN
ap-1815	10	19	(	(	PUNCT
ap-1815	10	20	e.g.	e.g.	ADV
ap-1815	10	21	,	,	PUNCT
ap-1815	10	22	sequential	sequential	ADJ
ap-1815	10	23	product	product	NOUN
ap-1815	10	24	of	of	ADP
ap-1815	10	25	elements	element	NOUN
ap-1815	10	26	)	)	PUNCT
ap-1815	10	27	.	.	PUNCT
ap-1815	11	1	2	2	X
ap-1815	11	2	.	.	X
ap-1815	11	3	basic	basic	ADJ
ap-1815	11	4	definitions	definition	NOUN
ap-1815	11	5	and	and	CCONJ
ap-1815	11	6	some	some	DET
ap-1815	11	7	known	know	VERB
ap-1815	11	8	facts	fact	NOUN
ap-1815	11	9	2.1	2.1	NUM
ap-1815	11	10	.	.	PUNCT
ap-1815	12	1	effect	effect	NOUN
ap-1815	12	2	algebras	algebra	NOUN
ap-1815	12	3	and	and	CCONJ
ap-1815	12	4	generalized	generalized	ADJ
ap-1815	12	5	effect	effect	NOUN
ap-1815	12	6	algebras	algebra	NOUN
ap-1815	12	7	definition	definition	NOUN
ap-1815	12	8	2.1	2.1	NUM
ap-1815	12	9	.	.	PUNCT
ap-1815	13	1	[	[	X
ap-1815	13	2	3	3	X
ap-1815	13	3	]	]	PUNCT
ap-1815	13	4	a	a	DET
ap-1815	13	5	partial	partial	ADJ
ap-1815	13	6	algebra	algebra	NOUN
ap-1815	13	7	(	(	PUNCT
ap-1815	13	8	e;⊕	e;⊕	ADJ
ap-1815	13	9	,	,	PUNCT
ap-1815	13	10	0	0	NUM
ap-1815	13	11	,	,	PUNCT
ap-1815	13	12	1	1	NUM
ap-1815	13	13	)	)	PUNCT
ap-1815	13	14	is	be	AUX
ap-1815	13	15	called	call	VERB
ap-1815	13	16	an	an	DET
ap-1815	13	17	effect	effect	NOUN
ap-1815	13	18	algebra	algebra	NOUN
ap-1815	13	19	if	if	SCONJ
ap-1815	13	20	0,1	0,1	NUM
ap-1815	13	21	are	be	AUX
ap-1815	13	22	two	two	NUM
ap-1815	13	23	distinguished	distinguished	ADJ
ap-1815	13	24	elements	element	NOUN
ap-1815	13	25	and	and	CCONJ
ap-1815	13	26	⊕	⊕	PROPN
ap-1815	13	27	is	be	AUX
ap-1815	13	28	a	a	DET
ap-1815	13	29	partially	partially	ADV
ap-1815	13	30	defined	define	VERB
ap-1815	13	31	binary	binary	ADJ
ap-1815	13	32	operation	operation	NOUN
ap-1815	13	33	on	on	ADP
ap-1815	13	34	e	e	PROPN
ap-1815	13	35	which	which	PRON
ap-1815	13	36	satisfies	satisfy	VERB
ap-1815	13	37	the	the	DET
ap-1815	13	38	following	follow	VERB
ap-1815	13	39	conditions	condition	NOUN
ap-1815	13	40	for	for	ADP
ap-1815	13	41	any	any	DET
ap-1815	13	42	x	x	NOUN
ap-1815	13	43	,	,	PUNCT
ap-1815	13	44	y	y	PROPN
ap-1815	13	45	,	,	PUNCT
ap-1815	13	46	z	z	NOUN
ap-1815	13	47	∈	∈	PROPN
ap-1815	14	1	e	e	NOUN
ap-1815	14	2	:	:	PUNCT
ap-1815	14	3	(	(	PUNCT
ap-1815	14	4	e1	e1	NOUN
ap-1815	14	5	)	)	PUNCT
ap-1815	14	6	x⊕	x⊕	PROPN
ap-1815	14	7	y	y	PROPN
ap-1815	14	8	=	=	SYM
ap-1815	14	9	y	y	PROPN
ap-1815	14	10	⊕	⊕	PROPN
ap-1815	14	11	x	x	PUNCT
ap-1815	15	1	if	if	SCONJ
ap-1815	15	2	x⊕	x⊕	PROPN
ap-1815	15	3	y	y	PROPN
ap-1815	15	4	is	be	AUX
ap-1815	15	5	defined	define	VERB
ap-1815	15	6	,	,	PUNCT
ap-1815	15	7	(	(	PUNCT
ap-1815	15	8	e2	e2	PROPN
ap-1815	15	9	)	)	PUNCT
ap-1815	15	10	(	(	PUNCT
ap-1815	15	11	x⊕	x⊕	PROPN
ap-1815	15	12	y)⊕	y)⊕	NOUN
ap-1815	15	13	z	z	NOUN
ap-1815	15	14	=	=	SYM
ap-1815	15	15	x⊕	x⊕	PROPN
ap-1815	15	16	(	(	PUNCT
ap-1815	15	17	y	y	PROPN
ap-1815	15	18	⊕	⊕	PROPN
ap-1815	15	19	z	z	PROPN
ap-1815	15	20	)	)	PUNCT
ap-1815	15	21	if	if	SCONJ
ap-1815	15	22	one	one	NUM
ap-1815	15	23	side	side	NOUN
ap-1815	15	24	is	be	AUX
ap-1815	15	25	defined	define	VERB
ap-1815	15	26	,	,	PUNCT
ap-1815	15	27	(	(	PUNCT
ap-1815	15	28	e3	e3	NOUN
ap-1815	15	29	)	)	PUNCT
ap-1815	15	30	for	for	ADP
ap-1815	15	31	every	every	DET
ap-1815	15	32	x	x	SYM
ap-1815	15	33	∈	∈	PROPN
ap-1815	15	34	e	e	NOUN
ap-1815	15	35	there	there	PRON
ap-1815	15	36	exists	exist	VERB
ap-1815	15	37	a	a	DET
ap-1815	15	38	unique	unique	ADJ
ap-1815	15	39	y	y	PROPN
ap-1815	15	40	∈	∈	PROPN
ap-1815	15	41	e	e	NOUN
ap-1815	15	42	such	such	ADJ
ap-1815	15	43	that	that	SCONJ
ap-1815	15	44	x⊕	x⊕	PROPN
ap-1815	15	45	y	y	PROPN
ap-1815	15	46	=	=	SYM
ap-1815	15	47	1	1	NUM
ap-1815	15	48	(	(	PUNCT
ap-1815	15	49	we	we	PRON
ap-1815	15	50	put	put	VERB
ap-1815	15	51	x′	x′	PROPN
ap-1815	16	1	=	=	SYM
ap-1815	16	2	y	y	PROPN
ap-1815	17	1	and	and	CCONJ
ap-1815	17	2	say	say	VERB
ap-1815	17	3	that	that	SCONJ
ap-1815	17	4	x′	x′	PROPN
ap-1815	17	5	is	be	AUX
ap-1815	17	6	a	a	DET
ap-1815	17	7	supplement	supplement	NOUN
ap-1815	17	8	of	of	ADP
ap-1815	17	9	x	x	NOUN
ap-1815	17	10	)	)	PUNCT
ap-1815	17	11	,	,	PUNCT
ap-1815	17	12	(	(	PUNCT
ap-1815	17	13	e4	e4	PROPN
ap-1815	17	14	)	)	PUNCT
ap-1815	17	15	if	if	SCONJ
ap-1815	17	16	1⊕	1⊕	NUM
ap-1815	17	17	x	x	SYM
ap-1815	17	18	is	be	AUX
ap-1815	17	19	defined	define	VERB
ap-1815	17	20	then	then	ADV
ap-1815	17	21	x	x	X
ap-1815	17	22	=	=	NOUN
ap-1815	17	23	0	0	X
ap-1815	17	24	.	.	PUNCT
ap-1815	18	1	we	we	PRON
ap-1815	18	2	often	often	ADV
ap-1815	18	3	denote	denote	VERB
ap-1815	18	4	the	the	DET
ap-1815	18	5	effect	effect	NOUN
ap-1815	18	6	algebra	algebra	NOUN
ap-1815	18	7	(	(	PUNCT
ap-1815	18	8	e;⊕	e;⊕	ADJ
ap-1815	18	9	,	,	PUNCT
ap-1815	18	10	0	0	NUM
ap-1815	18	11	,	,	PUNCT
ap-1815	18	12	1	1	NUM
ap-1815	18	13	)	)	PUNCT
ap-1815	18	14	briefly	briefly	ADV
ap-1815	18	15	by	by	ADP
ap-1815	18	16	e.	e.	PROPN
ap-1815	18	17	on	on	ADP
ap-1815	18	18	every	every	DET
ap-1815	18	19	effect	effect	NOUN
ap-1815	18	20	algebra	algebra	NOUN
ap-1815	18	21	e	e	NOUN
ap-1815	18	22	the	the	DET
ap-1815	18	23	partial	partial	ADJ
ap-1815	18	24	order	order	NOUN
ap-1815	18	25	≤	≤	NOUN
ap-1815	18	26	,	,	PUNCT
ap-1815	18	27	binary	binary	NOUN
ap-1815	18	28	relation	relation	NOUN
ap-1815	18	29	⊥	⊥	PROPN
ap-1815	18	30	and	and	CCONJ
ap-1815	18	31	partial	partial	ADJ
ap-1815	18	32	binary	binary	ADJ
ap-1815	18	33	operation	operation	NOUN
ap-1815	18	34	can	can	AUX
ap-1815	18	35	be	be	AUX
ap-1815	18	36	introduced	introduce	VERB
ap-1815	18	37	as	as	SCONJ
ap-1815	18	38	follows	follow	VERB
ap-1815	18	39	:	:	PUNCT
ap-1815	18	40	x	x	SYM
ap-1815	18	41	≤	≤	X
ap-1815	18	42	y	y	PROPN
ap-1815	18	43	and	and	CCONJ
ap-1815	18	44	x	x	PROPN
ap-1815	18	45	⊥	⊥	PROPN
ap-1815	18	46	z	z	PROPN
ap-1815	18	47	and	and	CCONJ
ap-1815	18	48	y	y	PROPN
ap-1815	18	49	x	x	PUNCT
ap-1815	19	1	=	=	PUNCT
ap-1815	19	2	z	z	PROPN
ap-1815	19	3	iff	iff	PROPN
ap-1815	19	4	x⊕	x⊕	PROPN
ap-1815	19	5	z	z	PROPN
ap-1815	19	6	is	be	AUX
ap-1815	19	7	defined	define	VERB
ap-1815	19	8	and	and	CCONJ
ap-1815	19	9	x⊕	x⊕	PROPN
ap-1815	19	10	z	z	PROPN
ap-1815	20	1	=	=	PUNCT
ap-1815	20	2	y.	y.	NOUN
ap-1815	20	3	generalizations	generalization	NOUN
ap-1815	20	4	of	of	ADP
ap-1815	20	5	effect	effect	NOUN
ap-1815	20	6	algebras	algebra	NOUN
ap-1815	20	7	(	(	PUNCT
ap-1815	20	8	i.e.	i.e.	X
ap-1815	20	9	without	without	ADP
ap-1815	20	10	a	a	DET
ap-1815	20	11	top	top	ADJ
ap-1815	20	12	element	element	NOUN
ap-1815	20	13	1	1	NUM
ap-1815	20	14	)	)	PUNCT
ap-1815	20	15	have	have	AUX
ap-1815	20	16	been	be	AUX
ap-1815	20	17	introduced	introduce	VERB
ap-1815	20	18	and	and	CCONJ
ap-1815	20	19	studied	study	VERB
ap-1815	20	20	in	in	ADP
ap-1815	20	21	[	[	X
ap-1815	20	22	3	3	NUM
ap-1815	20	23	]	]	PUNCT
ap-1815	20	24	,	,	PUNCT
ap-1815	20	25	[	[	X
ap-1815	20	26	5	5	NUM
ap-1815	20	27	]	]	PUNCT
ap-1815	20	28	,	,	PUNCT
ap-1815	20	29	[	[	X
ap-1815	20	30	6	6	NUM
ap-1815	20	31	]	]	PUNCT
ap-1815	20	32	and	and	CCONJ
ap-1815	20	33	[	[	X
ap-1815	20	34	9	9	NUM
ap-1815	20	35	]	]	PUNCT
ap-1815	20	36	.	.	PUNCT
ap-1815	21	1	definition	definition	NOUN
ap-1815	21	2	2.2	2.2	NUM
ap-1815	21	3	.	.	PUNCT
ap-1815	22	1	(	(	PUNCT
ap-1815	22	2	1	1	NUM
ap-1815	22	3	.	.	PUNCT
ap-1815	22	4	)	)	PUNCT
ap-1815	23	1	a	a	DET
ap-1815	23	2	generalized	generalized	ADJ
ap-1815	23	3	effect	effect	NOUN
ap-1815	23	4	algebra	algebra	NOUN
ap-1815	23	5	(	(	PUNCT
ap-1815	23	6	e	e	NOUN
ap-1815	23	7	,	,	PUNCT
ap-1815	23	8	⊕	⊕	PROPN
ap-1815	23	9	,	,	PUNCT
ap-1815	23	10	0	0	NUM
ap-1815	23	11	)	)	PUNCT
ap-1815	23	12	is	be	AUX
ap-1815	23	13	a	a	DET
ap-1815	23	14	set	set	NOUN
ap-1815	23	15	e	e	NOUN
ap-1815	23	16	with	with	ADP
ap-1815	23	17	an	an	DET
ap-1815	23	18	element	element	NOUN
ap-1815	23	19	0	0	NUM
ap-1815	23	20	∈	∈	PROPN
ap-1815	23	21	e	e	NOUN
ap-1815	23	22	and	and	CCONJ
ap-1815	23	23	a	a	DET
ap-1815	23	24	partial	partial	ADJ
ap-1815	23	25	binary	binary	NOUN
ap-1815	23	26	operation	operation	NOUN
ap-1815	23	27	⊕	⊕	PROPN
ap-1815	23	28	satisfying	satisfy	VERB
ap-1815	23	29	for	for	ADP
ap-1815	23	30	any	any	DET
ap-1815	23	31	x	x	NOUN
ap-1815	23	32	,	,	PUNCT
ap-1815	23	33	y	y	PROPN
ap-1815	23	34	,	,	PUNCT
ap-1815	23	35	z	z	NOUN
ap-1815	23	36	∈	∈	PROPN
ap-1815	23	37	e	e	NOUN
ap-1815	23	38	the	the	DET
ap-1815	23	39	conditions	condition	NOUN
ap-1815	23	40	(	(	PUNCT
ap-1815	23	41	ge1	ge1	NOUN
ap-1815	23	42	)	)	PUNCT
ap-1815	23	43	x⊕	x⊕	PROPN
ap-1815	24	1	y	y	PROPN
ap-1815	24	2	=	=	SYM
ap-1815	24	3	y	y	PROPN
ap-1815	24	4	⊕	⊕	PROPN
ap-1815	24	5	x	x	PUNCT
ap-1815	25	1	if	if	SCONJ
ap-1815	25	2	one	one	NUM
ap-1815	25	3	side	side	NOUN
ap-1815	25	4	is	be	AUX
ap-1815	25	5	defined	define	VERB
ap-1815	25	6	,	,	PUNCT
ap-1815	25	7	(	(	PUNCT
ap-1815	25	8	ge2	ge2	PROPN
ap-1815	25	9	)	)	PUNCT
ap-1815	25	10	(	(	PUNCT
ap-1815	25	11	x	x	PROPN
ap-1815	25	12	⊕	⊕	PROPN
ap-1815	25	13	y	y	PROPN
ap-1815	25	14	)	)	PUNCT
ap-1815	25	15	⊕	⊕	PROPN
ap-1815	25	16	z	z	PUNCT
ap-1815	26	1	=	=	PUNCT
ap-1815	26	2	x	x	SYM
ap-1815	26	3	⊕	⊕	PROPN
ap-1815	26	4	(	(	PUNCT
ap-1815	26	5	y	y	PROPN
ap-1815	26	6	⊕	⊕	PROPN
ap-1815	26	7	z	z	PROPN
ap-1815	26	8	)	)	PUNCT
ap-1815	26	9	if	if	SCONJ
ap-1815	26	10	one	one	NUM
ap-1815	26	11	side	side	NOUN
ap-1815	26	12	is	be	AUX
ap-1815	26	13	defined	define	VERB
ap-1815	26	14	,	,	PUNCT
ap-1815	26	15	(	(	PUNCT
ap-1815	26	16	ge3	ge3	NOUN
ap-1815	26	17	)	)	PUNCT
ap-1815	26	18	if	if	SCONJ
ap-1815	26	19	x⊕	x⊕	PROPN
ap-1815	26	20	y	y	PROPN
ap-1815	26	21	=	=	SYM
ap-1815	26	22	x⊕	x⊕	PROPN
ap-1815	26	23	z	z	PROPN
ap-1815	26	24	then	then	ADV
ap-1815	26	25	y	y	PROPN
ap-1815	26	26	=	=	SYM
ap-1815	26	27	z	z	PROPN
ap-1815	26	28	,	,	PUNCT
ap-1815	26	29	(	(	PUNCT
ap-1815	26	30	ge4	ge4	PROPN
ap-1815	26	31	)	)	PUNCT
ap-1815	26	32	if	if	SCONJ
ap-1815	26	33	x⊕	x⊕	PROPN
ap-1815	26	34	y	y	PROPN
ap-1815	26	35	=	=	PUNCT
ap-1815	26	36	0	0	PUNCT
ap-1815	27	1	then	then	ADV
ap-1815	27	2	x	x	X
ap-1815	27	3	=	=	SYM
ap-1815	27	4	y	y	PROPN
ap-1815	27	5	=	=	SYM
ap-1815	27	6	0	0	PROPN
ap-1815	27	7	,	,	PUNCT
ap-1815	27	8	(	(	PUNCT
ap-1815	27	9	ge5	ge5	PROPN
ap-1815	27	10	)	)	PUNCT
ap-1815	27	11	x⊕	x⊕	PROPN
ap-1815	27	12	0	0	PUNCT
ap-1815	28	1	=	=	PUNCT
ap-1815	28	2	x	x	PROPN
ap-1815	28	3	for	for	ADP
ap-1815	28	4	all	all	DET
ap-1815	28	5	x	x	SYM
ap-1815	28	6	∈	∈	PROPN
ap-1815	28	7	e.	e.	PROPN
ap-1815	28	8	(	(	PUNCT
ap-1815	28	9	2	2	NUM
ap-1815	28	10	.	.	PUNCT
ap-1815	28	11	)	)	PUNCT
ap-1815	28	12	define	define	VERB
ap-1815	28	13	a	a	DET
ap-1815	28	14	binary	binary	ADJ
ap-1815	28	15	relation	relation	NOUN
ap-1815	28	16	≤	≤	NUM
ap-1815	28	17	on	on	ADP
ap-1815	28	18	e	e	X
ap-1815	28	19	by	by	ADP
ap-1815	28	20	x	x	PROPN
ap-1815	28	21	≤	≤	PROPN
ap-1815	28	22	y	y	PROPN
ap-1815	28	23	iff	iff	PROPN
ap-1815	28	24	for	for	ADP
ap-1815	28	25	some	some	DET
ap-1815	28	26	z	z	NOUN
ap-1815	28	27	∈	∈	PROPN
ap-1815	28	28	e	e	PROPN
ap-1815	28	29	,	,	PUNCT
ap-1815	28	30	x⊕	x⊕	PROPN
ap-1815	28	31	z	z	PUNCT
ap-1815	29	1	=	=	PUNCT
ap-1815	29	2	y.	y.	NOUN
ap-1815	29	3	the	the	DET
ap-1815	29	4	significant	significant	ADJ
ap-1815	29	5	property	property	NOUN
ap-1815	29	6	of	of	ADP
ap-1815	29	7	a	a	DET
ap-1815	29	8	generalized	generalized	ADJ
ap-1815	29	9	effect	effect	NOUN
ap-1815	29	10	algebra	algebra	NOUN
ap-1815	29	11	(	(	PUNCT
ap-1815	29	12	e;⊕	e;⊕	ADJ
ap-1815	29	13	,	,	PUNCT
ap-1815	29	14	0	0	NUM
ap-1815	29	15	)	)	PUNCT
ap-1815	29	16	is	be	AUX
ap-1815	29	17	that	that	SCONJ
ap-1815	29	18	every	every	DET
ap-1815	29	19	interval	interval	NOUN
ap-1815	29	20	[	[	X
ap-1815	29	21	0	0	NUM
ap-1815	29	22	,	,	PUNCT
ap-1815	29	23	q	q	NOUN
ap-1815	29	24	]	]	X
ap-1815	29	25	,	,	PUNCT
ap-1815	29	26	for	for	ADP
ap-1815	29	27	q	q	PROPN
ap-1815	29	28	∈	∈	PROPN
ap-1815	29	29	e	e	NOUN
ap-1815	29	30	,	,	PUNCT
ap-1815	29	31	q	q	X
ap-1815	29	32	6=	6=	NUM
ap-1815	29	33	0	0	NUM
ap-1815	29	34	,	,	PUNCT
ap-1815	29	35	is	be	AUX
ap-1815	29	36	an	an	DET
ap-1815	29	37	effect	effect	NOUN
ap-1815	29	38	algebra	algebra	NOUN
ap-1815	29	39	with	with	ADP
ap-1815	29	40	⊕	⊕	PROPN
ap-1815	29	41	restricted	restrict	VERB
ap-1815	29	42	to	to	ADP
ap-1815	29	43	[	[	X
ap-1815	29	44	0	0	NUM
ap-1815	29	45	,	,	PUNCT
ap-1815	29	46	q	q	NOUN
ap-1815	29	47	]	]	X
ap-1815	29	48	.	.	PUNCT
ap-1815	30	1	308	308	NUM
ap-1815	30	2	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	ADJ
ap-1815	30	3	vol	vol	NOUN
ap-1815	30	4	.	.	PUNCT
ap-1815	31	1	53	53	NUM
ap-1815	31	2	no	no	NOUN
ap-1815	31	3	.	.	PUNCT
ap-1815	32	1	3/2013	3/2013	PROPN
ap-1815	32	2	inherited	inherit	VERB
ap-1815	32	3	properties	property	NOUN
ap-1815	32	4	of	of	ADP
ap-1815	32	5	effect	effect	NOUN
ap-1815	32	6	algebras	algebra	VERB
ap-1815	32	7	every	every	DET
ap-1815	32	8	effect	effect	NOUN
ap-1815	32	9	algebra	algebra	NOUN
ap-1815	32	10	e	e	NOUN
ap-1815	32	11	is	be	AUX
ap-1815	32	12	also	also	ADV
ap-1815	32	13	a	a	DET
ap-1815	32	14	generalized	generalized	ADJ
ap-1815	32	15	effect	effect	NOUN
ap-1815	32	16	algebra	algebra	NOUN
ap-1815	32	17	and	and	CCONJ
ap-1815	32	18	a	a	DET
ap-1815	32	19	generalized	generalized	ADJ
ap-1815	32	20	effect	effect	NOUN
ap-1815	32	21	algebra	algebra	NOUN
ap-1815	32	22	is	be	AUX
ap-1815	32	23	also	also	ADV
ap-1815	32	24	an	an	DET
ap-1815	32	25	effect	effect	NOUN
ap-1815	32	26	algebra	algebra	NOUN
ap-1815	32	27	iff	iff	VERB
ap-1815	32	28	it	it	PRON
ap-1815	32	29	includes	include	VERB
ap-1815	32	30	the	the	DET
ap-1815	32	31	top	top	ADJ
ap-1815	32	32	element	element	NOUN
ap-1815	32	33	.	.	PUNCT
ap-1815	33	1	definition	definition	NOUN
ap-1815	33	2	2.3	2.3	NUM
ap-1815	33	3	.	.	PUNCT
ap-1815	34	1	a	a	DET
ap-1815	34	2	nonempty	nonempty	NOUN
ap-1815	34	3	subset	subset	VERB
ap-1815	34	4	q	q	NOUN
ap-1815	34	5	of	of	ADP
ap-1815	34	6	an	an	DET
ap-1815	34	7	effect	effect	NOUN
ap-1815	34	8	algebra	algebra	NOUN
ap-1815	34	9	(	(	PUNCT
ap-1815	34	10	generalized	generalized	ADJ
ap-1815	34	11	effect	effect	NOUN
ap-1815	34	12	algebra	algebra	NOUN
ap-1815	34	13	)	)	PUNCT
ap-1815	34	14	e	e	NOUN
ap-1815	34	15	is	be	AUX
ap-1815	34	16	called	call	VERB
ap-1815	34	17	a	a	DET
ap-1815	34	18	subeffect	subeffect	NOUN
ap-1815	34	19	algebra	algebra	NOUN
ap-1815	34	20	(	(	PUNCT
ap-1815	34	21	sub	sub	ADJ
ap-1815	34	22	-	-	ADJ
ap-1815	34	23	generalized	generalized	ADJ
ap-1815	34	24	effect	effect	NOUN
ap-1815	34	25	algebra	algebra	NOUN
ap-1815	34	26	)	)	PUNCT
ap-1815	34	27	of	of	ADP
ap-1815	34	28	e	e	PROPN
ap-1815	34	29	iff	iff	PROPN
ap-1815	34	30	:	:	PUNCT
ap-1815	34	31	(	(	PUNCT
ap-1815	34	32	1	1	NUM
ap-1815	34	33	.	.	PUNCT
ap-1815	34	34	)	)	PUNCT
ap-1815	35	1	if	if	SCONJ
ap-1815	35	2	at	at	ADV
ap-1815	35	3	least	least	ADV
ap-1815	35	4	two	two	NUM
ap-1815	35	5	of	of	ADP
ap-1815	35	6	the	the	DET
ap-1815	35	7	elements	element	NOUN
ap-1815	35	8	x	x	NOUN
ap-1815	35	9	,	,	PUNCT
ap-1815	35	10	y	y	PROPN
ap-1815	35	11	,	,	PUNCT
ap-1815	35	12	z	z	NOUN
ap-1815	35	13	∈	∈	PROPN
ap-1815	35	14	e	e	X
ap-1815	35	15	with	with	ADP
ap-1815	35	16	x⊕	x⊕	PROPN
ap-1815	35	17	y	y	PROPN
ap-1815	35	18	=	=	PUNCT
ap-1815	35	19	z	z	NOUN
ap-1815	35	20	are	be	AUX
ap-1815	35	21	in	in	ADP
ap-1815	35	22	q	q	PROPN
ap-1815	35	23	then	then	ADV
ap-1815	35	24	all	all	DET
ap-1815	35	25	x	x	NOUN
ap-1815	35	26	,	,	PUNCT
ap-1815	35	27	y	y	PROPN
ap-1815	35	28	,	,	PUNCT
ap-1815	35	29	z	z	PROPN
ap-1815	35	30	are	be	AUX
ap-1815	35	31	in	in	ADP
ap-1815	35	32	q	q	NOUN
ap-1815	35	33	;	;	PUNCT
ap-1815	35	34	(	(	PUNCT
ap-1815	35	35	2	2	NUM
ap-1815	35	36	.	.	PUNCT
ap-1815	35	37	)	)	PUNCT
ap-1815	36	1	1	1	NUM
ap-1815	36	2	∈	∈	NOUN
ap-1815	36	3	q	q	NOUN
ap-1815	36	4	when	when	SCONJ
ap-1815	36	5	e	e	NOUN
ap-1815	36	6	is	be	AUX
ap-1815	36	7	an	an	DET
ap-1815	36	8	effect	effect	NOUN
ap-1815	36	9	algebra	algebra	NOUN
ap-1815	36	10	.	.	PUNCT
ap-1815	37	1	we	we	PRON
ap-1815	37	2	say	say	VERB
ap-1815	37	3	that	that	SCONJ
ap-1815	37	4	a	a	DET
ap-1815	37	5	finite	finite	ADJ
ap-1815	37	6	system	system	NOUN
ap-1815	37	7	f	f	PROPN
ap-1815	37	8	=	=	SYM
ap-1815	37	9	(	(	PUNCT
ap-1815	37	10	xk)nk=1	xk)nk=1	NOUN
ap-1815	37	11	of	of	ADP
ap-1815	37	12	not	not	PART
ap-1815	37	13	necessarily	necessarily	ADV
ap-1815	37	14	different	different	ADJ
ap-1815	37	15	elements	element	NOUN
ap-1815	37	16	of	of	ADP
ap-1815	37	17	an	an	DET
ap-1815	37	18	effect	effect	NOUN
ap-1815	37	19	algebra	algebra	NOUN
ap-1815	37	20	(	(	PUNCT
ap-1815	37	21	e;⊕	e;⊕	ADJ
ap-1815	37	22	,	,	PUNCT
ap-1815	37	23	0	0	NUM
ap-1815	37	24	,	,	PUNCT
ap-1815	37	25	1	1	NUM
ap-1815	37	26	)	)	PUNCT
ap-1815	37	27	is	be	AUX
ap-1815	37	28	orthogonal	orthogonal	ADJ
ap-1815	38	1	if	if	SCONJ
ap-1815	38	2	x1	x1	PROPN
ap-1815	38	3	⊕	⊕	PROPN
ap-1815	38	4	x2	x2	PROPN
ap-1815	38	5	⊕	⊕	PROPN
ap-1815	38	6	·	·	PUNCT
ap-1815	38	7	·	·	PUNCT
ap-1815	38	8	·	·	PUNCT
ap-1815	38	9	⊕	⊕	PROPN
ap-1815	38	10	xn	xn	PROPN
ap-1815	38	11	(	(	PUNCT
ap-1815	38	12	written	write	VERB
ap-1815	38	13	n⊕	n⊕	PROPN
ap-1815	38	14	k=1	k=1	PUNCT
ap-1815	38	15	xk	xk	PROPN
ap-1815	38	16	or	or	CCONJ
ap-1815	38	17	⊕	⊕	PROPN
ap-1815	38	18	f	f	PROPN
ap-1815	38	19	)	)	PUNCT
ap-1815	38	20	exists	exist	VERB
ap-1815	38	21	in	in	ADP
ap-1815	38	22	e.	e.	PROPN
ap-1815	38	23	here	here	ADV
ap-1815	38	24	we	we	PRON
ap-1815	38	25	define	define	VERB
ap-1815	38	26	x1⊕x2⊕	x1⊕x2⊕	PROPN
ap-1815	38	27	·	·	PUNCT
ap-1815	38	28	·	·	PUNCT
ap-1815	38	29	·	·	PUNCT
ap-1815	38	30	⊕xn	⊕xn	PUNCT
ap-1815	38	31	=	=	SYM
ap-1815	38	32	(	(	PUNCT
ap-1815	38	33	x1⊕x2⊕	x1⊕x2⊕	PROPN
ap-1815	38	34	·	·	PUNCT
ap-1815	38	35	·	·	PUNCT
ap-1815	38	36	·	·	PUNCT
ap-1815	38	37	⊕xn−1)⊕xn	⊕xn−1)⊕xn	NOUN
ap-1815	39	1	supposing	suppose	VERB
ap-1815	39	2	that	that	SCONJ
ap-1815	39	3	n−1⊕	n−1⊕	ADJ
ap-1815	39	4	k=1	k=1	PROPN
ap-1815	39	5	xk	xk	PROPN
ap-1815	39	6	is	be	AUX
ap-1815	39	7	defined	define	VERB
ap-1815	39	8	and	and	CCONJ
ap-1815	39	9	n−1⊕	n−1⊕	ADJ
ap-1815	39	10	k=1	k=1	PROPN
ap-1815	39	11	xk	xk	PROPN
ap-1815	39	12	≤	≤	PROPN
ap-1815	39	13	x′n	x′n	PROPN
ap-1815	39	14	.	.	PUNCT
ap-1815	40	1	we	we	PRON
ap-1815	40	2	also	also	ADV
ap-1815	40	3	define	define	VERB
ap-1815	40	4	⊕	⊕	PROPN
ap-1815	40	5	∅	∅	NOUN
ap-1815	40	6	=	=	NOUN
ap-1815	40	7	0	0	X
ap-1815	40	8	.	.	PUNCT
ap-1815	41	1	an	an	DET
ap-1815	41	2	arbitrary	arbitrary	ADJ
ap-1815	41	3	system	system	NOUN
ap-1815	41	4	g	g	NOUN
ap-1815	41	5	=	=	SYM
ap-1815	41	6	(	(	PUNCT
ap-1815	41	7	xκ)κ∈h	xκ)κ∈h	NUM
ap-1815	41	8	of	of	ADP
ap-1815	41	9	not	not	PART
ap-1815	41	10	necessarily	necessarily	ADV
ap-1815	41	11	different	different	ADJ
ap-1815	41	12	elements	element	NOUN
ap-1815	41	13	of	of	ADP
ap-1815	41	14	e	e	PROPN
ap-1815	41	15	is	be	AUX
ap-1815	41	16	called	call	VERB
ap-1815	41	17	orthogonal	orthogonal	ADJ
ap-1815	41	18	if	if	SCONJ
ap-1815	41	19	⊕	⊕	PROPN
ap-1815	41	20	k	k	PROPN
ap-1815	41	21	exists	exist	VERB
ap-1815	41	22	for	for	ADP
ap-1815	41	23	every	every	DET
ap-1815	41	24	finite	finite	NOUN
ap-1815	41	25	k	k	PROPN
ap-1815	41	26	⊆	⊆	NUM
ap-1815	41	27	g.	g.	NOUN
ap-1815	41	28	we	we	PRON
ap-1815	41	29	say	say	VERB
ap-1815	41	30	that	that	SCONJ
ap-1815	41	31	for	for	ADP
ap-1815	41	32	an	an	DET
ap-1815	41	33	orthogonal	orthogonal	ADJ
ap-1815	41	34	system	system	NOUN
ap-1815	41	35	g	g	NOUN
ap-1815	41	36	=	=	SYM
ap-1815	41	37	(	(	PUNCT
ap-1815	41	38	xκ)κ∈h	xκ)κ∈h	NUM
ap-1815	41	39	the	the	DET
ap-1815	41	40	element	element	NOUN
ap-1815	41	41	⊕	⊕	PROPN
ap-1815	41	42	g	g	PROPN
ap-1815	41	43	(	(	PUNCT
ap-1815	41	44	more	more	ADV
ap-1815	41	45	precisely	precisely	ADV
ap-1815	41	46	⊕	⊕	NOUN
ap-1815	41	47	e	e	NOUN
ap-1815	41	48	g	g	NOUN
ap-1815	41	49	)	)	PUNCT
ap-1815	41	50	exists	exist	VERB
ap-1815	41	51	iff∨{⊕	iff∨{⊕	NOUN
ap-1815	41	52	k	k	PROPN
ap-1815	42	1	|	|	ADV
ap-1815	42	2	k	k	PROPN
ap-1815	42	3	⊆	⊆	NUM
ap-1815	42	4	g	g	NOUN
ap-1815	42	5	is	be	AUX
ap-1815	42	6	finite	finite	ADJ
ap-1815	42	7	}	}	PUNCT
ap-1815	42	8	exists	exist	VERB
ap-1815	42	9	in	in	ADP
ap-1815	42	10	e	e	NOUN
ap-1815	42	11	,	,	PUNCT
ap-1815	42	12	and	and	CCONJ
ap-1815	42	13	then	then	ADV
ap-1815	42	14	we	we	PRON
ap-1815	42	15	put	put	VERB
ap-1815	42	16	⊕	⊕	PROPN
ap-1815	42	17	g	g	NOUN
ap-1815	42	18	=	=	PUNCT
ap-1815	42	19	∨{⊕	∨{⊕	PROPN
ap-1815	42	20	k	k	PROPN
ap-1815	43	1	|	|	ADV
ap-1815	43	2	k	k	PROPN
ap-1815	43	3	⊆	⊆	NUM
ap-1815	43	4	g	g	NOUN
ap-1815	43	5	is	be	AUX
ap-1815	43	6	finite	finite	ADJ
ap-1815	43	7	}	}	PUNCT
ap-1815	43	8	.	.	PUNCT
ap-1815	44	1	(	(	PUNCT
ap-1815	44	2	here	here	ADV
ap-1815	44	3	we	we	PRON
ap-1815	44	4	write	write	VERB
ap-1815	44	5	g1	g1	PROPN
ap-1815	44	6	⊆	⊆	PRON
ap-1815	44	7	g	g	PROPN
ap-1815	44	8	iff	iff	NOUN
ap-1815	44	9	there	there	PRON
ap-1815	44	10	is	be	VERB
ap-1815	44	11	h1	h1	PROPN
ap-1815	44	12	⊆	⊆	X
ap-1815	44	13	h	h	NOUN
ap-1815	44	14	such	such	ADJ
ap-1815	44	15	that	that	SCONJ
ap-1815	44	16	g1	g1	PROPN
ap-1815	44	17	=	=	SYM
ap-1815	44	18	(	(	PUNCT
ap-1815	44	19	xκ)κ∈h1	xκ)κ∈h1	PROPN
ap-1815	44	20	)	)	PUNCT
ap-1815	44	21	.	.	PUNCT
ap-1815	45	1	2.2	2.2	NUM
ap-1815	45	2	.	.	PUNCT
ap-1815	45	3	topologies	topology	NOUN
ap-1815	45	4	on	on	ADP
ap-1815	45	5	ordered	order	VERB
ap-1815	45	6	sets	set	NOUN
ap-1815	45	7	definition	definition	NOUN
ap-1815	45	8	2.4	2.4	NUM
ap-1815	45	9	.	.	PUNCT
ap-1815	46	1	(	(	PUNCT
ap-1815	46	2	1	1	NUM
ap-1815	46	3	.	.	PUNCT
ap-1815	46	4	)	)	PUNCT
ap-1815	47	1	a	a	DET
ap-1815	47	2	preordered	preordere	VERB
ap-1815	47	3	set	set	NOUN
ap-1815	47	4	(	(	PUNCT
ap-1815	47	5	λ;≤	λ;≤	PROPN
ap-1815	47	6	)	)	PUNCT
ap-1815	47	7	is	be	AUX
ap-1815	47	8	called	call	VERB
ap-1815	47	9	a	a	DET
ap-1815	47	10	directed	direct	VERB
ap-1815	47	11	(	(	PUNCT
ap-1815	47	12	upwards	upwards	ADV
ap-1815	47	13	)	)	PUNCT
ap-1815	47	14	set	set	NOUN
ap-1815	47	15	of	of	ADP
ap-1815	47	16	indices	index	NOUN
ap-1815	47	17	if	if	SCONJ
ap-1815	47	18	the	the	DET
ap-1815	47	19	following	follow	VERB
ap-1815	47	20	conditions	condition	NOUN
ap-1815	47	21	are	be	AUX
ap-1815	47	22	satisfied	satisfied	ADJ
ap-1815	47	23	:	:	PUNCT
ap-1815	47	24	(	(	PUNCT
ap-1815	47	25	a	a	X
ap-1815	47	26	)	)	PUNCT
ap-1815	47	27	α	α	PROPN
ap-1815	47	28	≤	≤	PUNCT
ap-1815	47	29	α	α	X
ap-1815	47	30	,	,	PUNCT
ap-1815	47	31	(	(	PUNCT
ap-1815	47	32	b	b	X
ap-1815	47	33	)	)	PUNCT
ap-1815	47	34	α	α	NOUN
ap-1815	47	35	≤	≤	NOUN
ap-1815	47	36	β	β	X
ap-1815	47	37	,	,	PUNCT
ap-1815	47	38	β	β	X
ap-1815	47	39	≤	≤	ADV
ap-1815	47	40	γ	γ	NOUN
ap-1815	47	41	implies	imply	VERB
ap-1815	47	42	α	α	NOUN
ap-1815	47	43	≤	≤	NUM
ap-1815	47	44	γ	γ	X
ap-1815	47	45	,	,	PUNCT
ap-1815	47	46	(	(	PUNCT
ap-1815	47	47	c	c	NOUN
ap-1815	47	48	)	)	PUNCT
ap-1815	47	49	for	for	ADP
ap-1815	47	50	all	all	DET
ap-1815	47	51	α	α	NOUN
ap-1815	47	52	,	,	PUNCT
ap-1815	47	53	β	β	X
ap-1815	47	54	∈	∈	PROPN
ap-1815	47	55	λ	λ	NOUN
ap-1815	47	56	there	there	PRON
ap-1815	47	57	exists	exist	VERB
ap-1815	47	58	γ	γ	PROPN
ap-1815	47	59	∈	∈	PROPN
ap-1815	47	60	λ	λ	NOUN
ap-1815	47	61	such	such	ADJ
ap-1815	47	62	that	that	SCONJ
ap-1815	47	63	α	α	X
ap-1815	47	64	,	,	PUNCT
ap-1815	47	65	β	β	X
ap-1815	47	66	≤	≤	NUM
ap-1815	47	67	γ	γ	X
ap-1815	47	68	.	.	PUNCT
ap-1815	47	69	a	a	DET
ap-1815	47	70	net	net	NOUN
ap-1815	47	71	(	(	PUNCT
ap-1815	47	72	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	47	73	is	be	AUX
ap-1815	47	74	a	a	DET
ap-1815	47	75	family	family	NOUN
ap-1815	47	76	of	of	ADP
ap-1815	47	77	not	not	PART
ap-1815	47	78	necessary	necessary	ADJ
ap-1815	47	79	different	different	ADJ
ap-1815	47	80	elements	element	NOUN
ap-1815	47	81	which	which	PRON
ap-1815	47	82	have	have	VERB
ap-1815	47	83	indices	index	NOUN
ap-1815	47	84	from	from	ADP
ap-1815	47	85	a	a	DET
ap-1815	47	86	directed	direct	VERB
ap-1815	47	87	set	set	NOUN
ap-1815	47	88	of	of	ADP
ap-1815	47	89	indices	index	NOUN
ap-1815	47	90	λ	λ	PROPN
ap-1815	47	91	.	.	PUNCT
ap-1815	48	1	(	(	PUNCT
ap-1815	48	2	2	2	NUM
ap-1815	48	3	.	.	PUNCT
ap-1815	48	4	)	)	PUNCT
ap-1815	49	1	a	a	DET
ap-1815	49	2	net	net	NOUN
ap-1815	49	3	(	(	PUNCT
ap-1815	49	4	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	49	5	of	of	ADP
ap-1815	49	6	elements	element	NOUN
ap-1815	49	7	of	of	ADP
ap-1815	49	8	a	a	DET
ap-1815	49	9	poset	poset	NOUN
ap-1815	49	10	(	(	PUNCT
ap-1815	49	11	p	p	NOUN
ap-1815	49	12	;	;	PUNCT
ap-1815	49	13	≤	≤	NUM
ap-1815	49	14	)	)	PUNCT
ap-1815	49	15	is	be	AUX
ap-1815	49	16	increasingly	increasingly	ADV
ap-1815	49	17	directed	direct	VERB
ap-1815	49	18	if	if	SCONJ
ap-1815	49	19	aα	aα	NOUN
ap-1815	49	20	≤	≤	NUM
ap-1815	49	21	aβ	aβ	VERB
ap-1815	49	22	for	for	ADP
ap-1815	49	23	all	all	DET
ap-1815	49	24	α	α	NOUN
ap-1815	49	25	,	,	PUNCT
ap-1815	49	26	β	β	X
ap-1815	49	27	∈	∈	NOUN
ap-1815	49	28	λ	λ	NOUN
ap-1815	49	29	such	such	ADJ
ap-1815	49	30	that	that	SCONJ
ap-1815	49	31	α	α	PROPN
ap-1815	49	32	≤	≤	NOUN
ap-1815	49	33	β	β	NOUN
ap-1815	49	34	,	,	PUNCT
ap-1815	49	35	and	and	CCONJ
ap-1815	49	36	then	then	ADV
ap-1815	49	37	we	we	PRON
ap-1815	49	38	write	write	VERB
ap-1815	49	39	aα	aα	NOUN
ap-1815	49	40	↑.	↑.	INTJ
ap-1815	49	41	if	if	SCONJ
ap-1815	49	42	moreover	moreover	ADV
ap-1815	49	43	a	a	DET
ap-1815	49	44	=	=	SYM
ap-1815	49	45	∨	∨	X
ap-1815	49	46	{	{	PUNCT
ap-1815	49	47	aα	aα	NOUN
ap-1815	50	1	|	|	ADV
ap-1815	50	2	α	α	PROPN
ap-1815	50	3	∈	∈	PROPN
ap-1815	50	4	λ	λ	NOUN
ap-1815	50	5	}	}	PUNCT
ap-1815	50	6	we	we	PRON
ap-1815	50	7	write	write	VERB
ap-1815	50	8	aα	aα	PROPN
ap-1815	50	9	↑	↑	PROPN
ap-1815	50	10	a	a	PRON
ap-1815	51	1	and	and	CCONJ
ap-1815	51	2	we	we	PRON
ap-1815	51	3	call	call	VERB
ap-1815	51	4	such	such	DET
ap-1815	51	5	a	a	DET
ap-1815	51	6	net	net	NOUN
ap-1815	51	7	increasing	increase	VERB
ap-1815	51	8	to	to	PART
ap-1815	51	9	a.	a.	VERB
ap-1815	51	10	the	the	DET
ap-1815	51	11	meaning	meaning	NOUN
ap-1815	51	12	of	of	ADP
ap-1815	51	13	aα	aα	NOUN
ap-1815	51	14	↓	↓	NOUN
ap-1815	51	15	and	and	CCONJ
ap-1815	51	16	aα	aα	PROPN
ap-1815	51	17	↓	↓	NOUN
ap-1815	51	18	a	a	PRON
ap-1815	51	19	is	be	AUX
ap-1815	51	20	dual	dual	ADJ
ap-1815	51	21	(	(	PUNCT
ap-1815	51	22	decreasingly	decreasingly	ADV
ap-1815	51	23	directed	direct	VERB
ap-1815	51	24	or	or	CCONJ
ap-1815	51	25	filtered	filter	VERB
ap-1815	51	26	)	)	PUNCT
ap-1815	51	27	.	.	PUNCT
ap-1815	52	1	(	(	PUNCT
ap-1815	52	2	3	3	NUM
ap-1815	52	3	.	.	PUNCT
ap-1815	52	4	)	)	PUNCT
ap-1815	53	1	a	a	DET
ap-1815	53	2	net	net	NOUN
ap-1815	53	3	(	(	PUNCT
ap-1815	53	4	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	53	5	of	of	ADP
ap-1815	53	6	elements	element	NOUN
ap-1815	53	7	of	of	ADP
ap-1815	53	8	a	a	DET
ap-1815	53	9	poset	poset	NOUN
ap-1815	53	10	(	(	PUNCT
ap-1815	53	11	p	p	NOUN
ap-1815	53	12	;	;	PUNCT
ap-1815	53	13	≤	≤	NUM
ap-1815	53	14	)	)	PUNCT
ap-1815	53	15	order	order	NOUN
ap-1815	53	16	converges	converge	NOUN
ap-1815	53	17	(	(	PUNCT
ap-1815	53	18	(	(	PUNCT
ap-1815	53	19	o)-converges	o)-converge	NOUN
ap-1815	53	20	,	,	PUNCT
ap-1815	53	21	for	for	ADP
ap-1815	53	22	short	short	ADJ
ap-1815	53	23	)	)	PUNCT
ap-1815	53	24	to	to	ADP
ap-1815	53	25	a	a	DET
ap-1815	53	26	point	point	NOUN
ap-1815	53	27	a	a	DET
ap-1815	53	28	∈	∈	NOUN
ap-1815	53	29	p	p	NOUN
ap-1815	53	30	if	if	SCONJ
ap-1815	53	31	there	there	PRON
ap-1815	53	32	are	be	VERB
ap-1815	53	33	nets	net	NOUN
ap-1815	53	34	(	(	PUNCT
ap-1815	53	35	uα)α∈λ	uα)α∈λ	NUM
ap-1815	53	36	and	and	CCONJ
ap-1815	53	37	(	(	PUNCT
ap-1815	53	38	vα)α∈λ	vα)α∈λ	NUM
ap-1815	53	39	of	of	ADP
ap-1815	53	40	elements	element	NOUN
ap-1815	53	41	of	of	ADP
ap-1815	53	42	p	p	NOUN
ap-1815	53	43	such	such	ADJ
ap-1815	53	44	that	that	SCONJ
ap-1815	53	45	a	a	DET
ap-1815	53	46	↑	↑	NOUN
ap-1815	53	47	uα	uα	PROPN
ap-1815	53	48	≤	≤	NUM
ap-1815	53	49	aα	aα	NOUN
ap-1815	53	50	≤	≤	NUM
ap-1815	53	51	vα	vα	ADP
ap-1815	53	52	↓	↓	NOUN
ap-1815	53	53	a.	a.	NOUN
ap-1815	53	54	we	we	PRON
ap-1815	53	55	write	write	VERB
ap-1815	53	56	aα	aα	NOUN
ap-1815	53	57	(	(	PUNCT
ap-1815	53	58	o)−→	o)−→	PROPN
ap-1815	53	59	a	a	PRON
ap-1815	53	60	in	in	ADP
ap-1815	53	61	p	p	NOUN
ap-1815	53	62	(	(	PUNCT
ap-1815	53	63	or	or	CCONJ
ap-1815	53	64	briefly	briefly	ADV
ap-1815	53	65	aα	aα	NOUN
ap-1815	53	66	(	(	PUNCT
ap-1815	53	67	o)−→	o)−→	PROPN
ap-1815	53	68	a	a	X
ap-1815	53	69	)	)	PUNCT
ap-1815	53	70	.	.	PUNCT
ap-1815	54	1	definition	definition	NOUN
ap-1815	54	2	2.5	2.5	NUM
ap-1815	54	3	.	.	PUNCT
ap-1815	55	1	the	the	DET
ap-1815	55	2	order	order	NOUN
ap-1815	55	3	topology	topology	NOUN
ap-1815	55	4	(	(	PUNCT
ap-1815	55	5	denoted	denote	VERB
ap-1815	55	6	by	by	ADP
ap-1815	55	7	τp0	τp0	NOUN
ap-1815	55	8	or	or	CCONJ
ap-1815	55	9	shortly	shortly	ADV
ap-1815	55	10	τ0	τ0	NOUN
ap-1815	55	11	)	)	PUNCT
ap-1815	55	12	on	on	ADP
ap-1815	55	13	a	a	DET
ap-1815	55	14	poset	poset	NOUN
ap-1815	55	15	(	(	PUNCT
ap-1815	55	16	p	p	NOUN
ap-1815	55	17	;	;	PUNCT
ap-1815	55	18	≤	≤	NUM
ap-1815	55	19	)	)	PUNCT
ap-1815	55	20	is	be	AUX
ap-1815	55	21	the	the	DET
ap-1815	55	22	finest	fine	ADJ
ap-1815	55	23	(	(	PUNCT
ap-1815	55	24	strongest	strong	ADJ
ap-1815	55	25	)	)	PUNCT
ap-1815	55	26	topology	topology	NOUN
ap-1815	55	27	on	on	ADP
ap-1815	55	28	p	p	PRON
ap-1815	55	29	such	such	ADJ
ap-1815	55	30	that	that	PRON
ap-1815	55	31	for	for	ADP
ap-1815	55	32	every	every	DET
ap-1815	55	33	net	net	NOUN
ap-1815	55	34	(	(	PUNCT
ap-1815	55	35	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	55	36	of	of	ADP
ap-1815	55	37	elements	element	NOUN
ap-1815	55	38	of	of	ADP
ap-1815	55	39	p	p	NOUN
ap-1815	55	40	,	,	PUNCT
ap-1815	55	41	aα	aα	PROPN
ap-1815	55	42	(	(	PUNCT
ap-1815	55	43	o)−→	o)−→	PROPN
ap-1815	55	44	a	a	PRON
ap-1815	55	45	in	in	ADP
ap-1815	55	46	p	p	NOUN
ap-1815	55	47	=	=	NOUN
ap-1815	55	48	⇒	⇒	NOUN
ap-1815	55	49	aα	aα	NOUN
ap-1815	55	50	τp	τp	NOUN
ap-1815	55	51	0−→	0−→	PROPN
ap-1815	55	52	a	a	PRON
ap-1815	55	53	,	,	PUNCT
ap-1815	55	54	where	where	SCONJ
ap-1815	55	55	aα	aα	NOUN
ap-1815	55	56	τp	τp	NOUN
ap-1815	55	57	0−→	0−→	PROPN
ap-1815	55	58	a	a	DET
ap-1815	55	59	denotes	denote	NOUN
ap-1815	55	60	that	that	PRON
ap-1815	55	61	(	(	PUNCT
ap-1815	55	62	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	55	63	converges	converge	VERB
ap-1815	55	64	to	to	ADP
ap-1815	55	65	a	a	DET
ap-1815	55	66	∈	∈	ADJ
ap-1815	55	67	p	p	NOUN
ap-1815	55	68	in	in	ADP
ap-1815	55	69	the	the	DET
ap-1815	55	70	topological	topological	ADJ
ap-1815	55	71	space	space	NOUN
ap-1815	55	72	(	(	PUNCT
ap-1815	55	73	p	p	NOUN
ap-1815	55	74	,	,	PUNCT
ap-1815	55	75	τp0	τp0	NOUN
ap-1815	55	76	)	)	PUNCT
ap-1815	55	77	.	.	PUNCT
ap-1815	56	1	clearly	clearly	ADV
ap-1815	56	2	,	,	PUNCT
ap-1815	56	3	aα	aα	PROPN
ap-1815	56	4	↑	↑	PROPN
ap-1815	56	5	a	a	DET
ap-1815	56	6	⇒	⇒	NOUN
ap-1815	56	7	aα	aα	NOUN
ap-1815	56	8	(	(	PUNCT
ap-1815	56	9	o)−→	o)−→	PROPN
ap-1815	56	10	a	a	DET
ap-1815	56	11	because	because	SCONJ
ap-1815	56	12	a	a	DET
ap-1815	56	13	↑	↑	PROPN
ap-1815	56	14	aα	aα	NOUN
ap-1815	56	15	≤	≤	NUM
ap-1815	56	16	aα	aα	NOUN
ap-1815	56	17	≤	≤	NOUN
ap-1815	56	18	a	a	DET
ap-1815	56	19	↓	↓	NOUN
ap-1815	56	20	a	a	DET
ap-1815	56	21	and	and	CCONJ
ap-1815	56	22	aα	aα	NOUN
ap-1815	56	23	↓	↓	NOUN
ap-1815	56	24	a	a	DET
ap-1815	56	25	⇒	⇒	PROPN
ap-1815	56	26	aα	aα	NOUN
ap-1815	56	27	(	(	PUNCT
ap-1815	56	28	o)−→	o)−→	PROPN
ap-1815	56	29	a	a	PRON
ap-1815	56	30	,	,	PUNCT
ap-1815	56	31	because	because	SCONJ
ap-1815	56	32	a	a	DET
ap-1815	56	33	↑	↑	NOUN
ap-1815	56	34	a	a	DET
ap-1815	56	35	≤	≤	NUM
ap-1815	56	36	aα	aα	NOUN
ap-1815	56	37	≤	≤	NUM
ap-1815	56	38	aα	aα	NOUN
ap-1815	56	39	↓	↓	NOUN
ap-1815	56	40	a	a	PRON
ap-1815	56	41	(	(	PUNCT
ap-1815	56	42	see	see	VERB
ap-1815	56	43	[	[	X
ap-1815	56	44	7	7	NUM
ap-1815	56	45	]	]	PUNCT
ap-1815	56	46	,	,	PUNCT
ap-1815	57	1	[	[	X
ap-1815	57	2	8	8	NUM
ap-1815	57	3	]	]	PUNCT
ap-1815	57	4	,	,	PUNCT
ap-1815	57	5	[	[	X
ap-1815	57	6	12],[13	12],[13	NOUN
ap-1815	57	7	]	]	PUNCT
ap-1815	57	8	)	)	PUNCT
ap-1815	57	9	.	.	PUNCT
ap-1815	58	1	theorem	theorem	VERB
ap-1815	58	2	2.6	2.6	NUM
ap-1815	58	3	(	(	PUNCT
ap-1815	58	4	[	[	X
ap-1815	58	5	11	11	NUM
ap-1815	58	6	,	,	PUNCT
ap-1815	58	7	theorem	theorem	VERB
ap-1815	58	8	2.1.21	2.1.21	NUM
ap-1815	58	9	]	]	X
ap-1815	58	10	)	)	PUNCT
ap-1815	58	11	.	.	PUNCT
ap-1815	59	1	let	let	AUX
ap-1815	59	2	(	(	PUNCT
ap-1815	59	3	p,≤	p,≤	VERB
ap-1815	59	4	)	)	PUNCT
ap-1815	59	5	be	be	VERB
ap-1815	59	6	a	a	DET
ap-1815	59	7	poset	poset	NOUN
ap-1815	59	8	and	and	CCONJ
ap-1815	59	9	f	f	NOUN
ap-1815	59	10	⊆	⊆	NUM
ap-1815	59	11	p	p	NOUN
ap-1815	59	12	.	.	PUNCT
ap-1815	60	1	then	then	ADV
ap-1815	60	2	f	f	PROPN
ap-1815	60	3	is	be	AUX
ap-1815	60	4	τ0	τ0	NOUN
ap-1815	60	5	-	-	PUNCT
ap-1815	60	6	closed	close	VERB
ap-1815	60	7	iff	iff	NOUN
ap-1815	60	8	for	for	ADP
ap-1815	60	9	every	every	DET
ap-1815	60	10	net	net	NOUN
ap-1815	60	11	(	(	PUNCT
ap-1815	60	12	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	60	13	of	of	ADP
ap-1815	60	14	elements	element	NOUN
ap-1815	60	15	of	of	ADP
ap-1815	60	16	p	p	NOUN
ap-1815	60	17	,	,	PUNCT
ap-1815	60	18	(	(	PUNCT
ap-1815	60	19	cs	cs	PROPN
ap-1815	60	20	)	)	PUNCT
ap-1815	60	21	(	(	PUNCT
ap-1815	60	22	aα	aα	NOUN
ap-1815	60	23	∈	∈	PROPN
ap-1815	61	1	f	f	PROPN
ap-1815	61	2	,	,	PUNCT
ap-1815	61	3	α	α	PROPN
ap-1815	61	4	∈	∈	PROPN
ap-1815	61	5	λ	λ	PROPN
ap-1815	61	6	,	,	PUNCT
ap-1815	61	7	aα	aα	PROPN
ap-1815	61	8	(	(	PUNCT
ap-1815	61	9	o)−→	o)−→	PROPN
ap-1815	61	10	a)⇒	a)⇒	PROPN
ap-1815	61	11	x	x	SYM
ap-1815	61	12	∈	∈	PROPN
ap-1815	61	13	f	f	X
ap-1815	61	14	.	.	PUNCT
ap-1815	62	1	2.3	2.3	NUM
ap-1815	62	2	.	.	PUNCT
ap-1815	63	1	morphisms	morphism	NOUN
ap-1815	63	2	,	,	PUNCT
ap-1815	63	3	embeddings	embedding	NOUN
ap-1815	63	4	and	and	CCONJ
ap-1815	63	5	isomorphisms	isomorphism	NOUN
ap-1815	63	6	of	of	ADP
ap-1815	63	7	effect	effect	NOUN
ap-1815	63	8	algebras	algebra	NOUN
ap-1815	63	9	recall	recall	VERB
ap-1815	63	10	the	the	DET
ap-1815	63	11	following	follow	VERB
ap-1815	63	12	definitions	definition	NOUN
ap-1815	63	13	,	,	PUNCT
ap-1815	63	14	needed	need	VERB
ap-1815	63	15	in	in	ADP
ap-1815	63	16	what	what	PRON
ap-1815	63	17	follows	follow	VERB
ap-1815	63	18	.	.	PUNCT
ap-1815	64	1	definition	definition	NOUN
ap-1815	64	2	2.7	2.7	NUM
ap-1815	64	3	(	(	PUNCT
ap-1815	64	4	[	[	X
ap-1815	64	5	2	2	NUM
ap-1815	64	6	,	,	PUNCT
ap-1815	64	7	18	18	NUM
ap-1815	64	8	]	]	PUNCT
ap-1815	64	9	)	)	PUNCT
ap-1815	64	10	.	.	PUNCT
ap-1815	65	1	let	let	AUX
ap-1815	65	2	(	(	PUNCT
ap-1815	65	3	e1;⊕1	e1;⊕1	PROPN
ap-1815	65	4	,	,	PUNCT
ap-1815	65	5	01	01	NUM
ap-1815	65	6	,	,	PUNCT
ap-1815	65	7	11	11	NUM
ap-1815	65	8	)	)	PUNCT
ap-1815	65	9	,	,	PUNCT
ap-1815	65	10	(	(	PUNCT
ap-1815	65	11	e2	e2	PROPN
ap-1815	65	12	;	;	PUNCT
ap-1815	65	13	⊕2	⊕2	PROPN
ap-1815	65	14	,	,	PUNCT
ap-1815	65	15	02	02	NUM
ap-1815	65	16	,	,	PUNCT
ap-1815	65	17	12	12	NUM
ap-1815	65	18	)	)	PUNCT
ap-1815	65	19	be	be	AUX
ap-1815	65	20	effect	effect	NOUN
ap-1815	65	21	algebras	algebra	NOUN
ap-1815	65	22	.	.	PUNCT
ap-1815	66	1	a	a	DET
ap-1815	66	2	mapping	mapping	NOUN
ap-1815	66	3	ϕ	ϕ	NOUN
ap-1815	66	4	:	:	PUNCT
ap-1815	66	5	e1	e1	PROPN
ap-1815	66	6	→	→	SYM
ap-1815	66	7	e2	e2	PROPN
ap-1815	66	8	is	be	AUX
ap-1815	66	9	called	call	VERB
ap-1815	66	10	(	(	PUNCT
ap-1815	66	11	1	1	NUM
ap-1815	66	12	.	.	PUNCT
ap-1815	66	13	)	)	PUNCT
ap-1815	67	1	a	a	DET
ap-1815	67	2	morphism	morphism	NOUN
ap-1815	67	3	,	,	PUNCT
ap-1815	67	4	if	if	SCONJ
ap-1815	67	5	(	(	PUNCT
ap-1815	67	6	a	a	NOUN
ap-1815	67	7	)	)	PUNCT
ap-1815	67	8	ϕ(01	ϕ(01	NOUN
ap-1815	67	9	)	)	PUNCT
ap-1815	67	10	=	=	SYM
ap-1815	67	11	02	02	NUM
ap-1815	67	12	,	,	PUNCT
ap-1815	67	13	ϕ(11	ϕ(11	NOUN
ap-1815	67	14	)	)	PUNCT
ap-1815	67	15	=	=	SYM
ap-1815	67	16	12	12	NUM
ap-1815	67	17	,	,	PUNCT
ap-1815	67	18	(	(	PUNCT
ap-1815	67	19	b	b	NOUN
ap-1815	67	20	)	)	PUNCT
ap-1815	67	21	for	for	ADP
ap-1815	67	22	all	all	DET
ap-1815	67	23	a	a	DET
ap-1815	67	24	,	,	PUNCT
ap-1815	67	25	b	b	PROPN
ap-1815	67	26	∈	∈	PROPN
ap-1815	67	27	e1	e1	NOUN
ap-1815	67	28	:	:	PUNCT
ap-1815	67	29	if	if	SCONJ
ap-1815	67	30	a⊕1b	a⊕1b	PROPN
ap-1815	67	31	exists	exist	VERB
ap-1815	67	32	then	then	ADV
ap-1815	67	33	ϕ(a)⊕2ϕ(b	ϕ(a)⊕2ϕ(b	PROPN
ap-1815	67	34	)	)	PUNCT
ap-1815	67	35	exists	exist	VERB
ap-1815	67	36	,	,	PUNCT
ap-1815	67	37	in	in	ADP
ap-1815	67	38	which	which	DET
ap-1815	67	39	case	case	NOUN
ap-1815	67	40	ϕ(a⊕1	ϕ(a⊕1	NOUN
ap-1815	67	41	b	b	NOUN
ap-1815	67	42	)	)	PUNCT
ap-1815	67	43	=	=	SYM
ap-1815	67	44	ϕ(a)⊕2	ϕ(a)⊕2	PROPN
ap-1815	67	45	ϕ(b	ϕ(b	PROPN
ap-1815	67	46	)	)	PUNCT
ap-1815	67	47	,	,	PUNCT
ap-1815	67	48	(	(	PUNCT
ap-1815	67	49	2	2	NUM
ap-1815	67	50	.	.	PUNCT
ap-1815	67	51	)	)	PUNCT
ap-1815	68	1	an	an	DET
ap-1815	68	2	ordering	ordering	NOUN
ap-1815	68	3	morphism	morphism	NOUN
ap-1815	68	4	,	,	PUNCT
ap-1815	68	5	if	if	SCONJ
ap-1815	68	6	it	it	PRON
ap-1815	68	7	is	be	AUX
ap-1815	68	8	a	a	DET
ap-1815	68	9	morphism	morphism	NOUN
ap-1815	68	10	and	and	CCONJ
ap-1815	68	11	,	,	PUNCT
ap-1815	68	12	for	for	ADP
ap-1815	68	13	all	all	DET
ap-1815	68	14	a	a	DET
ap-1815	68	15	,	,	PUNCT
ap-1815	68	16	b	b	PROPN
ap-1815	68	17	∈	∈	PROPN
ap-1815	68	18	e1	e1	PROPN
ap-1815	68	19	,	,	PUNCT
ap-1815	68	20	a	a	DET
ap-1815	68	21	≤1	≤1	PROPN
ap-1815	68	22	b	b	PROPN
ap-1815	68	23	iff	iff	PROPN
ap-1815	68	24	ϕ(a	ϕ(a	PROPN
ap-1815	68	25	)	)	PUNCT
ap-1815	68	26	≤2	≤2	NOUN
ap-1815	68	27	ϕ(b	ϕ(b	PROPN
ap-1815	68	28	)	)	PUNCT
ap-1815	68	29	,	,	PUNCT
ap-1815	68	30	(	(	PUNCT
ap-1815	68	31	3	3	X
ap-1815	68	32	.	.	PUNCT
ap-1815	68	33	)	)	PUNCT
ap-1815	69	1	an	an	DET
ap-1815	69	2	embedding	embed	VERB
ap-1815	69	3	(	(	PUNCT
ap-1815	69	4	also	also	ADV
ap-1815	69	5	called	call	VERB
ap-1815	69	6	a	a	DET
ap-1815	69	7	monomorphism	monomorphism	NOUN
ap-1815	69	8	)	)	PUNCT
ap-1815	69	9	,	,	PUNCT
ap-1815	69	10	if	if	SCONJ
ap-1815	69	11	ϕ	ϕ	NOUN
ap-1815	69	12	is	be	AUX
ap-1815	69	13	injective	injective	ADJ
ap-1815	69	14	and	and	CCONJ
ap-1815	69	15	(	(	PUNCT
ap-1815	69	16	a	a	NOUN
ap-1815	69	17	)	)	PUNCT
ap-1815	69	18	ϕ(01	ϕ(01	NOUN
ap-1815	69	19	)	)	PUNCT
ap-1815	69	20	=	=	SYM
ap-1815	69	21	02	02	NUM
ap-1815	69	22	,	,	PUNCT
ap-1815	69	23	ϕ(11	ϕ(11	NOUN
ap-1815	69	24	)	)	PUNCT
ap-1815	69	25	=	=	SYM
ap-1815	69	26	12	12	NUM
ap-1815	69	27	,	,	PUNCT
ap-1815	69	28	(	(	PUNCT
ap-1815	69	29	c	c	NOUN
ap-1815	69	30	)	)	PUNCT
ap-1815	69	31	for	for	ADP
ap-1815	69	32	all	all	DET
ap-1815	69	33	a	a	DET
ap-1815	69	34	,	,	PUNCT
ap-1815	69	35	b	b	PROPN
ap-1815	69	36	∈	∈	PROPN
ap-1815	69	37	e1	e1	NOUN
ap-1815	69	38	:	:	PUNCT
ap-1815	69	39	a⊕1	a⊕1	PROPN
ap-1815	69	40	b	b	PROPN
ap-1815	69	41	exists	exist	VERB
ap-1815	69	42	iff	iff	PROPN
ap-1815	69	43	ϕ(a)⊕2	ϕ(a)⊕2	PROPN
ap-1815	69	44	ϕ(b	ϕ(b	PROPN
ap-1815	69	45	)	)	PUNCT
ap-1815	69	46	exists	exist	VERB
ap-1815	69	47	,	,	PUNCT
ap-1815	69	48	in	in	ADP
ap-1815	69	49	which	which	DET
ap-1815	69	50	case	case	NOUN
ap-1815	69	51	ϕ(a⊕1	ϕ(a⊕1	NOUN
ap-1815	69	52	b	b	NOUN
ap-1815	69	53	)	)	PUNCT
ap-1815	69	54	=	=	SYM
ap-1815	69	55	ϕ(a)⊕2	ϕ(a)⊕2	PROPN
ap-1815	69	56	ϕ(b	ϕ(b	PROPN
ap-1815	69	57	)	)	PUNCT
ap-1815	69	58	,	,	PUNCT
ap-1815	69	59	(	(	PUNCT
ap-1815	69	60	4	4	NUM
ap-1815	69	61	.	.	PUNCT
ap-1815	69	62	)	)	PUNCT
ap-1815	69	63	an	an	DET
ap-1815	69	64	isomorphism	isomorphism	NOUN
ap-1815	69	65	,	,	PUNCT
ap-1815	69	66	if	if	SCONJ
ap-1815	69	67	ϕ	ϕ	NOUN
ap-1815	69	68	is	be	AUX
ap-1815	69	69	bijective	bijective	ADJ
ap-1815	69	70	embedding	embed	VERB
ap-1815	69	71	,	,	PUNCT
ap-1815	69	72	(	(	PUNCT
ap-1815	69	73	5	5	NUM
ap-1815	69	74	.	.	PUNCT
ap-1815	69	75	)	)	PUNCT
ap-1815	69	76	a	a	DET
ap-1815	69	77	positive	positive	ADJ
ap-1815	69	78	operator	operator	NOUN
ap-1815	69	79	valued	value	VERB
ap-1815	69	80	state	state	NOUN
ap-1815	69	81	(	(	PUNCT
ap-1815	69	82	povs	povs	PROPN
ap-1815	69	83	for	for	ADP
ap-1815	69	84	short	short	ADJ
ap-1815	69	85	)	)	PUNCT
ap-1815	69	86	on	on	ADP
ap-1815	69	87	e1	e1	PROPN
ap-1815	69	88	iff	iff	PROPN
ap-1815	69	89	ϕ	ϕ	PROPN
ap-1815	69	90	is	be	AUX
ap-1815	69	91	a	a	DET
ap-1815	69	92	morphism	morphism	NOUN
ap-1815	69	93	into	into	ADP
ap-1815	69	94	e2	e2	PROPN
ap-1815	69	95	=	=	PUNCT
ap-1815	69	96	e(h	e(h	PROPN
ap-1815	69	97	)	)	PUNCT
ap-1815	69	98	for	for	ADP
ap-1815	69	99	some	some	DET
ap-1815	69	100	complex	complex	ADJ
ap-1815	69	101	hilbert	hilbert	NOUN
ap-1815	69	102	space	space	NOUN
ap-1815	69	103	h	h	NOUN
ap-1815	69	104	,	,	PUNCT
ap-1815	69	105	(	(	PUNCT
ap-1815	69	106	6	6	NUM
ap-1815	69	107	.	.	PUNCT
ap-1815	69	108	)	)	PUNCT
ap-1815	70	1	a	a	DET
ap-1815	70	2	hilbert	hilbert	NOUN
ap-1815	70	3	space	space	NOUN
ap-1815	70	4	effect	effect	NOUN
ap-1815	70	5	-	-	PUNCT
ap-1815	70	6	representation	representation	NOUN
ap-1815	70	7	of	of	ADP
ap-1815	70	8	e1	e1	PROPN
ap-1815	70	9	iff	iff	PROPN
ap-1815	70	10	ϕ	ϕ	PROPN
ap-1815	70	11	is	be	AUX
ap-1815	70	12	an	an	DET
ap-1815	70	13	embedding	embedding	NOUN
ap-1815	70	14	into	into	ADP
ap-1815	70	15	e2	e2	PROPN
ap-1815	70	16	=	=	PUNCT
ap-1815	70	17	e(h	e(h	PROPN
ap-1815	70	18	)	)	PUNCT
ap-1815	70	19	for	for	ADP
ap-1815	70	20	some	some	DET
ap-1815	70	21	complex	complex	ADJ
ap-1815	70	22	hilbert	hilbert	NOUN
ap-1815	70	23	space	space	NOUN
ap-1815	70	24	h.	h.	PROPN
ap-1815	70	25	clearly	clearly	ADV
ap-1815	70	26	,	,	PUNCT
ap-1815	70	27	every	every	DET
ap-1815	70	28	embedding	embed	VERB
ap-1815	70	29	ϕ	ϕ	NOUN
ap-1815	70	30	is	be	AUX
ap-1815	70	31	an	an	DET
ap-1815	70	32	isomorphism	isomorphism	NOUN
ap-1815	70	33	of	of	ADP
ap-1815	70	34	effect	effect	NOUN
ap-1815	70	35	algebras	algebras	PROPN
ap-1815	70	36	e1	e1	PROPN
ap-1815	70	37	and	and	CCONJ
ap-1815	70	38	ϕ(e1	ϕ(e1	NOUN
ap-1815	70	39	)	)	PUNCT
ap-1815	70	40	;	;	PUNCT
ap-1815	70	41	ϕ(e1	ϕ(e1	NOUN
ap-1815	70	42	)	)	PUNCT
ap-1815	70	43	is	be	AUX
ap-1815	70	44	a	a	DET
ap-1815	70	45	sub	sub	ADJ
ap-1815	70	46	-	-	ADJ
ap-1815	70	47	effect	effect	ADJ
ap-1815	70	48	algebra	algebra	NOUN
ap-1815	70	49	of	of	ADP
ap-1815	70	50	e2	e2	PROPN
ap-1815	70	51	;	;	PUNCT
ap-1815	70	52	and	and	CCONJ
ap-1815	70	53	a	a	DET
ap-1815	70	54	composition	composition	NOUN
ap-1815	70	55	of	of	ADP
ap-1815	70	56	morphisms	morphism	NOUN
ap-1815	70	57	(	(	PUNCT
ap-1815	70	58	embeddings	embedding	NOUN
ap-1815	70	59	,	,	PUNCT
ap-1815	70	60	isomorphisms	isomorphism	NOUN
ap-1815	70	61	)	)	PUNCT
ap-1815	70	62	is	be	AUX
ap-1815	70	63	again	again	ADV
ap-1815	70	64	a	a	DET
ap-1815	70	65	morphism	morphism	NOUN
ap-1815	70	66	(	(	PUNCT
ap-1815	70	67	embedding	embed	VERB
ap-1815	70	68	,	,	PUNCT
ap-1815	70	69	isomorphism	isomorphism	NOUN
ap-1815	70	70	)	)	PUNCT
ap-1815	70	71	.	.	PUNCT
ap-1815	71	1	every	every	DET
ap-1815	71	2	morphism	morphism	NOUN
ap-1815	71	3	of	of	ADP
ap-1815	71	4	effect	effect	NOUN
ap-1815	71	5	algebras	algebra	NOUN
ap-1815	71	6	preserves	preserve	VERB
ap-1815	71	7	supplements	supplement	NOUN
ap-1815	71	8	.	.	PUNCT
ap-1815	72	1	recall	recall	VERB
ap-1815	72	2	that	that	PRON
ap-1815	72	3	ϕ	ϕ	NOUN
ap-1815	72	4	is	be	AUX
ap-1815	72	5	an	an	DET
ap-1815	72	6	isomorphism	isomorphism	NOUN
ap-1815	72	7	of	of	ADP
ap-1815	72	8	effect	effect	NOUN
ap-1815	72	9	algebras	algebra	VERB
ap-1815	72	10	iff	iff	PROPN
ap-1815	72	11	ϕ	ϕ	PROPN
ap-1815	72	12	is	be	AUX
ap-1815	72	13	bijective	bijective	ADJ
ap-1815	72	14	and	and	CCONJ
ap-1815	72	15	both	both	DET
ap-1815	72	16	ϕ	ϕ	NOUN
ap-1815	72	17	and	and	CCONJ
ap-1815	72	18	ϕ−1	ϕ−1	PROPN
ap-1815	72	19	are	be	AUX
ap-1815	72	20	morphisms	morphism	NOUN
ap-1815	72	21	of	of	ADP
ap-1815	72	22	effect	effect	NOUN
ap-1815	72	23	algebras	algebra	NOUN
ap-1815	72	24	.	.	PUNCT
ap-1815	73	1	lemma	lemma	PROPN
ap-1815	73	2	2.8	2.8	NUM
ap-1815	73	3	.	.	PUNCT
ap-1815	74	1	let	let	VERB
ap-1815	74	2	(	(	PUNCT
ap-1815	74	3	e1;⊕1	e1;⊕1	PROPN
ap-1815	74	4	,	,	PUNCT
ap-1815	74	5	01	01	NUM
ap-1815	74	6	,	,	PUNCT
ap-1815	74	7	11	11	NUM
ap-1815	74	8	)	)	PUNCT
ap-1815	74	9	and	and	CCONJ
ap-1815	74	10	(	(	PUNCT
ap-1815	74	11	e2;⊕2	e2;⊕2	PROPN
ap-1815	74	12	,	,	PUNCT
ap-1815	74	13	02	02	NUM
ap-1815	74	14	,	,	PUNCT
ap-1815	74	15	12	12	NUM
ap-1815	74	16	)	)	PUNCT
ap-1815	74	17	be	be	VERB
ap-1815	74	18	effect	effect	NOUN
ap-1815	74	19	algebras	algebra	NOUN
ap-1815	74	20	and	and	CCONJ
ap-1815	74	21	let	let	VERB
ap-1815	74	22	ϕ	ϕ	NOUN
ap-1815	74	23	:	:	PUNCT
ap-1815	74	24	e1	e1	PROPN
ap-1815	74	25	→	→	SYM
ap-1815	74	26	e2	e2	PROPN
ap-1815	74	27	be	be	AUX
ap-1815	74	28	a	a	DET
ap-1815	74	29	morphism	morphism	NOUN
ap-1815	74	30	of	of	ADP
ap-1815	74	31	effect	effect	NOUN
ap-1815	74	32	algebras	algebra	VERB
ap-1815	74	33	.	.	PUNCT
ap-1815	75	1	then	then	ADV
ap-1815	75	2	ϕ	ϕ	PROPN
ap-1815	75	3	is	be	AUX
ap-1815	75	4	order	order	NOUN
ap-1815	75	5	-	-	PUNCT
ap-1815	75	6	preserving	preserve	VERB
ap-1815	75	7	309	309	NUM
ap-1815	75	8	j.	j.	PROPN
ap-1815	75	9	paseka	paseka	PROPN
ap-1815	75	10	,	,	PUNCT
ap-1815	75	11	z.	z.	PROPN
ap-1815	75	12	riečanová	riečanová	PROPN
ap-1815	75	13	acta	acta	PROPN
ap-1815	75	14	polytechnica	polytechnica	PROPN
ap-1815	75	15	and	and	CCONJ
ap-1815	75	16	,	,	PUNCT
ap-1815	75	17	for	for	ADP
ap-1815	75	18	any	any	DET
ap-1815	75	19	orthogonal	orthogonal	ADJ
ap-1815	75	20	system	system	NOUN
ap-1815	75	21	g	g	NOUN
ap-1815	75	22	=	=	SYM
ap-1815	75	23	(	(	PUNCT
ap-1815	75	24	xκ)κ∈h	xκ)κ∈h	NUM
ap-1815	75	25	of	of	ADP
ap-1815	75	26	not	not	PART
ap-1815	75	27	necessarily	necessarily	ADV
ap-1815	75	28	different	different	ADJ
ap-1815	75	29	elements	element	NOUN
ap-1815	75	30	of	of	ADP
ap-1815	75	31	e1	e1	NOUN
ap-1815	75	32	,	,	PUNCT
ap-1815	75	33	the	the	DET
ap-1815	75	34	system	system	NOUN
ap-1815	75	35	ϕ(g	ϕ(g	PROPN
ap-1815	75	36	)	)	PUNCT
ap-1815	76	1	=	=	SYM
ap-1815	76	2	(	(	PUNCT
ap-1815	76	3	ϕ(xκ	ϕ(xκ	NUM
ap-1815	76	4	)	)	PUNCT
ap-1815	76	5	)	)	PUNCT
ap-1815	77	1	κ∈h	κ∈h	PROPN
ap-1815	77	2	is	be	AUX
ap-1815	77	3	again	again	ADV
ap-1815	77	4	orthogonal	orthogonal	ADJ
ap-1815	77	5	.	.	PUNCT
ap-1815	78	1	proof	proof	NOUN
ap-1815	78	2	.	.	PUNCT
ap-1815	79	1	assume	assume	VERB
ap-1815	79	2	that	that	SCONJ
ap-1815	79	3	a	a	DET
ap-1815	79	4	,	,	PUNCT
ap-1815	79	5	b	b	PROPN
ap-1815	79	6	∈	∈	PROPN
ap-1815	79	7	e1	e1	PROPN
ap-1815	79	8	,	,	PUNCT
ap-1815	79	9	a	a	DET
ap-1815	79	10	≤1	≤1	PROPN
ap-1815	79	11	b.	b.	PROPN
ap-1815	79	12	then	then	ADV
ap-1815	79	13	there	there	PRON
ap-1815	79	14	is	be	VERB
ap-1815	79	15	an	an	DET
ap-1815	79	16	element	element	NOUN
ap-1815	79	17	c	c	PROPN
ap-1815	79	18	∈	∈	PROPN
ap-1815	79	19	e1	e1	NOUN
ap-1815	79	20	such	such	ADJ
ap-1815	79	21	that	that	SCONJ
ap-1815	79	22	a	a	DET
ap-1815	79	23	⊕1	⊕1	PROPN
ap-1815	79	24	c	c	X
ap-1815	79	25	=	=	PUNCT
ap-1815	79	26	b.	b.	PROPN
ap-1815	80	1	it	it	PRON
ap-1815	80	2	follows	follow	VERB
ap-1815	80	3	that	that	SCONJ
ap-1815	80	4	ϕ(b	ϕ(b	PROPN
ap-1815	80	5	)	)	PUNCT
ap-1815	81	1	=	=	NOUN
ap-1815	82	1	ϕ(a⊕1	ϕ(a⊕1	NOUN
ap-1815	82	2	c	c	X
ap-1815	82	3	)	)	PUNCT
ap-1815	82	4	=	=	SYM
ap-1815	82	5	ϕ(a)⊕2	ϕ(a)⊕2	PRON
ap-1815	82	6	ϕ(c	ϕ(c	PROPN
ap-1815	82	7	)	)	PUNCT
ap-1815	83	1	≥2	≥2	PROPN
ap-1815	83	2	ϕ(a	ϕ(a	NOUN
ap-1815	83	3	)	)	PUNCT
ap-1815	83	4	.	.	PUNCT
ap-1815	84	1	now	now	ADV
ap-1815	84	2	,	,	PUNCT
ap-1815	84	3	let	let	VERB
ap-1815	84	4	l	l	NOUN
ap-1815	84	5	⊆	⊆	NUM
ap-1815	84	6	ϕ(g	ϕ(g	PROPN
ap-1815	84	7	)	)	PUNCT
ap-1815	84	8	be	be	AUX
ap-1815	84	9	finite	finite	ADJ
ap-1815	84	10	.	.	PUNCT
ap-1815	85	1	then	then	ADV
ap-1815	85	2	there	there	PRON
ap-1815	85	3	is	be	VERB
ap-1815	85	4	a	a	DET
ap-1815	85	5	finite	finite	NOUN
ap-1815	85	6	subset	subset	NOUN
ap-1815	85	7	f	f	PROPN
ap-1815	86	1	⊆	⊆	NUM
ap-1815	86	2	h	h	NOUN
ap-1815	86	3	such	such	ADJ
ap-1815	86	4	that	that	DET
ap-1815	86	5	l	l	NOUN
ap-1815	86	6	=	=	PUNCT
ap-1815	86	7	(	(	PUNCT
ap-1815	86	8	ϕ(xκ))κ∈f	ϕ(xκ))κ∈f	INTJ
ap-1815	86	9	.	.	PUNCT
ap-1815	87	1	put	put	VERB
ap-1815	87	2	k	k	PROPN
ap-1815	87	3	=	=	PUNCT
ap-1815	87	4	(	(	PUNCT
ap-1815	87	5	xκ)κ∈f	xκ)κ∈f	PROPN
ap-1815	87	6	.	.	PUNCT
ap-1815	88	1	then	then	ADV
ap-1815	88	2	⊕	⊕	PROPN
ap-1815	88	3	e1	e1	PROPN
ap-1815	88	4	k	k	PROPN
ap-1815	88	5	exists	exist	VERB
ap-1815	88	6	and	and	CCONJ
ap-1815	88	7	hence	hence	ADV
ap-1815	88	8	⊕	⊕	PROPN
ap-1815	88	9	e2	e2	PROPN
ap-1815	88	10	l	l	PROPN
ap-1815	88	11	exists	exist	VERB
ap-1815	88	12	and	and	CCONJ
ap-1815	88	13	⊕	⊕	PROPN
ap-1815	88	14	e2	e2	PROPN
ap-1815	88	15	l	l	PROPN
ap-1815	89	1	=	=	SYM
ap-1815	89	2	ϕ	ϕ	X
ap-1815	89	3	(	(	PUNCT
ap-1815	89	4	⊕	⊕	PROPN
ap-1815	89	5	e1	e1	PROPN
ap-1815	89	6	k	k	PROPN
ap-1815	89	7	)	)	PUNCT
ap-1815	89	8	.	.	PUNCT
ap-1815	90	1	it	it	PRON
ap-1815	90	2	follows	follow	VERB
ap-1815	90	3	that	that	SCONJ
ap-1815	90	4	ϕ(g	ϕ(g	PROPN
ap-1815	90	5	)	)	PUNCT
ap-1815	90	6	=(	=(	NOUN
ap-1815	90	7	ϕ(xκ	ϕ(xκ	NUM
ap-1815	90	8	)	)	PUNCT
ap-1815	90	9	)	)	PUNCT
ap-1815	90	10	κ∈h	κ∈h	PROPN
ap-1815	90	11	is	be	AUX
ap-1815	90	12	orthogonal	orthogonal	ADJ
ap-1815	90	13	.	.	PUNCT
ap-1815	91	1	proposition	proposition	NOUN
ap-1815	91	2	2.9	2.9	NUM
ap-1815	91	3	.	.	PUNCT
ap-1815	92	1	let	let	AUX
ap-1815	92	2	(	(	PUNCT
ap-1815	92	3	e1;⊕1	e1;⊕1	PROPN
ap-1815	92	4	,	,	PUNCT
ap-1815	92	5	01	01	NUM
ap-1815	92	6	,	,	PUNCT
ap-1815	92	7	11	11	NUM
ap-1815	92	8	)	)	PUNCT
ap-1815	92	9	and	and	CCONJ
ap-1815	92	10	(	(	PUNCT
ap-1815	92	11	e2;⊕2	e2;⊕2	PROPN
ap-1815	92	12	,	,	PUNCT
ap-1815	92	13	02	02	NUM
ap-1815	92	14	,	,	PUNCT
ap-1815	92	15	12	12	NUM
ap-1815	92	16	)	)	PUNCT
ap-1815	92	17	be	be	VERB
ap-1815	92	18	effect	effect	NOUN
ap-1815	92	19	algebras	algebra	NOUN
ap-1815	92	20	and	and	CCONJ
ap-1815	92	21	let	let	VERB
ap-1815	92	22	ϕ	ϕ	NOUN
ap-1815	92	23	:	:	PUNCT
ap-1815	92	24	e1	e1	PROPN
ap-1815	92	25	→	→	SYM
ap-1815	92	26	e2	e2	PROPN
ap-1815	92	27	be	be	AUX
ap-1815	92	28	a	a	DET
ap-1815	92	29	morphism	morphism	NOUN
ap-1815	92	30	of	of	ADP
ap-1815	92	31	effect	effect	NOUN
ap-1815	92	32	algebras	algebra	VERB
ap-1815	92	33	.	.	PUNCT
ap-1815	93	1	then	then	ADV
ap-1815	93	2	the	the	DET
ap-1815	93	3	following	follow	VERB
ap-1815	93	4	conditions	condition	NOUN
ap-1815	93	5	are	be	AUX
ap-1815	93	6	equivalent	equivalent	ADJ
ap-1815	93	7	:	:	PUNCT
ap-1815	93	8	(	(	PUNCT
ap-1815	93	9	1	1	NUM
ap-1815	93	10	.	.	PUNCT
ap-1815	93	11	)	)	PUNCT
ap-1815	94	1	ϕ	ϕ	NOUN
ap-1815	94	2	is	be	AUX
ap-1815	94	3	an	an	DET
ap-1815	94	4	ordering	ordering	NOUN
ap-1815	94	5	morphism	morphism	NOUN
ap-1815	94	6	.	.	PUNCT
ap-1815	95	1	(	(	PUNCT
ap-1815	95	2	2	2	NUM
ap-1815	95	3	.	.	PUNCT
ap-1815	95	4	)	)	PUNCT
ap-1815	96	1	ϕ	ϕ	NOUN
ap-1815	96	2	is	be	AUX
ap-1815	96	3	an	an	DET
ap-1815	96	4	embedding	embedding	NOUN
ap-1815	96	5	.	.	PUNCT
ap-1815	97	1	proof	proof	NOUN
ap-1815	97	2	.	.	PUNCT
ap-1815	98	1	assume	assume	VERB
ap-1815	98	2	that	that	SCONJ
ap-1815	98	3	a	a	DET
ap-1815	98	4	,	,	PUNCT
ap-1815	98	5	b	b	PROPN
ap-1815	98	6	∈	∈	PROPN
ap-1815	98	7	e1	e1	PROPN
ap-1815	98	8	.	.	PUNCT
ap-1815	99	1	then	then	ADV
ap-1815	99	2	a	a	DET
ap-1815	99	3	≤1	≤1	PROPN
ap-1815	99	4	b	b	PROPN
ap-1815	99	5	′	′	NUM
ap-1815	99	6	iff	iff	PROPN
ap-1815	99	7	a⊕1	a⊕1	PROPN
ap-1815	99	8	b	b	PROPN
ap-1815	99	9	exists	exist	VERB
ap-1815	99	10	and	and	CCONJ
ap-1815	99	11	ϕ(a	ϕ(a	NOUN
ap-1815	99	12	)	)	PUNCT
ap-1815	99	13	≤1	≤1	PROPN
ap-1815	99	14	ϕ(b′	ϕ(b′	PROPN
ap-1815	99	15	)	)	PUNCT
ap-1815	99	16	iff	iff	PROPN
ap-1815	99	17	ϕ(a	ϕ(a	PROPN
ap-1815	99	18	)	)	PUNCT
ap-1815	100	1	≤1	≤1	PROPN
ap-1815	100	2	ϕ(b)′	ϕ(b)′	PROPN
ap-1815	100	3	iff	iff	PROPN
ap-1815	100	4	ϕ(a)⊕1	ϕ(a)⊕1	PROPN
ap-1815	100	5	ϕ(b	ϕ(b	PROPN
ap-1815	100	6	)	)	PUNCT
ap-1815	100	7	exists	exist	VERB
ap-1815	100	8	.	.	PUNCT
ap-1815	101	1	hence	hence	ADV
ap-1815	101	2	ϕ	ϕ	PROPN
ap-1815	101	3	is	be	AUX
ap-1815	101	4	an	an	DET
ap-1815	101	5	ordering	ordering	NOUN
ap-1815	101	6	morphism	morphism	NOUN
ap-1815	101	7	iff	iff	PROPN
ap-1815	101	8	ϕ	ϕ	PROPN
ap-1815	101	9	is	be	AUX
ap-1815	101	10	an	an	DET
ap-1815	101	11	embedding	embed	VERB
ap-1815	101	12	.	.	PUNCT
ap-1815	102	1	theorem	theorem	ADJ
ap-1815	102	2	2.10	2.10	NUM
ap-1815	102	3	.	.	PUNCT
ap-1815	103	1	let	let	VERB
ap-1815	103	2	(	(	PUNCT
ap-1815	103	3	e;⊕	e;⊕	ADJ
ap-1815	103	4	,	,	PUNCT
ap-1815	103	5	0	0	NUM
ap-1815	103	6	,	,	PUNCT
ap-1815	103	7	1	1	NUM
ap-1815	103	8	)	)	PUNCT
ap-1815	103	9	be	be	AUX
ap-1815	103	10	an	an	DET
ap-1815	103	11	effect	effect	NOUN
ap-1815	103	12	algebra	algebra	NOUN
ap-1815	103	13	and	and	CCONJ
ap-1815	103	14	let	let	VERB
ap-1815	103	15	h	h	NOUN
ap-1815	103	16	be	be	AUX
ap-1815	103	17	some	some	DET
ap-1815	103	18	complex	complex	ADJ
ap-1815	103	19	hilbert	hilbert	NOUN
ap-1815	103	20	space	space	NOUN
ap-1815	103	21	.	.	PUNCT
ap-1815	104	1	for	for	ADP
ap-1815	104	2	a	a	DET
ap-1815	104	3	map	map	NOUN
ap-1815	104	4	ϕ	ϕ	NOUN
ap-1815	104	5	:	:	PUNCT
ap-1815	104	6	e	e	X
ap-1815	104	7	→	→	SYM
ap-1815	104	8	e(h	e(h	PROPN
ap-1815	104	9	)	)	PUNCT
ap-1815	104	10	the	the	DET
ap-1815	104	11	following	follow	VERB
ap-1815	104	12	conditions	condition	NOUN
ap-1815	104	13	are	be	AUX
ap-1815	104	14	equivalent	equivalent	ADJ
ap-1815	104	15	:	:	PUNCT
ap-1815	104	16	(	(	PUNCT
ap-1815	104	17	1	1	NUM
ap-1815	104	18	.	.	PUNCT
ap-1815	104	19	)	)	PUNCT
ap-1815	105	1	ϕ	ϕ	NOUN
ap-1815	105	2	is	be	AUX
ap-1815	105	3	an	an	DET
ap-1815	105	4	ordering	order	VERB
ap-1815	105	5	positive	positive	ADJ
ap-1815	105	6	operator	operator	NOUN
ap-1815	105	7	valued	value	VERB
ap-1815	105	8	state	state	NOUN
ap-1815	105	9	.	.	PUNCT
ap-1815	106	1	(	(	PUNCT
ap-1815	106	2	2	2	NUM
ap-1815	106	3	.	.	PUNCT
ap-1815	106	4	)	)	PUNCT
ap-1815	107	1	ϕ	ϕ	NOUN
ap-1815	107	2	is	be	AUX
ap-1815	107	3	an	an	DET
ap-1815	107	4	embedding	embed	VERB
ap-1815	107	5	.	.	PUNCT
ap-1815	108	1	(	(	PUNCT
ap-1815	108	2	3	3	NUM
ap-1815	108	3	.	.	PUNCT
ap-1815	108	4	)	)	PUNCT
ap-1815	109	1	ϕ	ϕ	NOUN
ap-1815	109	2	is	be	AUX
ap-1815	109	3	a	a	DET
ap-1815	109	4	hilbert	hilbert	NOUN
ap-1815	109	5	space	space	NOUN
ap-1815	109	6	effect	effect	NOUN
ap-1815	109	7	-	-	PUNCT
ap-1815	109	8	representation	representation	NOUN
ap-1815	109	9	of	of	ADP
ap-1815	109	10	e	e	PROPN
ap-1815	109	11	in	in	ADP
ap-1815	109	12	h.	h.	PROPN
ap-1815	109	13	proof	proof	NOUN
ap-1815	109	14	.	.	PUNCT
ap-1815	110	1	the	the	DET
ap-1815	110	2	equivalence	equivalence	NOUN
ap-1815	110	3	between	between	ADP
ap-1815	110	4	(	(	PUNCT
ap-1815	110	5	1	1	NUM
ap-1815	110	6	.	.	PUNCT
ap-1815	110	7	)	)	PUNCT
ap-1815	111	1	and	and	CCONJ
ap-1815	111	2	(	(	PUNCT
ap-1815	111	3	2	2	NUM
ap-1815	111	4	.	.	PUNCT
ap-1815	111	5	)	)	PUNCT
ap-1815	111	6	follows	follow	VERB
ap-1815	111	7	from	from	ADP
ap-1815	111	8	proposition	proposition	NOUN
ap-1815	111	9	2.9	2.9	NUM
ap-1815	111	10	,	,	PUNCT
ap-1815	111	11	the	the	DET
ap-1815	111	12	equivalence	equivalence	NOUN
ap-1815	111	13	between	between	ADP
ap-1815	111	14	(	(	PUNCT
ap-1815	111	15	2	2	NUM
ap-1815	111	16	.	.	PUNCT
ap-1815	111	17	)	)	PUNCT
ap-1815	112	1	and	and	CCONJ
ap-1815	112	2	(	(	PUNCT
ap-1815	112	3	3	3	NUM
ap-1815	112	4	.	.	PUNCT
ap-1815	112	5	)	)	PUNCT
ap-1815	112	6	follows	follow	VERB
ap-1815	112	7	from	from	ADP
ap-1815	112	8	definition	definition	NOUN
ap-1815	112	9	2.7	2.7	NUM
ap-1815	112	10	,	,	PUNCT
ap-1815	112	11	(	(	PUNCT
ap-1815	112	12	2	2	NUM
ap-1815	112	13	.	.	NUM
ap-1815	112	14	)	)	PUNCT
ap-1815	112	15	,	,	PUNCT
ap-1815	112	16	(	(	PUNCT
ap-1815	112	17	5	5	NUM
ap-1815	112	18	.	.	PUNCT
ap-1815	112	19	)	)	PUNCT
ap-1815	113	1	and	and	CCONJ
ap-1815	113	2	(	(	PUNCT
ap-1815	113	3	6	6	NUM
ap-1815	113	4	.	.	PUNCT
ap-1815	113	5	)	)	PUNCT
ap-1815	113	6	.	.	PUNCT
ap-1815	114	1	definition	definition	NOUN
ap-1815	114	2	2.11	2.11	NUM
ap-1815	114	3	(	(	PUNCT
ap-1815	114	4	[	[	X
ap-1815	114	5	2	2	NUM
ap-1815	114	6	,	,	PUNCT
ap-1815	114	7	14	14	NUM
ap-1815	114	8	,	,	PUNCT
ap-1815	114	9	18	18	NUM
ap-1815	114	10	]	]	PUNCT
ap-1815	114	11	)	)	PUNCT
ap-1815	114	12	.	.	PUNCT
ap-1815	115	1	(	(	PUNCT
ap-1815	115	2	1	1	NUM
ap-1815	115	3	.	.	PUNCT
ap-1815	115	4	)	)	PUNCT
ap-1815	116	1	a	a	DET
ap-1815	116	2	map	map	NOUN
ap-1815	116	3	ω	ω	NOUN
ap-1815	116	4	:	:	PUNCT
ap-1815	116	5	e	e	X
ap-1815	116	6	→	→	PUNCT
ap-1815	116	7	[	[	X
ap-1815	116	8	0	0	NUM
ap-1815	116	9	,	,	PUNCT
ap-1815	116	10	1	1	NUM
ap-1815	116	11	]	]	SYM
ap-1815	116	12	⊆	⊆	NUM
ap-1815	116	13	r	r	NOUN
ap-1815	116	14	is	be	AUX
ap-1815	116	15	a	a	DET
ap-1815	116	16	state	state	NOUN
ap-1815	116	17	on	on	ADP
ap-1815	116	18	an	an	DET
ap-1815	116	19	effect	effect	NOUN
ap-1815	116	20	algebra	algebra	NOUN
ap-1815	116	21	e	e	NOUN
ap-1815	116	22	if	if	SCONJ
ap-1815	116	23	ω(0	ω(0	PROPN
ap-1815	116	24	)	)	PUNCT
ap-1815	116	25	=	=	SYM
ap-1815	116	26	0	0	NUM
ap-1815	116	27	,	,	PUNCT
ap-1815	116	28	ω(1	ω(1	NOUN
ap-1815	116	29	)	)	PUNCT
ap-1815	116	30	=	=	SYM
ap-1815	116	31	1	1	NUM
ap-1815	116	32	and	and	CCONJ
ap-1815	116	33	ω(x	ω(x	PROPN
ap-1815	116	34	⊕	⊕	PROPN
ap-1815	116	35	y	y	PROPN
ap-1815	116	36	)	)	PUNCT
ap-1815	116	37	=	=	SYM
ap-1815	116	38	ω(x	ω(x	NOUN
ap-1815	116	39	)	)	PUNCT
ap-1815	116	40	+	+	CCONJ
ap-1815	116	41	ω(y	ω(y	NOUN
ap-1815	116	42	)	)	PUNCT
ap-1815	116	43	whenever	whenever	SCONJ
ap-1815	116	44	x	x	SYM
ap-1815	116	45	≤	≤	ADV
ap-1815	116	46	y′	y′	NUM
ap-1815	116	47	,	,	PUNCT
ap-1815	116	48	x	x	PRON
ap-1815	116	49	,	,	PUNCT
ap-1815	116	50	y	y	PROPN
ap-1815	116	51	∈	∈	PROPN
ap-1815	116	52	e.	e.	PROPN
ap-1815	116	53	(	(	PUNCT
ap-1815	116	54	2	2	NUM
ap-1815	116	55	.	.	PUNCT
ap-1815	116	56	)	)	PUNCT
ap-1815	117	1	a	a	DET
ap-1815	117	2	setm	setm	NOUN
ap-1815	117	3	of	of	ADP
ap-1815	117	4	states	state	NOUN
ap-1815	117	5	on	on	ADP
ap-1815	117	6	an	an	DET
ap-1815	117	7	effect	effect	NOUN
ap-1815	117	8	algebra	algebra	NOUN
ap-1815	117	9	e	e	NOUN
ap-1815	117	10	is	be	AUX
ap-1815	117	11	called	call	VERB
ap-1815	117	12	an	an	DET
ap-1815	117	13	ordering	ordering	NOUN
ap-1815	117	14	set	set	NOUN
ap-1815	117	15	of	of	ADP
ap-1815	117	16	states	state	NOUN
ap-1815	117	17	if	if	SCONJ
ap-1815	117	18	for	for	ADP
ap-1815	117	19	any	any	DET
ap-1815	117	20	a	a	NOUN
ap-1815	117	21	,	,	PUNCT
ap-1815	117	22	b	b	X
ap-1815	117	23	∈	∈	PROPN
ap-1815	117	24	e	e	NOUN
ap-1815	117	25	the	the	DET
ap-1815	117	26	condition	condition	NOUN
ap-1815	117	27	a	a	DET
ap-1815	117	28	≤	≤	NUM
ap-1815	117	29	b	b	NUM
ap-1815	117	30	iff	iff	PROPN
ap-1815	117	31	ω(a	ω(a	PROPN
ap-1815	117	32	)	)	PUNCT
ap-1815	117	33	≤	≤	NOUN
ap-1815	117	34	ω(b	ω(b	NOUN
ap-1815	117	35	)	)	PUNCT
ap-1815	117	36	for	for	ADP
ap-1815	117	37	all	all	DET
ap-1815	117	38	ω	ω	NUM
ap-1815	117	39	∈	∈	PROPN
ap-1815	117	40	m	m	NOUN
ap-1815	117	41	,	,	PUNCT
ap-1815	117	42	is	be	AUX
ap-1815	117	43	satisfied	satisfied	ADJ
ap-1815	117	44	.	.	PUNCT
ap-1815	118	1	(	(	PUNCT
ap-1815	118	2	3	3	NUM
ap-1815	118	3	.	.	PUNCT
ap-1815	118	4	)	)	PUNCT
ap-1815	119	1	a	a	DET
ap-1815	119	2	state	state	NOUN
ap-1815	119	3	ω	ω	PROPN
ap-1815	119	4	on	on	ADP
ap-1815	119	5	an	an	DET
ap-1815	119	6	effect	effect	NOUN
ap-1815	119	7	algebra	algebra	NOUN
ap-1815	119	8	e	e	NOUN
ap-1815	119	9	is	be	AUX
ap-1815	119	10	called	call	VERB
ap-1815	119	11	σadditive	σadditive	ADJ
ap-1815	119	12	if	if	SCONJ
ap-1815	119	13	,	,	PUNCT
ap-1815	119	14	for	for	ADP
ap-1815	119	15	every	every	DET
ap-1815	119	16	countable	countable	ADJ
ap-1815	119	17	net	net	NOUN
ap-1815	119	18	(	(	PUNCT
ap-1815	119	19	xn)n∈n	xn)n∈n	NUM
ap-1815	119	20	of	of	ADP
ap-1815	119	21	elements	element	NOUN
ap-1815	119	22	of	of	ADP
ap-1815	119	23	e	e	NOUN
ap-1815	119	24	,	,	PUNCT
ap-1815	119	25	xn	xn	PROPN
ap-1815	119	26	↑	↑	NOUN
ap-1815	119	27	x	x	PUNCT
ap-1815	120	1	=	=	NOUN
ap-1815	120	2	⇒	⇒	NOUN
ap-1815	120	3	ω(xn)→	ω(xn)→	NUM
ap-1815	120	4	ω(x	ω(x	NOUN
ap-1815	120	5	)	)	PUNCT
ap-1815	120	6	.	.	PUNCT
ap-1815	121	1	(	(	PUNCT
ap-1815	121	2	4	4	NUM
ap-1815	121	3	.	.	PUNCT
ap-1815	121	4	)	)	PUNCT
ap-1815	122	1	a	a	DET
ap-1815	122	2	state	state	NOUN
ap-1815	122	3	ω	ω	PROPN
ap-1815	122	4	on	on	ADP
ap-1815	122	5	an	an	DET
ap-1815	122	6	effect	effect	NOUN
ap-1815	122	7	algebra	algebra	NOUN
ap-1815	122	8	e	e	NOUN
ap-1815	122	9	is	be	AUX
ap-1815	122	10	called	call	VERB
ap-1815	122	11	(	(	PUNCT
ap-1815	122	12	o)continuous	o)continuous	ADJ
ap-1815	122	13	(	(	PUNCT
ap-1815	122	14	order	order	NOUN
ap-1815	122	15	-	-	PUNCT
ap-1815	122	16	continuous	continuous	ADJ
ap-1815	122	17	)	)	PUNCT
ap-1815	122	18	if	if	SCONJ
ap-1815	122	19	,	,	PUNCT
ap-1815	122	20	for	for	ADP
ap-1815	122	21	every	every	DET
ap-1815	122	22	net	net	NOUN
ap-1815	122	23	(	(	PUNCT
ap-1815	122	24	xα)α∈λ	xα)α∈λ	NUM
ap-1815	122	25	of	of	ADP
ap-1815	122	26	elements	element	NOUN
ap-1815	122	27	of	of	ADP
ap-1815	122	28	e	e	NOUN
ap-1815	122	29	,	,	PUNCT
ap-1815	122	30	xα	xα	INTJ
ap-1815	122	31	(	(	PUNCT
ap-1815	122	32	o)−→	o)−→	PROPN
ap-1815	122	33	x	x	PUNCT
ap-1815	122	34	implies	imply	VERB
ap-1815	122	35	ω(xα)→	ω(xα)→	NUM
ap-1815	122	36	ω(x	ω(x	NOUN
ap-1815	122	37	)	)	PUNCT
ap-1815	122	38	(	(	PUNCT
ap-1815	122	39	equivalently	equivalently	ADV
ap-1815	122	40	xα	xα	ADP
ap-1815	122	41	↑	↑	PROPN
ap-1815	122	42	x	x	PROPN
ap-1815	122	43	implies	imply	VERB
ap-1815	122	44	ω(xα	ω(xα	NOUN
ap-1815	122	45	)	)	PUNCT
ap-1815	122	46	↑	↑	PROPN
ap-1815	122	47	ω(x	ω(x	NOUN
ap-1815	122	48	)	)	PUNCT
ap-1815	122	49	)	)	PUNCT
ap-1815	122	50	.	.	PUNCT
ap-1815	123	1	(	(	PUNCT
ap-1815	123	2	5	5	NUM
ap-1815	123	3	.	.	PUNCT
ap-1815	123	4	)	)	PUNCT
ap-1815	124	1	a	a	DET
ap-1815	124	2	state	state	NOUN
ap-1815	124	3	ω	ω	PROPN
ap-1815	124	4	on	on	ADP
ap-1815	124	5	an	an	DET
ap-1815	124	6	effect	effect	NOUN
ap-1815	124	7	algebra	algebra	NOUN
ap-1815	124	8	e	e	NOUN
ap-1815	124	9	is	be	AUX
ap-1815	124	10	called	call	VERB
ap-1815	124	11	completely	completely	ADV
ap-1815	124	12	additive	additive	ADJ
ap-1815	124	13	if	if	SCONJ
ap-1815	124	14	for	for	ADP
ap-1815	124	15	any	any	DET
ap-1815	124	16	orthogonal	orthogonal	ADJ
ap-1815	124	17	system	system	NOUN
ap-1815	124	18	(	(	PUNCT
ap-1815	124	19	xκ)κ∈h	xκ)κ∈h	NUM
ap-1815	124	20	of	of	ADP
ap-1815	124	21	not	not	PART
ap-1815	124	22	necessarily	necessarily	ADV
ap-1815	124	23	different	different	ADJ
ap-1815	124	24	elements	element	NOUN
ap-1815	124	25	of	of	ADP
ap-1815	124	26	e	e	NOUN
ap-1815	124	27	such	such	ADJ
ap-1815	124	28	that	that	SCONJ
ap-1815	124	29	⊕	⊕	PROPN
ap-1815	124	30	{	{	PUNCT
ap-1815	125	1	xκ	xκ	NOUN
ap-1815	126	1	|	|	ADV
ap-1815	126	2	κ	κ	PROPN
ap-1815	126	3	∈	∈	PROPN
ap-1815	126	4	h	h	NOUN
ap-1815	126	5	}	}	PUNCT
ap-1815	126	6	exists	exist	VERB
ap-1815	126	7	,	,	PUNCT
ap-1815	126	8	ω	ω	PROPN
ap-1815	126	9	(	(	PUNCT
ap-1815	126	10	⊕	⊕	PROPN
ap-1815	126	11	{	{	PUNCT
ap-1815	127	1	xκ	xκ	NOUN
ap-1815	127	2	|	|	ADV
ap-1815	127	3	κ	κ	PROPN
ap-1815	127	4	∈	∈	PROPN
ap-1815	127	5	h	h	NOUN
ap-1815	127	6	}	}	PUNCT
ap-1815	127	7	)	)	PUNCT
ap-1815	128	1	=	=	PUNCT
ap-1815	128	2	∑	∑	PUNCT
ap-1815	128	3	{	{	PUNCT
ap-1815	128	4	ω(xκ	ω(xκ	X
ap-1815	128	5	)	)	PUNCT
ap-1815	128	6	|	|	ADV
ap-1815	128	7	κ	κ	PROPN
ap-1815	128	8	∈	∈	PROPN
ap-1815	128	9	h	h	NOUN
ap-1815	128	10	}	}	PUNCT
ap-1815	128	11	=	=	SYM
ap-1815	128	12	sup	sup	NOUN
ap-1815	128	13	{	{	PUNCT
ap-1815	128	14	∑	∑	DET
ap-1815	128	15	{	{	PUNCT
ap-1815	128	16	ω(xκ	ω(xκ	X
ap-1815	128	17	)	)	PUNCT
ap-1815	128	18	|	|	ADV
ap-1815	128	19	κ	κ	X
ap-1815	128	20	∈	∈	PROPN
ap-1815	129	1	f	f	NOUN
ap-1815	129	2	}	}	PUNCT
ap-1815	129	3	|	|	ADV
ap-1815	129	4	f	f	PROPN
ap-1815	129	5	⊆	⊆	NUM
ap-1815	129	6	h	h	NOUN
ap-1815	129	7	,	,	PUNCT
ap-1815	129	8	f	f	PROPN
ap-1815	129	9	finite	finite	PROPN
ap-1815	129	10	set	set	PROPN
ap-1815	129	11	}	}	PUNCT
ap-1815	129	12	.	.	PUNCT
ap-1815	130	1	it	it	PRON
ap-1815	130	2	follows	follow	VERB
ap-1815	130	3	that	that	SCONJ
ap-1815	130	4	states	state	NOUN
ap-1815	130	5	on	on	ADP
ap-1815	130	6	effect	effect	NOUN
ap-1815	130	7	algebras	algebra	NOUN
ap-1815	130	8	are	be	AUX
ap-1815	130	9	exactly	exactly	ADV
ap-1815	130	10	morphisms	morphisms	ADJ
ap-1815	130	11	from	from	ADP
ap-1815	130	12	them	they	PRON
ap-1815	130	13	into	into	ADP
ap-1815	130	14	[	[	X
ap-1815	130	15	0	0	NUM
ap-1815	130	16	,	,	PUNCT
ap-1815	130	17	1	1	NUM
ap-1815	130	18	]	]	PUNCT
ap-1815	130	19	.	.	PUNCT
ap-1815	131	1	note	note	VERB
ap-1815	131	2	that	that	SCONJ
ap-1815	131	3	any	any	PRON
ap-1815	131	4	(	(	PUNCT
ap-1815	131	5	o)continuous	o)continuous	ADJ
ap-1815	131	6	state	state	NOUN
ap-1815	131	7	is	be	AUX
ap-1815	131	8	completely	completely	ADV
ap-1815	131	9	additive	additive	ADJ
ap-1815	131	10	and	and	CCONJ
ap-1815	131	11	also	also	ADV
ap-1815	131	12	any	any	DET
ap-1815	131	13	completely	completely	ADV
ap-1815	131	14	additive	additive	ADJ
ap-1815	131	15	state	state	NOUN
ap-1815	131	16	is	be	AUX
ap-1815	131	17	σ	σ	NOUN
ap-1815	131	18	-	-	PUNCT
ap-1815	131	19	additive	additive	NOUN
ap-1815	131	20	.	.	PUNCT
ap-1815	132	1	moreover	moreover	ADV
ap-1815	132	2	,	,	PUNCT
ap-1815	132	3	it	it	PRON
ap-1815	132	4	was	be	AUX
ap-1815	132	5	proved	prove	VERB
ap-1815	132	6	in	in	ADP
ap-1815	132	7	[	[	X
ap-1815	132	8	18	18	NUM
ap-1815	132	9	]	]	PUNCT
ap-1815	132	10	that	that	SCONJ
ap-1815	132	11	,	,	PUNCT
ap-1815	132	12	for	for	ADP
ap-1815	132	13	an	an	DET
ap-1815	132	14	effect	effect	NOUN
ap-1815	132	15	algebra	algebra	NOUN
ap-1815	132	16	e	e	NOUN
ap-1815	132	17	,	,	PUNCT
ap-1815	132	18	there	there	PRON
ap-1815	132	19	exists	exist	VERB
ap-1815	132	20	a	a	DET
ap-1815	132	21	complex	complex	ADJ
ap-1815	132	22	hilbert	hilbert	NOUN
ap-1815	132	23	space	space	NOUN
ap-1815	132	24	h	h	NOUN
ap-1815	132	25	such	such	ADJ
ap-1815	132	26	that	that	SCONJ
ap-1815	132	27	e	e	PROPN
ap-1815	132	28	has	have	AUX
ap-1815	132	29	a	a	DET
ap-1815	132	30	hilbert	hilbert	NOUN
ap-1815	132	31	space	space	NOUN
ap-1815	132	32	effect	effect	NOUN
ap-1815	132	33	-	-	PUNCT
ap-1815	132	34	representation	representation	NOUN
ap-1815	132	35	into	into	ADP
ap-1815	132	36	e(h	e(h	PROPN
ap-1815	132	37	)	)	PUNCT
ap-1815	132	38	=	=	PUNCT
ap-1815	133	1	[	[	X
ap-1815	133	2	0	0	NUM
ap-1815	133	3	,	,	PUNCT
ap-1815	133	4	i]b+(h	i]b+(h	PROPN
ap-1815	133	5	)	)	PUNCT
ap-1815	133	6	,	,	PUNCT
ap-1815	133	7	where	where	SCONJ
ap-1815	133	8	b+(h	b+(h	NUM
ap-1815	133	9	)	)	PUNCT
ap-1815	133	10	are	be	AUX
ap-1815	133	11	positive	positive	ADJ
ap-1815	133	12	bounded	bounded	ADJ
ap-1815	133	13	operators	operator	NOUN
ap-1815	133	14	on	on	ADP
ap-1815	133	15	h	h	PROPN
ap-1815	133	16	iff	iff	PROPN
ap-1815	133	17	there	there	PRON
ap-1815	133	18	exists	exist	VERB
ap-1815	133	19	an	an	DET
ap-1815	133	20	ordering	order	VERB
ap-1815	133	21	setm	setm	NOUN
ap-1815	133	22	of	of	ADP
ap-1815	133	23	states	state	NOUN
ap-1815	133	24	on	on	ADP
ap-1815	133	25	e	e	NOUN
ap-1815	133	26	and	and	CCONJ
ap-1815	133	27	then	then	ADV
ap-1815	133	28	h	h	PROPN
ap-1815	133	29	=	=	PUNCT
ap-1815	133	30	l2(m	l2(m	PROPN
ap-1815	133	31	)	)	PUNCT
ap-1815	133	32	.	.	PUNCT
ap-1815	134	1	3	3	X
ap-1815	134	2	.	.	X
ap-1815	134	3	basic	basic	ADJ
ap-1815	134	4	properties	property	NOUN
ap-1815	134	5	of	of	ADP
ap-1815	134	6	isomorphisms	isomorphism	NOUN
ap-1815	134	7	of	of	ADP
ap-1815	134	8	effect	effect	NOUN
ap-1815	134	9	algebras	algebra	NOUN
ap-1815	134	10	and	and	CCONJ
ap-1815	134	11	operator	operator	NOUN
ap-1815	134	12	representations	representation	NOUN
ap-1815	134	13	roughly	roughly	ADV
ap-1815	134	14	speaking	speak	VERB
ap-1815	134	15	,	,	PUNCT
ap-1815	134	16	the	the	DET
ap-1815	134	17	operator	operator	NOUN
ap-1815	134	18	representations	representation	NOUN
ap-1815	134	19	of	of	ADP
ap-1815	134	20	abstract	abstract	ADJ
ap-1815	134	21	effect	effect	NOUN
ap-1815	134	22	algebras	algebra	NOUN
ap-1815	134	23	(	(	PUNCT
ap-1815	134	24	if	if	SCONJ
ap-1815	134	25	they	they	PRON
ap-1815	134	26	exist	exist	VERB
ap-1815	134	27	)	)	PUNCT
ap-1815	134	28	are	be	AUX
ap-1815	134	29	their	their	PRON
ap-1815	134	30	isomorphisms	isomorphism	NOUN
ap-1815	134	31	with	with	ADP
ap-1815	134	32	operator	operator	NOUN
ap-1815	134	33	effect	effect	NOUN
ap-1815	134	34	algebras	algebra	NOUN
ap-1815	134	35	in	in	ADP
ap-1815	134	36	some	some	DET
ap-1815	134	37	complex	complex	ADJ
ap-1815	134	38	hilbert	hilbert	NOUN
ap-1815	134	39	space	space	NOUN
ap-1815	134	40	h.	h.	PROPN
ap-1815	134	41	more	more	ADV
ap-1815	134	42	precisely	precisely	ADV
ap-1815	134	43	,	,	PUNCT
ap-1815	134	44	they	they	PRON
ap-1815	134	45	are	be	AUX
ap-1815	134	46	their	their	PRON
ap-1815	134	47	isomorphisms	isomorphism	NOUN
ap-1815	134	48	with	with	ADP
ap-1815	134	49	sub	sub	ADJ
ap-1815	134	50	-	-	ADJ
ap-1815	134	51	effect	effect	ADJ
ap-1815	134	52	algebras	algebra	NOUN
ap-1815	134	53	of	of	ADP
ap-1815	134	54	the	the	DET
ap-1815	134	55	standard	standard	ADJ
ap-1815	134	56	hilbert	hilbert	NOUN
ap-1815	134	57	space	space	NOUN
ap-1815	134	58	effect	effect	NOUN
ap-1815	134	59	algebra	algebra	VERB
ap-1815	134	60	e(h	e(h	PROPN
ap-1815	134	61	)	)	PUNCT
ap-1815	134	62	.	.	PUNCT
ap-1815	135	1	in	in	ADP
ap-1815	135	2	such	such	DET
ap-1815	135	3	a	a	DET
ap-1815	135	4	case	case	NOUN
ap-1815	135	5	it	it	PRON
ap-1815	135	6	may	may	AUX
ap-1815	135	7	be	be	AUX
ap-1815	135	8	interesting	interesting	ADJ
ap-1815	135	9	to	to	PART
ap-1815	135	10	know	know	VERB
ap-1815	135	11	which	which	DET
ap-1815	135	12	properties	property	NOUN
ap-1815	135	13	of	of	ADP
ap-1815	135	14	the	the	DET
ap-1815	135	15	initial	initial	ADJ
ap-1815	135	16	effect	effect	NOUN
ap-1815	135	17	algebras	algebra	NOUN
ap-1815	135	18	are	be	AUX
ap-1815	135	19	inherited	inherit	VERB
ap-1815	135	20	for	for	ADP
ap-1815	135	21	those	those	DET
ap-1815	135	22	isomorphic	isomorphic	ADJ
ap-1815	135	23	operator	operator	NOUN
ap-1815	135	24	effect	effect	NOUN
ap-1815	135	25	algebras	algebra	VERB
ap-1815	135	26	.	.	PUNCT
ap-1815	136	1	let	let	VERB
ap-1815	136	2	us	we	PRON
ap-1815	136	3	start	start	VERB
ap-1815	136	4	our	our	PRON
ap-1815	136	5	considerations	consideration	NOUN
ap-1815	136	6	with	with	ADP
ap-1815	136	7	properties	property	NOUN
ap-1815	136	8	of	of	ADP
ap-1815	136	9	two	two	NUM
ap-1815	136	10	isomorphic	isomorphic	ADJ
ap-1815	136	11	abstract	abstract	ADJ
ap-1815	136	12	effect	effect	NOUN
ap-1815	136	13	algebras	algebra	NOUN
ap-1815	136	14	.	.	PUNCT
ap-1815	136	15	theorem	theorem	VERB
ap-1815	136	16	3.1	3.1	NUM
ap-1815	136	17	.	.	PUNCT
ap-1815	137	1	let	let	VERB
ap-1815	137	2	(	(	PUNCT
ap-1815	137	3	e1;⊕1	e1;⊕1	PROPN
ap-1815	137	4	,	,	PUNCT
ap-1815	137	5	01	01	NUM
ap-1815	137	6	,	,	PUNCT
ap-1815	137	7	11	11	NUM
ap-1815	137	8	)	)	PUNCT
ap-1815	137	9	and	and	CCONJ
ap-1815	137	10	(	(	PUNCT
ap-1815	137	11	e2;⊕2	e2;⊕2	PROPN
ap-1815	137	12	,	,	PUNCT
ap-1815	137	13	02	02	NUM
ap-1815	137	14	,	,	PUNCT
ap-1815	137	15	12	12	NUM
ap-1815	137	16	)	)	PUNCT
ap-1815	137	17	be	be	VERB
ap-1815	137	18	effect	effect	NOUN
ap-1815	137	19	algebras	algebra	NOUN
ap-1815	137	20	and	and	CCONJ
ap-1815	137	21	let	let	VERB
ap-1815	137	22	ϕ	ϕ	NOUN
ap-1815	137	23	:	:	PUNCT
ap-1815	137	24	e1	e1	PROPN
ap-1815	137	25	→	→	SYM
ap-1815	137	26	e2	e2	PROPN
ap-1815	137	27	be	be	AUX
ap-1815	137	28	an	an	DET
ap-1815	137	29	isomorphism	isomorphism	NOUN
ap-1815	137	30	of	of	ADP
ap-1815	137	31	effect	effect	NOUN
ap-1815	137	32	algebras	algebra	VERB
ap-1815	137	33	.	.	PUNCT
ap-1815	138	1	then	then	ADV
ap-1815	138	2	(	(	PUNCT
ap-1815	138	3	1	1	NUM
ap-1815	138	4	.	.	PUNCT
ap-1815	138	5	)	)	PUNCT
ap-1815	138	6	for	for	ADP
ap-1815	138	7	all	all	DET
ap-1815	138	8	a	a	DET
ap-1815	138	9	,	,	PUNCT
ap-1815	138	10	b	b	PROPN
ap-1815	138	11	∈	∈	PROPN
ap-1815	138	12	e1	e1	PROPN
ap-1815	138	13	,	,	PUNCT
ap-1815	138	14	a	a	DET
ap-1815	138	15	≤1	≤1	PROPN
ap-1815	138	16	b	b	PROPN
ap-1815	138	17	if	if	SCONJ
ap-1815	138	18	and	and	CCONJ
ap-1815	138	19	only	only	ADV
ap-1815	138	20	if	if	SCONJ
ap-1815	138	21	ϕ(a	ϕ(a	NOUN
ap-1815	138	22	)	)	PUNCT
ap-1815	138	23	≤2	≤2	NOUN
ap-1815	138	24	ϕ(b	ϕ(b	PROPN
ap-1815	138	25	)	)	PUNCT
ap-1815	138	26	.	.	PUNCT
ap-1815	139	1	(	(	PUNCT
ap-1815	139	2	2	2	NUM
ap-1815	139	3	.	.	PUNCT
ap-1815	139	4	)	)	PUNCT
ap-1815	139	5	for	for	ADP
ap-1815	139	6	all	all	DET
ap-1815	139	7	s	s	PART
ap-1815	139	8	⊆	⊆	NUM
ap-1815	139	9	e1	e1	NOUN
ap-1815	139	10	,	,	PUNCT
ap-1815	139	11	∨	∨	NUM
ap-1815	139	12	e1	e1	PROPN
ap-1815	139	13	s	s	PART
ap-1815	139	14	exists	exist	VERB
ap-1815	139	15	if	if	SCONJ
ap-1815	139	16	and	and	CCONJ
ap-1815	139	17	only	only	ADV
ap-1815	139	18	if	if	SCONJ
ap-1815	139	19	∨	∨	PROPN
ap-1815	139	20	e2	e2	PROPN
ap-1815	139	21	ϕ(s	ϕ(s	PROPN
ap-1815	139	22	)	)	PUNCT
ap-1815	139	23	exists	exist	VERB
ap-1815	139	24	,	,	PUNCT
ap-1815	139	25	in	in	ADP
ap-1815	139	26	which	which	DET
ap-1815	139	27	case	case	NOUN
ap-1815	139	28	∨	∨	NUM
ap-1815	139	29	e2	e2	PROPN
ap-1815	139	30	ϕ(s	ϕ(s	PROPN
ap-1815	139	31	)	)	PUNCT
ap-1815	140	1	=	=	SYM
ap-1815	140	2	ϕ	ϕ	X
ap-1815	140	3	(	(	PUNCT
ap-1815	140	4	∨	∨	NUM
ap-1815	140	5	e1	e1	PROPN
ap-1815	140	6	s	s	PART
ap-1815	140	7	)	)	PUNCT
ap-1815	140	8	.	.	PUNCT
ap-1815	141	1	(	(	PUNCT
ap-1815	141	2	3	3	NUM
ap-1815	141	3	.	.	PUNCT
ap-1815	141	4	)	)	PUNCT
ap-1815	141	5	for	for	ADP
ap-1815	141	6	any	any	DET
ap-1815	141	7	increasingly	increasingly	ADV
ap-1815	141	8	directed	direct	VERB
ap-1815	141	9	net	net	NOUN
ap-1815	141	10	(	(	PUNCT
ap-1815	141	11	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	141	12	of	of	ADP
ap-1815	141	13	elements	element	NOUN
ap-1815	141	14	of	of	ADP
ap-1815	141	15	e1	e1	PROPN
ap-1815	141	16	and	and	CCONJ
ap-1815	141	17	a	a	DET
ap-1815	141	18	∈	∈	PROPN
ap-1815	141	19	e1	e1	NOUN
ap-1815	141	20	,	,	PUNCT
ap-1815	141	21	aα	aα	NOUN
ap-1815	141	22	↑	↑	NOUN
ap-1815	141	23	a	a	DET
ap-1815	141	24	if	if	NOUN
ap-1815	142	1	and	and	CCONJ
ap-1815	142	2	only	only	ADV
ap-1815	142	3	if	if	SCONJ
ap-1815	142	4	ϕ(aα	ϕ(aα	PROPN
ap-1815	142	5	)	)	PUNCT
ap-1815	142	6	↑	↑	PROPN
ap-1815	143	1	ϕ(a	ϕ(a	NOUN
ap-1815	143	2	)	)	PUNCT
ap-1815	143	3	.	.	PUNCT
ap-1815	144	1	(	(	PUNCT
ap-1815	144	2	4	4	NUM
ap-1815	144	3	.	.	PUNCT
ap-1815	144	4	)	)	PUNCT
ap-1815	145	1	for	for	ADP
ap-1815	145	2	any	any	DET
ap-1815	145	3	decreasingly	decreasingly	ADV
ap-1815	145	4	directed	direct	VERB
ap-1815	145	5	net	net	NOUN
ap-1815	145	6	(	(	PUNCT
ap-1815	145	7	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	145	8	of	of	ADP
ap-1815	145	9	elements	element	NOUN
ap-1815	145	10	of	of	ADP
ap-1815	145	11	e1	e1	PROPN
ap-1815	145	12	and	and	CCONJ
ap-1815	145	13	a	a	DET
ap-1815	145	14	∈	∈	PROPN
ap-1815	145	15	e1	e1	NOUN
ap-1815	145	16	,	,	PUNCT
ap-1815	145	17	aα	aα	NOUN
ap-1815	145	18	↓	↓	NOUN
ap-1815	145	19	a	a	DET
ap-1815	145	20	if	if	NOUN
ap-1815	145	21	and	and	CCONJ
ap-1815	145	22	only	only	ADV
ap-1815	145	23	if	if	SCONJ
ap-1815	145	24	ϕ(aα	ϕ(aα	PROPN
ap-1815	145	25	)	)	PUNCT
ap-1815	145	26	↓	↓	NOUN
ap-1815	146	1	ϕ(a	ϕ(a	NOUN
ap-1815	146	2	)	)	PUNCT
ap-1815	146	3	.	.	PUNCT
ap-1815	147	1	(	(	PUNCT
ap-1815	147	2	5	5	NUM
ap-1815	147	3	.	.	PUNCT
ap-1815	147	4	)	)	PUNCT
ap-1815	148	1	for	for	ADP
ap-1815	148	2	any	any	DET
ap-1815	148	3	net	net	NOUN
ap-1815	148	4	(	(	PUNCT
ap-1815	148	5	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	148	6	of	of	ADP
ap-1815	148	7	elements	element	NOUN
ap-1815	148	8	of	of	ADP
ap-1815	148	9	e1	e1	PROPN
ap-1815	148	10	and	and	CCONJ
ap-1815	148	11	a	a	DET
ap-1815	148	12	∈	∈	PROPN
ap-1815	148	13	e1	e1	NOUN
ap-1815	148	14	,	,	PUNCT
ap-1815	148	15	aα	aα	PROPN
ap-1815	148	16	(	(	PUNCT
ap-1815	148	17	o)1−−→	o)1−−→	VERB
ap-1815	148	18	a	a	DET
ap-1815	148	19	if	if	NOUN
ap-1815	148	20	and	and	CCONJ
ap-1815	148	21	only	only	ADV
ap-1815	148	22	if	if	SCONJ
ap-1815	148	23	ϕ(aα	ϕ(aα	NOUN
ap-1815	148	24	)	)	PUNCT
ap-1815	148	25	(	(	PUNCT
ap-1815	148	26	o)2−−→	o)2−−→	NOUN
ap-1815	148	27	ϕ(a	ϕ(a	NOUN
ap-1815	148	28	)	)	PUNCT
ap-1815	148	29	.	.	PUNCT
ap-1815	149	1	(	(	PUNCT
ap-1815	149	2	6	6	NUM
ap-1815	149	3	.	.	PUNCT
ap-1815	149	4	)	)	PUNCT
ap-1815	149	5	for	for	ADP
ap-1815	149	6	subsets	subset	NOUN
ap-1815	149	7	and	and	CCONJ
ap-1815	149	8	nets	net	NOUN
ap-1815	149	9	of	of	ADP
ap-1815	149	10	elements	element	NOUN
ap-1815	149	11	of	of	ADP
ap-1815	149	12	e1	e1	PROPN
ap-1815	149	13	and	and	CCONJ
ap-1815	149	14	e2	e2	VERB
ap-1815	149	15	the	the	DET
ap-1815	149	16	following	following	ADJ
ap-1815	149	17	statements	statement	NOUN
ap-1815	149	18	are	be	AUX
ap-1815	149	19	satisfied	satisfied	ADJ
ap-1815	149	20	:	:	PUNCT
ap-1815	149	21	•	•	ADP
ap-1815	149	22	for	for	ADP
ap-1815	149	23	all	all	DET
ap-1815	149	24	f	f	PROPN
ap-1815	149	25	⊆	⊆	NUM
ap-1815	149	26	e1	e1	NOUN
ap-1815	149	27	,	,	PUNCT
ap-1815	149	28	f	f	PROPN
ap-1815	149	29	is	be	AUX
ap-1815	149	30	τe1	τe1	ADJ
ap-1815	149	31	0	0	PUNCT
ap-1815	149	32	-closed	-close	VERB
ap-1815	149	33	if	if	SCONJ
ap-1815	149	34	and	and	CCONJ
ap-1815	149	35	only	only	ADV
ap-1815	149	36	if	if	SCONJ
ap-1815	149	37	ϕ(f	ϕ(f	NOUN
ap-1815	149	38	)	)	PUNCT
ap-1815	149	39	is	be	AUX
ap-1815	149	40	τe2	τe2	ADJ
ap-1815	149	41	0	0	NUM
ap-1815	150	1	-closed	-close	VERB
ap-1815	150	2	.	.	PUNCT
ap-1815	150	3	•	•	NUM
ap-1815	150	4	for	for	ADP
ap-1815	150	5	all	all	DET
ap-1815	150	6	u	u	PROPN
ap-1815	150	7	⊆	⊆	NUM
ap-1815	150	8	e1	e1	NOUN
ap-1815	150	9	,	,	PUNCT
ap-1815	150	10	u	u	NOUN
ap-1815	150	11	is	be	AUX
ap-1815	150	12	τe1	τe1	ADJ
ap-1815	150	13	0	0	NUM
ap-1815	150	14	-open	-open	NOUN
ap-1815	150	15	if	if	SCONJ
ap-1815	150	16	and	and	CCONJ
ap-1815	150	17	only	only	ADV
ap-1815	150	18	if	if	SCONJ
ap-1815	150	19	ϕ(u	ϕ(u	PROPN
ap-1815	150	20	)	)	PUNCT
ap-1815	150	21	is	be	AUX
ap-1815	150	22	τe2	τe2	ADJ
ap-1815	150	23	0	0	NUM
ap-1815	150	24	-open	-open	NOUN
ap-1815	150	25	.	.	NOUN
ap-1815	150	26	•	•	NUM
ap-1815	150	27	for	for	ADP
ap-1815	150	28	any	any	DET
ap-1815	150	29	net	net	NOUN
ap-1815	150	30	(	(	PUNCT
ap-1815	150	31	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	150	32	of	of	ADP
ap-1815	150	33	elements	element	NOUN
ap-1815	150	34	of	of	ADP
ap-1815	150	35	e1	e1	PROPN
ap-1815	150	36	and	and	CCONJ
ap-1815	150	37	a	a	DET
ap-1815	150	38	∈	∈	PROPN
ap-1815	150	39	e1	e1	NOUN
ap-1815	150	40	,	,	PUNCT
ap-1815	150	41	aα	aα	NOUN
ap-1815	150	42	τ	τ	PROPN
ap-1815	150	43	e1	e1	NOUN
ap-1815	150	44	0−−→	0−−→	NOUN
ap-1815	151	1	a	a	DET
ap-1815	151	2	if	if	NOUN
ap-1815	152	1	and	and	CCONJ
ap-1815	152	2	only	only	ADV
ap-1815	152	3	if	if	SCONJ
ap-1815	152	4	ϕ(aα	ϕ(aα	PROPN
ap-1815	153	1	)	)	PUNCT
ap-1815	153	2	τ	τ	PROPN
ap-1815	153	3	e2	e2	NOUN
ap-1815	153	4	0−−→	0−−→	NUM
ap-1815	153	5	ϕ(a	ϕ(a	NOUN
ap-1815	153	6	)	)	PUNCT
ap-1815	153	7	.	.	PUNCT
ap-1815	154	1	310	310	NUM
ap-1815	154	2	vol	vol	NOUN
ap-1815	154	3	.	.	PUNCT
ap-1815	155	1	53	53	NUM
ap-1815	155	2	no	no	NOUN
ap-1815	155	3	.	.	PUNCT
ap-1815	156	1	3/2013	3/2013	PROPN
ap-1815	156	2	inherited	inherit	VERB
ap-1815	156	3	properties	property	NOUN
ap-1815	156	4	of	of	ADP
ap-1815	156	5	effect	effect	NOUN
ap-1815	156	6	algebras	algebra	NOUN
ap-1815	156	7	proof	proof	NOUN
ap-1815	156	8	.	.	PUNCT
ap-1815	157	1	(	(	PUNCT
ap-1815	157	2	1	1	NUM
ap-1815	157	3	.	.	PUNCT
ap-1815	157	4	)	)	PUNCT
ap-1815	157	5	assume	assume	VERB
ap-1815	157	6	that	that	SCONJ
ap-1815	157	7	a	a	DET
ap-1815	157	8	,	,	PUNCT
ap-1815	157	9	b	b	PROPN
ap-1815	157	10	∈	∈	PROPN
ap-1815	157	11	e1	e1	NOUN
ap-1815	157	12	.	.	PUNCT
ap-1815	158	1	if	if	SCONJ
ap-1815	158	2	a	a	DET
ap-1815	158	3	≤1	≤1	PROPN
ap-1815	158	4	b.	b.	PROPN
ap-1815	158	5	from	from	ADP
ap-1815	158	6	lemma	lemma	PROPN
ap-1815	158	7	2.8	2.8	NUM
ap-1815	158	8	we	we	PRON
ap-1815	158	9	have	have	VERB
ap-1815	158	10	that	that	DET
ap-1815	158	11	ϕ(a	ϕ(a	NOUN
ap-1815	158	12	)	)	PUNCT
ap-1815	158	13	≤2	≤2	NOUN
ap-1815	158	14	ϕ(b	ϕ(b	PROPN
ap-1815	158	15	)	)	PUNCT
ap-1815	158	16	.	.	PUNCT
ap-1815	159	1	conversely	conversely	ADV
ap-1815	159	2	,	,	PUNCT
ap-1815	159	3	let	let	VERB
ap-1815	159	4	ϕ(a	ϕ(a	NOUN
ap-1815	159	5	)	)	PUNCT
ap-1815	159	6	≤2	≤2	NOUN
ap-1815	159	7	ϕ(b).then	ϕ(b).then	ADV
ap-1815	159	8	again	again	ADV
ap-1815	159	9	by	by	ADP
ap-1815	159	10	lemma	lemma	PROPN
ap-1815	159	11	2.8	2.8	NUM
ap-1815	159	12	applied	apply	VERB
ap-1815	159	13	to	to	ADP
ap-1815	159	14	ϕ−1	ϕ−1	PROPN
ap-1815	159	15	we	we	PRON
ap-1815	159	16	get	get	VERB
ap-1815	159	17	a	a	DET
ap-1815	159	18	=	=	PUNCT
ap-1815	159	19	ϕ−1(ϕ(a	ϕ−1(ϕ(a	NOUN
ap-1815	159	20	)	)	PUNCT
ap-1815	159	21	)	)	PUNCT
ap-1815	160	1	≤1	≤1	PROPN
ap-1815	160	2	ϕ	ϕ	PROPN
ap-1815	160	3	−1(ϕ(b	−1(ϕ(b	NOUN
ap-1815	160	4	)	)	PUNCT
ap-1815	160	5	)	)	PUNCT
ap-1815	161	1	=	=	SYM
ap-1815	161	2	b.	b.	PROPN
ap-1815	161	3	(	(	PUNCT
ap-1815	161	4	2	2	NUM
ap-1815	161	5	.	.	PUNCT
ap-1815	161	6	)	)	PUNCT
ap-1815	161	7	assume	assume	VERB
ap-1815	161	8	that	that	SCONJ
ap-1815	161	9	s	s	VERB
ap-1815	161	10	⊆	⊆	NUM
ap-1815	161	11	e1	e1	NOUN
ap-1815	161	12	such	such	ADJ
ap-1815	161	13	that	that	SCONJ
ap-1815	161	14	∨	∨	NUM
ap-1815	161	15	e1	e1	PROPN
ap-1815	161	16	s	s	PART
ap-1815	161	17	exists	exist	VERB
ap-1815	161	18	.	.	PUNCT
ap-1815	162	1	let	let	VERB
ap-1815	162	2	us	we	PRON
ap-1815	162	3	put	put	VERB
ap-1815	162	4	a	a	DET
ap-1815	162	5	=	=	SYM
ap-1815	162	6	∨	∨	NOUN
ap-1815	162	7	e1	e1	PROPN
ap-1815	162	8	s.	s.	PROPN
ap-1815	162	9	then	then	ADV
ap-1815	162	10	,	,	PUNCT
ap-1815	162	11	for	for	SCONJ
ap-1815	162	12	all	all	DET
ap-1815	162	13	s	s	PART
ap-1815	162	14	∈	∈	PROPN
ap-1815	162	15	s	s	NOUN
ap-1815	162	16	,	,	PUNCT
ap-1815	162	17	s	s	VERB
ap-1815	162	18	≤	≤	NOUN
ap-1815	162	19	a	a	PRON
ap-1815	163	1	and	and	CCONJ
ap-1815	163	2	we	we	PRON
ap-1815	163	3	get	get	VERB
ap-1815	163	4	from	from	ADP
ap-1815	163	5	lemma	lemma	PROPN
ap-1815	163	6	2.8	2.8	NUM
ap-1815	163	7	that	that	DET
ap-1815	163	8	ϕ(s	ϕ(s	PROPN
ap-1815	163	9	)	)	PUNCT
ap-1815	163	10	≤1	≤1	PROPN
ap-1815	163	11	ϕ(a	ϕ(a	NOUN
ap-1815	163	12	)	)	PUNCT
ap-1815	163	13	.	.	PUNCT
ap-1815	164	1	hence	hence	ADV
ap-1815	164	2	ϕ(a	ϕ(a	NOUN
ap-1815	164	3	)	)	PUNCT
ap-1815	164	4	is	be	AUX
ap-1815	164	5	an	an	DET
ap-1815	164	6	upper	upper	ADJ
ap-1815	164	7	bound	bound	NOUN
ap-1815	164	8	of	of	ADP
ap-1815	164	9	ϕ(s	ϕ(s	PROPN
ap-1815	164	10	)	)	PUNCT
ap-1815	164	11	for	for	ADP
ap-1815	164	12	all	all	DET
ap-1815	164	13	s	s	PROPN
ap-1815	164	14	∈	∈	PROPN
ap-1815	164	15	s.	s.	PROPN
ap-1815	164	16	let	let	VERB
ap-1815	164	17	d	d	X
ap-1815	164	18	=	=	SYM
ap-1815	164	19	ϕ(c	ϕ(c	PROPN
ap-1815	164	20	)	)	PUNCT
ap-1815	164	21	∈	∈	PROPN
ap-1815	164	22	e2	e2	PROPN
ap-1815	164	23	,	,	PUNCT
ap-1815	164	24	c	c	PROPN
ap-1815	164	25	∈	∈	PROPN
ap-1815	164	26	e1	e1	PROPN
ap-1815	164	27	be	be	VERB
ap-1815	164	28	an	an	DET
ap-1815	164	29	upper	upper	ADJ
ap-1815	164	30	bound	bound	NOUN
ap-1815	164	31	of	of	ADP
ap-1815	164	32	ϕ(s	ϕ(s	PROPN
ap-1815	164	33	)	)	PUNCT
ap-1815	164	34	for	for	ADP
ap-1815	164	35	all	all	DET
ap-1815	164	36	s	s	PROPN
ap-1815	164	37	∈	∈	PROPN
ap-1815	164	38	s.	s.	PROPN
ap-1815	164	39	then	then	ADV
ap-1815	164	40	c	c	PROPN
ap-1815	164	41	∈	∈	PROPN
ap-1815	164	42	e1	e1	PROPN
ap-1815	164	43	is	be	AUX
ap-1815	164	44	an	an	DET
ap-1815	164	45	upper	upper	ADJ
ap-1815	164	46	bound	bound	NOUN
ap-1815	164	47	of	of	ADP
ap-1815	164	48	s	s	PRON
ap-1815	164	49	for	for	ADP
ap-1815	164	50	all	all	PRON
ap-1815	165	1	s	s	PART
ap-1815	165	2	∈	∈	NOUN
ap-1815	165	3	s	s	NOUN
ap-1815	165	4	by	by	ADP
ap-1815	165	5	part	part	NOUN
ap-1815	165	6	1	1	NUM
ap-1815	165	7	.	.	PUNCT
ap-1815	166	1	this	this	DET
ap-1815	166	2	yields	yield	NOUN
ap-1815	166	3	that	that	PRON
ap-1815	166	4	a	a	DET
ap-1815	166	5	≤	≤	ADJ
ap-1815	166	6	c.	c.	NOUN
ap-1815	166	7	therefore	therefore	ADV
ap-1815	166	8	by	by	ADP
ap-1815	166	9	lemma	lemma	PROPN
ap-1815	166	10	2.8	2.8	NUM
ap-1815	166	11	we	we	PRON
ap-1815	166	12	get	get	VERB
ap-1815	166	13	ϕ(a	ϕ(a	NOUN
ap-1815	166	14	)	)	PUNCT
ap-1815	166	15	≤	≤	NUM
ap-1815	166	16	ϕ(c	ϕ(c	NOUN
ap-1815	166	17	)	)	PUNCT
ap-1815	167	1	=	=	SYM
ap-1815	167	2	d	d	NOUN
ap-1815	167	3	,	,	PUNCT
ap-1815	167	4	i.e.	i.e.	X
ap-1815	167	5	∨	∨	NUM
ap-1815	167	6	e2	e2	PROPN
ap-1815	167	7	ϕ(s	ϕ(s	PROPN
ap-1815	167	8	)	)	PUNCT
ap-1815	167	9	=	=	SYM
ap-1815	168	1	ϕ	ϕ	PROPN
ap-1815	168	2	(	(	PUNCT
ap-1815	168	3	∨	∨	NUM
ap-1815	168	4	e1	e1	PROPN
ap-1815	168	5	s	s	NOUN
ap-1815	168	6	)	)	PUNCT
ap-1815	168	7	.	.	PUNCT
ap-1815	169	1	the	the	DET
ap-1815	169	2	converse	converse	PROPN
ap-1815	169	3	implication	implication	NOUN
ap-1815	169	4	follows	follow	VERB
ap-1815	169	5	by	by	ADP
ap-1815	169	6	the	the	DET
ap-1815	169	7	same	same	ADJ
ap-1815	169	8	considerations	consideration	NOUN
ap-1815	169	9	as	as	SCONJ
ap-1815	169	10	were	be	AUX
ap-1815	169	11	applied	apply	VERB
ap-1815	169	12	above	above	ADV
ap-1815	169	13	to	to	ADP
ap-1815	169	14	ϕ−1	ϕ−1	PROPN
ap-1815	169	15	and	and	CCONJ
ap-1815	169	16	the	the	DET
ap-1815	169	17	assumption	assumption	NOUN
ap-1815	169	18	that	that	SCONJ
ap-1815	169	19	∨	∨	PROPN
ap-1815	169	20	e2	e2	PROPN
ap-1815	169	21	ϕ(s	ϕ(s	PROPN
ap-1815	169	22	)	)	PUNCT
ap-1815	169	23	exists	exist	VERB
ap-1815	169	24	.	.	PUNCT
ap-1815	170	1	(	(	PUNCT
ap-1815	170	2	3	3	NUM
ap-1815	170	3	.	.	PUNCT
ap-1815	170	4	)	)	PUNCT
ap-1815	170	5	assume	assume	VERB
ap-1815	170	6	α	α	NOUN
ap-1815	170	7	≤	≤	NUM
ap-1815	171	1	β	β	X
ap-1815	171	2	,	,	PUNCT
ap-1815	171	3	α	α	X
ap-1815	171	4	,	,	PUNCT
ap-1815	171	5	β	β	X
ap-1815	171	6	∈	∈	PROPN
ap-1815	171	7	λ	λ	PROPN
ap-1815	171	8	.	.	PUNCT
ap-1815	172	1	then	then	ADV
ap-1815	172	2	aα	aα	NOUN
ap-1815	172	3	≤1	≤1	ADJ
ap-1815	173	1	aβ	aβ	INTJ
ap-1815	173	2	and	and	CCONJ
ap-1815	173	3	by	by	ADP
ap-1815	173	4	lemma	lemma	PROPN
ap-1815	173	5	2.8	2.8	NUM
ap-1815	173	6	we	we	PRON
ap-1815	173	7	obtain	obtain	VERB
ap-1815	173	8	that	that	DET
ap-1815	173	9	ϕ(aα	ϕ(aα	NOUN
ap-1815	173	10	)	)	PUNCT
ap-1815	173	11	≤2	≤2	NOUN
ap-1815	173	12	ϕ(aβ	ϕ(aβ	NOUN
ap-1815	173	13	)	)	PUNCT
ap-1815	173	14	.	.	PUNCT
ap-1815	174	1	it	it	PRON
ap-1815	174	2	follows	follow	VERB
ap-1815	174	3	that	that	SCONJ
ap-1815	174	4	(	(	PUNCT
ap-1815	174	5	ϕ(aα))α∈λ	ϕ(aα))α∈λ	NOUN
ap-1815	174	6	is	be	AUX
ap-1815	174	7	an	an	DET
ap-1815	174	8	increasingly	increasingly	ADV
ap-1815	174	9	directed	direct	VERB
ap-1815	174	10	net	net	NOUN
ap-1815	174	11	.	.	PUNCT
ap-1815	175	1	assume	assume	VERB
ap-1815	175	2	now	now	ADV
ap-1815	175	3	that	that	SCONJ
ap-1815	175	4	aα	aα	PROPN
ap-1815	175	5	↑	↑	NOUN
ap-1815	175	6	a.	a.	NOUN
ap-1815	175	7	then	then	ADV
ap-1815	175	8	by	by	ADP
ap-1815	175	9	(	(	PUNCT
ap-1815	175	10	2	2	NUM
ap-1815	175	11	.	.	PUNCT
ap-1815	175	12	)	)	PUNCT
ap-1815	176	1	we	we	PRON
ap-1815	176	2	obtain	obtain	VERB
ap-1815	176	3	ϕ(aα	ϕ(aα	PROPN
ap-1815	176	4	)	)	PUNCT
ap-1815	176	5	↑	↑	PROPN
ap-1815	176	6	ϕ(a	ϕ(a	NOUN
ap-1815	176	7	)	)	PUNCT
ap-1815	176	8	.	.	PUNCT
ap-1815	177	1	the	the	DET
ap-1815	177	2	converse	converse	PROPN
ap-1815	177	3	implication	implication	NOUN
ap-1815	177	4	follows	follow	VERB
ap-1815	177	5	by	by	ADP
ap-1815	177	6	the	the	DET
ap-1815	177	7	same	same	ADJ
ap-1815	177	8	considerations	consideration	NOUN
ap-1815	177	9	as	as	SCONJ
ap-1815	177	10	were	be	AUX
ap-1815	177	11	applied	apply	VERB
ap-1815	177	12	above	above	ADV
ap-1815	177	13	to	to	ADP
ap-1815	177	14	ϕ−1	ϕ−1	PROPN
ap-1815	177	15	and	and	CCONJ
ap-1815	177	16	ϕ(aα	ϕ(aα	PROPN
ap-1815	177	17	)	)	PUNCT
ap-1815	177	18	↑	↑	PROPN
ap-1815	177	19	ϕ(a	ϕ(a	NOUN
ap-1815	177	20	)	)	PUNCT
ap-1815	177	21	.	.	PUNCT
ap-1815	178	1	(	(	PUNCT
ap-1815	178	2	4	4	NUM
ap-1815	178	3	.	.	PUNCT
ap-1815	178	4	)	)	PUNCT
ap-1815	179	1	it	it	PRON
ap-1815	179	2	follows	follow	VERB
ap-1815	179	3	by	by	ADP
ap-1815	179	4	the	the	DET
ap-1815	179	5	considerations	consideration	NOUN
ap-1815	179	6	dual	dual	ADJ
ap-1815	179	7	to	to	ADP
ap-1815	179	8	them	they	PRON
ap-1815	179	9	in	in	ADP
ap-1815	179	10	(	(	PUNCT
ap-1815	179	11	3	3	NUM
ap-1815	179	12	.	.	NUM
ap-1815	179	13	)	)	PUNCT
ap-1815	179	14	.	.	PUNCT
ap-1815	180	1	(	(	PUNCT
ap-1815	180	2	5	5	NUM
ap-1815	180	3	.	.	PUNCT
ap-1815	180	4	)	)	PUNCT
ap-1815	180	5	assume	assume	VERB
ap-1815	180	6	first	first	ADV
ap-1815	180	7	that	that	DET
ap-1815	180	8	aα	aα	PROPN
ap-1815	180	9	(	(	PUNCT
ap-1815	180	10	o)1−−→	o)1−−→	VERB
ap-1815	180	11	a.	a.	NOUN
ap-1815	180	12	hence	hence	ADV
ap-1815	180	13	there	there	PRON
ap-1815	180	14	are	be	VERB
ap-1815	180	15	nets	net	NOUN
ap-1815	180	16	(	(	PUNCT
ap-1815	180	17	uα)α∈λ	uα)α∈λ	NUM
ap-1815	180	18	and	and	CCONJ
ap-1815	180	19	(	(	PUNCT
ap-1815	180	20	vα)α∈λ	vα)α∈λ	NUM
ap-1815	180	21	of	of	ADP
ap-1815	180	22	elements	element	NOUN
ap-1815	180	23	of	of	ADP
ap-1815	180	24	e1	e1	NOUN
ap-1815	180	25	such	such	ADJ
ap-1815	180	26	that	that	SCONJ
ap-1815	180	27	a	a	DET
ap-1815	180	28	↑	↑	NOUN
ap-1815	180	29	uα	uα	PROPN
ap-1815	180	30	≤	≤	NUM
ap-1815	180	31	aα	aα	NOUN
ap-1815	180	32	≤	≤	NUM
ap-1815	180	33	vα	vα	ADP
ap-1815	180	34	↓	↓	PROPN
ap-1815	180	35	a.	a.	NOUN
ap-1815	180	36	from	from	ADP
ap-1815	180	37	lemma	lemma	PROPN
ap-1815	180	38	2.8	2.8	NUM
ap-1815	180	39	and	and	CCONJ
ap-1815	180	40	(	(	PUNCT
ap-1815	180	41	3	3	NUM
ap-1815	180	42	.	.	PUNCT
ap-1815	180	43	)	)	PUNCT
ap-1815	181	1	and	and	CCONJ
ap-1815	181	2	(	(	PUNCT
ap-1815	181	3	4	4	NUM
ap-1815	181	4	.	.	PUNCT
ap-1815	181	5	)	)	PUNCT
ap-1815	182	1	we	we	PRON
ap-1815	182	2	obtain	obtain	VERB
ap-1815	182	3	nets	net	NOUN
ap-1815	182	4	(	(	PUNCT
ap-1815	182	5	ϕ(uα))α∈λ	ϕ(uα))α∈λ	PROPN
ap-1815	182	6	and	and	CCONJ
ap-1815	182	7	(	(	PUNCT
ap-1815	182	8	ϕ(vα))α∈λ	ϕ(vα))α∈λ	PROPN
ap-1815	182	9	of	of	ADP
ap-1815	182	10	elements	element	NOUN
ap-1815	182	11	of	of	ADP
ap-1815	182	12	e2	e2	PROPN
ap-1815	182	13	such	such	ADJ
ap-1815	182	14	that	that	SCONJ
ap-1815	182	15	ϕ(a	ϕ(a	NOUN
ap-1815	182	16	)	)	PUNCT
ap-1815	182	17	↑	↑	PROPN
ap-1815	182	18	ϕ(uα	ϕ(uα	PROPN
ap-1815	182	19	)	)	PUNCT
ap-1815	182	20	≤	≤	NUM
ap-1815	182	21	ϕ(aα	ϕ(aα	PROPN
ap-1815	182	22	)	)	PUNCT
ap-1815	182	23	≤	≤	NUM
ap-1815	182	24	ϕ(vα	ϕ(vα	PROPN
ap-1815	182	25	)	)	PUNCT
ap-1815	182	26	↓	↓	NOUN
ap-1815	182	27	ϕ(a	ϕ(a	NOUN
ap-1815	182	28	)	)	PUNCT
ap-1815	182	29	.	.	PUNCT
ap-1815	183	1	it	it	PRON
ap-1815	183	2	follows	follow	VERB
ap-1815	183	3	that	that	SCONJ
ap-1815	183	4	ϕ(aα	ϕ(aα	X
ap-1815	183	5	)	)	PUNCT
ap-1815	183	6	(	(	PUNCT
ap-1815	183	7	o)2−−→	o)2−−→	NOUN
ap-1815	183	8	ϕ(a	ϕ(a	NOUN
ap-1815	183	9	)	)	PUNCT
ap-1815	183	10	.	.	PUNCT
ap-1815	184	1	the	the	DET
ap-1815	184	2	converse	converse	PROPN
ap-1815	184	3	implication	implication	NOUN
ap-1815	184	4	follows	follow	VERB
ap-1815	184	5	by	by	ADP
ap-1815	184	6	the	the	DET
ap-1815	184	7	same	same	ADJ
ap-1815	184	8	considerations	consideration	NOUN
ap-1815	184	9	as	as	ADP
ap-1815	184	10	above	above	ADV
ap-1815	184	11	applied	apply	VERB
ap-1815	184	12	to	to	ADP
ap-1815	184	13	ϕ−1	ϕ−1	PROPN
ap-1815	184	14	and	and	CCONJ
ap-1815	184	15	the	the	DET
ap-1815	184	16	net	net	NOUN
ap-1815	184	17	(	(	PUNCT
ap-1815	184	18	ϕ(aα))α∈λ	ϕ(aα))α∈λ	NOUN
ap-1815	184	19	of	of	ADP
ap-1815	184	20	elements	element	NOUN
ap-1815	184	21	of	of	ADP
ap-1815	184	22	e2	e2	PROPN
ap-1815	184	23	and	and	CCONJ
ap-1815	184	24	ϕ(a	ϕ(a	PROPN
ap-1815	184	25	)	)	PUNCT
ap-1815	184	26	∈	∈	PROPN
ap-1815	184	27	e2	e2	PROPN
ap-1815	184	28	such	such	ADJ
ap-1815	184	29	that	that	PRON
ap-1815	184	30	ϕ(aα	ϕ(aα	NOUN
ap-1815	184	31	)	)	PUNCT
ap-1815	184	32	(	(	PUNCT
ap-1815	184	33	o)2−−→	o)2−−→	NOUN
ap-1815	184	34	ϕ(a	ϕ(a	NOUN
ap-1815	184	35	)	)	PUNCT
ap-1815	184	36	.	.	PUNCT
ap-1815	185	1	(	(	PUNCT
ap-1815	185	2	6	6	NUM
ap-1815	185	3	.	.	PUNCT
ap-1815	185	4	)	)	PUNCT
ap-1815	185	5	assume	assume	VERB
ap-1815	185	6	that	that	SCONJ
ap-1815	185	7	f	f	PROPN
ap-1815	185	8	⊆	⊆	NUM
ap-1815	185	9	e1	e1	PROPN
ap-1815	185	10	.	.	PUNCT
ap-1815	186	1	then	then	ADV
ap-1815	186	2	f	f	PROPN
ap-1815	186	3	is	be	AUX
ap-1815	186	4	τe1	τe1	ADJ
ap-1815	186	5	0	0	PUNCT
ap-1815	186	6	-closed	-close	VERB
ap-1815	186	7	if	if	SCONJ
ap-1815	186	8	and	and	CCONJ
ap-1815	186	9	only	only	ADV
ap-1815	186	10	if	if	SCONJ
ap-1815	186	11	by	by	ADP
ap-1815	186	12	theorem	theorem	NOUN
ap-1815	186	13	2.6	2.6	NUM
ap-1815	186	14	for	for	ADP
ap-1815	186	15	every	every	DET
ap-1815	186	16	net	net	NOUN
ap-1815	186	17	(	(	PUNCT
ap-1815	186	18	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	186	19	of	of	ADP
ap-1815	186	20	elements	element	NOUN
ap-1815	186	21	of	of	ADP
ap-1815	186	22	e1	e1	NOUN
ap-1815	186	23	it	it	PRON
ap-1815	186	24	holds	hold	VERB
ap-1815	186	25	(	(	PUNCT
ap-1815	186	26	aα	aα	NOUN
ap-1815	186	27	∈	∈	PROPN
ap-1815	186	28	f	f	PROPN
ap-1815	186	29	,	,	PUNCT
ap-1815	186	30	α	α	PROPN
ap-1815	186	31	∈	∈	PROPN
ap-1815	186	32	λ	λ	PROPN
ap-1815	186	33	,	,	PUNCT
ap-1815	186	34	aα	aα	PROPN
ap-1815	186	35	(	(	PUNCT
ap-1815	186	36	o)1−−→	o)1−−→	AUX
ap-1815	186	37	a	a	PRON
ap-1815	186	38	)	)	PUNCT
ap-1815	186	39	⇒	⇒	NOUN
ap-1815	186	40	a	a	DET
ap-1815	186	41	∈	∈	ADJ
ap-1815	186	42	f	f	NOUN
ap-1815	187	1	if	if	SCONJ
ap-1815	187	2	and	and	CCONJ
ap-1815	187	3	only	only	ADV
ap-1815	187	4	if	if	SCONJ
ap-1815	187	5	by	by	ADP
ap-1815	187	6	(	(	PUNCT
ap-1815	187	7	5	5	NUM
ap-1815	187	8	.	.	PUNCT
ap-1815	187	9	)	)	PUNCT
ap-1815	188	1	for	for	ADP
ap-1815	188	2	every	every	DET
ap-1815	188	3	net	net	NOUN
ap-1815	188	4	(	(	PUNCT
ap-1815	188	5	ϕ(aα))α∈λ	ϕ(aα))α∈λ	NOUN
ap-1815	188	6	of	of	ADP
ap-1815	188	7	elements	element	NOUN
ap-1815	188	8	of	of	ADP
ap-1815	188	9	e2	e2	PROPN
ap-1815	188	10	it	it	PRON
ap-1815	188	11	holds	hold	VERB
ap-1815	188	12	(	(	PUNCT
ap-1815	188	13	ϕ(aα	ϕ(aα	ADJ
ap-1815	188	14	)	)	PUNCT
ap-1815	188	15	∈	∈	PROPN
ap-1815	188	16	ϕ(f	ϕ(f	PROPN
ap-1815	188	17	)	)	PUNCT
ap-1815	188	18	,	,	PUNCT
ap-1815	188	19	α	α	PROPN
ap-1815	188	20	∈	∈	PROPN
ap-1815	188	21	λ	λ	PROPN
ap-1815	188	22	,	,	PUNCT
ap-1815	188	23	ϕ(aα	ϕ(aα	PROPN
ap-1815	188	24	)	)	PUNCT
ap-1815	188	25	(	(	PUNCT
ap-1815	188	26	o)2−−→	o)2−−→	NOUN
ap-1815	188	27	ϕ(a	ϕ(a	NOUN
ap-1815	188	28	)	)	PUNCT
ap-1815	188	29	)	)	PUNCT
ap-1815	188	30	⇒	⇒	PROPN
ap-1815	188	31	ϕ(a	ϕ(a	PROPN
ap-1815	188	32	)	)	PUNCT
ap-1815	188	33	∈	∈	PROPN
ap-1815	188	34	ϕ(f	ϕ(f	NOUN
ap-1815	188	35	)	)	PUNCT
ap-1815	188	36	if	if	SCONJ
ap-1815	188	37	and	and	CCONJ
ap-1815	188	38	only	only	ADV
ap-1815	188	39	if	if	SCONJ
ap-1815	188	40	for	for	ADP
ap-1815	188	41	every	every	DET
ap-1815	188	42	net	net	NOUN
ap-1815	188	43	(	(	PUNCT
ap-1815	188	44	bα)α∈λ	bα)α∈λ	NUM
ap-1815	188	45	of	of	ADP
ap-1815	188	46	elements	element	NOUN
ap-1815	188	47	of	of	ADP
ap-1815	188	48	e2	e2	PROPN
ap-1815	188	49	it	it	PRON
ap-1815	188	50	holds	hold	VERB
ap-1815	188	51	(	(	PUNCT
ap-1815	188	52	bα	bα	NOUN
ap-1815	188	53	∈	∈	PROPN
ap-1815	188	54	ϕ(f	ϕ(f	PROPN
ap-1815	188	55	)	)	PUNCT
ap-1815	188	56	,	,	PUNCT
ap-1815	188	57	α	α	PROPN
ap-1815	188	58	∈	∈	PROPN
ap-1815	188	59	λ	λ	PROPN
ap-1815	188	60	,	,	PUNCT
ap-1815	188	61	bα	bα	PROPN
ap-1815	188	62	(	(	PUNCT
ap-1815	188	63	o)2−−→	o)2−−→	NOUN
ap-1815	188	64	b	b	NOUN
ap-1815	188	65	)	)	PUNCT
ap-1815	188	66	⇒	⇒	NOUN
ap-1815	188	67	b	b	X
ap-1815	188	68	∈	∈	PROPN
ap-1815	188	69	ϕ(f	ϕ(f	X
ap-1815	188	70	)	)	PUNCT
ap-1815	189	1	if	if	SCONJ
ap-1815	189	2	and	and	CCONJ
ap-1815	189	3	only	only	ADV
ap-1815	189	4	if	if	SCONJ
ap-1815	189	5	by	by	SCONJ
ap-1815	189	6	theorem	theorem	NOUN
ap-1815	189	7	2.6	2.6	NUM
ap-1815	189	8	ϕ(f	ϕ(f	X
ap-1815	189	9	)	)	PUNCT
ap-1815	189	10	is	be	AUX
ap-1815	189	11	τe2	τe2	ADJ
ap-1815	189	12	0	0	NUM
ap-1815	189	13	-closed	-closed	ADJ
ap-1815	189	14	.	.	PUNCT
ap-1815	190	1	now	now	ADV
ap-1815	190	2	,	,	PUNCT
ap-1815	190	3	let	let	VERB
ap-1815	190	4	us	we	PRON
ap-1815	190	5	assume	assume	VERB
ap-1815	190	6	that	that	SCONJ
ap-1815	190	7	u	u	PROPN
ap-1815	190	8	⊆	⊆	NUM
ap-1815	190	9	e1	e1	NOUN
ap-1815	190	10	.	.	PUNCT
ap-1815	191	1	then	then	ADV
ap-1815	191	2	u	u	NOUN
ap-1815	191	3	is	be	AUX
ap-1815	191	4	τe1	τe1	ADJ
ap-1815	191	5	0	0	PUNCT
ap-1815	191	6	open	open	ADJ
ap-1815	191	7	if	if	SCONJ
ap-1815	191	8	and	and	CCONJ
ap-1815	191	9	only	only	ADV
ap-1815	191	10	if	if	SCONJ
ap-1815	191	11	f	f	PROPN
ap-1815	191	12	=	=	SYM
ap-1815	191	13	e1	e1	PROPN
ap-1815	191	14	\	\	NOUN
ap-1815	191	15	u	u	NOUN
ap-1815	191	16	is	be	AUX
ap-1815	191	17	τe1	τe1	ADJ
ap-1815	191	18	0	0	PUNCT
ap-1815	192	1	-closed	-close	VERB
ap-1815	192	2	if	if	SCONJ
ap-1815	192	3	and	and	CCONJ
ap-1815	192	4	only	only	ADV
ap-1815	192	5	if	if	SCONJ
ap-1815	192	6	ϕ(f	ϕ(f	NOUN
ap-1815	192	7	)	)	PUNCT
ap-1815	193	1	=	=	PUNCT
ap-1815	193	2	ϕ(e1	ϕ(e1	NOUN
ap-1815	193	3	\	\	X
ap-1815	193	4	u	u	NOUN
ap-1815	193	5	)	)	PUNCT
ap-1815	193	6	=	=	SYM
ap-1815	193	7	ϕ(e1	ϕ(e1	NOUN
ap-1815	193	8	)	)	PUNCT
ap-1815	193	9	\	\	NOUN
ap-1815	193	10	ϕ(u	ϕ(u	NOUN
ap-1815	193	11	)	)	PUNCT
ap-1815	193	12	=	=	SYM
ap-1815	193	13	e2	e2	PROPN
ap-1815	193	14	\	\	PROPN
ap-1815	193	15	ϕ(u	ϕ(u	PROPN
ap-1815	193	16	)	)	PUNCT
ap-1815	193	17	is	be	AUX
ap-1815	193	18	τe2	τe2	ADJ
ap-1815	193	19	0	0	PUNCT
ap-1815	193	20	-closed	-close	VERB
ap-1815	193	21	if	if	SCONJ
ap-1815	193	22	and	and	CCONJ
ap-1815	193	23	only	only	ADV
ap-1815	193	24	if	if	SCONJ
ap-1815	193	25	ϕ(u	ϕ(u	PROPN
ap-1815	193	26	)	)	PUNCT
ap-1815	193	27	is	be	AUX
ap-1815	193	28	τe2	τe2	ADJ
ap-1815	193	29	0	0	NUM
ap-1815	193	30	open	open	ADJ
ap-1815	193	31	.	.	PUNCT
ap-1815	194	1	in	in	ADP
ap-1815	194	2	what	what	PRON
ap-1815	194	3	remains	remain	VERB
ap-1815	194	4	,	,	PUNCT
ap-1815	194	5	we	we	PRON
ap-1815	194	6	will	will	AUX
ap-1815	194	7	assume	assume	VERB
ap-1815	194	8	that	that	SCONJ
ap-1815	194	9	we	we	PRON
ap-1815	194	10	have	have	VERB
ap-1815	194	11	a	a	DET
ap-1815	194	12	net	net	NOUN
ap-1815	194	13	(	(	PUNCT
ap-1815	194	14	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	194	15	of	of	ADP
ap-1815	194	16	elements	element	NOUN
ap-1815	194	17	of	of	ADP
ap-1815	194	18	e1	e1	PROPN
ap-1815	194	19	and	and	CCONJ
ap-1815	194	20	a	a	DET
ap-1815	194	21	∈	∈	PROPN
ap-1815	194	22	e1	e1	NOUN
ap-1815	194	23	,	,	PUNCT
ap-1815	194	24	aα	aα	NOUN
ap-1815	194	25	τ	τ	PROPN
ap-1815	194	26	e1	e1	NOUN
ap-1815	194	27	0−−→	0−−→	NUM
ap-1815	194	28	a.	a.	NOUN
ap-1815	194	29	let	let	VERB
ap-1815	194	30	us	we	PRON
ap-1815	194	31	check	check	VERB
ap-1815	194	32	that	that	SCONJ
ap-1815	194	33	ϕ(aα	ϕ(aα	X
ap-1815	194	34	)	)	PUNCT
ap-1815	194	35	τ	τ	PROPN
ap-1815	194	36	e2	e2	NOUN
ap-1815	194	37	0−−→	0−−→	NUM
ap-1815	194	38	ϕ(a	ϕ(a	NOUN
ap-1815	194	39	)	)	PUNCT
ap-1815	194	40	.	.	PUNCT
ap-1815	195	1	assume	assume	VERB
ap-1815	195	2	that	that	SCONJ
ap-1815	195	3	we	we	PRON
ap-1815	195	4	have	have	VERB
ap-1815	195	5	a	a	DET
ap-1815	195	6	τe2	τe2	ADJ
ap-1815	195	7	0	0	NUM
ap-1815	195	8	-open	-open	NOUN
ap-1815	195	9	set	set	VERB
ap-1815	195	10	v	v	ADP
ap-1815	195	11	⊆	⊆	NUM
ap-1815	195	12	e2	e2	NOUN
ap-1815	195	13	such	such	ADJ
ap-1815	195	14	that	that	SCONJ
ap-1815	195	15	ϕ(a	ϕ(a	NOUN
ap-1815	195	16	)	)	PUNCT
ap-1815	195	17	∈	∈	PROPN
ap-1815	195	18	v	v	NOUN
ap-1815	195	19	.	.	PUNCT
ap-1815	196	1	since	since	SCONJ
ap-1815	196	2	v	v	NOUN
ap-1815	196	3	=	=	SYM
ap-1815	196	4	ϕ(u	ϕ(u	PROPN
ap-1815	196	5	)	)	PUNCT
ap-1815	196	6	and	and	CCONJ
ap-1815	196	7	u	u	X
ap-1815	196	8	=	=	PROPN
ap-1815	196	9	ϕ−1(v	ϕ−1(v	PROPN
ap-1815	196	10	)	)	PUNCT
ap-1815	196	11	for	for	ADP
ap-1815	196	12	some	some	DET
ap-1815	196	13	τe1	τe1	ADJ
ap-1815	196	14	0	0	NUM
ap-1815	196	15	-open	-open	NOUN
ap-1815	196	16	subset	subset	NOUN
ap-1815	196	17	u	u	NOUN
ap-1815	196	18	⊆	⊆	NUM
ap-1815	196	19	e1	e1	NOUN
ap-1815	196	20	we	we	PRON
ap-1815	196	21	get	get	VERB
ap-1815	196	22	that	that	PRON
ap-1815	196	23	a	a	DET
ap-1815	196	24	∈	∈	PROPN
ap-1815	196	25	u	u	NOUN
ap-1815	196	26	.	.	PUNCT
ap-1815	197	1	hence	hence	ADV
ap-1815	197	2	there	there	PRON
ap-1815	197	3	is	be	VERB
ap-1815	197	4	an	an	DET
ap-1815	197	5	index	index	NOUN
ap-1815	197	6	α0	α0	PROPN
ap-1815	197	7	∈	∈	PROPN
ap-1815	197	8	λ	λ	NOUN
ap-1815	197	9	such	such	ADJ
ap-1815	197	10	that	that	DET
ap-1815	197	11	aα	aα	NOUN
ap-1815	197	12	∈	∈	PROPN
ap-1815	197	13	u	u	NOUN
ap-1815	197	14	for	for	ADP
ap-1815	197	15	all	all	PRON
ap-1815	197	16	α	α	DET
ap-1815	197	17	≥	≥	NOUN
ap-1815	197	18	α0	α0	ADJ
ap-1815	197	19	.	.	PUNCT
ap-1815	198	1	it	it	PRON
ap-1815	198	2	follows	follow	VERB
ap-1815	198	3	that	that	SCONJ
ap-1815	198	4	ϕ(aα	ϕ(aα	X
ap-1815	198	5	)	)	PUNCT
ap-1815	198	6	∈	∈	PROPN
ap-1815	198	7	ϕ(u	ϕ(u	NOUN
ap-1815	198	8	)	)	PUNCT
ap-1815	199	1	=	=	SYM
ap-1815	199	2	v	v	NOUN
ap-1815	199	3	for	for	ADP
ap-1815	199	4	all	all	PRON
ap-1815	199	5	α	α	DET
ap-1815	199	6	≥	≥	NOUN
ap-1815	199	7	α0	α0	VERB
ap-1815	199	8	.	.	PUNCT
ap-1815	200	1	therefore	therefore	ADV
ap-1815	200	2	ϕ(aα	ϕ(aα	X
ap-1815	200	3	)	)	PUNCT
ap-1815	200	4	τ	τ	PROPN
ap-1815	200	5	e2	e2	NOUN
ap-1815	200	6	0−−→	0−−→	NUM
ap-1815	200	7	ϕ(a	ϕ(a	NOUN
ap-1815	200	8	)	)	PUNCT
ap-1815	200	9	.	.	PUNCT
ap-1815	201	1	the	the	DET
ap-1815	201	2	converse	converse	PROPN
ap-1815	201	3	implication	implication	NOUN
ap-1815	201	4	goes	go	VERB
ap-1815	201	5	the	the	DET
ap-1815	201	6	same	same	ADJ
ap-1815	201	7	way	way	NOUN
ap-1815	201	8	.	.	PUNCT
ap-1815	202	1	recall	recall	VERB
ap-1815	202	2	that	that	PRON
ap-1815	202	3	theorem	theorem	VERB
ap-1815	202	4	3.1	3.1	NUM
ap-1815	202	5	can	can	AUX
ap-1815	202	6	be	be	AUX
ap-1815	202	7	stated	state	VERB
ap-1815	202	8	and	and	CCONJ
ap-1815	202	9	proved	prove	VERB
ap-1815	202	10	entirely	entirely	ADV
ap-1815	202	11	for	for	ADP
ap-1815	202	12	posets	poset	NOUN
ap-1815	202	13	.	.	PUNCT
ap-1815	203	1	theorem	theorem	ADJ
ap-1815	203	2	3.2	3.2	NUM
ap-1815	203	3	has	have	VERB
ap-1815	203	4	to	to	PART
ap-1815	203	5	be	be	AUX
ap-1815	203	6	stated	state	VERB
ap-1815	203	7	and	and	CCONJ
ap-1815	203	8	proved	prove	VERB
ap-1815	203	9	for	for	ADP
ap-1815	203	10	effect	effect	NOUN
ap-1815	203	11	algebras	algebra	NOUN
ap-1815	203	12	.	.	PUNCT
ap-1815	204	1	effect	effect	PROPN
ap-1815	204	2	algebras	algebra	NOUN
ap-1815	204	3	are	be	AUX
ap-1815	204	4	suitable	suitable	ADJ
ap-1815	204	5	algebraic	algebraic	ADJ
ap-1815	204	6	structures	structure	NOUN
ap-1815	204	7	to	to	PART
ap-1815	204	8	be	be	AUX
ap-1815	204	9	carriers	carrier	NOUN
ap-1815	204	10	of	of	ADP
ap-1815	204	11	states	state	NOUN
ap-1815	204	12	or	or	CCONJ
ap-1815	204	13	probability	probability	NOUN
ap-1815	204	14	measures	measure	NOUN
ap-1815	204	15	(	(	PUNCT
ap-1815	204	16	σ	σ	NOUN
ap-1815	204	17	-	-	PUNCT
ap-1815	204	18	additive	additive	ADJ
ap-1815	204	19	states	state	NOUN
ap-1815	204	20	)	)	PUNCT
ap-1815	204	21	also	also	ADV
ap-1815	204	22	in	in	ADP
ap-1815	204	23	cases	case	NOUN
ap-1815	204	24	when	when	SCONJ
ap-1815	204	25	events	event	NOUN
ap-1815	204	26	may	may	AUX
ap-1815	204	27	be	be	AUX
ap-1815	204	28	unsharp	unsharp	ADJ
ap-1815	204	29	or	or	CCONJ
ap-1815	204	30	some	some	DET
ap-1815	204	31	pairs	pair	NOUN
ap-1815	204	32	of	of	ADP
ap-1815	204	33	events	event	NOUN
ap-1815	204	34	are	be	AUX
ap-1815	204	35	noncompatible	noncompatible	ADJ
ap-1815	204	36	.	.	PUNCT
ap-1815	205	1	theorem	theorem	VERB
ap-1815	205	2	3.2	3.2	NUM
ap-1815	205	3	.	.	PUNCT
ap-1815	206	1	let	let	VERB
ap-1815	206	2	(	(	PUNCT
ap-1815	206	3	e1;⊕1	e1;⊕1	PROPN
ap-1815	206	4	,	,	PUNCT
ap-1815	206	5	01	01	NUM
ap-1815	206	6	,	,	PUNCT
ap-1815	206	7	11	11	NUM
ap-1815	206	8	)	)	PUNCT
ap-1815	206	9	and	and	CCONJ
ap-1815	206	10	(	(	PUNCT
ap-1815	206	11	e2;⊕2	e2;⊕2	PROPN
ap-1815	206	12	,	,	PUNCT
ap-1815	206	13	02	02	NUM
ap-1815	206	14	,	,	PUNCT
ap-1815	206	15	12	12	NUM
ap-1815	206	16	)	)	PUNCT
ap-1815	206	17	be	be	VERB
ap-1815	206	18	effect	effect	NOUN
ap-1815	206	19	algebras	algebra	NOUN
ap-1815	206	20	and	and	CCONJ
ap-1815	206	21	let	let	VERB
ap-1815	206	22	ϕ	ϕ	NOUN
ap-1815	206	23	:	:	PUNCT
ap-1815	206	24	e1	e1	PROPN
ap-1815	206	25	→	→	SYM
ap-1815	206	26	e2	e2	PROPN
ap-1815	206	27	be	be	AUX
ap-1815	206	28	an	an	DET
ap-1815	206	29	isomorphism	isomorphism	NOUN
ap-1815	206	30	of	of	ADP
ap-1815	206	31	effect	effect	NOUN
ap-1815	206	32	algebras	algebra	NOUN
ap-1815	206	33	.	.	PUNCT
ap-1815	207	1	then	then	ADV
ap-1815	207	2	,	,	PUNCT
ap-1815	207	3	for	for	ADP
ap-1815	207	4	any	any	DET
ap-1815	207	5	mapping	mapping	NOUN
ap-1815	207	6	ω	ω	NOUN
ap-1815	207	7	:	:	PUNCT
ap-1815	207	8	e2	e2	PROPN
ap-1815	207	9	→	→	PUNCT
ap-1815	208	1	[	[	X
ap-1815	208	2	0	0	NUM
ap-1815	208	3	,	,	PUNCT
ap-1815	208	4	1	1	NUM
ap-1815	208	5	]	]	PUNCT
ap-1815	208	6	,	,	PUNCT
ap-1815	208	7	(	(	PUNCT
ap-1815	208	8	1	1	NUM
ap-1815	208	9	.	.	PUNCT
ap-1815	208	10	)	)	PUNCT
ap-1815	209	1	ω	ω	PROPN
ap-1815	209	2	is	be	AUX
ap-1815	209	3	a	a	DET
ap-1815	209	4	state	state	NOUN
ap-1815	209	5	on	on	ADP
ap-1815	209	6	e2	e2	PROPN
ap-1815	209	7	if	if	SCONJ
ap-1815	209	8	and	and	CCONJ
ap-1815	209	9	only	only	ADV
ap-1815	209	10	if	if	SCONJ
ap-1815	209	11	ω	ω	NUM
ap-1815	209	12	◦	◦	NOUN
ap-1815	209	13	ϕ	ϕ	NOUN
ap-1815	209	14	is	be	AUX
ap-1815	209	15	a	a	DET
ap-1815	209	16	state	state	NOUN
ap-1815	209	17	on	on	ADP
ap-1815	209	18	e1	e1	PROPN
ap-1815	209	19	.	.	PUNCT
ap-1815	210	1	(	(	PUNCT
ap-1815	210	2	2	2	NUM
ap-1815	210	3	.	.	PUNCT
ap-1815	210	4	)	)	PUNCT
ap-1815	211	1	ω	ω	PROPN
ap-1815	211	2	is	be	AUX
ap-1815	211	3	an	an	DET
ap-1815	211	4	(	(	PUNCT
ap-1815	211	5	o)-continuous	o)-continuous	ADJ
ap-1815	211	6	state	state	NOUN
ap-1815	211	7	on	on	ADP
ap-1815	211	8	e2	e2	PROPN
ap-1815	211	9	if	if	SCONJ
ap-1815	211	10	and	and	CCONJ
ap-1815	211	11	only	only	ADV
ap-1815	211	12	if	if	SCONJ
ap-1815	211	13	ω	ω	NUM
ap-1815	211	14	◦	◦	NOUN
ap-1815	211	15	ϕ	ϕ	NOUN
ap-1815	211	16	is	be	AUX
ap-1815	211	17	an	an	DET
ap-1815	211	18	(	(	PUNCT
ap-1815	211	19	o)-continuous	o)-continuous	ADJ
ap-1815	211	20	state	state	NOUN
ap-1815	211	21	on	on	ADP
ap-1815	211	22	e1	e1	PROPN
ap-1815	211	23	.	.	PUNCT
ap-1815	212	1	(	(	PUNCT
ap-1815	212	2	3	3	NUM
ap-1815	212	3	.	.	PUNCT
ap-1815	212	4	)	)	PUNCT
ap-1815	213	1	ω	ω	PROPN
ap-1815	213	2	is	be	AUX
ap-1815	213	3	a	a	DET
ap-1815	213	4	σ	σ	NOUN
ap-1815	213	5	-	-	PUNCT
ap-1815	213	6	additive	additive	ADJ
ap-1815	213	7	state	state	NOUN
ap-1815	213	8	on	on	ADP
ap-1815	213	9	e2	e2	PROPN
ap-1815	213	10	if	if	SCONJ
ap-1815	213	11	and	and	CCONJ
ap-1815	213	12	only	only	ADV
ap-1815	213	13	if	if	SCONJ
ap-1815	213	14	ω	ω	NUM
ap-1815	213	15	◦	◦	NOUN
ap-1815	213	16	ϕ	ϕ	NOUN
ap-1815	213	17	is	be	AUX
ap-1815	213	18	a	a	DET
ap-1815	213	19	σ	σ	NOUN
ap-1815	213	20	-	-	PUNCT
ap-1815	213	21	additive	additive	ADJ
ap-1815	213	22	state	state	NOUN
ap-1815	213	23	on	on	ADP
ap-1815	213	24	e1	e1	PROPN
ap-1815	213	25	.	.	PUNCT
ap-1815	214	1	(	(	PUNCT
ap-1815	214	2	4	4	NUM
ap-1815	214	3	.	.	PUNCT
ap-1815	214	4	)	)	PUNCT
ap-1815	215	1	ω	ω	PROPN
ap-1815	215	2	is	be	AUX
ap-1815	215	3	a	a	DET
ap-1815	215	4	completely	completely	ADV
ap-1815	215	5	additive	additive	ADJ
ap-1815	215	6	state	state	NOUN
ap-1815	215	7	on	on	ADP
ap-1815	215	8	e2	e2	PROPN
ap-1815	215	9	if	if	SCONJ
ap-1815	215	10	and	and	CCONJ
ap-1815	215	11	only	only	ADV
ap-1815	215	12	if	if	SCONJ
ap-1815	215	13	ω	ω	NUM
ap-1815	215	14	◦	◦	NOUN
ap-1815	215	15	ϕ	ϕ	NOUN
ap-1815	215	16	is	be	AUX
ap-1815	215	17	a	a	DET
ap-1815	215	18	completely	completely	ADV
ap-1815	215	19	additive	additive	ADJ
ap-1815	215	20	state	state	NOUN
ap-1815	215	21	on	on	ADP
ap-1815	215	22	e1	e1	NOUN
ap-1815	215	23	.	.	PUNCT
ap-1815	216	1	proof	proof	NOUN
ap-1815	216	2	.	.	PUNCT
ap-1815	217	1	(	(	PUNCT
ap-1815	217	2	1	1	NUM
ap-1815	217	3	.	.	PUNCT
ap-1815	217	4	)	)	PUNCT
ap-1815	217	5	let	let	VERB
ap-1815	217	6	ω	ω	NOUN
ap-1815	217	7	be	be	AUX
ap-1815	217	8	a	a	DET
ap-1815	217	9	state	state	NOUN
ap-1815	217	10	on	on	ADP
ap-1815	217	11	e2	e2	PROPN
ap-1815	217	12	.	.	PUNCT
ap-1815	218	1	then	then	ADV
ap-1815	218	2	the	the	DET
ap-1815	218	3	composition	composition	NOUN
ap-1815	218	4	ω	ω	PROPN
ap-1815	218	5	◦	◦	NOUN
ap-1815	218	6	ϕ	ϕ	NOUN
ap-1815	218	7	is	be	AUX
ap-1815	218	8	a	a	DET
ap-1815	218	9	morphism	morphism	NOUN
ap-1815	218	10	from	from	ADP
ap-1815	218	11	e1	e1	PROPN
ap-1815	218	12	to	to	ADP
ap-1815	218	13	[	[	X
ap-1815	218	14	0	0	NUM
ap-1815	218	15	,	,	PUNCT
ap-1815	218	16	1	1	NUM
ap-1815	218	17	]	]	PUNCT
ap-1815	218	18	,	,	PUNCT
ap-1815	218	19	hence	hence	ADV
ap-1815	218	20	a	a	DET
ap-1815	218	21	state	state	NOUN
ap-1815	218	22	.	.	PUNCT
ap-1815	219	1	conversely	conversely	ADV
ap-1815	219	2	,	,	PUNCT
ap-1815	219	3	let	let	VERB
ap-1815	219	4	ω	ω	NUM
ap-1815	219	5	◦	◦	NOUN
ap-1815	219	6	ϕ	ϕ	NOUN
ap-1815	219	7	be	be	AUX
ap-1815	219	8	a	a	DET
ap-1815	219	9	state	state	NOUN
ap-1815	219	10	on	on	ADP
ap-1815	219	11	e1	e1	PROPN
ap-1815	219	12	.	.	PUNCT
ap-1815	220	1	then	then	ADV
ap-1815	220	2	ω	ω	X
ap-1815	220	3	=	=	SYM
ap-1815	220	4	(	(	PUNCT
ap-1815	220	5	ω	ω	NUM
ap-1815	220	6	◦	◦	PROPN
ap-1815	220	7	ϕ	ϕ	NOUN
ap-1815	220	8	)	)	PUNCT
ap-1815	220	9	◦	◦	NOUN
ap-1815	220	10	ϕ−1	ϕ−1	PUNCT
ap-1815	220	11	is	be	AUX
ap-1815	220	12	a	a	DET
ap-1815	220	13	morphism	morphism	NOUN
ap-1815	220	14	from	from	ADP
ap-1815	220	15	e2	e2	PROPN
ap-1815	220	16	to	to	ADP
ap-1815	220	17	[	[	X
ap-1815	220	18	0	0	NUM
ap-1815	220	19	,	,	PUNCT
ap-1815	220	20	1	1	NUM
ap-1815	220	21	]	]	PUNCT
ap-1815	220	22	.	.	PUNCT
ap-1815	221	1	(	(	PUNCT
ap-1815	221	2	2	2	NUM
ap-1815	221	3	.	.	PUNCT
ap-1815	221	4	)	)	PUNCT
ap-1815	221	5	let	let	VERB
ap-1815	221	6	ω	ω	NOUN
ap-1815	221	7	be	be	AUX
ap-1815	221	8	an	an	DET
ap-1815	221	9	(	(	PUNCT
ap-1815	221	10	o)-continuous	o)-continuous	ADJ
ap-1815	221	11	state	state	NOUN
ap-1815	221	12	on	on	ADP
ap-1815	221	13	e2	e2	PROPN
ap-1815	221	14	.	.	PUNCT
ap-1815	222	1	assume	assume	VERB
ap-1815	222	2	that	that	SCONJ
ap-1815	222	3	(	(	PUNCT
ap-1815	222	4	aα)α∈λ	aα)α∈λ	PROPN
ap-1815	222	5	is	be	AUX
ap-1815	222	6	an	an	DET
ap-1815	222	7	increasingly	increasingly	ADV
ap-1815	222	8	directed	direct	VERB
ap-1815	222	9	net	net	NOUN
ap-1815	222	10	of	of	ADP
ap-1815	222	11	elements	element	NOUN
ap-1815	222	12	of	of	ADP
ap-1815	222	13	e1	e1	PROPN
ap-1815	222	14	and	and	CCONJ
ap-1815	222	15	that	that	SCONJ
ap-1815	222	16	a	a	DET
ap-1815	222	17	∈	∈	NOUN
ap-1815	222	18	e1	e1	NOUN
ap-1815	222	19	such	such	ADJ
ap-1815	222	20	that	that	DET
ap-1815	222	21	aα	aα	PROPN
ap-1815	222	22	↑	↑	PROPN
ap-1815	222	23	a.	a.	NOUN
ap-1815	222	24	from	from	ADP
ap-1815	222	25	theorem	theorem	VERB
ap-1815	222	26	3.1	3.1	NUM
ap-1815	222	27	we	we	PRON
ap-1815	222	28	obtain	obtain	VERB
ap-1815	222	29	that	that	DET
ap-1815	222	30	ϕ(aα	ϕ(aα	X
ap-1815	222	31	)	)	PUNCT
ap-1815	222	32	↑	↑	PROPN
ap-1815	222	33	ϕ(a	ϕ(a	NOUN
ap-1815	222	34	)	)	PUNCT
ap-1815	222	35	in	in	ADP
ap-1815	222	36	e2	e2	PROPN
ap-1815	222	37	.	.	PUNCT
ap-1815	223	1	since	since	SCONJ
ap-1815	223	2	ω	ω	PROPN
ap-1815	223	3	is	be	AUX
ap-1815	223	4	(	(	PUNCT
ap-1815	223	5	o)-continuous	o)-continuous	ADJ
ap-1815	223	6	we	we	PRON
ap-1815	223	7	have	have	VERB
ap-1815	223	8	that	that	PRON
ap-1815	223	9	ω	ω	PROPN
ap-1815	223	10	(	(	PUNCT
ap-1815	223	11	ϕ(aα	ϕ(aα	PROPN
ap-1815	223	12	)	)	PUNCT
ap-1815	223	13	)	)	PUNCT
ap-1815	224	1	↑	↑	PROPN
ap-1815	224	2	ω	ω	PROPN
ap-1815	224	3	(	(	PUNCT
ap-1815	224	4	ϕ(a	ϕ(a	NOUN
ap-1815	224	5	)	)	PUNCT
ap-1815	224	6	)	)	PUNCT
ap-1815	224	7	.	.	PUNCT
ap-1815	225	1	hence	hence	ADV
ap-1815	225	2	,	,	PUNCT
ap-1815	225	3	by	by	ADP
ap-1815	225	4	(	(	PUNCT
ap-1815	225	5	1	1	NUM
ap-1815	225	6	.	.	NUM
ap-1815	225	7	)	)	PUNCT
ap-1815	225	8	,	,	PUNCT
ap-1815	225	9	ω	ω	X
ap-1815	225	10	◦	◦	NOUN
ap-1815	225	11	ϕ	ϕ	NOUN
ap-1815	225	12	is	be	AUX
ap-1815	225	13	an	an	DET
ap-1815	225	14	(	(	PUNCT
ap-1815	225	15	o)-continuous	o)-continuous	ADJ
ap-1815	225	16	state	state	NOUN
ap-1815	225	17	on	on	ADP
ap-1815	225	18	e1	e1	PROPN
ap-1815	225	19	.	.	PUNCT
ap-1815	226	1	the	the	DET
ap-1815	226	2	converse	converse	PROPN
ap-1815	226	3	implication	implication	NOUN
ap-1815	226	4	follows	follow	VERB
ap-1815	226	5	by	by	ADP
ap-1815	226	6	the	the	DET
ap-1815	226	7	same	same	ADJ
ap-1815	226	8	considerations	consideration	NOUN
ap-1815	226	9	as	as	ADP
ap-1815	226	10	above	above	ADV
ap-1815	226	11	applied	apply	VERB
ap-1815	226	12	to	to	ADP
ap-1815	226	13	ϕ−1	ϕ−1	PROPN
ap-1815	226	14	and	and	CCONJ
ap-1815	226	15	the	the	DET
ap-1815	226	16	(	(	PUNCT
ap-1815	226	17	o)continuous	o)continuous	ADJ
ap-1815	226	18	state	state	NOUN
ap-1815	226	19	ω	ω	PROPN
ap-1815	226	20	◦	◦	NOUN
ap-1815	226	21	ϕ	ϕ	NOUN
ap-1815	226	22	on	on	ADP
ap-1815	226	23	e1	e1	PROPN
ap-1815	226	24	.	.	PUNCT
ap-1815	227	1	(	(	PUNCT
ap-1815	227	2	3	3	NUM
ap-1815	227	3	.	.	PUNCT
ap-1815	227	4	)	)	PUNCT
ap-1815	228	1	it	it	PRON
ap-1815	228	2	follows	follow	VERB
ap-1815	228	3	by	by	ADP
ap-1815	228	4	literally	literally	ADV
ap-1815	228	5	the	the	DET
ap-1815	228	6	same	same	ADJ
ap-1815	228	7	considerations	consideration	NOUN
ap-1815	228	8	as	as	ADP
ap-1815	228	9	in	in	ADP
ap-1815	228	10	(	(	PUNCT
ap-1815	228	11	2	2	NUM
ap-1815	228	12	.	.	PUNCT
ap-1815	228	13	)	)	PUNCT
ap-1815	229	1	applied	apply	VERB
ap-1815	229	2	to	to	ADP
ap-1815	229	3	any	any	DET
ap-1815	229	4	countable	countable	ADJ
ap-1815	229	5	increasingly	increasingly	ADV
ap-1815	229	6	directed	direct	VERB
ap-1815	229	7	net	net	NOUN
ap-1815	229	8	.	.	PUNCT
ap-1815	230	1	(	(	PUNCT
ap-1815	230	2	4	4	NUM
ap-1815	230	3	.	.	PUNCT
ap-1815	230	4	)	)	PUNCT
ap-1815	230	5	let	let	VERB
ap-1815	230	6	ω	ω	NOUN
ap-1815	230	7	be	be	AUX
ap-1815	230	8	a	a	DET
ap-1815	230	9	completely	completely	ADV
ap-1815	230	10	additive	additive	ADJ
ap-1815	230	11	state	state	NOUN
ap-1815	230	12	on	on	ADP
ap-1815	230	13	e2	e2	PROPN
ap-1815	230	14	.	.	PUNCT
ap-1815	231	1	assume	assume	VERB
ap-1815	231	2	that	that	SCONJ
ap-1815	231	3	(	(	PUNCT
ap-1815	231	4	xκ)κ∈h	xκ)κ∈h	NUM
ap-1815	231	5	is	be	AUX
ap-1815	231	6	an	an	DET
ap-1815	231	7	orthogonal	orthogonal	ADJ
ap-1815	231	8	system	system	NOUN
ap-1815	231	9	of	of	ADP
ap-1815	231	10	not	not	PART
ap-1815	231	11	necessarily	necessarily	ADV
ap-1815	231	12	different	different	ADJ
ap-1815	231	13	elements	element	NOUN
ap-1815	231	14	of	of	ADP
ap-1815	231	15	e1	e1	NOUN
ap-1815	231	16	such	such	ADJ
ap-1815	231	17	that	that	SCONJ
ap-1815	231	18	⊕	⊕	PROPN
ap-1815	231	19	e1	e1	PROPN
ap-1815	231	20	{	{	PUNCT
ap-1815	231	21	xκ	xκ	NOUN
ap-1815	231	22	|	|	ADV
ap-1815	231	23	κ	κ	PROPN
ap-1815	231	24	∈	∈	PROPN
ap-1815	231	25	h	h	NOUN
ap-1815	231	26	}	}	PUNCT
ap-1815	231	27	exists	exist	VERB
ap-1815	231	28	.	.	PUNCT
ap-1815	232	1	then	then	ADV
ap-1815	232	2	by	by	ADP
ap-1815	232	3	lemma	lemma	PROPN
ap-1815	232	4	2.8	2.8	NUM
ap-1815	232	5	we	we	PRON
ap-1815	232	6	get	get	VERB
ap-1815	232	7	that	that	PRON
ap-1815	232	8	(	(	PUNCT
ap-1815	232	9	ϕ(xκ	ϕ(xκ	NUM
ap-1815	232	10	)	)	PUNCT
ap-1815	232	11	)	)	PUNCT
ap-1815	233	1	κ∈h	κ∈h	PROPN
ap-1815	233	2	is	be	AUX
ap-1815	233	3	an	an	DET
ap-1815	233	4	orthogonal	orthogonal	ADJ
ap-1815	233	5	system	system	NOUN
ap-1815	233	6	in	in	ADP
ap-1815	233	7	e2	e2	PROPN
ap-1815	233	8	and	and	CCONJ
ap-1815	233	9	by	by	ADP
ap-1815	233	10	theorem	theorem	NOUN
ap-1815	233	11	3.1	3.1	NUM
ap-1815	233	12	we	we	PRON
ap-1815	233	13	obtain	obtain	VERB
ap-1815	233	14	that	that	DET
ap-1815	233	15	ϕ	ϕ	NOUN
ap-1815	233	16	(	(	PUNCT
ap-1815	233	17	⊕	⊕	NOUN
ap-1815	233	18	e1	e1	PROPN
ap-1815	233	19	{	{	PUNCT
ap-1815	233	20	xκ	xκ	X
ap-1815	233	21	∣∣κ	∣∣κ	PROPN
ap-1815	233	22	∈	∈	PROPN
ap-1815	233	23	h	h	NOUN
ap-1815	233	24	}	}	PUNCT
ap-1815	233	25	)	)	PUNCT
ap-1815	233	26	=	=	PUNCT
ap-1815	233	27	⊕	⊕	PROPN
ap-1815	233	28	e2	e2	PROPN
ap-1815	233	29	{	{	PUNCT
ap-1815	233	30	ϕ(xκ	ϕ(xκ	NUM
ap-1815	233	31	)	)	PUNCT
ap-1815	233	32	∣∣κ	∣∣κ	PROPN
ap-1815	233	33	∈	∈	PROPN
ap-1815	233	34	h	h	NOUN
ap-1815	233	35	}	}	PUNCT
ap-1815	233	36	.	.	PUNCT
ap-1815	234	1	since	since	SCONJ
ap-1815	234	2	ω	ω	PROPN
ap-1815	234	3	is	be	AUX
ap-1815	234	4	a	a	DET
ap-1815	234	5	completely	completely	ADV
ap-1815	234	6	additive	additive	ADJ
ap-1815	234	7	state	state	NOUN
ap-1815	234	8	on	on	ADP
ap-1815	234	9	e2	e2	PROPN
ap-1815	234	10	we	we	PRON
ap-1815	234	11	311	311	NUM
ap-1815	234	12	j.	j.	PROPN
ap-1815	234	13	paseka	paseka	PROPN
ap-1815	234	14	,	,	PUNCT
ap-1815	234	15	z.	z.	PROPN
ap-1815	234	16	riečanová	riečanová	PROPN
ap-1815	234	17	acta	acta	PROPN
ap-1815	234	18	polytechnica	polytechnica	PROPN
ap-1815	234	19	have	have	VERB
ap-1815	234	20	that	that	PRON
ap-1815	234	21	(	(	PUNCT
ap-1815	234	22	ω	ω	NUM
ap-1815	234	23	◦	◦	NOUN
ap-1815	234	24	ϕ	ϕ	NOUN
ap-1815	234	25	)	)	PUNCT
ap-1815	234	26	(	(	PUNCT
ap-1815	234	27	⊕	⊕	NOUN
ap-1815	234	28	e1	e1	PROPN
ap-1815	234	29	{	{	PUNCT
ap-1815	234	30	xκ	xκ	NOUN
ap-1815	234	31	|	|	ADV
ap-1815	234	32	κ	κ	PROPN
ap-1815	234	33	∈	∈	PROPN
ap-1815	234	34	h	h	NOUN
ap-1815	234	35	}	}	PUNCT
ap-1815	234	36	)	)	PUNCT
ap-1815	235	1	=	=	SYM
ap-1815	235	2	ω	ω	PROPN
ap-1815	235	3	(	(	PUNCT
ap-1815	235	4	⊕	⊕	PROPN
ap-1815	235	5	e2	e2	PROPN
ap-1815	235	6	{	{	PUNCT
ap-1815	235	7	ϕ(xκ	ϕ(xκ	NUM
ap-1815	235	8	)	)	PUNCT
ap-1815	235	9	∣∣κ	∣∣κ	PROPN
ap-1815	235	10	∈	∈	PROPN
ap-1815	235	11	h	h	NOUN
ap-1815	235	12	}	}	PUNCT
ap-1815	235	13	)	)	PUNCT
ap-1815	235	14	=	=	PUNCT
ap-1815	236	1	∑	∑	PROPN
ap-1815	236	2	{	{	PUNCT
ap-1815	236	3	ω	ω	PROPN
ap-1815	236	4	(	(	PUNCT
ap-1815	236	5	ϕ(xκ	ϕ(xκ	NUM
ap-1815	236	6	)	)	PUNCT
ap-1815	236	7	)	)	PUNCT
ap-1815	236	8	∣∣κ	∣∣κ	PROPN
ap-1815	236	9	∈	∈	PROPN
ap-1815	236	10	h	h	NOUN
ap-1815	236	11	}	}	PUNCT
ap-1815	236	12	=	=	SYM
ap-1815	236	13	sup	sup	NOUN
ap-1815	236	14	{	{	PUNCT
ap-1815	236	15	∑	∑	PROPN
ap-1815	236	16	{	{	PUNCT
ap-1815	236	17	(	(	PUNCT
ap-1815	236	18	ω	ω	INTJ
ap-1815	236	19	◦	◦	NOUN
ap-1815	236	20	ϕ)(xκ	ϕ)(xκ	NUM
ap-1815	236	21	)	)	PUNCT
ap-1815	236	22	∣∣κ	∣∣κ	PROPN
ap-1815	236	23	∈	∈	NOUN
ap-1815	236	24	f	f	X
ap-1815	236	25	}	}	PUNCT
ap-1815	236	26	∣∣∣f	∣∣∣f	PROPN
ap-1815	236	27	⊆	⊆	NUM
ap-1815	236	28	h	h	NOUN
ap-1815	236	29	,	,	PUNCT
ap-1815	236	30	f	f	PROPN
ap-1815	236	31	finite	finite	PROPN
ap-1815	236	32	set	set	PROPN
ap-1815	236	33	}	}	PUNCT
ap-1815	236	34	.	.	PUNCT
ap-1815	237	1	the	the	DET
ap-1815	237	2	converse	converse	PROPN
ap-1815	237	3	implication	implication	NOUN
ap-1815	237	4	follows	follow	VERB
ap-1815	237	5	by	by	ADP
ap-1815	237	6	the	the	DET
ap-1815	237	7	same	same	ADJ
ap-1815	237	8	considerations	consideration	NOUN
ap-1815	237	9	as	as	ADP
ap-1815	237	10	above	above	ADV
ap-1815	237	11	applied	apply	VERB
ap-1815	237	12	to	to	ADP
ap-1815	237	13	ϕ−1	ϕ−1	PROPN
ap-1815	237	14	.	.	PROPN
ap-1815	238	1	4	4	NUM
ap-1815	238	2	.	.	X
ap-1815	239	1	some	some	DET
ap-1815	239	2	properties	property	NOUN
ap-1815	239	3	of	of	ADP
ap-1815	239	4	operator	operator	NOUN
ap-1815	239	5	effect	effect	NOUN
ap-1815	239	6	algebras	algebra	NOUN
ap-1815	239	7	that	that	SCONJ
ap-1815	239	8	need	need	AUX
ap-1815	239	9	not	not	PART
ap-1815	239	10	be	be	AUX
ap-1815	239	11	preserved	preserve	VERB
ap-1815	239	12	by	by	ADP
ap-1815	239	13	effect	effect	NOUN
ap-1815	239	14	algebraic	algebraic	PROPN
ap-1815	239	15	isomorphisms	isomorphism	VERB
ap-1815	239	16	we	we	PRON
ap-1815	239	17	see	see	VERB
ap-1815	239	18	,	,	PUNCT
ap-1815	239	19	in	in	ADP
ap-1815	239	20	section	section	NOUN
ap-1815	239	21	3	3	NUM
ap-1815	239	22	,	,	PUNCT
ap-1815	239	23	that	that	SCONJ
ap-1815	239	24	isomorphism	isomorphism	NOUN
ap-1815	239	25	of	of	ADP
ap-1815	239	26	effect	effect	NOUN
ap-1815	239	27	algebras	algebra	NOUN
ap-1815	239	28	preserves	preserve	VERB
ap-1815	239	29	those	those	DET
ap-1815	239	30	properties	property	NOUN
ap-1815	239	31	of	of	ADP
ap-1815	239	32	effect	effect	NOUN
ap-1815	239	33	algebras	algebra	NOUN
ap-1815	239	34	which	which	PRON
ap-1815	239	35	depend	depend	VERB
ap-1815	239	36	only	only	ADV
ap-1815	239	37	on	on	ADP
ap-1815	239	38	the	the	DET
ap-1815	239	39	⊕-operation	⊕-operation	NOUN
ap-1815	239	40	or	or	CCONJ
ap-1815	239	41	on	on	ADP
ap-1815	239	42	the	the	DET
ap-1815	239	43	partial	partial	ADJ
ap-1815	239	44	order	order	NOUN
ap-1815	239	45	that	that	PRON
ap-1815	239	46	is	be	AUX
ap-1815	239	47	derived	derive	VERB
ap-1815	239	48	from	from	ADP
ap-1815	239	49	⊕.	⊕.	ADV
ap-1815	239	50	on	on	ADP
ap-1815	239	51	the	the	DET
ap-1815	239	52	other	other	ADJ
ap-1815	239	53	hand	hand	NOUN
ap-1815	239	54	,	,	PUNCT
ap-1815	239	55	there	there	PRON
ap-1815	239	56	are	be	VERB
ap-1815	239	57	properties	property	NOUN
ap-1815	239	58	of	of	ADP
ap-1815	239	59	effect	effect	NOUN
ap-1815	239	60	algebras	algebra	NOUN
ap-1815	239	61	for	for	ADP
ap-1815	239	62	which	which	PRON
ap-1815	239	63	the	the	DET
ap-1815	239	64	preservation	preservation	NOUN
ap-1815	239	65	of	of	ADP
ap-1815	239	66	the	the	DET
ap-1815	239	67	⊕-operation	⊕-operation	NOUN
ap-1815	239	68	by	by	ADP
ap-1815	239	69	isomorphisms	isomorphisms	PROPN
ap-1815	239	70	is	be	AUX
ap-1815	239	71	not	not	PART
ap-1815	239	72	substantional	substantional	ADJ
ap-1815	239	73	.	.	PUNCT
ap-1815	240	1	for	for	ADP
ap-1815	240	2	operator	operator	NOUN
ap-1815	240	3	effect	effect	NOUN
ap-1815	240	4	algebras	algebra	VERB
ap-1815	240	5	it	it	PRON
ap-1815	240	6	is	be	AUX
ap-1815	240	7	,	,	PUNCT
ap-1815	240	8	e.g.	e.g.	ADV
ap-1815	240	9	,	,	PUNCT
ap-1815	240	10	boundedness	boundedness	NOUN
ap-1815	240	11	or	or	CCONJ
ap-1815	240	12	self	self	NOUN
ap-1815	240	13	-	-	PUNCT
ap-1815	240	14	adjointness	adjointness	NOUN
ap-1815	240	15	of	of	ADP
ap-1815	240	16	operators	operator	NOUN
ap-1815	240	17	(	(	PUNCT
ap-1815	240	18	elements	element	NOUN
ap-1815	240	19	of	of	ADP
ap-1815	240	20	operator	operator	NOUN
ap-1815	240	21	effect	effect	NOUN
ap-1815	240	22	algebras	algebra	NOUN
ap-1815	240	23	)	)	PUNCT
ap-1815	240	24	.	.	PUNCT
ap-1815	241	1	definition	definition	NOUN
ap-1815	241	2	4.1	4.1	NUM
ap-1815	241	3	.	.	PUNCT
ap-1815	242	1	[	[	X
ap-1815	242	2	4	4	X
ap-1815	242	3	]	]	PUNCT
ap-1815	242	4	a	a	DET
ap-1815	242	5	sequential	sequential	ADJ
ap-1815	242	6	effect	effect	NOUN
ap-1815	242	7	algebra	algebra	NOUN
ap-1815	242	8	is	be	AUX
ap-1815	242	9	a	a	DET
ap-1815	242	10	partial	partial	ADJ
ap-1815	242	11	algebra	algebra	NOUN
ap-1815	242	12	(	(	PUNCT
ap-1815	242	13	e	e	NOUN
ap-1815	242	14	;	;	PUNCT
ap-1815	242	15	◦	◦	NOUN
ap-1815	242	16	,	,	PUNCT
ap-1815	242	17	⊕	⊕	PROPN
ap-1815	242	18	,	,	PUNCT
ap-1815	242	19	0	0	NUM
ap-1815	242	20	,	,	PUNCT
ap-1815	242	21	1	1	NUM
ap-1815	242	22	)	)	PUNCT
ap-1815	242	23	such	such	ADJ
ap-1815	242	24	that	that	SCONJ
ap-1815	242	25	(	(	PUNCT
ap-1815	242	26	e;⊕	e;⊕	ADJ
ap-1815	242	27	,	,	PUNCT
ap-1815	242	28	0	0	NUM
ap-1815	242	29	,	,	PUNCT
ap-1815	242	30	1	1	NUM
ap-1815	242	31	)	)	PUNCT
ap-1815	242	32	is	be	AUX
ap-1815	242	33	an	an	DET
ap-1815	242	34	effect	effect	NOUN
ap-1815	242	35	algebra	algebra	NOUN
ap-1815	242	36	and	and	CCONJ
ap-1815	242	37	◦	◦	NOUN
ap-1815	242	38	is	be	AUX
ap-1815	242	39	another	another	DET
ap-1815	242	40	binary	binary	ADJ
ap-1815	242	41	operation	operation	NOUN
ap-1815	242	42	(	(	PUNCT
ap-1815	242	43	called	call	VERB
ap-1815	242	44	a	a	DET
ap-1815	242	45	sequential	sequential	ADJ
ap-1815	242	46	product	product	NOUN
ap-1815	242	47	)	)	PUNCT
ap-1815	242	48	defined	define	VERB
ap-1815	242	49	on	on	ADP
ap-1815	242	50	e	e	X
ap-1815	242	51	satisfying	satisfy	VERB
ap-1815	242	52	:	:	PUNCT
ap-1815	242	53	(	(	PUNCT
ap-1815	242	54	sea1	sea1	PROPN
ap-1815	242	55	)	)	PUNCT
ap-1815	242	56	the	the	DET
ap-1815	242	57	map	map	NOUN
ap-1815	243	1	b	b	PROPN
ap-1815	243	2	7→	7→	NUM
ap-1815	243	3	a	a	DET
ap-1815	243	4	◦	◦	NOUN
ap-1815	243	5	b	b	NOUN
ap-1815	243	6	is	be	AUX
ap-1815	243	7	additive	additive	ADJ
ap-1815	243	8	for	for	ADP
ap-1815	243	9	each	each	DET
ap-1815	243	10	a	a	DET
ap-1815	243	11	∈	∈	PROPN
ap-1815	243	12	e	e	NOUN
ap-1815	243	13	,	,	PUNCT
ap-1815	243	14	that	that	ADV
ap-1815	243	15	is	is	ADV
ap-1815	243	16	,	,	PUNCT
ap-1815	243	17	if	if	SCONJ
ap-1815	243	18	b	b	PROPN
ap-1815	243	19	⊥	⊥	PROPN
ap-1815	243	20	c	c	PROPN
ap-1815	243	21	,	,	PUNCT
ap-1815	243	22	then	then	ADV
ap-1815	243	23	a	a	DET
ap-1815	243	24	◦	◦	NOUN
ap-1815	243	25	b	b	NOUN
ap-1815	243	26	⊥	⊥	NOUN
ap-1815	243	27	a	a	DET
ap-1815	243	28	◦	◦	NOUN
ap-1815	243	29	c	c	NOUN
ap-1815	243	30	and	and	CCONJ
ap-1815	243	31	a	a	DET
ap-1815	243	32	◦	◦	NOUN
ap-1815	243	33	(	(	PUNCT
ap-1815	243	34	b⊕	b⊕	NOUN
ap-1815	243	35	c	c	NOUN
ap-1815	243	36	)	)	PUNCT
ap-1815	243	37	=	=	SYM
ap-1815	243	38	a	a	DET
ap-1815	243	39	◦	◦	NOUN
ap-1815	243	40	b⊕	b⊕	NOUN
ap-1815	243	41	a	a	DET
ap-1815	243	42	◦	◦	NOUN
ap-1815	243	43	c.	c.	NOUN
ap-1815	243	44	(	(	PUNCT
ap-1815	243	45	sea2	sea2	PROPN
ap-1815	243	46	)	)	PUNCT
ap-1815	243	47	1	1	NUM
ap-1815	243	48	◦	◦	NOUN
ap-1815	243	49	a	a	DET
ap-1815	243	50	=	=	NOUN
ap-1815	243	51	a	a	PRON
ap-1815	243	52	for	for	ADP
ap-1815	243	53	each	each	DET
ap-1815	243	54	a	a	DET
ap-1815	243	55	∈	∈	PROPN
ap-1815	243	56	e.	e.	PROPN
ap-1815	243	57	(	(	PUNCT
ap-1815	243	58	sea3	sea3	PROPN
ap-1815	243	59	)	)	PUNCT
ap-1815	243	60	if	if	SCONJ
ap-1815	243	61	a	a	DET
ap-1815	243	62	◦	◦	NOUN
ap-1815	243	63	b	b	NOUN
ap-1815	243	64	=	=	SYM
ap-1815	243	65	0	0	NUM
ap-1815	243	66	,	,	PUNCT
ap-1815	243	67	then	then	ADV
ap-1815	243	68	a	a	DET
ap-1815	243	69	◦	◦	NOUN
ap-1815	243	70	b	b	NOUN
ap-1815	243	71	=	=	SYM
ap-1815	243	72	b	b	PROPN
ap-1815	243	73	◦	◦	NOUN
ap-1815	243	74	a.	a.	NOUN
ap-1815	243	75	(	(	PUNCT
ap-1815	243	76	sea4	sea4	PROPN
ap-1815	243	77	)	)	PUNCT
ap-1815	243	78	if	if	SCONJ
ap-1815	243	79	a	a	DET
ap-1815	243	80	◦	◦	NOUN
ap-1815	243	81	b	b	NOUN
ap-1815	244	1	=	=	SYM
ap-1815	244	2	b	b	PROPN
ap-1815	244	3	◦	◦	NOUN
ap-1815	244	4	a	a	X
ap-1815	244	5	,	,	PUNCT
ap-1815	244	6	then	then	ADV
ap-1815	244	7	a	a	DET
ap-1815	244	8	◦	◦	NOUN
ap-1815	244	9	b′	b′	NUM
ap-1815	244	10	=	=	SYM
ap-1815	244	11	b′	b′	NUM
ap-1815	244	12	◦	◦	NOUN
ap-1815	244	13	a	a	PRON
ap-1815	244	14	and	and	CCONJ
ap-1815	244	15	a	a	DET
ap-1815	244	16	◦	◦	NOUN
ap-1815	244	17	(	(	PUNCT
ap-1815	244	18	b	b	X
ap-1815	244	19	◦	◦	NOUN
ap-1815	244	20	c	c	NOUN
ap-1815	244	21	)	)	PUNCT
ap-1815	244	22	=	=	SYM
ap-1815	245	1	(	(	PUNCT
ap-1815	245	2	a	a	DET
ap-1815	245	3	◦	◦	NOUN
ap-1815	245	4	b	b	NOUN
ap-1815	245	5	)	)	PUNCT
ap-1815	245	6	◦	◦	NOUN
ap-1815	245	7	c	c	NOUN
ap-1815	245	8	for	for	ADP
ap-1815	245	9	each	each	DET
ap-1815	245	10	c	c	PROPN
ap-1815	245	11	∈	∈	PROPN
ap-1815	245	12	e.	e.	PROPN
ap-1815	245	13	(	(	PUNCT
ap-1815	245	14	sea5	sea5	PROPN
ap-1815	245	15	)	)	PUNCT
ap-1815	245	16	if	if	SCONJ
ap-1815	245	17	c	c	NOUN
ap-1815	245	18	◦	◦	VERB
ap-1815	245	19	a	a	DET
ap-1815	245	20	=	=	NOUN
ap-1815	245	21	a	a	DET
ap-1815	245	22	◦	◦	NOUN
ap-1815	245	23	c	c	PROPN
ap-1815	245	24	and	and	CCONJ
ap-1815	245	25	c	c	PROPN
ap-1815	245	26	◦	◦	NOUN
ap-1815	245	27	b	b	PROPN
ap-1815	245	28	=	=	SYM
ap-1815	245	29	b	b	PROPN
ap-1815	245	30	◦	◦	NOUN
ap-1815	245	31	c	c	X
ap-1815	245	32	,	,	PUNCT
ap-1815	245	33	then	then	ADV
ap-1815	245	34	c	c	PROPN
ap-1815	245	35	◦	◦	NOUN
ap-1815	245	36	(	(	PUNCT
ap-1815	245	37	a	a	DET
ap-1815	245	38	◦	◦	NOUN
ap-1815	245	39	b	b	NOUN
ap-1815	245	40	)	)	PUNCT
ap-1815	245	41	=	=	SYM
ap-1815	245	42	(	(	PUNCT
ap-1815	245	43	a	a	DET
ap-1815	245	44	◦	◦	NOUN
ap-1815	245	45	b	b	NOUN
ap-1815	245	46	)	)	PUNCT
ap-1815	245	47	◦	◦	NOUN
ap-1815	245	48	c	c	PROPN
ap-1815	245	49	and	and	CCONJ
ap-1815	245	50	c	c	PROPN
ap-1815	245	51	◦	◦	NOUN
ap-1815	245	52	(	(	PUNCT
ap-1815	245	53	a⊕	a⊕	NOUN
ap-1815	245	54	b	b	NOUN
ap-1815	245	55	)	)	PUNCT
ap-1815	245	56	=	=	SYM
ap-1815	245	57	(	(	PUNCT
ap-1815	245	58	a⊕	a⊕	PROPN
ap-1815	245	59	b	b	X
ap-1815	245	60	)	)	PUNCT
ap-1815	245	61	◦	◦	NOUN
ap-1815	245	62	c	c	NOUN
ap-1815	246	1	whenever	whenever	SCONJ
ap-1815	246	2	a	a	DET
ap-1815	246	3	⊥	⊥	PROPN
ap-1815	246	4	b.	b.	PROPN
ap-1815	246	5	assume	assume	VERB
ap-1815	246	6	that	that	SCONJ
ap-1815	246	7	(	(	PUNCT
ap-1815	246	8	e1	e1	NOUN
ap-1815	246	9	;	;	PUNCT
ap-1815	246	10	◦	◦	NOUN
ap-1815	246	11	1,⊕1	1,⊕1	NUM
ap-1815	246	12	,	,	PUNCT
ap-1815	246	13	01	01	NUM
ap-1815	246	14	,	,	PUNCT
ap-1815	246	15	11	11	NUM
ap-1815	246	16	)	)	PUNCT
ap-1815	246	17	and	and	CCONJ
ap-1815	246	18	(	(	PUNCT
ap-1815	246	19	e2	e2	PROPN
ap-1815	246	20	;	;	PUNCT
ap-1815	246	21	◦	◦	NOUN
ap-1815	246	22	2,⊕2	2,⊕2	NUM
ap-1815	246	23	,	,	PUNCT
ap-1815	246	24	02	02	NUM
ap-1815	246	25	,	,	PUNCT
ap-1815	246	26	12	12	NUM
ap-1815	246	27	)	)	PUNCT
ap-1815	246	28	are	be	AUX
ap-1815	246	29	sequential	sequential	ADJ
ap-1815	246	30	effect	effect	NOUN
ap-1815	246	31	algebras	algebra	NOUN
ap-1815	246	32	.	.	PUNCT
ap-1815	247	1	a	a	DET
ap-1815	247	2	mapping	mapping	NOUN
ap-1815	247	3	ϕ	ϕ	NOUN
ap-1815	247	4	:	:	PUNCT
ap-1815	247	5	e1	e1	PROPN
ap-1815	247	6	→	→	SYM
ap-1815	247	7	e2	e2	PROPN
ap-1815	247	8	is	be	AUX
ap-1815	247	9	called	call	VERB
ap-1815	247	10	a	a	DET
ap-1815	247	11	sequential	sequential	ADJ
ap-1815	247	12	effect	effect	NOUN
ap-1815	247	13	algebraic	algebraic	ADJ
ap-1815	247	14	morphism	morphism	NOUN
ap-1815	247	15	if	if	SCONJ
ap-1815	247	16	ϕ	ϕ	NOUN
ap-1815	247	17	is	be	AUX
ap-1815	247	18	a	a	DET
ap-1815	247	19	morphism	morphism	NOUN
ap-1815	247	20	of	of	ADP
ap-1815	247	21	the	the	DET
ap-1815	247	22	effect	effect	NOUN
ap-1815	247	23	algebra	algebra	NOUN
ap-1815	247	24	e1	e1	VERB
ap-1815	247	25	into	into	ADP
ap-1815	247	26	the	the	DET
ap-1815	247	27	effect	effect	NOUN
ap-1815	247	28	algebra	algebra	NOUN
ap-1815	247	29	e2	e2	PROPN
ap-1815	247	30	and	and	CCONJ
ap-1815	247	31	,	,	PUNCT
ap-1815	247	32	for	for	ADP
ap-1815	247	33	all	all	DET
ap-1815	247	34	a	a	DET
ap-1815	247	35	,	,	PUNCT
ap-1815	247	36	b	b	PROPN
ap-1815	247	37	∈	∈	PROPN
ap-1815	247	38	e1	e1	NOUN
ap-1815	247	39	,	,	PUNCT
ap-1815	247	40	ϕ(a	ϕ(a	NOUN
ap-1815	247	41	◦	◦	NOUN
ap-1815	247	42	1	1	NUM
ap-1815	247	43	b	b	NOUN
ap-1815	247	44	)	)	PUNCT
ap-1815	247	45	=	=	SYM
ap-1815	247	46	ϕ(a	ϕ(a	NOUN
ap-1815	247	47	)	)	PUNCT
ap-1815	247	48	◦	◦	NOUN
ap-1815	247	49	2	2	NUM
ap-1815	247	50	ϕ(b	ϕ(b	NUM
ap-1815	247	51	)	)	PUNCT
ap-1815	247	52	.	.	PUNCT
ap-1815	248	1	in	in	ADP
ap-1815	248	2	what	what	PRON
ap-1815	248	3	follows	follow	VERB
ap-1815	248	4	we	we	PRON
ap-1815	248	5	will	will	AUX
ap-1815	248	6	assume	assume	VERB
ap-1815	248	7	that	that	SCONJ
ap-1815	248	8	h	h	NOUN
ap-1815	248	9	is	be	AUX
ap-1815	248	10	an	an	DET
ap-1815	248	11	infinitedimensional	infinitedimensional	ADJ
ap-1815	248	12	complex	complex	ADJ
ap-1815	248	13	hilbert	hilbert	NOUN
ap-1815	248	14	space	space	NOUN
ap-1815	248	15	,	,	PUNCT
ap-1815	248	16	i.e.	i.e.	X
ap-1815	248	17	,	,	PUNCT
ap-1815	248	18	a	a	DET
ap-1815	248	19	linear	linear	ADJ
ap-1815	248	20	space	space	NOUN
ap-1815	248	21	with	with	ADP
ap-1815	248	22	inner	inner	ADJ
ap-1815	248	23	product	product	NOUN
ap-1815	248	24	(	(	PUNCT
ap-1815	248	25	·	·	PUNCT
ap-1815	248	26	,	,	PUNCT
ap-1815	248	27	·	·	PUNCT
ap-1815	248	28	)	)	PUNCT
ap-1815	248	29	which	which	PRON
ap-1815	248	30	is	be	AUX
ap-1815	248	31	complete	complete	ADJ
ap-1815	248	32	in	in	ADP
ap-1815	248	33	the	the	DET
ap-1815	248	34	induced	induce	VERB
ap-1815	248	35	metric	metric	NOUN
ap-1815	248	36	.	.	PUNCT
ap-1815	249	1	the	the	DET
ap-1815	249	2	term	term	NOUN
ap-1815	249	3	dimension	dimension	NOUN
ap-1815	249	4	ofh	ofh	PROPN
ap-1815	249	5	is	be	AUX
ap-1815	249	6	defined	define	VERB
ap-1815	249	7	as	as	ADP
ap-1815	249	8	the	the	DET
ap-1815	249	9	cardinality	cardinality	NOUN
ap-1815	249	10	of	of	ADP
ap-1815	249	11	any	any	DET
ap-1815	249	12	orthonormal	orthonormal	ADJ
ap-1815	249	13	basis	basis	NOUN
ap-1815	249	14	of	of	ADP
ap-1815	249	15	h	h	PROPN
ap-1815	249	16	(	(	PUNCT
ap-1815	249	17	see	see	VERB
ap-1815	249	18	[	[	X
ap-1815	249	19	1	1	NUM
ap-1815	249	20	]	]	NUM
ap-1815	249	21	)	)	PUNCT
ap-1815	249	22	.	.	PUNCT
ap-1815	250	1	moreover	moreover	ADV
ap-1815	250	2	,	,	PUNCT
ap-1815	250	3	we	we	PRON
ap-1815	250	4	will	will	AUX
ap-1815	250	5	assume	assume	VERB
ap-1815	250	6	that	that	SCONJ
ap-1815	250	7	all	all	PRON
ap-1815	250	8	considered	consider	VERB
ap-1815	250	9	linear	linear	PROPN
ap-1815	250	10	operators	operator	NOUN
ap-1815	250	11	a	a	DET
ap-1815	250	12	(	(	PUNCT
ap-1815	250	13	i.e.	i.e.	X
ap-1815	250	14	linear	linear	PROPN
ap-1815	250	15	maps	map	VERB
ap-1815	250	16	a	a	DET
ap-1815	250	17	:	:	PUNCT
ap-1815	250	18	d(a	d(a	PROPN
ap-1815	250	19	)	)	PUNCT
ap-1815	250	20	→	→	SYM
ap-1815	250	21	h	h	X
ap-1815	250	22	)	)	PUNCT
ap-1815	250	23	have	have	VERB
ap-1815	250	24	a	a	DET
ap-1815	250	25	domain	domain	NOUN
ap-1815	250	26	d(a	d(a	PROPN
ap-1815	250	27	)	)	PUNCT
ap-1815	250	28	that	that	PRON
ap-1815	250	29	is	be	AUX
ap-1815	250	30	a	a	DET
ap-1815	250	31	linear	linear	ADJ
ap-1815	250	32	subspace	subspace	NOUN
ap-1815	250	33	dense	dense	ADJ
ap-1815	250	34	in	in	ADP
ap-1815	250	35	h	h	NOUN
ap-1815	250	36	with	with	ADP
ap-1815	250	37	respect	respect	NOUN
ap-1815	250	38	to	to	ADP
ap-1815	250	39	the	the	DET
ap-1815	250	40	metric	metric	ADJ
ap-1815	250	41	topology	topology	NOUN
ap-1815	250	42	induced	induce	VERB
ap-1815	250	43	by	by	ADP
ap-1815	250	44	the	the	DET
ap-1815	250	45	inner	inner	ADJ
ap-1815	250	46	product	product	NOUN
ap-1815	250	47	on	on	ADP
ap-1815	250	48	h	h	PROPN
ap-1815	250	49	(	(	PUNCT
ap-1815	250	50	i.e.	i.e.	X
ap-1815	250	51	,	,	PUNCT
ap-1815	250	52	d(a	d(a	PROPN
ap-1815	250	53	)	)	PUNCT
ap-1815	251	1	=	=	SYM
ap-1815	251	2	h	h	NOUN
ap-1815	251	3	)	)	PUNCT
ap-1815	251	4	.	.	PUNCT
ap-1815	252	1	recall	recall	VERB
ap-1815	252	2	that	that	SCONJ
ap-1815	252	3	a	a	DET
ap-1815	252	4	linear	linear	ADJ
ap-1815	252	5	operator	operator	NOUN
ap-1815	252	6	a	a	PRON
ap-1815	252	7	is	be	AUX
ap-1815	252	8	called	call	VERB
ap-1815	252	9	positive	positive	ADJ
ap-1815	252	10	(	(	PUNCT
ap-1815	252	11	denoted	denote	VERB
ap-1815	252	12	by	by	ADP
ap-1815	252	13	a	a	DET
ap-1815	252	14	≥	≥	NOUN
ap-1815	252	15	0	0	NUM
ap-1815	252	16	)	)	PUNCT
ap-1815	252	17	iff	iff	NOUN
ap-1815	252	18	(	(	PUNCT
ap-1815	252	19	x	x	NOUN
ap-1815	252	20	,	,	PUNCT
ap-1815	252	21	ax	ax	NOUN
ap-1815	252	22	)	)	PUNCT
ap-1815	252	23	≥	≥	NOUN
ap-1815	252	24	0	0	NUM
ap-1815	252	25	for	for	ADP
ap-1815	252	26	all	all	DET
ap-1815	252	27	x	x	SYM
ap-1815	252	28	∈	∈	PROPN
ap-1815	252	29	d(a	d(a	PROPN
ap-1815	252	30	)	)	PUNCT
ap-1815	252	31	,	,	PUNCT
ap-1815	252	32	hence	hence	ADV
ap-1815	252	33	a	a	PRON
ap-1815	252	34	is	be	AUX
ap-1815	252	35	also	also	ADV
ap-1815	252	36	symmetric	symmetric	ADJ
ap-1815	252	37	,	,	PUNCT
ap-1815	252	38	meaning	mean	VERB
ap-1815	252	39	that	that	SCONJ
ap-1815	252	40	(	(	PUNCT
ap-1815	252	41	y	y	NOUN
ap-1815	252	42	,	,	PUNCT
ap-1815	252	43	ax	ax	NOUN
ap-1815	252	44	)	)	PUNCT
ap-1815	252	45	=	=	SYM
ap-1815	252	46	(	(	PUNCT
ap-1815	252	47	ay	ay	INTJ
ap-1815	252	48	,	,	PUNCT
ap-1815	252	49	x	x	NOUN
ap-1815	252	50	)	)	PUNCT
ap-1815	252	51	for	for	ADP
ap-1815	252	52	all	all	DET
ap-1815	252	53	x	x	NOUN
ap-1815	252	54	,	,	PUNCT
ap-1815	252	55	y	y	PROPN
ap-1815	252	56	∈	∈	PROPN
ap-1815	252	57	d(a	d(a	PROPN
ap-1815	252	58	)	)	PUNCT
ap-1815	252	59	(	(	PUNCT
ap-1815	252	60	see	see	VERB
ap-1815	252	61	[	[	X
ap-1815	252	62	1	1	X
ap-1815	252	63	]	]	PUNCT
ap-1815	252	64	for	for	ADP
ap-1815	252	65	more	more	ADJ
ap-1815	252	66	details	detail	NOUN
ap-1815	252	67	)	)	PUNCT
ap-1815	252	68	.	.	PUNCT
ap-1815	253	1	recall	recall	VERB
ap-1815	253	2	that	that	PRON
ap-1815	253	3	a	a	DET
ap-1815	253	4	:	:	PUNCT
ap-1815	253	5	d(a	d(a	PROPN
ap-1815	253	6	)	)	PUNCT
ap-1815	253	7	→	→	SYM
ap-1815	253	8	h	h	NOUN
ap-1815	253	9	is	be	AUX
ap-1815	253	10	called	call	VERB
ap-1815	253	11	a	a	DET
ap-1815	253	12	bounded	bounded	ADJ
ap-1815	253	13	operator	operator	NOUN
ap-1815	253	14	if	if	SCONJ
ap-1815	253	15	there	there	PRON
ap-1815	253	16	exists	exist	VERB
ap-1815	253	17	a	a	DET
ap-1815	253	18	real	real	ADJ
ap-1815	253	19	constant	constant	ADJ
ap-1815	253	20	c	c	PROPN
ap-1815	253	21	≥	≥	NOUN
ap-1815	253	22	0	0	NUM
ap-1815	253	23	such	such	ADJ
ap-1815	253	24	that	that	SCONJ
ap-1815	253	25	‖ax‖	‖ax‖	ADJ
ap-1815	253	26	≤	≤	NOUN
ap-1815	253	27	c‖x‖	c‖x‖	NOUN
ap-1815	253	28	for	for	ADP
ap-1815	253	29	all	all	DET
ap-1815	253	30	x	x	SYM
ap-1815	253	31	∈	∈	PROPN
ap-1815	253	32	d(a	d(a	PROPN
ap-1815	253	33	)	)	PUNCT
ap-1815	253	34	.	.	PUNCT
ap-1815	254	1	gudder	gudder	VERB
ap-1815	254	2	[	[	X
ap-1815	254	3	4	4	X
ap-1815	254	4	]	]	PUNCT
ap-1815	254	5	showed	show	VERB
ap-1815	254	6	that	that	SCONJ
ap-1815	254	7	,	,	PUNCT
ap-1815	254	8	for	for	ADP
ap-1815	254	9	any	any	DET
ap-1815	254	10	standard	standard	ADJ
ap-1815	254	11	hilbert	hilbert	NOUN
ap-1815	254	12	space	space	NOUN
ap-1815	254	13	effect	effect	NOUN
ap-1815	254	14	algebra	algebra	VERB
ap-1815	254	15	e(h	e(h	PROPN
ap-1815	254	16	)	)	PUNCT
ap-1815	254	17	on	on	ADP
ap-1815	254	18	a	a	DET
ap-1815	254	19	complex	complex	ADJ
ap-1815	254	20	hilbert	hilbert	NOUN
ap-1815	254	21	space	space	NOUN
ap-1815	254	22	h	h	NOUN
ap-1815	254	23	,	,	PUNCT
ap-1815	254	24	there	there	PRON
ap-1815	254	25	is	be	VERB
ap-1815	254	26	a	a	DET
ap-1815	254	27	binary	binary	ADJ
ap-1815	254	28	operation	operation	NOUN
ap-1815	254	29	◦	◦	NOUN
ap-1815	254	30	defined	define	VERB
ap-1815	254	31	by	by	ADP
ap-1815	254	32	b	b	PROPN
ap-1815	254	33	◦	◦	NOUN
ap-1815	254	34	c	c	NOUN
ap-1815	255	1	=	=	SYM
ap-1815	255	2	b	b	X
ap-1815	255	3	1	1	NUM
ap-1815	255	4	2cb	2cb	NOUN
ap-1815	255	5	1	1	NUM
ap-1815	255	6	2	2	NUM
ap-1815	255	7	for	for	ADP
ap-1815	255	8	all	all	DET
ap-1815	255	9	b	b	NOUN
ap-1815	255	10	,	,	PUNCT
ap-1815	255	11	c	c	PROPN
ap-1815	255	12	∈	∈	PROPN
ap-1815	255	13	e(h	e(h	PROPN
ap-1815	255	14	)	)	PUNCT
ap-1815	255	15	such	such	ADJ
ap-1815	255	16	that	that	SCONJ
ap-1815	255	17	it	it	PRON
ap-1815	255	18	satisfies	satisfy	VERB
ap-1815	255	19	conditions	condition	NOUN
ap-1815	255	20	(	(	PUNCT
ap-1815	255	21	sea1)–(sea5	sea1)–(sea5	NOUN
ap-1815	255	22	)	)	PUNCT
ap-1815	255	23	,	,	PUNCT
ap-1815	255	24	and	and	CCONJ
ap-1815	255	25	so	so	ADV
ap-1815	255	26	it	it	PRON
ap-1815	255	27	is	be	AUX
ap-1815	255	28	a	a	DET
ap-1815	255	29	sequential	sequential	ADJ
ap-1815	255	30	product	product	NOUN
ap-1815	255	31	of	of	ADP
ap-1815	255	32	e(h	e(h	PROPN
ap-1815	255	33	)	)	PUNCT
ap-1815	255	34	.	.	PUNCT
ap-1815	256	1	liu	liu	PROPN
ap-1815	256	2	weihua	weihua	PROPN
ap-1815	256	3	and	and	CCONJ
ap-1815	256	4	wu	wu	PROPN
ap-1815	256	5	junde	junde	NOUN
ap-1815	256	6	in	in	ADP
ap-1815	256	7	[	[	X
ap-1815	256	8	10	10	NUM
ap-1815	256	9	,	,	PUNCT
ap-1815	256	10	theorem	theorem	VERB
ap-1815	256	11	4.3	4.3	NUM
ap-1815	256	12	]	]	PUNCT
ap-1815	256	13	proved	prove	VERB
ap-1815	256	14	that	that	SCONJ
ap-1815	256	15	there	there	PRON
ap-1815	256	16	is	be	VERB
ap-1815	256	17	a	a	DET
ap-1815	256	18	binary	binary	ADJ
ap-1815	256	19	operation	operation	NOUN
ap-1815	256	20	◦	◦	NOUN
ap-1815	256	21	i	i	PRON
ap-1815	256	22	on	on	ADP
ap-1815	256	23	e(h	e(h	PROPN
ap-1815	256	24	)	)	PUNCT
ap-1815	256	25	such	such	ADJ
ap-1815	256	26	that	that	SCONJ
ap-1815	256	27	it	it	PRON
ap-1815	256	28	satisfies	satisfy	VERB
ap-1815	256	29	conditions	condition	NOUN
ap-1815	256	30	(	(	PUNCT
ap-1815	256	31	sea1	sea1	PROPN
ap-1815	256	32	)	)	PUNCT
ap-1815	256	33	–	–	PUNCT
ap-1815	256	34	(	(	PUNCT
ap-1815	256	35	sea5	sea5	PROPN
ap-1815	256	36	)	)	PUNCT
ap-1815	256	37	and	and	CCONJ
ap-1815	256	38	◦	◦	NOUN
ap-1815	256	39	i	i	PROPN
ap-1815	256	40	6=	6=	PROPN
ap-1815	256	41	◦	◦	NOUN
ap-1815	256	42	.	.	PUNCT
ap-1815	257	1	this	this	PRON
ap-1815	257	2	yields	yield	VERB
ap-1815	257	3	the	the	DET
ap-1815	257	4	following	following	NOUN
ap-1815	257	5	.	.	PUNCT
ap-1815	258	1	theorem	theorem	VERB
ap-1815	258	2	4.2	4.2	NUM
ap-1815	258	3	.	.	PUNCT
ap-1815	259	1	let	let	VERB
ap-1815	259	2	h	h	PRON
ap-1815	259	3	be	be	AUX
ap-1815	259	4	a	a	DET
ap-1815	259	5	complex	complex	ADJ
ap-1815	259	6	hilbert	hilbert	NOUN
ap-1815	259	7	space	space	NOUN
ap-1815	259	8	.	.	PUNCT
ap-1815	260	1	then	then	ADV
ap-1815	260	2	there	there	PRON
ap-1815	260	3	are	be	VERB
ap-1815	260	4	sequential	sequential	ADJ
ap-1815	260	5	operator	operator	NOUN
ap-1815	260	6	effect	effect	NOUN
ap-1815	260	7	algebras	algebra	NOUN
ap-1815	260	8	(	(	PUNCT
ap-1815	260	9	e(h	e(h	PROPN
ap-1815	260	10	)	)	PUNCT
ap-1815	260	11	;	;	PUNCT
ap-1815	260	12	◦	◦	NOUN
ap-1815	260	13	,	,	PUNCT
ap-1815	260	14	⊕	⊕	PROPN
ap-1815	260	15	,	,	PUNCT
ap-1815	260	16	0	0	NUM
ap-1815	260	17	,	,	PUNCT
ap-1815	260	18	1	1	NUM
ap-1815	260	19	)	)	PUNCT
ap-1815	260	20	and	and	CCONJ
ap-1815	260	21	(	(	PUNCT
ap-1815	260	22	e(h	e(h	PROPN
ap-1815	260	23	)	)	PUNCT
ap-1815	260	24	;	;	PUNCT
ap-1815	260	25	◦	◦	NOUN
ap-1815	260	26	i,⊕	i,⊕	NOUN
ap-1815	260	27	,	,	PUNCT
ap-1815	260	28	0	0	NUM
ap-1815	260	29	,	,	PUNCT
ap-1815	260	30	1	1	NUM
ap-1815	260	31	)	)	PUNCT
ap-1815	260	32	that	that	PRON
ap-1815	260	33	are	be	AUX
ap-1815	260	34	isomorphic	isomorphic	ADJ
ap-1815	260	35	as	as	ADP
ap-1815	260	36	effect	effect	NOUN
ap-1815	260	37	algebras	algebra	NOUN
ap-1815	260	38	but	but	CCONJ
ap-1815	260	39	the	the	DET
ap-1815	260	40	respective	respective	ADJ
ap-1815	260	41	effect	effect	NOUN
ap-1815	260	42	algebraic	algebraic	ADJ
ap-1815	260	43	isomorphism	isomorphism	NOUN
ap-1815	260	44	does	do	AUX
ap-1815	260	45	not	not	PART
ap-1815	260	46	preserve	preserve	VERB
ap-1815	260	47	the	the	DET
ap-1815	260	48	sequential	sequential	ADJ
ap-1815	260	49	product	product	NOUN
ap-1815	260	50	.	.	PUNCT
ap-1815	261	1	proof	proof	NOUN
ap-1815	261	2	.	.	PUNCT
ap-1815	262	1	evidently	evidently	ADV
ap-1815	262	2	,	,	PUNCT
ap-1815	262	3	ide(h	ide(h	PROPN
ap-1815	262	4	)	)	PUNCT
ap-1815	262	5	is	be	AUX
ap-1815	262	6	an	an	DET
ap-1815	262	7	effect	effect	NOUN
ap-1815	262	8	algebraic	algebraic	ADJ
ap-1815	262	9	isomorphism	isomorphism	NOUN
ap-1815	262	10	.	.	PUNCT
ap-1815	263	1	from	from	ADP
ap-1815	263	2	[	[	X
ap-1815	263	3	10	10	NUM
ap-1815	263	4	,	,	PUNCT
ap-1815	263	5	theorem	theorem	VERB
ap-1815	263	6	4.3	4.3	NUM
ap-1815	263	7	]	]	PUNCT
ap-1815	263	8	we	we	PRON
ap-1815	263	9	know	know	VERB
ap-1815	263	10	that	that	SCONJ
ap-1815	263	11	there	there	PRON
ap-1815	263	12	are	be	VERB
ap-1815	263	13	a	a	DET
ap-1815	263	14	,	,	PUNCT
ap-1815	263	15	b	b	PROPN
ap-1815	263	16	∈	∈	PROPN
ap-1815	263	17	e(h	e(h	PROPN
ap-1815	263	18	)	)	PUNCT
ap-1815	263	19	such	such	ADJ
ap-1815	263	20	that	that	SCONJ
ap-1815	263	21	a	a	DET
ap-1815	263	22	◦	◦	NOUN
ap-1815	263	23	b	b	NOUN
ap-1815	263	24	6=	6=	ADP
ap-1815	263	25	a	a	DET
ap-1815	263	26	◦	◦	NOUN
ap-1815	263	27	i	i	PROPN
ap-1815	263	28	b.	b.	NOUN
ap-1815	263	29	hence	hence	ADV
ap-1815	263	30	ide(h)(a	ide(h)(a	NUM
ap-1815	263	31	◦	◦	NOUN
ap-1815	263	32	b	b	NOUN
ap-1815	263	33	)	)	PUNCT
ap-1815	263	34	=	=	PUNCT
ap-1815	263	35	a	a	DET
ap-1815	263	36	◦	◦	NOUN
ap-1815	263	37	b	b	NOUN
ap-1815	263	38	6=	6=	ADP
ap-1815	263	39	a	a	DET
ap-1815	263	40	◦	◦	NOUN
ap-1815	263	41	i	i	NOUN
ap-1815	263	42	b	b	NOUN
ap-1815	263	43	=	=	SYM
ap-1815	263	44	ide(h)(a	ide(h)(a	NOUN
ap-1815	263	45	)	)	PUNCT
ap-1815	263	46	◦	◦	NOUN
ap-1815	263	47	i	i	PRON
ap-1815	263	48	ide(h)(b	ide(h)(b	ADJ
ap-1815	263	49	)	)	PUNCT
ap-1815	263	50	.	.	PUNCT
ap-1815	264	1	let	let	VERB
ap-1815	264	2	v(h	v(h	NOUN
ap-1815	264	3	)	)	PUNCT
ap-1815	264	4	be	be	AUX
ap-1815	264	5	the	the	DET
ap-1815	264	6	set	set	NOUN
ap-1815	264	7	of	of	ADP
ap-1815	264	8	all	all	DET
ap-1815	264	9	positive	positive	ADJ
ap-1815	264	10	linear	linear	NOUN
ap-1815	264	11	operators	operator	NOUN
ap-1815	264	12	densely	densely	ADV
ap-1815	264	13	defined	define	VERB
ap-1815	264	14	in	in	ADP
ap-1815	264	15	an	an	DET
ap-1815	264	16	infinite	infinite	ADJ
ap-1815	264	17	-	-	PUNCT
ap-1815	264	18	dimensional	dimensional	ADJ
ap-1815	264	19	complex	complex	ADJ
ap-1815	264	20	hilbert	hilbert	NOUN
ap-1815	264	21	space	space	NOUN
ap-1815	264	22	h	h	NOUN
ap-1815	264	23	and	and	CCONJ
ap-1815	264	24	the	the	DET
ap-1815	264	25	domain	domain	NOUN
ap-1815	264	26	d(b	d(b	X
ap-1815	264	27	)	)	PUNCT
ap-1815	265	1	=	=	SYM
ap-1815	265	2	h	h	NOUN
ap-1815	265	3	for	for	ADP
ap-1815	265	4	every	every	DET
ap-1815	265	5	bounded	bounded	ADJ
ap-1815	265	6	operator	operator	NOUN
ap-1815	265	7	b.	b.	PROPN
ap-1815	265	8	to	to	ADP
ap-1815	265	9	every	every	DET
ap-1815	265	10	such	such	ADJ
ap-1815	265	11	linear	linear	ADJ
ap-1815	265	12	operator	operator	NOUN
ap-1815	265	13	with	with	ADP
ap-1815	265	14	d(a	d(a	PROPN
ap-1815	265	15	)	)	PUNCT
ap-1815	266	1	=	=	NOUN
ap-1815	267	1	h	h	NOUN
ap-1815	267	2	there	there	PRON
ap-1815	267	3	exists	exist	VERB
ap-1815	267	4	the	the	DET
ap-1815	267	5	adjoint	adjoint	NOUN
ap-1815	267	6	operator	operator	NOUN
ap-1815	267	7	a∗	a∗	NOUN
ap-1815	267	8	of	of	ADP
ap-1815	267	9	a	a	DET
ap-1815	267	10	such	such	ADJ
ap-1815	267	11	that	that	DET
ap-1815	267	12	d(a∗	d(a∗	NOUN
ap-1815	267	13	)	)	PUNCT
ap-1815	267	14	=	=	SYM
ap-1815	267	15	{	{	PUNCT
ap-1815	268	1	y	y	PROPN
ap-1815	268	2	∈	∈	PROPN
ap-1815	268	3	h	h	NOUN
ap-1815	269	1	|	|	ADV
ap-1815	269	2	there	there	PRON
ap-1815	269	3	exists	exist	VERB
ap-1815	269	4	y∗	y∗	PROPN
ap-1815	269	5	∈	∈	PROPN
ap-1815	269	6	h	h	NOUN
ap-1815	269	7	such	such	ADJ
ap-1815	269	8	that	that	PRON
ap-1815	269	9	(	(	PUNCT
ap-1815	269	10	y∗	y∗	ADV
ap-1815	269	11	,	,	PUNCT
ap-1815	269	12	x	x	NOUN
ap-1815	269	13	)	)	PUNCT
ap-1815	269	14	=	=	SYM
ap-1815	270	1	(	(	PUNCT
ap-1815	270	2	y	y	NOUN
ap-1815	270	3	,	,	PUNCT
ap-1815	270	4	ax	ax	NOUN
ap-1815	270	5	)	)	PUNCT
ap-1815	270	6	for	for	ADP
ap-1815	270	7	every	every	DET
ap-1815	270	8	x	x	PROPN
ap-1815	270	9	∈	∈	PROPN
ap-1815	270	10	d(a	d(a	PROPN
ap-1815	270	11	)	)	PUNCT
ap-1815	270	12	}	}	PUNCT
ap-1815	270	13	and	and	CCONJ
ap-1815	270	14	a∗y	a∗y	NUM
ap-1815	270	15	=	=	PUNCT
ap-1815	270	16	y∗	y∗	ADV
ap-1815	270	17	for	for	ADP
ap-1815	270	18	every	every	DET
ap-1815	270	19	y	y	PROPN
ap-1815	270	20	∈	∈	PROPN
ap-1815	270	21	d(a∗	d(a∗	NUM
ap-1815	270	22	)	)	PUNCT
ap-1815	270	23	.	.	PUNCT
ap-1815	271	1	if	if	SCONJ
ap-1815	271	2	a∗	a∗	PROPN
ap-1815	271	3	=	=	PUNCT
ap-1815	271	4	a	a	PRON
ap-1815	271	5	then	then	ADV
ap-1815	271	6	a	a	PRON
ap-1815	271	7	is	be	AUX
ap-1815	271	8	called	call	VERB
ap-1815	271	9	self	self	NOUN
ap-1815	271	10	-	-	PUNCT
ap-1815	271	11	adjoint	adjoint	NOUN
ap-1815	271	12	.	.	PUNCT
ap-1815	272	1	an	an	DET
ap-1815	272	2	operator	operator	NOUN
ap-1815	272	3	a	a	PRON
ap-1815	272	4	:	:	PUNCT
ap-1815	272	5	d(a	d(a	PROPN
ap-1815	272	6	)	)	PUNCT
ap-1815	272	7	→	→	SYM
ap-1815	272	8	h	h	NOUN
ap-1815	272	9	is	be	AUX
ap-1815	272	10	called	call	VERB
ap-1815	272	11	closed	closed	ADJ
ap-1815	272	12	if	if	SCONJ
ap-1815	272	13	for	for	ADP
ap-1815	272	14	every	every	DET
ap-1815	272	15	sequence	sequence	NOUN
ap-1815	272	16	(	(	PUNCT
ap-1815	272	17	xn)n∈n	xn)n∈n	NUM
ap-1815	272	18	,	,	PUNCT
ap-1815	272	19	xn	xn	PROPN
ap-1815	272	20	∈	∈	PROPN
ap-1815	272	21	d(a	d(a	PROPN
ap-1815	272	22	)	)	PUNCT
ap-1815	272	23	,	,	PUNCT
ap-1815	272	24	such	such	ADJ
ap-1815	272	25	that	that	PRON
ap-1815	272	26	xn	xn	PUNCT
ap-1815	273	1	→	→	SYM
ap-1815	273	2	x	x	SYM
ap-1815	273	3	∈	∈	PROPN
ap-1815	273	4	h	h	NOUN
ap-1815	273	5	and	and	CCONJ
ap-1815	273	6	axn	axn	PROPN
ap-1815	273	7	→	→	SYM
ap-1815	273	8	y	y	PROPN
ap-1815	273	9	∈	∈	PROPN
ap-1815	273	10	h	h	NOUN
ap-1815	273	11	as	as	ADP
ap-1815	273	12	n	n	PROPN
ap-1815	273	13	→	→	SYM
ap-1815	273	14	∞	∞	NUM
ap-1815	273	15	one	one	NOUN
ap-1815	273	16	has	have	VERB
ap-1815	273	17	x	x	PROPN
ap-1815	273	18	∈	∈	PROPN
ap-1815	273	19	d(a	d(a	PROPN
ap-1815	273	20	)	)	PUNCT
ap-1815	273	21	and	and	CCONJ
ap-1815	273	22	ax	ax	NOUN
ap-1815	273	23	=	=	PUNCT
ap-1815	273	24	y.	y.	NOUN
ap-1815	273	25	since	since	SCONJ
ap-1815	273	26	every	every	DET
ap-1815	273	27	a	a	DET
ap-1815	273	28	∈	∈	PROPN
ap-1815	273	29	v(h	v(h	NOUN
ap-1815	273	30	)	)	PUNCT
ap-1815	273	31	is	be	AUX
ap-1815	273	32	symmetric	symmetric	ADJ
ap-1815	273	33	there	there	PRON
ap-1815	273	34	exists	exist	VERB
ap-1815	273	35	a	a	DET
ap-1815	273	36	closed	closed	ADJ
ap-1815	273	37	operator	operator	NOUN
ap-1815	273	38	a	a	DET
ap-1815	273	39	such	such	ADJ
ap-1815	273	40	that	that	SCONJ
ap-1815	273	41	a	a	DET
ap-1815	273	42	⊂	⊂	PROPN
ap-1815	273	43	a	a	PROPN
ap-1815	273	44	and	and	CCONJ
ap-1815	273	45	a	a	DET
ap-1815	273	46	⊂	⊂	PROPN
ap-1815	273	47	b	b	PROPN
ap-1815	273	48	for	for	ADP
ap-1815	273	49	every	every	DET
ap-1815	273	50	closed	closed	ADJ
ap-1815	273	51	operator	operator	NOUN
ap-1815	273	52	extending	extend	VERB
ap-1815	273	53	a.	a.	NOUN
ap-1815	273	54	moreover	moreover	ADV
ap-1815	273	55	a	a	PRON
ap-1815	273	56	is	be	AUX
ap-1815	273	57	again	again	ADV
ap-1815	273	58	symmetric	symmetric	ADJ
ap-1815	273	59	and	and	CCONJ
ap-1815	273	60	it	it	PRON
ap-1815	273	61	is	be	AUX
ap-1815	273	62	called	call	VERB
ap-1815	273	63	the	the	DET
ap-1815	273	64	closure	closure	NOUN
ap-1815	273	65	of	of	ADP
ap-1815	273	66	a.	a.	NOUN
ap-1815	273	67	a	a	DET
ap-1815	273	68	symmetric	symmetric	ADJ
ap-1815	273	69	operator	operator	NOUN
ap-1815	273	70	a	a	PRON
ap-1815	273	71	is	be	AUX
ap-1815	273	72	called	call	VERB
ap-1815	273	73	essentially	essentially	ADV
ap-1815	273	74	self	self	NOUN
ap-1815	273	75	-	-	PUNCT
ap-1815	273	76	adjoint	adjoint	NOUN
ap-1815	273	77	if	if	SCONJ
ap-1815	273	78	(	(	PUNCT
ap-1815	273	79	a	a	DET
ap-1815	273	80	)	)	PUNCT
ap-1815	273	81	∗	∗	NOUN
ap-1815	273	82	=	=	PUNCT
ap-1815	273	83	a	a	PROPN
ap-1815	274	1	and	and	CCONJ
ap-1815	274	2	then	then	ADV
ap-1815	274	3	a	a	PRON
ap-1815	274	4	is	be	AUX
ap-1815	274	5	a	a	DET
ap-1815	274	6	unique	unique	ADJ
ap-1815	274	7	self	self	NOUN
ap-1815	274	8	-	-	PUNCT
ap-1815	274	9	adjoint	adjoint	NOUN
ap-1815	274	10	extension	extension	NOUN
ap-1815	274	11	of	of	ADP
ap-1815	274	12	a	a	PRON
ap-1815	274	13	(	(	PUNCT
ap-1815	274	14	see	see	PROPN
ap-1815	274	15	[	[	X
ap-1815	274	16	1	1	NUM
ap-1815	274	17	,	,	PUNCT
ap-1815	274	18	p.	p.	NOUN
ap-1815	274	19	96	96	NUM
ap-1815	274	20	]	]	PUNCT
ap-1815	274	21	)	)	PUNCT
ap-1815	274	22	.	.	PUNCT
ap-1815	275	1	finally	finally	ADV
ap-1815	275	2	,	,	PUNCT
ap-1815	275	3	recall	recall	VERB
ap-1815	275	4	that	that	SCONJ
ap-1815	275	5	every	every	DET
ap-1815	275	6	a	a	DET
ap-1815	275	7	∈	∈	PROPN
ap-1815	275	8	v(h	v(h	NOUN
ap-1815	275	9	)	)	PUNCT
ap-1815	275	10	has	have	VERB
ap-1815	275	11	a	a	DET
ap-1815	275	12	positive	positive	ADJ
ap-1815	275	13	self	self	NOUN
ap-1815	275	14	-	-	PUNCT
ap-1815	275	15	adjoint	adjoint	NOUN
ap-1815	275	16	extension	extension	NOUN
ap-1815	275	17	â	â	ADP
ap-1815	275	18	called	call	VERB
ap-1815	275	19	friedrichs	friedrich	NOUN
ap-1815	275	20	’	'	PUNCT
ap-1815	275	21	extension	extension	NOUN
ap-1815	275	22	of	of	ADP
ap-1815	275	23	a	a	DET
ap-1815	275	24	(	(	PUNCT
ap-1815	275	25	see	see	NOUN
ap-1815	275	26	,	,	PUNCT
ap-1815	275	27	e.g.	e.g.	ADV
ap-1815	275	28	,	,	PUNCT
ap-1815	275	29	[	[	X
ap-1815	275	30	16	16	NUM
ap-1815	275	31	]	]	SYM
ap-1815	275	32	)	)	PUNCT
ap-1815	275	33	.	.	PUNCT
ap-1815	276	1	moreover	moreover	ADV
ap-1815	276	2	,	,	PUNCT
ap-1815	276	3	â	â	X
ap-1815	276	4	extends	extend	VERB
ap-1815	276	5	all	all	DET
ap-1815	276	6	symmetric	symmetric	ADJ
ap-1815	276	7	extensions	extension	NOUN
ap-1815	276	8	a′	a′	PROPN
ap-1815	276	9	of	of	ADP
ap-1815	276	10	a.	a.	NOUN
ap-1815	277	1	it	it	PRON
ap-1815	277	2	was	be	AUX
ap-1815	277	3	shown	show	VERB
ap-1815	277	4	in	in	ADP
ap-1815	277	5	[	[	X
ap-1815	277	6	15	15	NUM
ap-1815	277	7	,	,	PUNCT
ap-1815	277	8	theorem	theorem	VERB
ap-1815	277	9	1	1	NUM
ap-1815	277	10	]	]	PUNCT
ap-1815	277	11	that	that	SCONJ
ap-1815	277	12	,	,	PUNCT
ap-1815	277	13	for	for	ADP
ap-1815	277	14	any	any	DET
ap-1815	277	15	infinite	infinite	ADJ
ap-1815	277	16	-	-	PUNCT
ap-1815	277	17	dimensional	dimensional	ADJ
ap-1815	277	18	complex	complex	ADJ
ap-1815	277	19	hilbert	hilbert	NOUN
ap-1815	277	20	space	space	NOUN
ap-1815	277	21	h	h	NOUN
ap-1815	277	22	,	,	PUNCT
ap-1815	277	23	there	there	PRON
ap-1815	277	24	are	be	VERB
ap-1815	277	25	positive	positive	ADJ
ap-1815	277	26	unbounded	unbounded	ADJ
ap-1815	277	27	a	a	PRON
ap-1815	277	28	and	and	CCONJ
ap-1815	277	29	b	b	NOUN
ap-1815	277	30	such	such	ADJ
ap-1815	277	31	that	that	SCONJ
ap-1815	277	32	a	a	PRON
ap-1815	277	33	is	be	AUX
ap-1815	277	34	not	not	PART
ap-1815	277	35	essentially	essentially	ADV
ap-1815	277	36	self	self	NOUN
ap-1815	277	37	-	-	PUNCT
ap-1815	277	38	adjoint	adjoint	NOUN
ap-1815	277	39	and	and	CCONJ
ap-1815	277	40	b	b	NOUN
ap-1815	277	41	is	be	AUX
ap-1815	277	42	not	not	PART
ap-1815	277	43	closed	closed	ADJ
ap-1815	277	44	.	.	PUNCT
ap-1815	278	1	312	312	NUM
ap-1815	278	2	vol	vol	NOUN
ap-1815	278	3	.	.	PUNCT
ap-1815	279	1	53	53	NUM
ap-1815	279	2	no	no	NOUN
ap-1815	279	3	.	.	PUNCT
ap-1815	280	1	3/2013	3/2013	PROPN
ap-1815	280	2	inherited	inherit	VERB
ap-1815	280	3	properties	property	NOUN
ap-1815	280	4	of	of	ADP
ap-1815	280	5	effect	effect	NOUN
ap-1815	280	6	algebras	algebra	NOUN
ap-1815	280	7	furthermore	furthermore	ADV
ap-1815	280	8	,	,	PUNCT
ap-1815	280	9	let	let	VERB
ap-1815	280	10	v(h	v(h	NOUN
ap-1815	280	11	)	)	PUNCT
ap-1815	280	12	be	be	AUX
ap-1815	280	13	equipped	equip	VERB
ap-1815	280	14	with	with	ADP
ap-1815	280	15	the	the	DET
ap-1815	280	16	partial	partial	ADJ
ap-1815	280	17	sum	sum	NOUN
ap-1815	280	18	⊕	⊕	PROPN
ap-1815	280	19	such	such	ADJ
ap-1815	280	20	that	that	PRON
ap-1815	280	21	for	for	ADP
ap-1815	280	22	any	any	DET
ap-1815	280	23	a	a	DET
ap-1815	280	24	,	,	PUNCT
ap-1815	280	25	b	b	PROPN
ap-1815	280	26	∈	∈	PROPN
ap-1815	280	27	v(h	v(h	NOUN
ap-1815	280	28	)	)	PUNCT
ap-1815	280	29	the	the	DET
ap-1815	280	30	sum	sum	NOUN
ap-1815	280	31	a⊕b	a⊕b	NOUN
ap-1815	280	32	is	be	AUX
ap-1815	280	33	defined	define	VERB
ap-1815	280	34	iff	iff	PROPN
ap-1815	280	35	either	either	DET
ap-1815	280	36	one	one	NUM
ap-1815	280	37	of	of	ADP
ap-1815	280	38	a	a	PRON
ap-1815	280	39	,	,	PUNCT
ap-1815	280	40	b	b	PROPN
ap-1815	280	41	is	be	AUX
ap-1815	280	42	bounded	bound	VERB
ap-1815	280	43	or	or	CCONJ
ap-1815	280	44	d(a	d(a	PROPN
ap-1815	280	45	)	)	PUNCT
ap-1815	280	46	=	=	PUNCT
ap-1815	280	47	d(b	d(b	PROPN
ap-1815	280	48	)	)	PUNCT
ap-1815	280	49	.	.	PUNCT
ap-1815	281	1	then	then	ADV
ap-1815	281	2	we	we	PRON
ap-1815	281	3	set	set	VERB
ap-1815	281	4	a⊕b	a⊕b	NOUN
ap-1815	281	5	=	=	SYM
ap-1815	281	6	a+b	a+b	X
ap-1815	281	7	(	(	PUNCT
ap-1815	281	8	the	the	DET
ap-1815	281	9	usual	usual	ADJ
ap-1815	281	10	operator	operator	NOUN
ap-1815	281	11	sum	sum	NOUN
ap-1815	281	12	)	)	PUNCT
ap-1815	281	13	.	.	PUNCT
ap-1815	282	1	in	in	ADP
ap-1815	282	2	[	[	X
ap-1815	282	3	17	17	NUM
ap-1815	282	4	]	]	X
ap-1815	282	5	it	it	PRON
ap-1815	282	6	was	be	AUX
ap-1815	282	7	proved	prove	VERB
ap-1815	282	8	that	that	SCONJ
ap-1815	282	9	(	(	PUNCT
ap-1815	282	10	v(h);⊕	v(h);⊕	NOUN
ap-1815	282	11	,	,	PUNCT
ap-1815	282	12	0	0	NUM
ap-1815	282	13	)	)	PUNCT
ap-1815	282	14	is	be	AUX
ap-1815	282	15	a	a	DET
ap-1815	282	16	generalized	generalized	ADJ
ap-1815	282	17	effect	effect	NOUN
ap-1815	282	18	algebra	algebra	NOUN
ap-1815	282	19	.	.	PUNCT
ap-1815	283	1	theorem	theorem	VERB
ap-1815	283	2	4.3	4.3	NUM
ap-1815	283	3	(	(	PUNCT
ap-1815	283	4	[	[	X
ap-1815	283	5	17	17	NUM
ap-1815	283	6	,	,	PUNCT
ap-1815	283	7	theorem	theorem	VERB
ap-1815	283	8	2	2	NUM
ap-1815	283	9	]	]	PUNCT
ap-1815	283	10	,	,	PUNCT
ap-1815	284	1	[	[	X
ap-1815	284	2	18	18	NUM
ap-1815	284	3	,	,	PUNCT
ap-1815	284	4	theorem	theorem	VERB
ap-1815	284	5	7	7	NUM
ap-1815	284	6	]	]	PUNCT
ap-1815	284	7	)	)	PUNCT
ap-1815	284	8	.	.	PUNCT
ap-1815	285	1	for	for	ADP
ap-1815	285	2	every	every	DET
ap-1815	285	3	infinite	infinite	ADJ
ap-1815	285	4	-	-	PUNCT
ap-1815	285	5	dimensional	dimensional	ADJ
ap-1815	285	6	complex	complex	ADJ
ap-1815	285	7	hilbert	hilbert	NOUN
ap-1815	285	8	space	space	NOUN
ap-1815	285	9	h	h	NOUN
ap-1815	285	10	and	and	CCONJ
ap-1815	285	11	every	every	DET
ap-1815	285	12	q	q	PROPN
ap-1815	285	13	∈	∈	PROPN
ap-1815	285	14	v(h	v(h	NOUN
ap-1815	285	15	)	)	PUNCT
ap-1815	285	16	,	,	PUNCT
ap-1815	285	17	q	q	X
ap-1815	285	18	6=	6=	NUM
ap-1815	285	19	0	0	NUM
ap-1815	285	20	it	it	PRON
ap-1815	285	21	holds	hold	VERB
ap-1815	285	22	:	:	PUNCT
ap-1815	285	23	(	(	PUNCT
ap-1815	285	24	1	1	NUM
ap-1815	285	25	.	.	PUNCT
ap-1815	285	26	)	)	PUNCT
ap-1815	286	1	the	the	DET
ap-1815	286	2	interval	interval	NOUN
ap-1815	286	3	(	(	PUNCT
ap-1815	286	4	[	[	X
ap-1815	286	5	0	0	NUM
ap-1815	286	6	,	,	PUNCT
ap-1815	286	7	q]v(h);⊕q	q]v(h);⊕q	PROPN
ap-1815	286	8	,	,	PUNCT
ap-1815	286	9	0	0	NUM
ap-1815	286	10	,	,	PUNCT
ap-1815	286	11	q	q	NOUN
ap-1815	286	12	)	)	PUNCT
ap-1815	286	13	where	where	SCONJ
ap-1815	286	14	a	a	DET
ap-1815	286	15	⊕q	⊕q	NOUN
ap-1815	286	16	b	b	X
ap-1815	286	17	=	=	PUNCT
ap-1815	286	18	a	a	DET
ap-1815	286	19	+	+	PROPN
ap-1815	286	20	b	b	X
ap-1815	286	21	iff	iff	PROPN
ap-1815	286	22	a	a	DET
ap-1815	286	23	+	+	PROPN
ap-1815	286	24	b	b	NOUN
ap-1815	286	25	≤	≤	NUM
ap-1815	286	26	q	q	NOUN
ap-1815	286	27	,	,	PUNCT
ap-1815	286	28	for	for	ADP
ap-1815	286	29	any	any	DET
ap-1815	286	30	a	a	DET
ap-1815	286	31	,	,	PUNCT
ap-1815	286	32	b	b	PROPN
ap-1815	286	33	∈	∈	PROPN
ap-1815	286	34	[	[	X
ap-1815	286	35	0	0	NUM
ap-1815	286	36	,	,	PUNCT
ap-1815	286	37	q]v(h	q]v(h	NOUN
ap-1815	286	38	)	)	PUNCT
ap-1815	286	39	,	,	PUNCT
ap-1815	286	40	is	be	AUX
ap-1815	286	41	an	an	DET
ap-1815	286	42	effect	effect	NOUN
ap-1815	286	43	algebra	algebra	NOUN
ap-1815	286	44	and	and	CCONJ
ap-1815	286	45	mq	mq	NOUN
ap-1815	286	46	=	=	PROPN
ap-1815	286	47	{	{	PUNCT
ap-1815	286	48	ωx	ωx	PROPN
ap-1815	286	49	|	|	ADV
ap-1815	286	50	x	x	SYM
ap-1815	286	51	∈	∈	PROPN
ap-1815	286	52	d(q	d(q	PROPN
ap-1815	286	53	)	)	PUNCT
ap-1815	286	54	,	,	PUNCT
ap-1815	286	55	(	(	PUNCT
ap-1815	286	56	x	x	X
ap-1815	286	57	,	,	PUNCT
ap-1815	286	58	qx	qx	PROPN
ap-1815	286	59	)	)	PUNCT
ap-1815	286	60	>	>	X
ap-1815	286	61	0	0	NUM
ap-1815	286	62	}	}	PUNCT
ap-1815	286	63	is	be	AUX
ap-1815	286	64	an	an	DET
ap-1815	286	65	ordering	ordering	NOUN
ap-1815	286	66	set	set	NOUN
ap-1815	286	67	of	of	ADP
ap-1815	286	68	states	state	NOUN
ap-1815	286	69	on	on	ADP
ap-1815	286	70	[	[	X
ap-1815	286	71	0	0	NUM
ap-1815	286	72	,	,	PUNCT
ap-1815	286	73	q]v(h	q]v(h	NOUN
ap-1815	286	74	)	)	PUNCT
ap-1815	286	75	;	;	PUNCT
ap-1815	286	76	here	here	ADV
ap-1815	286	77	the	the	DET
ap-1815	286	78	mapping	mapping	NOUN
ap-1815	286	79	ωx	ωx	X
ap-1815	286	80	:	:	PUNCT
ap-1815	287	1	[	[	X
ap-1815	287	2	0	0	NUM
ap-1815	287	3	,	,	PUNCT
ap-1815	287	4	q]v(h	q]v(h	NOUN
ap-1815	287	5	)	)	PUNCT
ap-1815	287	6	→	→	PUNCT
ap-1815	288	1	[	[	X
ap-1815	288	2	0	0	NUM
ap-1815	288	3	,	,	PUNCT
ap-1815	288	4	1	1	NUM
ap-1815	288	5	]	]	SYM
ap-1815	288	6	⊆	⊆	NUM
ap-1815	288	7	r	r	NOUN
ap-1815	288	8	is	be	AUX
ap-1815	288	9	defined	define	VERB
ap-1815	288	10	for	for	ADP
ap-1815	288	11	every	every	DET
ap-1815	288	12	a	a	DET
ap-1815	288	13	∈	∈	NOUN
ap-1815	288	14	[	[	X
ap-1815	288	15	0	0	NUM
ap-1815	288	16	,	,	PUNCT
ap-1815	288	17	q]v(h	q]v(h	NOUN
ap-1815	288	18	)	)	PUNCT
ap-1815	288	19	by	by	ADP
ap-1815	288	20	ωx(a	ωx(a	NOUN
ap-1815	288	21	)	)	PUNCT
ap-1815	288	22	=	=	SYM
ap-1815	288	23	(	(	PUNCT
ap-1815	288	24	x	x	NOUN
ap-1815	288	25	,	,	PUNCT
ap-1815	288	26	ax	ax	NOUN
ap-1815	288	27	)	)	PUNCT
ap-1815	288	28	(	(	PUNCT
ap-1815	288	29	x	x	X
ap-1815	288	30	,	,	PUNCT
ap-1815	288	31	qx	qx	PROPN
ap-1815	288	32	)	)	PUNCT
ap-1815	288	33	.	.	PUNCT
ap-1815	289	1	(	(	PUNCT
ap-1815	289	2	2	2	NUM
ap-1815	289	3	.	.	PUNCT
ap-1815	289	4	)	)	PUNCT
ap-1815	290	1	the	the	DET
ap-1815	290	2	effect	effect	NOUN
ap-1815	290	3	algebra	algebra	NOUN
ap-1815	290	4	(	(	PUNCT
ap-1815	290	5	[	[	X
ap-1815	290	6	0	0	NUM
ap-1815	290	7	,	,	PUNCT
ap-1815	290	8	q]v(h);⊕q	q]v(h);⊕q	PROPN
ap-1815	290	9	,	,	PUNCT
ap-1815	290	10	0	0	NUM
ap-1815	290	11	,	,	PUNCT
ap-1815	290	12	q	q	PUNCT
ap-1815	290	13	)	)	PUNCT
ap-1815	290	14	can	can	AUX
ap-1815	290	15	be	be	AUX
ap-1815	290	16	embedded	embed	VERB
ap-1815	290	17	into	into	ADP
ap-1815	290	18	the	the	DET
ap-1815	290	19	standard	standard	ADJ
ap-1815	290	20	hilbert	hilbert	PROPN
ap-1815	290	21	effect	effect	NOUN
ap-1815	290	22	algebra	algebra	NOUN
ap-1815	290	23	e	e	PROPN
ap-1815	290	24	(	(	PUNCT
ap-1815	290	25	l2(mq	l2(mq	PROPN
ap-1815	290	26	)	)	PUNCT
ap-1815	290	27	)	)	PUNCT
ap-1815	290	28	.	.	PUNCT
ap-1815	291	1	we	we	PRON
ap-1815	291	2	denote	denote	VERB
ap-1815	291	3	the	the	DET
ap-1815	291	4	respective	respective	ADJ
ap-1815	291	5	embedding	embed	VERB
ap-1815	291	6	by	by	ADP
ap-1815	291	7	ϕq	ϕq	PROPN
ap-1815	291	8	.	.	PUNCT
ap-1815	292	1	therefore	therefore	ADV
ap-1815	292	2	we	we	PRON
ap-1815	292	3	obtain	obtain	VERB
ap-1815	292	4	the	the	DET
ap-1815	292	5	following	following	NOUN
ap-1815	292	6	theorem	theorem	NOUN
ap-1815	292	7	that	that	DET
ap-1815	292	8	boundedness	boundedness	NOUN
ap-1815	292	9	(	(	PUNCT
ap-1815	292	10	self	self	NOUN
ap-1815	292	11	-	-	PUNCT
ap-1815	292	12	adjointness	adjointness	NOUN
ap-1815	292	13	,	,	PUNCT
ap-1815	292	14	closedness	closedness	ADJ
ap-1815	292	15	,	,	PUNCT
ap-1815	292	16	essential	essential	ADJ
ap-1815	292	17	self	self	NOUN
ap-1815	292	18	-	-	PUNCT
ap-1815	292	19	adjointness	adjointness	NOUN
ap-1815	292	20	,	,	PUNCT
ap-1815	292	21	friedrichs	friedrich	NOUN
ap-1815	292	22	’	'	PUNCT
ap-1815	292	23	extension	extension	NOUN
ap-1815	292	24	)	)	PUNCT
ap-1815	292	25	of	of	ADP
ap-1815	292	26	operators	operator	NOUN
ap-1815	292	27	need	need	AUX
ap-1815	292	28	not	not	PART
ap-1815	292	29	be	be	AUX
ap-1815	292	30	preserved	preserve	VERB
ap-1815	292	31	by	by	ADP
ap-1815	292	32	effect	effect	NOUN
ap-1815	292	33	algebraic	algebraic	PROPN
ap-1815	292	34	isomorphisms	isomorphisms	PROPN
ap-1815	292	35	.	.	PUNCT
ap-1815	293	1	theorem	theorem	VERB
ap-1815	293	2	4.4	4.4	NUM
ap-1815	293	3	.	.	PUNCT
ap-1815	294	1	for	for	ADP
ap-1815	294	2	every	every	DET
ap-1815	294	3	infinite	infinite	ADJ
ap-1815	294	4	-	-	PUNCT
ap-1815	294	5	dimensional	dimensional	ADJ
ap-1815	294	6	complex	complex	ADJ
ap-1815	294	7	hilbert	hilbert	NOUN
ap-1815	294	8	space	space	NOUN
ap-1815	294	9	h	h	NOUN
ap-1815	294	10	and	and	CCONJ
ap-1815	294	11	every	every	DET
ap-1815	294	12	q	q	PROPN
ap-1815	294	13	∈	∈	PROPN
ap-1815	294	14	v(h	v(h	NOUN
ap-1815	294	15	)	)	PUNCT
ap-1815	294	16	,	,	PUNCT
ap-1815	294	17	q	q	X
ap-1815	294	18	6=	6=	ADP
ap-1815	294	19	0	0	NUM
ap-1815	294	20	unbounded	unbounded	ADJ
ap-1815	294	21	(	(	PUNCT
ap-1815	294	22	unbounded	unbounded	ADJ
ap-1815	294	23	and	and	CCONJ
ap-1815	294	24	non	non	ADJ
ap-1815	294	25	self	self	NOUN
ap-1815	294	26	-	-	PUNCT
ap-1815	294	27	adjoint	adjoint	NOUN
ap-1815	294	28	,	,	PUNCT
ap-1815	294	29	unbounded	unbounded	ADJ
ap-1815	294	30	and	and	CCONJ
ap-1815	294	31	non	non	ADJ
ap-1815	294	32	closed	closed	ADJ
ap-1815	294	33	,	,	PUNCT
ap-1815	294	34	unbounded	unbounded	ADJ
ap-1815	294	35	and	and	CCONJ
ap-1815	294	36	non	non	ADJ
ap-1815	294	37	essentially	essentially	ADV
ap-1815	294	38	self	self	NOUN
ap-1815	294	39	-	-	PUNCT
ap-1815	294	40	adjoint	adjoint	NOUN
ap-1815	294	41	,	,	PUNCT
ap-1815	294	42	unbounded	unbounded	ADJ
ap-1815	294	43	and	and	CCONJ
ap-1815	294	44	with	with	ADP
ap-1815	294	45	q̂	q̂	NUM
ap-1815	294	46	6=	6=	ADP
ap-1815	294	47	q	q	NOUN
ap-1815	294	48	respectively	respectively	ADV
ap-1815	294	49	)	)	PUNCT
ap-1815	294	50	we	we	PRON
ap-1815	294	51	have	have	VERB
ap-1815	294	52	an	an	DET
ap-1815	294	53	effect	effect	NOUN
ap-1815	294	54	algebraic	algebraic	ADJ
ap-1815	294	55	isomorphism	isomorphism	NOUN
ap-1815	294	56	ϕ−1	ϕ−1	ADP
ap-1815	294	57	q	q	NOUN
ap-1815	294	58	:	:	PUNCT
ap-1815	294	59	ϕq	ϕq	X
ap-1815	294	60	(	(	PUNCT
ap-1815	294	61	[	[	X
ap-1815	294	62	0	0	NUM
ap-1815	294	63	,	,	PUNCT
ap-1815	294	64	q]v(h	q]v(h	NOUN
ap-1815	294	65	)	)	PUNCT
ap-1815	294	66	)	)	PUNCT
ap-1815	295	1	→	→	PUNCT
ap-1815	295	2	[	[	X
ap-1815	295	3	0	0	NUM
ap-1815	295	4	,	,	PUNCT
ap-1815	295	5	q]v(h	q]v(h	NOUN
ap-1815	295	6	)	)	PUNCT
ap-1815	295	7	such	such	ADJ
ap-1815	295	8	that	that	SCONJ
ap-1815	295	9	ϕ−1	ϕ−1	PROPN
ap-1815	295	10	q	q	PROPN
ap-1815	295	11	does	do	AUX
ap-1815	295	12	not	not	PART
ap-1815	295	13	preserve	preserve	VERB
ap-1815	295	14	bounded	bounded	ADJ
ap-1815	295	15	operators	operator	NOUN
ap-1815	295	16	(	(	PUNCT
ap-1815	295	17	self	self	NOUN
ap-1815	295	18	-	-	PUNCT
ap-1815	295	19	adjoint	adjoint	NOUN
ap-1815	295	20	operators	operator	NOUN
ap-1815	295	21	,	,	PUNCT
ap-1815	295	22	closed	closed	ADJ
ap-1815	295	23	operators	operator	NOUN
ap-1815	295	24	,	,	PUNCT
ap-1815	295	25	non	non	X
ap-1815	295	26	essentially	essentially	ADV
ap-1815	295	27	self	self	NOUN
ap-1815	295	28	-	-	PUNCT
ap-1815	295	29	adjoint	adjoint	NOUN
ap-1815	295	30	operators	operator	NOUN
ap-1815	295	31	,	,	PUNCT
ap-1815	295	32	friedrichs	friedrich	NOUN
ap-1815	295	33	’	'	PUNCT
ap-1815	295	34	extension	extension	NOUN
ap-1815	295	35	,	,	PUNCT
ap-1815	295	36	respectively	respectively	ADV
ap-1815	295	37	)	)	PUNCT
ap-1815	295	38	.	.	PUNCT
ap-1815	296	1	proof	proof	NOUN
ap-1815	296	2	.	.	PUNCT
ap-1815	297	1	clearly	clearly	ADV
ap-1815	297	2	,	,	PUNCT
ap-1815	297	3	ϕq(q	ϕq(q	NUM
ap-1815	297	4	)	)	PUNCT
ap-1815	297	5	=	=	SYM
ap-1815	297	6	il2(mq	il2(mq	NOUN
ap-1815	297	7	)	)	PUNCT
ap-1815	297	8	∈	∈	PROPN
ap-1815	297	9	ϕq	ϕq	X
ap-1815	298	1	(	(	PUNCT
ap-1815	298	2	[	[	X
ap-1815	298	3	0	0	NUM
ap-1815	298	4	,	,	PUNCT
ap-1815	298	5	q]v(h	q]v(h	NOUN
ap-1815	298	6	)	)	PUNCT
ap-1815	298	7	)	)	PUNCT
ap-1815	298	8	.	.	PUNCT
ap-1815	299	1	note	note	VERB
ap-1815	299	2	that	that	SCONJ
ap-1815	299	3	il2(mq	il2(mq	NOUN
ap-1815	299	4	)	)	PUNCT
ap-1815	299	5	is	be	AUX
ap-1815	299	6	bounded	bound	VERB
ap-1815	299	7	and	and	CCONJ
ap-1815	299	8	positive	positive	ADJ
ap-1815	299	9	.	.	PUNCT
ap-1815	300	1	it	it	PRON
ap-1815	300	2	follows	follow	VERB
ap-1815	300	3	that	that	SCONJ
ap-1815	300	4	it	it	PRON
ap-1815	300	5	is	be	AUX
ap-1815	300	6	also	also	ADV
ap-1815	300	7	self	self	NOUN
ap-1815	300	8	-	-	PUNCT
ap-1815	300	9	adjoint	adjoint	NOUN
ap-1815	300	10	,	,	PUNCT
ap-1815	300	11	closed	closed	ADJ
ap-1815	300	12	,	,	PUNCT
ap-1815	300	13	essentially	essentially	ADV
ap-1815	300	14	self	self	NOUN
ap-1815	300	15	-	-	PUNCT
ap-1815	300	16	adjoint	adjoint	NOUN
ap-1815	300	17	and	and	CCONJ
ap-1815	300	18	it	it	PRON
ap-1815	300	19	coincides	coincide	VERB
ap-1815	300	20	with	with	ADP
ap-1815	300	21	its	its	PRON
ap-1815	300	22	friedrichs	friedrich	NOUN
ap-1815	300	23	’	'	PUNCT
ap-1815	300	24	extension	extension	NOUN
ap-1815	300	25	.	.	PUNCT
ap-1815	301	1	hence	hence	ADV
ap-1815	301	2	ϕ−1	ϕ−1	INTJ
ap-1815	301	3	q	q	PROPN
ap-1815	301	4	(	(	PUNCT
ap-1815	301	5	il2(mq	il2(mq	NOUN
ap-1815	301	6	)	)	PUNCT
ap-1815	301	7	)	)	PUNCT
ap-1815	302	1	=	=	PUNCT
ap-1815	302	2	q	q	ADJ
ap-1815	302	3	,	,	PUNCT
ap-1815	302	4	il2(mq	il2(mq	NOUN
ap-1815	302	5	)	)	PUNCT
ap-1815	302	6	is	be	AUX
ap-1815	302	7	bounded	bound	VERB
ap-1815	302	8	(	(	PUNCT
ap-1815	302	9	self	self	NOUN
ap-1815	302	10	-	-	PUNCT
ap-1815	302	11	adjoint	adjoint	NOUN
ap-1815	302	12	,	,	PUNCT
ap-1815	302	13	closed	closed	ADJ
ap-1815	302	14	,	,	PUNCT
ap-1815	302	15	essentially	essentially	ADV
ap-1815	302	16	self	self	NOUN
ap-1815	302	17	-	-	PUNCT
ap-1815	302	18	adjoint	adjoint	NOUN
ap-1815	302	19	and	and	CCONJ
ap-1815	302	20	it	it	PRON
ap-1815	302	21	coincides	coincide	VERB
ap-1815	302	22	with	with	ADP
ap-1815	302	23	its	its	PRON
ap-1815	302	24	friedrichs	friedrich	NOUN
ap-1815	302	25	’	'	PUNCT
ap-1815	302	26	extension	extension	NOUN
ap-1815	302	27	respectively	respectively	ADV
ap-1815	302	28	)	)	PUNCT
ap-1815	302	29	in	in	ADP
ap-1815	302	30	l2(mq	l2(mq	PROPN
ap-1815	302	31	)	)	PUNCT
ap-1815	302	32	and	and	CCONJ
ap-1815	302	33	q	q	NOUN
ap-1815	302	34	is	be	AUX
ap-1815	302	35	unbounded	unbounded	ADJ
ap-1815	302	36	(	(	PUNCT
ap-1815	302	37	unbounded	unbounded	ADJ
ap-1815	302	38	and	and	CCONJ
ap-1815	302	39	non	non	ADJ
ap-1815	302	40	self	self	NOUN
ap-1815	302	41	-	-	PUNCT
ap-1815	302	42	adjoint	adjoint	NOUN
ap-1815	302	43	,	,	PUNCT
ap-1815	302	44	unbounded	unbounded	ADJ
ap-1815	302	45	and	and	CCONJ
ap-1815	302	46	non	non	ADJ
ap-1815	302	47	closed	closed	ADJ
ap-1815	302	48	,	,	PUNCT
ap-1815	302	49	unbounded	unbounded	ADJ
ap-1815	302	50	and	and	CCONJ
ap-1815	302	51	non	non	ADJ
ap-1815	302	52	essentially	essentially	ADV
ap-1815	302	53	self	self	NOUN
ap-1815	302	54	-	-	PUNCT
ap-1815	302	55	adjoint	adjoint	NOUN
ap-1815	302	56	,	,	PUNCT
ap-1815	302	57	unbounded	unbounded	ADJ
ap-1815	302	58	and	and	CCONJ
ap-1815	302	59	with	with	ADP
ap-1815	302	60	q̂	q̂	NUM
ap-1815	302	61	6=	6=	ADP
ap-1815	302	62	q	q	X
ap-1815	302	63	,	,	PUNCT
ap-1815	302	64	respectively	respectively	ADV
ap-1815	302	65	)	)	PUNCT
ap-1815	302	66	.	.	PUNCT
ap-1815	303	1	acknowledgements	acknowledgement	NOUN
ap-1815	303	2	the	the	DET
ap-1815	303	3	first	first	ADJ
ap-1815	303	4	author	author	NOUN
ap-1815	303	5	acknowledges	acknowledge	VERB
ap-1815	303	6	support	support	NOUN
ap-1815	303	7	from	from	ADP
ap-1815	303	8	esf	esf	PROPN
ap-1815	303	9	project	project	PROPN
ap-1815	303	10	cz.1.07/2.3.00/20.0051	cz.1.07/2.3.00/20.0051	PROPN
ap-1815	303	11	algebraic	algebraic	ADJ
ap-1815	303	12	methods	method	NOUN
ap-1815	303	13	in	in	ADP
ap-1815	303	14	quantum	quantum	ADJ
ap-1815	303	15	logic	logic	NOUN
ap-1815	303	16	of	of	ADP
ap-1815	303	17	the	the	DET
ap-1815	303	18	masaryk	masaryk	PROPN
ap-1815	303	19	university	university	NOUN
ap-1815	303	20	.	.	PUNCT
ap-1815	304	1	the	the	DET
ap-1815	304	2	second	second	ADJ
ap-1815	304	3	author	author	NOUN
ap-1815	304	4	was	be	AUX
ap-1815	304	5	supported	support	VERB
ap-1815	304	6	by	by	ADP
ap-1815	304	7	grant	grant	PROPN
ap-1815	304	8	vega	vega	PROPN
ap-1815	304	9	1/0297/11	1/0297/11	NUM
ap-1815	304	10	of	of	ADP
ap-1815	304	11	the	the	DET
ap-1815	304	12	ministry	ministry	PROPN
ap-1815	304	13	of	of	ADP
ap-1815	304	14	education	education	NOUN
ap-1815	304	15	of	of	ADP
ap-1815	304	16	the	the	DET
ap-1815	304	17	slovak	slovak	ADJ
ap-1815	304	18	republic	republic	NOUN
ap-1815	304	19	,	,	PUNCT
ap-1815	304	20	and	and	CCONJ
ap-1815	304	21	by	by	ADP
ap-1815	304	22	project	project	NOUN
ap-1815	304	23	apvv0178/11	apvv0178/11	NOUN
ap-1815	304	24	.	.	PUNCT
ap-1815	305	1	references	reference	NOUN
ap-1815	305	2	[	[	X
ap-1815	305	3	1	1	X
ap-1815	305	4	]	]	PUNCT
ap-1815	305	5	j.	j.	PROPN
ap-1815	305	6	blank	blank	PROPN
ap-1815	305	7	,	,	PUNCT
ap-1815	305	8	p.	p.	PROPN
ap-1815	305	9	exner	exner	NOUN
ap-1815	305	10	,	,	PUNCT
ap-1815	305	11	m.	m.	NOUN
ap-1815	305	12	havlíček	havlíček	PROPN
ap-1815	305	13	,	,	PUNCT
ap-1815	305	14	hilbert	hilbert	NOUN
ap-1815	305	15	space	space	NOUN
ap-1815	305	16	operators	operator	NOUN
ap-1815	305	17	in	in	ADP
ap-1815	305	18	quantum	quantum	ADJ
ap-1815	305	19	physics	physics	NOUN
ap-1815	305	20	(	(	PUNCT
ap-1815	305	21	second	second	ADJ
ap-1815	305	22	edition	edition	NOUN
ap-1815	305	23	)	)	PUNCT
ap-1815	305	24	,	,	PUNCT
ap-1815	305	25	springer	springer	NOUN
ap-1815	305	26	,	,	PUNCT
ap-1815	305	27	2008	2008	NUM
ap-1815	305	28	.	.	PUNCT
ap-1815	306	1	[	[	X
ap-1815	306	2	2	2	X
ap-1815	306	3	]	]	PUNCT
ap-1815	306	4	dvurečenskij	dvurečenskij	PROPN
ap-1815	306	5	a.	a.	PROPN
ap-1815	306	6	,	,	PUNCT
ap-1815	306	7	pulmannová	pulmannová	PROPN
ap-1815	306	8	s.	s.	PROPN
ap-1815	306	9	,	,	PUNCT
ap-1815	306	10	new	new	ADJ
ap-1815	306	11	trends	trend	NOUN
ap-1815	306	12	in	in	ADP
ap-1815	306	13	quantum	quantum	ADJ
ap-1815	306	14	structures	structure	NOUN
ap-1815	306	15	,	,	PUNCT
ap-1815	306	16	kluwer	kluwer	NOUN
ap-1815	306	17	,	,	PUNCT
ap-1815	306	18	dordrecht	dordrecht	PROPN
ap-1815	306	19	,	,	PUNCT
ap-1815	306	20	the	the	DET
ap-1815	306	21	netherlands	netherlands	PROPN
ap-1815	306	22	,	,	PUNCT
ap-1815	306	23	2000	2000	NUM
ap-1815	306	24	.	.	PUNCT
ap-1815	307	1	[	[	X
ap-1815	307	2	3	3	X
ap-1815	307	3	]	]	X
ap-1815	307	4	foulis	foulis	PROPN
ap-1815	307	5	d.j	d.j	PROPN
ap-1815	307	6	.	.	PROPN
ap-1815	307	7	,	,	PUNCT
ap-1815	307	8	bennett	bennett	PROPN
ap-1815	307	9	m.k	m.k	PROPN
ap-1815	307	10	.	.	PROPN
ap-1815	307	11	,	,	PUNCT
ap-1815	307	12	effect	effect	NOUN
ap-1815	307	13	algebras	algebra	NOUN
ap-1815	307	14	and	and	CCONJ
ap-1815	307	15	unsharp	unsharp	ADJ
ap-1815	307	16	quantum	quantum	NOUN
ap-1815	307	17	logics	logic	NOUN
ap-1815	307	18	,	,	PUNCT
ap-1815	307	19	foundations	foundation	NOUN
ap-1815	307	20	of	of	ADP
ap-1815	307	21	physics	physics	NOUN
ap-1815	307	22	24	24	NUM
ap-1815	307	23	(	(	PUNCT
ap-1815	307	24	1994	1994	NUM
ap-1815	307	25	)	)	PUNCT
ap-1815	307	26	,	,	PUNCT
ap-1815	307	27	1331–1352	1331–1352	NUM
ap-1815	307	28	.	.	PUNCT
ap-1815	308	1	[	[	X
ap-1815	308	2	4	4	NUM
ap-1815	308	3	]	]	X
ap-1815	308	4	gudder	gudder	ADJ
ap-1815	308	5	s.	s.	PROPN
ap-1815	308	6	,	,	PUNCT
ap-1815	308	7	greechie	greechie	PROPN
ap-1815	308	8	r.	r.	PROPN
ap-1815	308	9	,	,	PUNCT
ap-1815	308	10	sequential	sequential	ADJ
ap-1815	308	11	products	product	NOUN
ap-1815	308	12	on	on	ADP
ap-1815	308	13	effect	effect	NOUN
ap-1815	308	14	algebras	algebra	NOUN
ap-1815	308	15	,	,	PUNCT
ap-1815	308	16	reports	report	NOUN
ap-1815	308	17	of	of	ADP
ap-1815	308	18	mathematical	mathematical	ADJ
ap-1815	308	19	physics	physics	NOUN
ap-1815	308	20	,	,	PUNCT
ap-1815	308	21	49	49	NUM
ap-1815	308	22	(	(	PUNCT
ap-1815	308	23	2002	2002	NUM
ap-1815	308	24	)	)	PUNCT
ap-1815	308	25	,	,	PUNCT
ap-1815	308	26	87–111	87–111	NUM
ap-1815	308	27	.	.	PUNCT
ap-1815	309	1	[	[	X
ap-1815	309	2	5	5	X
ap-1815	309	3	]	]	X
ap-1815	309	4	hedlíková	hedlíková	PROPN
ap-1815	309	5	j.	j.	PROPN
ap-1815	309	6	,	,	PUNCT
ap-1815	309	7	pulmannová	pulmannová	PROPN
ap-1815	309	8	s.	s.	PROPN
ap-1815	309	9	,	,	PUNCT
ap-1815	309	10	generalized	generalized	ADJ
ap-1815	309	11	difference	difference	NOUN
ap-1815	309	12	posets	poset	NOUN
ap-1815	309	13	and	and	CCONJ
ap-1815	309	14	orthoalgebras	orthoalgebra	NOUN
ap-1815	309	15	,	,	PUNCT
ap-1815	309	16	acta	acta	PROPN
ap-1815	309	17	math	math	PROPN
ap-1815	309	18	.	.	PUNCT
ap-1815	310	1	univ	univ	PROPN
ap-1815	310	2	.	.	PROPN
ap-1815	310	3	comenianae	comenianae	PROPN
ap-1815	310	4	lxv	lxv	PROPN
ap-1815	310	5	(	(	PUNCT
ap-1815	310	6	1996	1996	NUM
ap-1815	310	7	)	)	PUNCT
ap-1815	310	8	,	,	PUNCT
ap-1815	310	9	247–279	247–279	NUM
ap-1815	310	10	.	.	PUNCT
ap-1815	311	1	[	[	X
ap-1815	311	2	6	6	NUM
ap-1815	311	3	]	]	PUNCT
ap-1815	311	4	kalmbach	kalmbach	PROPN
ap-1815	311	5	g.	g.	PROPN
ap-1815	311	6	,	,	PUNCT
ap-1815	311	7	riečanová	riečanová	PROPN
ap-1815	311	8	z.	z.	PROPN
ap-1815	311	9	,	,	PUNCT
ap-1815	311	10	an	an	DET
ap-1815	311	11	axiomatization	axiomatization	NOUN
ap-1815	311	12	for	for	ADP
ap-1815	311	13	abelian	abelian	PROPN
ap-1815	311	14	relative	relative	PROPN
ap-1815	311	15	inverses	inverses	PROPN
ap-1815	311	16	,	,	PUNCT
ap-1815	311	17	demonstratio	demonstratio	PROPN
ap-1815	311	18	math	math	PROPN
ap-1815	311	19	.	.	PUNCT
ap-1815	312	1	27	27	NUM
ap-1815	312	2	(	(	PUNCT
ap-1815	312	3	1994	1994	NUM
ap-1815	312	4	)	)	PUNCT
ap-1815	312	5	,	,	PUNCT
ap-1815	312	6	769–780	769–780	NUM
ap-1815	312	7	.	.	PUNCT
ap-1815	313	1	[	[	X
ap-1815	313	2	7	7	X
ap-1815	313	3	]	]	X
ap-1815	313	4	kirchheimová	kirchheimová	PROPN
ap-1815	313	5	h.	h.	PROPN
ap-1815	313	6	,	,	PUNCT
ap-1815	313	7	some	some	DET
ap-1815	313	8	remarks	remark	NOUN
ap-1815	313	9	on	on	ADP
ap-1815	313	10	(	(	PUNCT
ap-1815	313	11	o)-converfengence	o)-converfengence	NOUN
ap-1815	313	12	,	,	PUNCT
ap-1815	313	13	proc	proc	NOUN
ap-1815	313	14	.	.	PUNCT
ap-1815	313	15	of	of	ADP
ap-1815	313	16	the	the	DET
ap-1815	313	17	first	first	ADJ
ap-1815	313	18	winter	winter	NOUN
ap-1815	313	19	school	school	NOUN
ap-1815	313	20	of	of	ADP
ap-1815	313	21	measure	measure	NOUN
ap-1815	313	22	theory	theory	NOUN
ap-1815	313	23	,	,	PUNCT
ap-1815	313	24	liptovský	liptovský	ADJ
ap-1815	313	25	ján	ján	PROPN
ap-1815	313	26	(	(	PUNCT
ap-1815	313	27	1990	1990	NUM
ap-1815	313	28	)	)	PUNCT
ap-1815	313	29	,	,	PUNCT
ap-1815	313	30	110–113	110–113	NUM
ap-1815	313	31	.	.	PUNCT
ap-1815	314	1	[	[	X
ap-1815	314	2	8	8	X
ap-1815	314	3	]	]	X
ap-1815	314	4	kirchheimová	kirchheimová	PROPN
ap-1815	314	5	h.	h.	PROPN
ap-1815	314	6	,	,	PUNCT
ap-1815	314	7	riečanová	riečanová	PROPN
ap-1815	314	8	z.	z.	PROPN
ap-1815	314	9	,	,	PUNCT
ap-1815	314	10	note	note	VERB
ap-1815	314	11	on	on	ADP
ap-1815	314	12	order	order	NOUN
ap-1815	314	13	convergence	convergence	NOUN
ap-1815	314	14	and	and	CCONJ
ap-1815	314	15	order	order	NOUN
ap-1815	314	16	topology	topology	NOUN
ap-1815	314	17	,	,	PUNCT
ap-1815	314	18	appendix	appendix	VERB
ap-1815	314	19	b	b	NOUN
ap-1815	314	20	,	,	PUNCT
ap-1815	314	21	in	in	ADP
ap-1815	314	22	riečan	riečan	NOUN
ap-1815	314	23	,	,	PUNCT
ap-1815	314	24	b.	b.	PROPN
ap-1815	314	25	,	,	PUNCT
ap-1815	314	26	neubrunn	neubrunn	PROPN
ap-1815	314	27	,	,	PUNCT
ap-1815	314	28	t.	t.	NOUN
ap-1815	314	29	,	,	PUNCT
ap-1815	314	30	measure	measure	NOUN
ap-1815	314	31	,	,	PUNCT
ap-1815	314	32	integral	integral	ADJ
ap-1815	314	33	and	and	CCONJ
ap-1815	314	34	order	order	NOUN
ap-1815	314	35	,	,	PUNCT
ap-1815	314	36	ister	ister	ADV
ap-1815	314	37	science	science	NOUN
ap-1815	314	38	(	(	PUNCT
ap-1815	314	39	bratislava	bratislava	NOUN
ap-1815	314	40	)	)	PUNCT
ap-1815	314	41	and	and	CCONJ
ap-1815	314	42	kluwer	kluwer	PROPN
ap-1815	314	43	academic	academic	ADJ
ap-1815	314	44	publishers	publisher	NOUN
ap-1815	314	45	(	(	PUNCT
ap-1815	314	46	dordrecht	dordrecht	PROPN
ap-1815	314	47	-	-	PUNCT
ap-1815	314	48	boston	boston	PROPN
ap-1815	314	49	-	-	PUNCT
ap-1815	314	50	london	london	PROPN
ap-1815	314	51	)	)	PUNCT
ap-1815	314	52	,	,	PUNCT
ap-1815	314	53	1997	1997	NUM
ap-1815	314	54	.	.	PUNCT
ap-1815	315	1	[	[	X
ap-1815	315	2	9	9	NUM
ap-1815	315	3	]	]	SYM
ap-1815	315	4	kôpka	kôpka	PROPN
ap-1815	315	5	f.	f.	PROPN
ap-1815	315	6	,	,	PUNCT
ap-1815	315	7	chovanec	chovanec	PROPN
ap-1815	315	8	f.	f.	PROPN
ap-1815	315	9	,	,	PUNCT
ap-1815	315	10	d	d	NOUN
ap-1815	315	11	-	-	PUNCT
ap-1815	315	12	posets	poset	NOUN
ap-1815	315	13	,	,	PUNCT
ap-1815	315	14	math	math	NOUN
ap-1815	315	15	.	.	PUNCT
ap-1815	316	1	slovaca	slovaca	NOUN
ap-1815	316	2	44	44	NUM
ap-1815	316	3	(	(	PUNCT
ap-1815	316	4	1994	1994	NUM
ap-1815	316	5	)	)	PUNCT
ap-1815	316	6	,	,	PUNCT
ap-1815	316	7	21–34	21–34	NUM
ap-1815	316	8	.	.	PUNCT
ap-1815	317	1	[	[	X
ap-1815	317	2	10	10	NUM
ap-1815	317	3	]	]	X
ap-1815	317	4	liu	liu	PROPN
ap-1815	317	5	,	,	PUNCT
ap-1815	317	6	w.	w.	PROPN
ap-1815	317	7	h.	h.	PROPN
ap-1815	317	8	,	,	PUNCT
ap-1815	317	9	wu	wu	PROPN
ap-1815	317	10	,	,	PUNCT
ap-1815	317	11	j.	j.	PROPN
ap-1815	317	12	d.	d.	PROPN
ap-1815	317	13	,	,	PUNCT
ap-1815	317	14	the	the	DET
ap-1815	317	15	uniqueness	uniqueness	ADJ
ap-1815	317	16	problem	problem	NOUN
ap-1815	317	17	of	of	ADP
ap-1815	317	18	sequence	sequence	NOUN
ap-1815	317	19	product	product	NOUN
ap-1815	317	20	on	on	ADP
ap-1815	317	21	operator	operator	NOUN
ap-1815	317	22	effect	effect	NOUN
ap-1815	317	23	algebra	algebra	VERB
ap-1815	317	24	e(h	e(h	PROPN
ap-1815	317	25	)	)	PUNCT
ap-1815	317	26	,	,	PUNCT
ap-1815	317	27	j.	j.	PROPN
ap-1815	317	28	phys	phys	PROPN
ap-1815	317	29	.	.	PUNCT
ap-1815	318	1	a	a	DET
ap-1815	318	2	:	:	PUNCT
ap-1815	318	3	math	math	NOUN
ap-1815	318	4	.	.	PUNCT
ap-1815	319	1	theor	theor	PROPN
ap-1815	319	2	.	.	PUNCT
ap-1815	320	1	42	42	NUM
ap-1815	320	2	(	(	PUNCT
ap-1815	320	3	2009	2009	NUM
ap-1815	320	4	)	)	PUNCT
ap-1815	320	5	.	.	PUNCT
ap-1815	321	1	[	[	X
ap-1815	321	2	11	11	NUM
ap-1815	321	3	]	]	PUNCT
ap-1815	321	4	mosná	mosná	PROPN
ap-1815	321	5	k.	k.	PROPN
ap-1815	321	6	,	,	PUNCT
ap-1815	321	7	paseka	paseka	SCONJ
ap-1815	321	8	j.	j.	PROPN
ap-1815	321	9	,	,	PUNCT
ap-1815	321	10	riečanová	riečanová	PROPN
ap-1815	321	11	z.	z.	PROPN
ap-1815	321	12	,	,	PUNCT
ap-1815	321	13	order	order	NOUN
ap-1815	321	14	convergence	convergence	NOUN
ap-1815	321	15	,	,	PUNCT
ap-1815	321	16	order	order	NOUN
ap-1815	321	17	and	and	CCONJ
ap-1815	321	18	interval	interval	NOUN
ap-1815	321	19	topologies	topology	NOUN
ap-1815	321	20	on	on	ADP
ap-1815	321	21	posets	poset	NOUN
ap-1815	321	22	and	and	CCONJ
ap-1815	321	23	lattice	lattice	ADJ
ap-1815	321	24	effect	effect	NOUN
ap-1815	321	25	algebras	algebra	NOUN
ap-1815	321	26	,	,	PUNCT
ap-1815	321	27	in	in	ADP
ap-1815	321	28	uncertainty	uncertainty	NOUN
ap-1815	321	29	2008	2008	NUM
ap-1815	321	30	,	,	PUNCT
ap-1815	321	31	bratislava	bratislava	PROPN
ap-1815	321	32	,	,	PUNCT
ap-1815	321	33	slovak	slovak	ADJ
ap-1815	321	34	republic	republic	NOUN
ap-1815	321	35	:	:	PUNCT
ap-1815	321	36	slovak	slovak	ADJ
ap-1815	321	37	university	university	NOUN
ap-1815	321	38	of	of	ADP
ap-1815	321	39	technology	technology	NOUN
ap-1815	321	40	in	in	ADP
ap-1815	321	41	bratislava	bratislava	PROPN
ap-1815	321	42	,	,	PUNCT
ap-1815	321	43	publishing	publish	VERB
ap-1815	321	44	house	house	PROPN
ap-1815	321	45	of	of	ADP
ap-1815	321	46	stu	stu	PROPN
ap-1815	321	47	,	,	PUNCT
ap-1815	321	48	(	(	PUNCT
ap-1815	321	49	2008	2008	NUM
ap-1815	321	50	)	)	PUNCT
ap-1815	321	51	,	,	PUNCT
ap-1815	321	52	45–62	45–62	NUM
ap-1815	321	53	,	,	PUNCT
ap-1815	321	54	[	[	X
ap-1815	321	55	12	12	NUM
ap-1815	321	56	]	]	X
ap-1815	321	57	olejček	olejček	PROPN
ap-1815	321	58	v.	v.	NOUN
ap-1815	321	59	,	,	PUNCT
ap-1815	321	60	order	order	NOUN
ap-1815	321	61	convergence	convergence	NOUN
ap-1815	321	62	and	and	CCONJ
ap-1815	321	63	order	order	NOUN
ap-1815	321	64	topology	topology	NOUN
ap-1815	321	65	on	on	ADP
ap-1815	321	66	a	a	DET
ap-1815	321	67	poset	poset	NOUN
ap-1815	321	68	,	,	PUNCT
ap-1815	321	69	int	int	NOUN
ap-1815	321	70	.	.	PUNCT
ap-1815	322	1	j.	j.	PROPN
ap-1815	322	2	theor	theor	PROPN
ap-1815	322	3	.	.	PUNCT
ap-1815	323	1	physics	physics	PROPN
ap-1815	323	2	38	38	NUM
ap-1815	323	3	(	(	PUNCT
ap-1815	323	4	1999	1999	NUM
ap-1815	323	5	)	)	PUNCT
ap-1815	323	6	,	,	PUNCT
ap-1815	323	7	557–561	557–561	NUM
ap-1815	323	8	.	.	PUNCT
ap-1815	324	1	[	[	X
ap-1815	324	2	13	13	NUM
ap-1815	324	3	]	]	X
ap-1815	324	4	olejček	olejček	PROPN
ap-1815	324	5	v.	v.	ADV
ap-1815	324	6	,	,	PUNCT
ap-1815	324	7	the	the	DET
ap-1815	324	8	order	order	NOUN
ap-1815	324	9	topology	topology	NOUN
ap-1815	324	10	on	on	ADP
ap-1815	324	11	a	a	DET
ap-1815	324	12	lattice	lattice	NOUN
ap-1815	324	13	and	and	CCONJ
ap-1815	324	14	its	its	PRON
ap-1815	324	15	macneille	macneille	PROPN
ap-1815	324	16	completion	completion	NOUN
ap-1815	324	17	,	,	PUNCT
ap-1815	324	18	int	int	NOUN
ap-1815	324	19	.	.	PUNCT
ap-1815	325	1	j.	j.	PROPN
ap-1815	325	2	theor	theor	PROPN
ap-1815	325	3	.	.	PUNCT
ap-1815	326	1	physics	physics	PROPN
ap-1815	326	2	,	,	PUNCT
ap-1815	326	3	39	39	NUM
ap-1815	326	4	(	(	PUNCT
ap-1815	326	5	2000	2000	NUM
ap-1815	326	6	)	)	PUNCT
ap-1815	326	7	,	,	PUNCT
ap-1815	326	8	801–803	801–803	NUM
ap-1815	326	9	.	.	PUNCT
ap-1815	327	1	[	[	X
ap-1815	327	2	14	14	NUM
ap-1815	327	3	]	]	PUNCT
ap-1815	327	4	paseka	paseka	SCONJ
ap-1815	327	5	j.	j.	PROPN
ap-1815	327	6	,	,	PUNCT
ap-1815	327	7	riečanová	riečanová	PROPN
ap-1815	327	8	z.	z.	PROPN
ap-1815	327	9	,	,	PUNCT
ap-1815	327	10	the	the	DET
ap-1815	327	11	inheritance	inheritance	NOUN
ap-1815	327	12	of	of	ADP
ap-1815	327	13	bde	bde	NOUN
ap-1815	327	14	-	-	PUNCT
ap-1815	327	15	property	property	NOUN
ap-1815	327	16	in	in	ADP
ap-1815	327	17	sharply	sharply	ADV
ap-1815	327	18	dominating	dominate	VERB
ap-1815	327	19	lattice	lattice	NOUN
ap-1815	327	20	effect	effect	NOUN
ap-1815	327	21	algebras	algebra	NOUN
ap-1815	327	22	and	and	CCONJ
ap-1815	327	23	(	(	PUNCT
ap-1815	327	24	o)-continuous	o)-continuous	ADJ
ap-1815	327	25	states	state	NOUN
ap-1815	327	26	,	,	PUNCT
ap-1815	327	27	soft	soft	ADJ
ap-1815	327	28	computing	computing	NOUN
ap-1815	327	29	,	,	PUNCT
ap-1815	327	30	15	15	NUM
ap-1815	327	31	(	(	PUNCT
ap-1815	327	32	2011	2011	NUM
ap-1815	327	33	)	)	PUNCT
ap-1815	327	34	,	,	PUNCT
ap-1815	327	35	543–555	543–555	NUM
ap-1815	327	36	.	.	PUNCT
ap-1815	328	1	[	[	X
ap-1815	328	2	15	15	NUM
ap-1815	328	3	]	]	PUNCT
ap-1815	328	4	paseka	paseka	ADP
ap-1815	328	5	j.	j.	PROPN
ap-1815	328	6	,	,	PUNCT
ap-1815	328	7	riečanová	riečanová	PROPN
ap-1815	328	8	z.	z.	PROPN
ap-1815	328	9	,	,	PUNCT
ap-1815	328	10	considerable	considerable	ADJ
ap-1815	328	11	sets	set	NOUN
ap-1815	328	12	of	of	ADP
ap-1815	328	13	linear	linear	PROPN
ap-1815	328	14	operators	operator	NOUN
ap-1815	328	15	in	in	ADP
ap-1815	328	16	hilbert	hilbert	PROPN
ap-1815	328	17	spaces	space	NOUN
ap-1815	328	18	as	as	ADP
ap-1815	328	19	operator	operator	NOUN
ap-1815	328	20	generalized	generalized	ADJ
ap-1815	328	21	effect	effect	NOUN
ap-1815	328	22	algebras	algebra	NOUN
ap-1815	328	23	,	,	PUNCT
ap-1815	328	24	foundations	foundation	NOUN
ap-1815	328	25	of	of	ADP
ap-1815	328	26	physics	physics	NOUN
ap-1815	328	27	,	,	PUNCT
ap-1815	328	28	41	41	NUM
ap-1815	328	29	(	(	PUNCT
ap-1815	328	30	2011	2011	NUM
ap-1815	328	31	)	)	PUNCT
ap-1815	328	32	,	,	PUNCT
ap-1815	328	33	1634–1647	1634–1647	NUM
ap-1815	328	34	.	.	PUNCT
ap-1815	329	1	[	[	X
ap-1815	329	2	16	16	NUM
ap-1815	329	3	]	]	PUNCT
ap-1815	329	4	reed	reed	NOUN
ap-1815	329	5	m.	m.	NOUN
ap-1815	329	6	,	,	PUNCT
ap-1815	329	7	simon	simon	PROPN
ap-1815	329	8	b.	b.	PROPN
ap-1815	329	9	,	,	PUNCT
ap-1815	329	10	methods	method	NOUN
ap-1815	329	11	of	of	ADP
ap-1815	329	12	modern	modern	ADJ
ap-1815	329	13	mathematical	mathematical	ADJ
ap-1815	329	14	physics	physics	PROPN
ap-1815	329	15	ii	ii	PROPN
ap-1815	329	16	,	,	PUNCT
ap-1815	329	17	fourier	fourier	ADJ
ap-1815	329	18	analysis	analysis	NOUN
ap-1815	329	19	,	,	PUNCT
ap-1815	329	20	self	self	NOUN
ap-1815	329	21	-	-	PUNCT
ap-1815	329	22	adjointness	adjointness	NOUN
ap-1815	329	23	,	,	PUNCT
ap-1815	329	24	academic	academic	ADJ
ap-1815	329	25	press	press	PROPN
ap-1815	329	26	new	new	PROPN
ap-1815	329	27	york	york	PROPN
ap-1815	329	28	,	,	PUNCT
ap-1815	329	29	san	san	PROPN
ap-1815	329	30	francisco	francisco	PROPN
ap-1815	329	31	,	,	PUNCT
ap-1815	329	32	london	london	PROPN
ap-1815	329	33	,	,	PUNCT
ap-1815	329	34	1975	1975	NUM
ap-1815	329	35	.	.	PUNCT
ap-1815	330	1	[	[	X
ap-1815	330	2	17	17	NUM
ap-1815	330	3	]	]	PUNCT
ap-1815	330	4	riečanová	riečanová	PROPN
ap-1815	330	5	z.	z.	PROPN
ap-1815	330	6	,	,	PUNCT
ap-1815	330	7	zajac	zajac	PROPN
ap-1815	330	8	m.	m.	PROPN
ap-1815	330	9	,	,	PUNCT
ap-1815	330	10	pulmannová	pulmannová	PROPN
ap-1815	330	11	s.	s.	PROPN
ap-1815	330	12	,	,	PUNCT
ap-1815	330	13	effect	effect	NOUN
ap-1815	330	14	algebras	algebra	NOUN
ap-1815	330	15	of	of	ADP
ap-1815	330	16	positive	positive	ADJ
ap-1815	330	17	linear	linear	PROPN
ap-1815	330	18	operators	operator	NOUN
ap-1815	330	19	densely	densely	ADV
ap-1815	330	20	defined	define	VERB
ap-1815	330	21	on	on	ADP
ap-1815	330	22	hilbert	hilbert	PROPN
ap-1815	330	23	spaces	space	NOUN
ap-1815	330	24	,	,	PUNCT
ap-1815	330	25	reports	report	NOUN
ap-1815	330	26	of	of	ADP
ap-1815	330	27	mathematical	mathematical	ADJ
ap-1815	330	28	physics	physics	NOUN
ap-1815	330	29	,	,	PUNCT
ap-1815	330	30	68	68	NUM
ap-1815	330	31	(	(	PUNCT
ap-1815	330	32	2011	2011	NUM
ap-1815	330	33	)	)	PUNCT
ap-1815	330	34	,	,	PUNCT
ap-1815	330	35	261–270	261–270	NUM
ap-1815	330	36	.	.	PUNCT
ap-1815	331	1	[	[	X
ap-1815	331	2	18	18	NUM
ap-1815	331	3	]	]	PUNCT
ap-1815	331	4	riečanová	riečanová	PROPN
ap-1815	331	5	z.	z.	PROPN
ap-1815	331	6	,	,	PUNCT
ap-1815	331	7	zajac	zajac	PROPN
ap-1815	331	8	m.	m.	PROPN
ap-1815	331	9	,	,	PUNCT
ap-1815	331	10	hilbert	hilbert	NOUN
ap-1815	331	11	space	space	NOUN
ap-1815	331	12	effect	effect	NOUN
ap-1815	331	13	-	-	PUNCT
ap-1815	331	14	representations	representation	NOUN
ap-1815	331	15	of	of	ADP
ap-1815	331	16	effect	effect	NOUN
ap-1815	331	17	algebras	algebra	NOUN
ap-1815	331	18	,	,	PUNCT
ap-1815	331	19	reports	report	NOUN
ap-1815	331	20	of	of	ADP
ap-1815	331	21	mathematical	mathematical	ADJ
ap-1815	331	22	physics	physics	NOUN
ap-1815	331	23	,	,	PUNCT
ap-1815	331	24	70	70	NUM
ap-1815	331	25	(	(	PUNCT
ap-1815	331	26	2012	2012	NUM
ap-1815	331	27	)	)	PUNCT
ap-1815	331	28	,	,	PUNCT
ap-1815	331	29	283–290	283–290	NUM
ap-1815	331	30	.	.	PUNCT
ap-1815	332	1	313	313	NUM
ap-1815	332	2	http://dx.doi.org/10.1007/bf02283036	http://dx.doi.org/10.1007/bf02283036	PROPN
ap-1815	332	3	http://dx.doi.org/10.1088/1751-8113/42/18/185206	http://dx.doi.org/10.1088/1751-8113/42/18/185206	NOUN
ap-1815	332	4	http://dx.doi.org/10.1088/1751-8113/42/18/185206	http://dx.doi.org/10.1088/1751-8113/42/18/185206	NOUN
ap-1815	332	5	doi	doi	NOUN
ap-1815	332	6	:	:	PUNCT
ap-1815	332	7	10.1007	10.1007	NUM
ap-1815	332	8	/	/	SYM
ap-1815	332	9	s00500	s00500	PROPN
ap-1815	332	10	-	-	PUNCT
ap-1815	332	11	010	010	NUM
ap-1815	332	12	-	-	PUNCT
ap-1815	332	13	0561	0561	NUM
ap-1815	332	14	-	-	PUNCT
ap-1815	332	15	7.doi	7.doi	NOUN
ap-1815	332	16	:	:	PUNCT
ap-1815	332	17	10.1007	10.1007	NUM
ap-1815	332	18	/	/	SYM
ap-1815	332	19	s00500	s00500	PROPN
ap-1815	332	20	-	-	PUNCT
ap-1815	332	21	010	010	NUM
ap-1815	332	22	-	-	PUNCT
ap-1815	332	23	0561	0561	NUM
ap-1815	332	24	-	-	PUNCT
ap-1815	332	25	7	7	NUM
ap-1815	332	26	doi	doi	NOUN
ap-1815	332	27	:	:	PUNCT
ap-1815	332	28	10.1007	10.1007	NUM
ap-1815	332	29	/	/	SYM
ap-1815	332	30	s10701	s10701	PROPN
ap-1815	332	31	-	-	PUNCT
ap-1815	332	32	011	011	NUM
ap-1815	332	33	-	-	PUNCT
ap-1815	332	34	9573	9573	NUM
ap-1815	332	35	-	-	PUNCT
ap-1815	332	36	0	0	NUM
ap-1815	332	37	acta	acta	PROPN
ap-1815	332	38	polytechnica	polytechnica	PROPN
ap-1815	332	39	53(3):308–313	53(3):308–313	NUM
ap-1815	332	40	,	,	PUNCT
ap-1815	332	41	2013	2013	NUM
ap-1815	332	42	1	1	NUM
ap-1815	332	43	introduction	introduction	NOUN
ap-1815	332	44	2	2	NUM
ap-1815	332	45	basic	basic	ADJ
ap-1815	332	46	definitions	definition	NOUN
ap-1815	332	47	and	and	CCONJ
ap-1815	332	48	some	some	DET
ap-1815	332	49	known	know	VERB
ap-1815	332	50	facts	fact	NOUN
ap-1815	332	51	2.1	2.1	NUM
ap-1815	332	52	effect	effect	NOUN
ap-1815	332	53	algebras	algebra	NOUN
ap-1815	332	54	and	and	CCONJ
ap-1815	332	55	generalized	generalized	ADJ
ap-1815	332	56	effect	effect	NOUN
ap-1815	332	57	algebras	algebra	VERB
ap-1815	332	58	2.2	2.2	NUM
ap-1815	332	59	topologies	topology	NOUN
ap-1815	332	60	on	on	ADP
ap-1815	332	61	ordered	order	VERB
ap-1815	332	62	sets	set	VERB
ap-1815	332	63	2.3	2.3	NUM
ap-1815	332	64	morphisms	morphism	NOUN
ap-1815	332	65	,	,	PUNCT
ap-1815	332	66	embeddings	embedding	NOUN
ap-1815	332	67	and	and	CCONJ
ap-1815	332	68	isomorphisms	isomorphism	NOUN
ap-1815	332	69	of	of	ADP
ap-1815	332	70	effect	effect	NOUN
ap-1815	332	71	algebras	algebra	VERB
ap-1815	332	72	3	3	NUM
ap-1815	332	73	basic	basic	ADJ
ap-1815	332	74	properties	property	NOUN
ap-1815	332	75	of	of	ADP
ap-1815	332	76	isomorphisms	isomorphism	NOUN
ap-1815	332	77	of	of	ADP
ap-1815	332	78	effect	effect	NOUN
ap-1815	332	79	algebras	algebra	NOUN
ap-1815	332	80	and	and	CCONJ
ap-1815	332	81	operator	operator	NOUN
ap-1815	332	82	representations	representation	VERB
ap-1815	332	83	4	4	NUM
ap-1815	332	84	some	some	DET
ap-1815	332	85	properties	property	NOUN
ap-1815	332	86	of	of	ADP
ap-1815	332	87	operator	operator	NOUN
ap-1815	332	88	effect	effect	NOUN
ap-1815	332	89	algebras	algebra	NOUN
ap-1815	332	90	that	that	SCONJ
ap-1815	332	91	need	need	AUX
ap-1815	332	92	not	not	PART
ap-1815	332	93	be	be	AUX
ap-1815	332	94	preserved	preserve	VERB
ap-1815	332	95	by	by	ADP
ap-1815	332	96	effect	effect	NOUN
ap-1815	332	97	algebraic	algebraic	PROPN
ap-1815	332	98	isomorphisms	isomorphism	VERB
ap-1815	332	99	acknowledgements	acknowledgement	VERB
ap-1815	332	100	references	reference	NOUN