Tensor products of D-posets and D-test spaces

vol. 34 (1994)

ON MATHEMATKAL

TENSOR

PRODUCTS

ANATOLIJ Mathematical

Institute,

Slovak (E-mail:

OF D-POSETS

Academy

AND D-TEST

and SYLVIA

DVURE~ENSKIJ

of Sciences,

[email protected]

No. 3

I’HYSIC’S

Stefsnikova

SPACES*

PULMANNOVA 49, SK-814 73 Bratisiava,

Slovakia

[email protected])

(Received March 3. 1994 -

Revised July 5, 1994)

WC introduce a state tensor product of difference poscts and a weight tensor product of D-test spaces which are more transuarcnt and easier to obtain than an algebraic tensor obtained in any separate product studied in [6],and. in addition, the statistical information entity remains the same in both coupled systems. I

1.

Introduction

In the axiomatic approach to quantum mechanics, the event structure of a physical system is identified with a quantum logic [23] or an orthoalgebra [ll], while in the case of classical mechanics with a Boolean algebra. Recently, there appeared a new axiomatic algebraic model, difference posets (D-posets, for short), introduced by Kbpka and Chovanec [16], which generalizes all the above mentioned models as well as the set of all effects (i.e. the system of all Hermitian operators A on a Hilbert space H with 0 5 A < I, which are important for modelling unsharp measurement in Hilbert space quantum mechanics [ 11). In [6], we presented operational statistics corresponding to difference posets. This approach is analogous to the operational statistics of Foulis and Randall [12, 211, however, it is more complicated than that because it allows one to describe the situation corresponding to fuzzy approach which is possible, for example, in Hilbert space quantum mechanics using effects. In addition, it unifies both the approach of Kolmogorov [15] to the foundations of probability theory and statistics and that of Dirac [2] and von Neumann [19] for orthodox quantum mechanics. Moreover, in [4, 61, we have introduced and studied tensor product of difference posets as well as of D-test spaces using purely algebraic notions. We have shown [4] that, for example, the tensor product of the orthoalgebra with itself, called the Phano plane, exists in the class of D-posets, while in the class of orthoalgebras it does not [9]. lY8O Mafhemath Subject Classijication (I985 Revision). 03G12, 81PlO. Key words and phrases. D-test space, D-test, D-algebraic test space, difference poset, orthomodular poset, quantum logic, orthoalgcbra, state tensor product, weight tensor product, tensor product of D-post%. *This rcscarch is supported by the grant G-229194 of the Slovak Academy of Sciences, Slovakia. v511

A. DVURE&NSKIJ

252

and S. PULMANNoVA

On the other hand, if the exact form of tensor products is not known, as e.g. for the Phano plane, we can formulate and prove the existence of a tensor product of D-test spaces or D-posets in the category of D-test spaces or D-posets having for difference posets. These forms of a full family of weights or states, similarly tensor products, the weight tensor product and the state tensor product, are in many important cases more elementary than the algebraic one and can be obtained directly. They are very useful since they save the whole statistical information involved in both separated entities and which is also the same as that in the algebraic tensor product.

2. Difference

posets

A D-poset, or a d@erence poser, is a partially ordered set L’ with partial ordering 5, the greatest element I, and with a partial binary operation k3: I, x L ----tL, called difference, such that, for n, b E I;, b t_, CLis defined if and only if a 5 b. and the following axioms hold for CL,b. c E L: (DPi) b 13 n < b; (DPii) b w (b G u) = a; (DPiii) a 5 h 5 c =+ c r b 5 C’6-4a and (c * u) :_I (c + b) = b ky a. The following

statements

have been

proved

in [ 161:

2.1. Let CL,b, c, d he elements qf a D-poset (i) 1 ~j 1 is the smullest element qf I,; denote it by 0. (ii) n c-30 = a. (iii) n tin = 0.

PROPOSITION

(iv) (v) (vi) (vii) (viii)

a 5 a 5 (1 5 b5

L. Then

b + b c; a = 0 H b = n. b =+ b (3 a = b ++ c1= 0. b 5 c + b c; n 5 c 6’ (I, and (c +: a) I-: (b b a) = c t? b. c, a 5 c :? b + b < c !:? CLand (C I _:b) c; u = (c H a) I? b.

n 5 b 5 c =s a 5 c E! (b 1-3a) and (c-r-4(b 9 u)) 1: a = c a b.

Remark 2.2 [18]: A poset L with the smallest and greatest elements 0 and 1, respectively, and with a partial binary operation b: L x I, 4 L such that b 6 a is defined iff CL5 b, and for a, b, c E L WC have (i) a @ 0 = tr; (ii) if u 5 b < c, then c-b 115 cc-4 rr and (c G CL)b: (c 1;: b) = b cm.n. is a D-poset. For any element n E L we put (,_L:= 1

(I.

t+ 5 aI. Two elements Then (i) nil = u; (ii) rc < b implies orthogonal, and we write n I b ifi n 5 bl (iff b 5 (I~).

a and

h of

C are

253

TENSOR PRODUCTS OF D-POSETS AND D-TEST SPACES

Now we present the situation when L is not a priori endowed with partial ordering, however, the following system, defined only as a partial algebra, will be a D-poset, too. PROPOSITION 2.3. Let L be a set which is endowed with two special elements 0 and 1 and equipped with a partial binary operation 0 satisfying the following conditions for all a, b, c E L: (01) (02)

a80 is defined, and a 8 0 = a. a 0 a is dejined.

(03)

Zf b0 a and c@ b are defined, then c0 (c 0 a) 0 (c 0 b) = b 0 a. (04) Zf 0 0 a is defined, then a = 0. (05)

a,(c@a)

0 (c0

b) are defined, and

18 a is defined.

Then (Oi) Zf b 0 a is dejined, then b 0 (b 0 a) is dejined and b 0 (b 0 a) = a. (Oii) Zf a 8 b = a 8 c, then b = c. (Oiii) a 8 a = 0 for any a E L. Zf we define an ordering 5 on L via a 5 b iff b 0 a is defined, then 5 is a partial ordering with the smallest and greatest elements 0 and 1, and L with respect to 5, 1, and 0 is a D-poset. Conversely, any D-poset satisjies all conditions of the proposition. Proof (Oi) From (01) and (03) we conclude that if b0a a, which proves (Oi). (b0a)=a00= (Oii) By (Oi), we have b = a0 (a0 b) = a0 (a0c) = c.

is defined,

then

(b00)

0

(Oiii) Since a00 and sea exist in L, from (03) we obtain (~~O)O(U@U) = ~~00, so that a@ (a@ a) = a80 which, by (Oii), means that u8u = 0. of 5. The From (02) we have a 5 a, and from (03) we have the transitivity antisymmetry follows from the following: Let a I b and b < a. Then b 0 a and a 0 b exist in L, and, by (03), we conclude (a 0 u) B (u 0 b) = b 0 a, so that 0 0 (a 0 b) is defined, and (04) entails that a 0 b = 0, and by symmetry, b 8 a = 0. Hence, by (Oi), a = b0(b0u) = be0 = b. that

The conditions (01) and (05) imply that 0 I a _< 1 for any a E L. This L with respect to 1, 5, and 8 is a D-poset. It is clear

that

any D-poset

satisfies

all conditions

of Proposition

2.3.

proves cl

If we omit in Proposition 2.3 the greatest element 1 and the condition (05), then 2 from Proposition 2.3 is also a partial ordering on L with 0 as the smallest element in L, and the conditions (DPi)-(DPiii) are satisfied as well as the properties (ii)-(viii) of Proposition 2.1 remain valid. We call this structure a D-poset without 1 (these structures in other setups have been studied in [13, 141). In any D-poset we have also the following left cancelation law: If a0b=c@b,

then

u=c.

(2.1)

254

A. DVUREtENSKIJ

This can be easily proved we have

as follows

and S. PULMANNOVA

[8, 51: Using

(viii) of Proposition

2.1 and (DPi),

( 1 YJ (u i ) b)) c> b = I ‘; CI = ( 1 ~3 (c i 6)) @ b = I - C. which implies u = c. In D-pose& without 1, this left cancelation law holds iff any two elements a and c in the left-hand side of (2.1) have a majorant in L. Now we introduce a partial binary operation +: L x L + L such that an element r = a ;f: b in L is defined iff CLI b. and for c we have b < c and u = c F 0. The partial operation q: is defined correctly because if there exists cl E L with b < cl and a = c-18 b, then, by (viii) of Proposition 2.1 and (DPii), we have (1 cr-/(c I-; b)) ‘_; b = I ‘7 c = (1 c? (c, c b)) c$ b = I pi cl. which

implies

c = cl. Moreover,

by [lg],

c = CL+: b = (a’ The operation 3 is commutative difference posets are orthomodular effects, and the interval [O, 11.

c-3b)l

= (b’ c: a)+

(2.2)

and associative [18]. Very important examples of posets (= quantum logics), orthoalgebras, sets of

EXAMPLE 2.4. An orthomodular poset (OMP), that is a partially ordered set L with an ordering 5, the smallest and greatest elements 0 and 1, respectively, and an orthocomplementation I: L + L such that (OMi) a” = a for any CL E L: (OMii) a V a’ = 1 for any 0 E L; (OMiii) if CL< b. then b’ < a’ ; (OMiv) if a 5 bl (and we write a _L b), then a v b E L; (OMv) if CL< b, then b = (1 V (CLV b’)i (orthomodular law), is a D-poset, when 0 c: n := b A a’. EXAMPLE 2.5. An orthoalgebra, that is a set L with two particular elements 0, I, and with a partial binary operation @: L x L + L such that for all a, b, c E L we have: (OAi) if a 2~ b c L. then b (1; CL E L and a cl: b = b i??a (commutativity); (OAii) if btlj c t L and CL8 (b+ c) t L. then (L$ b E L and (a 8 b) @ r E I,, and u & (b t3 c) = (n c%b) +j c (associativity); (OAiii) for any a E L there is a unique b E L such that CL61‘1 b is defined, and u (R b = 1 (orthocomplementation); (OAiv) if a ~13a is defined, then IL = 0 (consistency), is a D-poset if

(2.3)

b.- (I := (a + bl)+ where

bi

is a unique

element

c in L such that

If the assumptions of (OAii) (CL@ b) @ c = a or: (0 + c) in L.

are

satisfied,

bill_ c = 1. we write

c1CPb & c for

the

element

255

TENSOR PRODUCTS OF D-POSETS AND D-TEST SPACES

We note that if L is an orthomodular poset and a 63 b := u V b whenever a 1 b in L, then L with 0, I, @ is an orthoalgebra. The converse statement does not hold, in general. We recall that an orthoalgebra L is an OMP iff u I b implies a V b E L. By [18], we conclude that a D-poset L with 0,l and @. defined by (2.2) is an orthoalgebra if and only if a 2 1 8 a implies a = 0. Therefore, it is not hard to give many examples of D-posets which are not orthoalgebras; such ones are the sets of effects and the interval [O. I]: EXAMPLE 2.6. The set I(H) of all Hermitian operators A on H such that 0 < A 5 I, where I is the identity operator on H, is a difference poset which is not an orthoalgebra; a partial ordering 5 is defined via A 5 B iff (Ax, x) 5 (Bx, ST). 2 E H, and C=BeA iff (Bz,x)-(Ax,JJ)=(C’~~~),~EH. This ics [I].

set plays

an important

role for unsharp

measurements

of quantum

mechan-

EXAMPLE 2.7. Let the closed interval [O, I] be ordered by the natural ordering. Let g be any continuous, increasing mapping from [0, I] onto [0, l] such that g(0) = 0 and g(1) = 1 (called a generator). Define a partial binary operation 8, via b 8,9 a := g&(g(b)

- g(u)).

(2.4)

Then L with 5, I, and ey is a D-poset [lb]. In particular, if g = id,,,, 11%then b@id a =b-a. Conversely, by [17], any difference 8 on the D-poset [0, 11 is equal to some 8, defined by (2.4). We can define also a D-poset introducing another partial binary operation, @ which will imply 8 via (2.3). Such attempts are made in [lo, 201. PROPOSITION 2.8. Let L be a set with two special ped with a partial binary operation @ sati&ng, for (OAi)-(OAiii) and: (Ei) If 1 @ a is defined, then a = 0. tf we define a 5 b iff there exists c E L such that ordering on L, and if we put c = b 8 a iff a @ c = b! Conversely, for any D-poset L, the partial binary satisfies the conditions of the proposition. The set L with @, 0, 1 defined as that in Proposition [lo]. We recall that D-posets and effect algebras are primary notions are the difference 8 and the addition 3. @-Orthogonal Let F = {al,.

namely

elements 0 and 1 and equipall a, b, c E L, the conditions

a 6 c = b, then 5 is a partial then 8 is a difference on L. operation @ defined by (2.2)

2.8 is called an effect algebra the same things, where only @> respectively.

sums . , aT,,} be a finite ur @...

sequence

in L. We define

@ a?1 := (al @ .f. csiaT1--l) Bi a,,,

recursively

for n > 3 (3.1)

256

A. DVUREtENSKIJ

supposing that al@. .@cL,~_~and of @ in D-posets we conclude =a~ if n= 1, and (LI$.‘.63U, (ir,... ,i,,) of (1,. . . , n) and any

(al $. .@c~,,-l)@a~, exist in L. From the associativity that (3.1) is correctly defined. By definition we put ar$...@a,, = 0 if n = 0. Then for any permutation k with 1 5 X:< n we have

ar @...tt:u,, ar ‘F”‘@U,,

and S. PULMANNOVA

= [email protected],,l.

= (a, Q!? . k&z
(3.2) (3.3)

We say that a finite sequence F = {al,. , a,,} in L is @-otihogonaf if aI C$ . @ u,, exists in L. In this case we say that F has a @-sum, @y=, u,: defined via

6

a, =

CL]

~~~~~W,,.

(3.4)

I=1

It is clear that two elements u and b of I; are orthogonal, i.e. u I b, iff {u, b} is $-orthogonal. We recall that a D-poset L is an OMP iff a @ b 83 c exists in I, whenever u I bi c I a [lo]. different elements of L is An arbitrary system G = {uz}7tr of not necessarily $-orthogonal iff, for every finite subset F of I, the system {u,}~~F is @-orthogonal. If G = {ui}.ltl is @-orthogonal, so is any {u,},~.J for any J C I. A @-orthogonal system G = {u,}itr of L has a @-SLUR in L, written as ezEI ai. iff in L there exists the join @a, 1EI

:= V@%, F stF

(35)

where F runs over all finite subsets in I. In this case, we also write @ G : = ezEI CL?. It is evident that if G = {al,. ?CL,,} is @-orthogonal, then the @sums defined by (3.4) and (3.5) coincide. many i’s, then if G is We recall that if G = {u~}~~I and a, = u for infinitely @-orthogonal then a = 0. Indeed, let a,() = @G. then aI, = CL,,,(2 @ a7 = a cii (LO ztr\{7<>) which gives a = 0. We say that a D-poset L is a complete D-poset (a-D-poser) if, for any @-orthogonal system (any $-orthogonal sequence) G of L, there exists the @-sum, @G, in L. It is straightforward to verify that a D-poset L is a D-a-poset if, for any sequence {ai} in L with al 2 a? 5 , the join V,“=, a, exists in L. from Example 2.6 is @-orthogonal iff We recall that a system of effects {A,},,1 the system {Ai},tl is summable, for example in the weak topology, and A = CIEI A, D-poset. belongs to E(H). In this case, ezEIAt = CzEI A,. and E(H) is a complete Similarly, in Example 2.7, a system of real numbers {a,}, is @g-orthogonal iff the system {g(u,)},Er is summable and CatI g(q) 5 1. In this case, @ gu, 7EI D-poset with respect to ~13,. = s-‘(c,CI Lf(%)), and [0, l] is a complete A finite partition of 1 is any $-orthogonal finite sequence {a], , o,, } such that @_-, u, = I.

TENSOR PRODUCTSOF D-POSETSAND D-TEST SPACES 4. D-test

257

spaces

The following important notions and relationships between them have been motivated and introduced in the paper [6]. Here we present only basic notions and properties. Let X be a nonempty set, the elements of which are called outcomes. A function F: I -+ X is said to be of finite multiplicity if, for any x E X, F-‘(x) is of finite cardinal@. In what follows, we shall consider only functions of finite multiplicity. For arbitrary sets I and J and functions F E X’ and G E XJ define F =$G iff there is an injection a: I + J such that F = Go(T, i.e. F(i) = G(u(i)) for all i E 1. If F =$G and 0: I + J is a bijection, we say that F and G are equivalent, in symbols F N G, and in what follows, we shall identify functions which are equivalent.2 Therefore, we can define unambiguously a function FOG as follows: Let K = I’ U J’, where 1’ n J’ = 0 and 4: 1 + I’, 11: J + J’ are bijections. Then FOG is an element of XK such that F’G(k)

=

F(i) G(j)

if i E I, k = 4(i), if j E J, k = $(j).

We recall that the associativity (FljG)OH = FO(GOH) holds, and therefore for both sides we write FOGWI. It can be easily checked that + is a preorder which becomes a partial order on equivalence classes with respect to N . Let R(F) := {F(i): i E I} denote the range of F E X’. It is worth noting that two functions F and G are equivalent iff card F-l(x) = card G-‘(x) for any x E X. DEFINITION 4.1. Let ‘T C {F E X I: I E Z} be nonempty, where X # 0, and Z is a nonvoid family of index sets. We say that the pair (X, 7) is a D-test space iff the following three conditions are satisfied: (i) Any T E 7 is of finite multiplicity. (ii) For every x E X there is a T E 7 such that x E R(T). (iii) If S, T E 7 and S =$T, then 5’ N T. Any element of 7 is said to be a D-test. DEFINITION 4.2. Let (X, 7) be a D-test space. Let J be arbitrary and let G E XJ. We say that G is an event iff there is a D-test T E 7 such that G Z$T. Let us denote the set of all events in 7 by & = &(X,7). Clearly, 0 E 1. DEFINITION 4.3. Let (X,7) be a D-test space. We say that two events F and G are (i) orthogonal to each other, in symbols F I G, iff there is a D-test T E 7 such that FOG d T; (ii) local compZements of each other, in symbols F locG, iff there is a D-test T E 7 such that FljG N T; (iii) perspective with axis H iff they share a common local complement H. We write F %:H G or F z G if the axis is not emphasized. ’ Observe

that

if I = 0, then

X’

= (0).

258

A.

DEFINIIION

and S.

4.4. The D-test G i H.

space

(X, 7)

is D-ulgehruic

3,

for F, G’, H

E E. F E G

and F I H entail

We recall that some examples of D-algebraic D-test spaces, but not yet under this name, are studied in [4]. For simplicity, we usually refer to *Y, rather than to (41,7). as a D-test space, similarly, we put & = I(X). In what follows, let X be a D-algebraic D-test space. We note that N is an equivalence relation on I(X, 7). If E‘ E &. we define yfiliuted with F. r(F) := {G E E: G z F} and we refer to T(F) as the proposition The set IT = n(X) is called the logic of the D-test We define 0.1 t II by

space

:= {x(F):

F E E}

0 = 7r(0),

I = T(T).

where T is any D-test, and x(F)’ := n(G). where G is any F in E. Similarly we define n(F) 5 n(G) itf there is H E 1: h = T(G), H i F and G z F~JH. and in this case we put ir(H) The crucial relationships between D-algebraic D-test spaces following result [6]:

(4.1) local complement of such that n = z(F), = x(G) -- n(F). and D-posets is the

4.5. Let AY he u D-ulgehruic D-test space. Then the logic n(X) he orgunized into u D-poser, where 0. 1. 5. .--+ure us inclicuted uho~~.

THEOREM

cm

(4.1)

X.

qf’ _\i’

Conversely, let L be a D-poset. Then there exists a D-algebraic D-test space (X. 7) such that L is isomorphic to n(X). Indeed, let X = L \ (0) and let 7 be the set of all finite partitions of I in L consisting of nonzero elements, then (X. 7) is a D-algebraic D-test space. We recall that if all D-tests in a D-test space (X, 7) are injective functions, then (X. 7) is a test space in the sense of Foulis and Randall [12], in addition, n(X) is an orthoalgebra. Let P and I, be two D-poscts. A mapping $1: P + I, is said to be iff &( 1) = I. and 11 _L (1. 11.y t P. implies o(p) I o(q) and (i) a rnorphm 8(p $3 q) = @(II) tf; d(q); (ii) a monornorplzism iff 0 is a morphism and $+I) i d(y) iff r> I ‘I: monomorphism, and we say that P is (iii) an i.somorplzism iff Q is a surjective isomorphic to I,. In particular, if P is a D-poset and 1, = [O. I] is the D-poset from Example 2.7 with the natural difference, then any morphism ~1: P + [O. l] is said to be a state. 1s a cT-state or a completely udditive stute We recall that a mapping /I: L + or iff P(d3&1 u,) = C,,r /~(a ) wheneve!’ 7El n, exists in I,. and I is a countable arbitrary index set, and ~~11) = 1. It is worth mentioning that in Example 2.7 there is a unique state, namely, P(f) = g(f). where

g is a generator.

f E [(I. 11,

(4.3)

TENSOR

PRODUCTS

OF D-POSETS AND

D-TEST SPACES

259

If dimension of a Hilbert space H is greater than 2, then by generalized Gleason’s theorem [3], there is a one-to-one correspondence between the set of all completely additive states p on E(H) and the set of all von Neumann operators T on H such that A E E(H).

CL(A) = Tr(TA), 5. Tensor product

(4.4)

of D-posets

A tensor product of D-posets has been investigated in the paper [4], where a necessary and sufficient condition as well as a sufficient condition for the existence the of a tensor product have been presented. In the present section, we generalize former criterion. Let P, Q, L be D-pose&. A mapping p: P x Q --) L is called a bimorphism iff (i) a, b E P with a _L b, q E Q imply ,13(a,q) I P(b,q) and P(a @ b, 4) = P(a, q)@ P(b, q); (ii) c, d E Q with

c I d, p E P imply

P(p, c) I p(p, d) and

a(~, c CEd) = O(P, cl@

D(P, 4;

(iii) p(l? 1) = 1. If /3: P x Q + L is a bimorphism, morphisms. Therefore, for p E P and = P(1: 4’)

Also,

and

then ,!?(.. 1): P + L and p(l, .): Q + L are q E Q, we have P(P, 1)l = P(pi, I>, P(l> q>l

P(P, 0) = P(O, 4) = 0.

if a, b,p E P and c, d, q E Q, we have a I b * /?‘(a, q) F P(b, 9) and c 5 d *

P(P, c> I B(P> 0 DEFINITION 5.1. Let P and Q be difference posets. We say th,l~ ;I pair (R,T) consisting of a difference poset R and a bimorphism r: P x Q + R is a tensor product-’ of P and Q iff the following conditions are satisfied: (i) If L is a D-poset and B: P x Q + L is a bimorphism, there exists a morphism 4: R + L such that p = 4 o r. (ii) Every element of R is a finite orthogonal sum of elements of the form ~(p,q) with p E P, q E Q.

It is not hard to show that if a tensor product (R, 7) of P and Q exists, it is unique up to an isomorphism, i.e., if (R,T) and (R*, 7”) are tensor products of D-posets P and Q, then there is a unique isomorphism 4: R + R* such that cJ(+, q)) = T*(P, q) for all P E P, 4 E Q. Unless confusion threatens, we usually refer to P @ Q rather than to (P @ Q. @), and we shall write p @ q := @(p, q), p E P, q E Q, where 8 is a bimorphism. THEOREM 5.2. Let both D-posets P and Q possess at least one state. Then the tensor product of P and Q exists in the category of D-posets. In addition, for any state p on P and any state u on Q, there is a unique state /I @ u on P COQ such that

= PL(PMd, 3 Or

an

algebraictemor

product.

PEP,

q

E Q.

(5.1)

260

A. DVUREkENSKIJ

and S. PULMANNOVA

Proof: Let L = [0, l] be endowed with the natural ordering and the difference b 8 a := b - a, a, b E [0, 11. Then L is a D-poset. Choose two states p and v on P and Q, respectively, and define a mapping fiIIc,,: P x Q 4 L such that

&h? 4) = P(P) .4d,

PEP, q E Q.

(5.2)

which, by [4, Thm 7.21, is a necessary and sufficient Then P/L” is a bimorphism, condition for P and Q to admit a tensor product. from the definition of P @ Q it follows that there is Since & is a bimorphism, a morphism 4: P @ Q + [0, l] such that #(p Qtiq) = &,(p,q), p E I’, q E Q. But it of means that 4 is a state on P x Q with the desired property (5.1). The uniqueness 4 is clear due to the property of P cs Q that any element t t P :A Q is of the form 0 t = @:1,P,@% We note that in Theorem 7.3 of [4] it has been stated that if both a sufficient system of states,3 then P and Q admit the tensor product, shows that the assumption of the sufficiency of states is superfluous. Let P and Q be D-posets. We say that a couple K = (C, B). where family of D-posets and B is a nonvoid family of bimorphisms on P (i) for any 0 E B there exists a D-poset L E C such that 0: P x Q + any L E C there is a bimorphism p E B with 8: P x Q + L, is said to cluss for P and Q.

P and Q have Theorem 5.2 L is a nonvoid x Q such that L, and (ii) for be a consistent

DEFINITION 5.3. Let K = (l,B) be a consistent class for the D-posets P and Q. We say that a pair (T, 7) consisting of a difference poset T and a bimorphism 7: P x Q + T is a tensor product of P and Q in rhe cfuss K = (,C, I?) iff the following conditions are satisfied: (i) T E C, 7 E B. (ii) If L is a D-poset in if, and 3 is a bimorphism in B, ,O: P x Q --f L, there exists a morphism 4: T + L such that ,!I = 4 o 7. (iii) Every element of T is a finite $-orthogonal sum of elements of the form r(p,q) with p E P, q E Q.

Similarly as above, if a tensor product (T, r) of P and Q exists in the class K, it is unique up to isomorphism. It is clear that if C consists of all D-posets L for which there exists a bimorphism P: P x Q + L and B is the family of all those bimorphisms, then tensor products from Definition 5.1 and Definition 5.2 coincide. Moreover, let K, = (C,, B,), i = 1,2, be two consistent classes for P and Q such that Cr C: Cz and Br C &. If (T~,r~), i = 1,2, is a tensor product of P and Q, then there exists a morphism 4: T2 * TI such that 71 = do ~1. Suppose that Pr and Pl are nonempty families of states on difference posets P and Q. We set P := Pr x P? and, if X = (p, U) E P and (p, q) E P x Q> then XP! s) := P(P). u(q).

state

4 A system of states P on a D-poset 11 E P such that 11(n) # 0.

P is said

to he su~jkYen/ if, given

n # 0. a. t P. there

is a

TENSOR PRODUCTS OF D-POSETS AND D-TEST SPACES

261

We say that a family of states P on a D-poset L is separating iff p(u) = p(b) for any p E P implies a = b; (9; strong iff, for a,b E L, the statement “if ~(a) = 1, then p(b) = 1, p E P” implies a < b; (iii) unital iff, for any a E L, a # 0, there is I_LE P with ~(a) = 1; order determining iff the condition p(u) 5 p(b) for all p E P implies a 5 b; V fuZZ iff P is separating, and if X:=1 ,~(ai) = 1 for any p E P, then @Tz, ui (G exists in L and @y=“=, ui = 1. If P is a strong system of states, then P is (i) unital; (ii) separating, and (iii) order determining. If P is order determining, so is separating. It is not hard to show that if a D-poset L has a unital system of states, then L is an orthoalgebra: Indeed, let a -L a. If a # 0, there is ,LLE P with p(u) = 1, which gives a contradiction 1 > ~(a @ u) = ~(a) + ~(a) = 2. Now if an orthoalgebra has a strong system of states, then L is an OMP: a I b and a, b 5 c, then p(c) = 0 implies ~(a) = p(b) = 0 = ~(a $ b). Hence, cl 5 (u @ b)‘, which gives a $ b 5 c and u@b=uVb. If L is a D-poset and P is order determining, then P is full: Let CL, ,u(uz) = 1. Then, for any i # j, ,u(u~) 5 I, so that a, -L a3 and ui @a, E L. By induction we can show that @y=“=, ui E L, and the separateness of P implies @z, a, = 1. As a corollary we have that E(H) has no strong system of states whenever dimH > 1, but it has an order determining system of states and a full one (see formula (4.4)). Similarly, the D-poset L, from the Example 2.7, which is determined by a generator g, has the unique order determining and full system of states, namely P = {p}, where ,LLis determined by (4.3). THEOREM 5.4.Let PI and PZ be nonvoid systems of states on the D-posets P and Q, respectively. Let Cp be the set of all D-posets L such that there is a bimorphism K: P x Q + L, and the set P, := {p. ~3~V: ,LLE PI, v E Pz} is a full system of states on L, where p 18%v(K(I), q)) := &I) . v(q), p E P, q E Q, and let t?~ be the set of all these bimorphisms KS. Then K = (C,, t?p) is a consistent class for P and Q, and there exists the tensor product of P and Q in the class Ic.

Proof Let X be the subset of P x Q consisting of all pairs (p, q) with p # 0, q # 0. is a finite sequence of elements from X and X E P, If IV = ((Pl,ql>,...,(Pn,qn)) we put R.

with the understanding that if M = 0, then X(M) = 0. Define now the set F of all finite sequences T = {(pi, qz)}yzI of elements in X such that X(T) = 1 for any X E P. Since X(1,1) = 1, _7=is nonvoid. It is clear that if (p, q) E X, then from the set {(p, q), (p’, q), (p, ql), (p’, ql)} n X we can choose a finite sequence containing (p, q) and belonging to FT. Denote by &(_F) the set of all finite sequences {(pj, qj): j E J} such that J C 1 and {(pi, qi): i E I} E F. We put {(pi, qi): i E 0) = 0. It is evident that (X, .F) is a D-algebraic D-test space.

r\. 1X’URti’ENSKI.l

262

md S. IYJL.MANNOV~

For two events _A.II t I(F) we define A C=ZR iti X(A) = A(B) for any X t P. on E(7). and let T(A) := {B E &(_?J: I3 Z A}. Let Then E is an equivalence n(x) = {7r(A): A E E(F)}. w e organize U(X) into a poset by defining a partial ordering 5 on n(A7) as follows: 7i(_A) < n(B). where 11 = {(PI 1q~), , (pll. y,?)}. R = {(r,, .S]). . . (I‘,,, .s,,,)}, iff there is C’ = {($,,q;). . (p::y:)} E E(3) such that AI := {(PI.~l)~....(P,~.Qr,)~(l-l~.~l~)~”’.(/~:~(l:)} E E(F) and A(M) = X(B) for any X E P. Then 7r(ol) and T(T). where T t F, are the smallest and greatest elements in n(X). The difference operation t on n(X) is defined whenever 7r(il) 5 rr(B). and conditions for the -ir(B) 1 r(A) = r(C). where =I. R. C’ satistj’ the above mentioned partial ordering 5 Then 1:. is defined correctly, and U(X) is a difference poset. Define a mapping K,,: P x Q + ZI(AY) via

Then h-,, is, evidently, a himorphism. From the construction of any I E II(X) is of the form f = @:I=, K,,(~,.c/,). and the mapping defined by 11 ,::,,,, ~(~,(lj.q)) = i/(p) . n(q). p t P. q E Q. is a addition, P, ,, is a full system of states on If(X). Therefore, CT the set of all bimorphisms K such that K maps P x Q into some a full system of states on L. Then h1~ = (C p%BP) is a consistent

n(X) we see that 11~:x:,,~, v on n(X) state on n(x). In f lil. and let .13F be I; t Cp and P^ be class for P and Q.

We assert that (II(X). K,,) is a tensor product of P and Q in the class Kp. Choose 1, t CT and a himorphism K: P x Q -- L. Since P,, is full for L. it follows that it’ K,,(TI).q) = K,,(JI’. q’), then r;(l). q) = K(J)‘. (1’) and we can define a mapping 0: n ( s ) - 1, such that o(K,,(L~. q)) = K(I)>q)_ 11 E I’, q E Q. We claim that if we extend (2 to whole n(s) via o(t) = @:L, K(II,./I,), whenever 1 = @:I, ~,(a,,b,), then ++I ,is a well-defined morphism. Indeed, let I = @:I=, ~~,(a~.b,) = @‘,‘l, K,,(c,,. ~1~). Then t ’ 1x1s a form

t’

= @;.=, K,,( t/1,. lsk), and,

for all I/ E ‘PI and

11t Pl.

we have

TENSOR

It is easy to check of Theorem.

PRODUCTS

that

OF D-POSETS AND

4 is a morphism

263

D-TEST SPACES

in question,

which

proves

the assertion 0

The tensor product, (P@p Q, 8~) := (17(X), K,). of the D-posets P and Q in the class KF is said to be a state tensor product of P and Q with respect to the state system P = P, x Pz. Unless confusion threatens, we usually refer to P @‘p Q rather than to (P @p Q, By), and we shall write p @p q := @p(p, q). Analogically, we shall write p @p V, for the unique state on P @3pQ, where 1-1E PI, u E Pz, defined by /1 BP PROPOSITION

D-posets

P and

4) = AI))

.4d>

qEQ.

PEP,

(5.3)

= PmJ

1.

P E p,

P2(9)

= 1 @‘p 9,

Q,

4 E

P = PI x Pz2! are monomorphisms,

Proof:

It is clear

for any state +/I

@P

5.5. Let P, and Pz be order determining systems of states on the Q. Then the mappings ,‘31: P + P BP Q, P2: Q + P BP Q, defined Pi(P)

where

4P

BP

Similarly

that

Bt is a morphism.

p @P v, we have,

431(1)2)) = I

by (5.3)

+ 14~2)

5

1,

Suppose

~1@P ~PI(PI)

hence,

&I)

that

Pl(pt)

@ Pl(p2)) 5

.u(P~),

I

pl(p~).

Then,

= CL@w ~PI(PI))

which

means

+

PL 5 P$.

0

we deal with ,!?2.

PROPOSITION 5.6. Let PI and P2 be nonvoid systems of states on the D-posets P and Q, and let P = PI x P2, and let P @p Q be the state tensor product of P and Q taken in the category of D-posets. Then

(i) If PI and Pz Oep=Oorq=O. (ii) If PI and

are unital,

then

P2 are strong systems

P Q?P 4 = PI BP

41 # 0,

then

P

Q are orthoalgebras,

of states,

P = PI

(iii) If PI and Pi are the g-convex PC+ Q! where P’ = P,’ x Pi.

and

and

then

P and

and

p @p q =

Q are OMPs,

and if

4 = a.

hulls generated

by PI and PI. then

P@pQ

=

Proof: (i) If, for example, p # 0, there exists a state p E PI such that p(p) = 1. Then, for any state u E Pz, we have 0 = p @p ~(p @p q) = v(q), so that q = 0. (ii) Since by (i) p # 0,q # 0, we can find states p E PI and v E PI such that p(p) = 1 and v(q) = 1. Hence, p(p~) = 1 and v(ql) = 1, so that p < PI and q < 91. By symmetry we have p = pl and q = 41. (iii) It is clear that both Pi and Pi are strong systems of states. Using the construction of the D-algebraic D-test spaces from the proof of Theorem 5.4 for P’ and P, we see that any D-test T = {(pi, qi)}z,, pi E P, 4% E Q, i = 1> . ?n, corresponding to the system P’ is a D-test corresponding to the system P. Conversely, let T = {(pi, q2)}yz1 b e a D-test corresponding to the system P, i.e. CL, ,u(pi)v(q,) = 1 for any p E PI and any v E Pz. We choose any state p E Pi and u E Pi. Then p = C, c]p~j, u = CI, dkuk, where cj > 0, dk > 0 and Cj c3 = 1,

264 Ck &

A. DVURECENSKIJ = 1, cl,, E PI,

which proves = Pj x p:.

that

and S. PULMANNoVA

uk E Pz. Then

T

=

{(PL~Y,)l:l, is

a D-test

corresponding

to

the

system

P’ q

Remark 5.7. Similarly as for D-posets, we can define a tensor product of orthoalgebras (see, e.g. [9]) as well as a state tensor product of ones in the category of orthoalgebras. This is possible, for example, if Pt and P, are unital systems of states, since then, for any 0 # 11 E P, 0 # q E Q, there are p E Pt and L/ E Pz such that &MY) > l/2 (:;:)> and in this case, by [9, Thm 6.11, it is possible to show that II(X) from the proof of Theorem 5.4 is an orthoalgebra. In addition, all above results of the present section can be reformulated for orthoalgebras (with the condition (*)). Let L, be via (2.4) and 8 (i) IL(I) = g-states P on if C:L, generalize

the D-poset from Example 2.7 which is determined by a generator g let L be any D-poset. A mapping ~1: L + L, is said to be a g-stute 1: and (ii) I.L(Uci3b) = /~(a) CIJ,/I, ~1~h E L. We say that a family of a D-poset L is ,fulf iff P(U) = /I for any IL E P implies n = h, and

,I(w~) = I for any Theorem

/L

E

P.

then

& !,(I, exists /=I

from

Example

in L and

& !,c1, = 1. Now we /=I

5.4.

Suppose that g is a generator k.!~,~:[0, 11 x [0, I] + [0, I] via

fl I’),, b = &J(n)

. g(h)).

2.7. We define

n,

h

E

[O.

I].

a binary

operation

(5.3)

Then q, is commutative and associative, and [O, I] is a semigroup with respect to the “multiplication” !:!,, with the neutral element 1 and, for CL]~u~.b E [O, 11, we have ((I, kc’<,“2) 1: !, h = ((I, 1. !I h) ti’, (“2 i.)(, h), whenever

one

(5.4)

side of (5.4) exists in [0, I].

'THEOREM 5.8.Let g he a generator on [O. 11. Let P, and P2 be nonvoid systems g-states on the D-posets P and Q, respectivc!v. Let C$ be the set qf all D-posets p E PI, L such that there is a himorphism K: PxQ - L. and the set P? := {p#u: v E Pz} is a ,full gIstem of g-stutes on L, where @~L/(K(I). q)) := /&)&v(q), p E P, K ‘s. Then KY = (C$, a$) is y E Q, and let I3,g he the set of all these bimorphisms a consistent cluss ,f
qf

Proofi It is similar to the proof of Theorem 5.4, therefore, we outline here only main steps. Let X be the subset of P x Q consisting of all pairs (p, y) with 1-,# 0, of elements from X and q # 0. If M = {(Pl*41) ..... (P?l.rl,,)} is a finite sequence

265

TENSOR PRODUCTS OF D-POSETS AND D-TEST SPACES X = (,u, v), I* E Pt,

v E P2, we put

with the understanding

that

if M = 0, then

X(M) = 0.

Define now the set 3’9 of all finite sequences T = {(pi, qz)}yC, of elements in X such that X(T) = 1 for any X = (,u, v),,u E Pi, v E P2. Since X(1,1) = 1, 39 is nonvoid. Denote by 5(3,) the set of all finite sequences {(pj, qJ): j E J} such that J 2 1 and {(pi, qi): i E I} E 3g. We put {(pi,qi): i E 0) = 0. It is evident that (X,39) is a D-algebraic D-test space. For two events A, B E &(3g) we define A M B iff X(A) = X(B) for any X E P. Then M is an equivalence on 5(3,), and let n,(A) := {B E 5(3,): B z A}. Let f19(X) into a D-poset by Theorem 4.5. e or anize UX) = MA): A E E(3& W g Define

a mapping

P x Q + 17,(X)

n,:

via

?7({(P,4))) (P7dE XT

64P,4) =

0 {

Then

K, is, evidently,

any t E 17,(X)

(P, 4) 6 X.

a bimorphism.

is of the form

From the construction of n,(X) we see that 1L t = @ S~,(pi, qi), and the mapping p I%{, v on 17,(X)

defined by ~1@z, Y(K,,(P, q)) = p(p) 0, v(q), p E P, q addition, Pi, is a full system of g-states on lirg(X). be the set of all bimorphisms K such that K maps Pi be a full system of states on L. Then ICC = (C$, and Q.

E Q, is a g-state on 17,(X). In Therefore, C$, # 0, and let B$ P x Q into some L E C; and a$) is a consistent class for P

We assert that (II,(X),K.,) is a tensor product of P and Q in the class Kg. Choose L E C!$ and a bimorphism r;: P x Q -+ L. Since P, is full for L, it follows that if K~(P, q) = K&I’, q’), then n(p, q) = n(p’, q’) and we can define a mapping 4: IT,(X) + L such that c$(~Jp, q)) = ~(p, q), p E P, q E Q. It is easy to check that @Jis a well-defined mapping in question, which proves the assertion of Theorem. 0 PROPOSITION 5.9.Let the conditions of Theorem 5.8 be satisfied. Define p, = {g o p: p E P,}, i = 1,2. Then p, and & are nonempty systems of states on P and Q, respectively, In addition, the state tensor product of P and Q with respect to P1 x p2 and the tensor product from Theorem 5.8 are the same. Proof

It is evident

If M = {(~i,qi)):==, v(qi)

that

if ,u is a g-state

is a finite

sequence

on P, then of elements

g o p is a state from

X, then

on P. 6

gp(pi) 0,

i=l

= 1 iff 2 jl(p%) . V(gi) = 1, we conclude

3 = 3g,

and

similarly

E(3)

= E(3,)

i=l

and r(M)

= r,(M),

17(X) = II,(X).

•J

266

A. DVURE'?ENSKlJ and S. PULMANNOVA

6. Tensor

product

of D-test

spaces

The tensor product of D-algebraic D-test spaces and its relationships to the tensor product of the corresponding D-posets have been introduced and studied in [6]. In the present section, we give an analogue of the state tensor product of D-posets for D-test spaces. Let (X, I), (Y,S) be D-test spaces. Let 4: X + Y be a mapping. For E E X’ we denote by 4(E) the function @(E):=QoEEY’. that

is, &(E)(i)

i E I.

= d(E(i)).

A mapping 4: X + Y is called a morphism (of D-test spaces) if 4(T) E S for every T E 7. Evidently, if E, F E I(X) and ELF, then 4(E), 4(F) E E(Y), @(E&+6(F) and #(EilF) = ti(E)C@(F) (we identify events related by -). Also, E cz F implies Q(E) = b(F). A morphism 4: X + Y will be called an isomorphism iff q5 is bijective, and 4-l is a morphism. Let (X, I), (Y, S) be D-test spaces and let F t X’. G E Y“. We define a function FAG: IxJ+XxY via FkG(i,j)

:= (F(i),

(i,j)

G(j)),

Let (X, I), (Y,S) and (Z,U) be D-test spaces. (i) for any T E 7, S E S: P(TkS) E U; (ii) for any TI, T? E 7, S’,>SZ E S, E E f(X),

t I x J.

A mapping

p: X x Y + 2 such that

F E I(Y),

g(T, itF) z A(T2kF). and ‘(EkS,)

= $(EkS,);

will be called a bimolphism. Clearly, if E E &(X), G E I(Y), then P(EkG) E I(Z). E e X’, F t X~’ with I n J = 0, and let G E E(Y), EijF = H E XI’,‘. we get N(EoF),G)

= #(H(i),

for any E E E(X),

F, G E &(Y),

a(Ek(FirG))

1

FIG

G(J’)))(~.~,E.J~A

we obtain

= /!?(Ek F)OP(EkG).

6.1. Let (X, I), (Y,S), (Z,U) be D-algebraic D-test spaces and let 2 be a bimorphism. We say that a triple (Z,U, T) is a tensor product

DEFINITION

7: XXY

EIF, putting

G(j))}(i..l)~(~~./)xh.

= {@(H(i), G(j))]ci.,,~lx~lj(a(H(i), = /3(EkG)lj,!?(FkG). Similarly,

Let E, F E I(X), G E ZK. Then,

TENSOR

PRODUCTS

OF D-POSETS

AND

D-TEST

267

SPACES

of (X, 7) and (Y,S) in the category of D-algebraic D-test spaces iff the following conditions are satisfied: (i) for any D-algebraic D-test space (V, W) and a bimorphism K: X x Y + V, there is a morphism 4: 2 + V such that K = 4 0 7; (ii) to every test T E U, T E Z’, there exists a function F E (X x Y)I such that T = T(F).

If a tensor product of D-algebraic D-test spaces exists, it is unique up to isomorphism. Unless confusion threatens, we usually refer to the tensor product (X @ Y, 7 @ S, @), and by CC 8 y we denote the image of (x, y) under the bimorphism 8. Theorem 10.3 in [6] says that the tensor product of D-algebraic D-test spaces (X,7) and (Y,S) in the category of D-algebraic D-test spaces exists iff there is a D-algebraic D-test space (Z, U) and a bimorphism p: X x Y + 2. Let (X, 7) be a D-test space. A weight on X is a function w: X 4 [0, l] such that for every E E 7, E E X’, w(E) := xw(E(i))

= 1. iEI In other words, any weight on a D-test space is any morphism values in the interval [0, 11.

(6.1) on (X,7)

with

THEOREM 6.2. Let the systems of all weights on D-algebraic D-test spaces (X, 7) and (Y, S) be nonvoid. Then there is a tensor product (X BY, I@S, @), and for any couple of weights WI and w2 on (X, 7) and (Y, S) there is a unique weight WI ~3w2 on (X ~$3 Y, 7 @ S, 8) such that Wl @ w2(z

@ Y) = WI(X)

. W2(Y),

XEX,YEY.

(6.2)

Proof Let L = [0, 11, and endow L with the natural ordering and the difference tes = t-s iff 0 5 s 5 t 5 1. Then L with 1,8 is a complete D-poset. Let U be the set of all finite or countable sequences {ti}i in [0, l] such that ti # 0 and xi t, = 1. Then (L,U) is a D-algebraic D-test space. Let w1 and w2 be weights on (X, 7) and (Y, S), respectively, and define a mapping &w2(x, Y> = Wl(X) . W2(Y), x E X, y E Y. Then easy calculations prove that /3w,w, is a bimorphism from X x Y into [0, 11. According to [6, Thm 10.31, there is a tensor product (X @ Y,I@S,@) of (X,7) and (Y,S). Due to the basic property of the tensor product, there is a morphism C#Jfrom X 8 Y into [0, l] such that &,,,, = 4 o 18. We assert that $I is a weight on X 8 Y. Indeed, since if T = {T(i)}i is a D-test in X @Y, so 40 T is a D-test in [0, 11, which means that xi 4(T(i)) = 1. Let x E X, y E Y be given, then 4(x C$y) = Pw,wz(x, y) = WI(Z). w2(y) and 4 is a unique weight on X @Y with the property (6.2). 0

We note that the former statement generalizes that from [6], where it was assumed that systems of weights are suficienfl. 5 We say that 4x)

# 0.

a set of weights

/I on (X,7)

is sujjicienf

if, for every z t X, there

is an w E A with

268

A. DVUREi‘ENSKIJ and S.PULMANNOVA

Let (X, 7) be a D-test space and let n be a set of weights on it. We say that n is full if (i) for any finite index set I and F E X’, we have F E 7 iff w(F) = 1 for all w t A, and (ii) w(z) = w(w) for all ~cit’n implies 2 = y. Remark 6.3 (1) A full system of weights is sufficient. Indeed, if, for z E X, w(z) = 0 for all LJ E A, then for any test T, {r}ijT is a test, whence {z} N 0. (2) A D-test space with a full system of states is D-algebraic; the proof is similar to that of Theorem 8.2 in [6]. D-test space. We will say that (X, 7) is reduced (3) Let (X, 7) b e a D-algebraic if, for z,y E X, {z} z {y} implies T = y. It is easy to see that for every D-algebraic D-test space there is a reduced D-algebraic D-test space with the same D-poset II(X). Clearly, a D-test space with a full set of weights is reduced. Let (X, 7) and (Y,S) be D-algebraic D-test spaces. = (X, B), where X is a nonvoid family of D-algebraic nonvoid family of bimorphisms on X x YS such that (i) D-algebraic D-test space (U,U) in X such that Y: X x (U,U) E X there is a bimorphism ~3 E 13 with 3: X x consistent cluss for (X, 7) and (Y. S).

We say that a couple K D-test spaces and f? is a for any 13 E B there is a Y - [I, and (ii) for any Y - CT, is said to be a

DEFINITION 6.4.Let K = (X, f3) be a consistent class for D-algebraic D-test spaces (X, 7) and (Y,S). We say that the triple (Z,U, 7) is a tensor product of (X, 7) and (Y,S) in the cluss K if the following conditions are satisfied: (Z,U) belongs to X and T t B. (9 D-test space (‘c: W) in X and every bimorphism 13: (ii) For every D-algebraic X x Y ---) 17 from B, there is a morphism 4: Z + V such that ,B: 4 o 7. F E (X x Y)’ such that (iii) For any D-test T E U, T E Z’, there is a function T = 7(F). We note that, similarly as for the general show that if a tensor product of (X, 7) and up to an isomorphism.

tensor product of D-test spaces, we can (Y,S) exists in the class K, it is unique

THEOREM 6.5. Let (X, 7) und (Y, S) be D-algebraic D-tesf spaces having only finite D-tests with nonempty sets of weights AA- and Ay, respectively. Let X be the cluss of all D-algebraic D-test spaces (V, W) such that there is a bimorphism I%: x x Y + V, V = r;(X x Y). und the set A, := {u/yix6 w2: LU’J c d2y. UJj? t A,}, where wI c!a3,w~(K(x, y)) = WI(S) . Q(Y), (V, W), where w=

(

r;(F):

FE(XXY)‘.

(.z, y) t X x Y, is a full set of weights on

c cjl @, L~~(K(F(~))) = 1. v((w,, iu’?) E A, ItI

x &

>

(I is uny finite index set), and B be the set of all these bimorphisms 6’s. Then K = (X, I?) is u consistent cluss ,for (X, 7) und (Y, S), und there is a tensor product of (X, 7) and (Y S) in the cluss K.

TENSOR

PRODUCTS

OF D-POSETS

AND

D-TEST

Proof: For (zr, y), (u, w) E X x Y, put (CC, y) = (u,v) for all wt E Ax and w2 E Ay. Let 2 = (X x Y)= be ,LL:X x Y + Z be the canonical mapping. Then, for any we have a well-defined mapping WI @WI: 2 + [0, 11 such Wl @ wz(z)

whenever Put

z = ~(z, y) for some

U = {F E 2’:

cur

(2,~)

g~~(F(i))

=

269

SPACES

iff wt(5)4y) = wt(u)wz(~) the quotient space and let WI E AX and any w2 E AY that

WI (x> . f-JZ(Y>

E X x Y. = 1, /7’((wl,w2) t AX x Al,,

IVmite

I}.

iEI

Then (2, U) is a D-algebraic D-test space with a full system of weights A := {WI @Iwz: WI E A,, w2 E A,}. We claim that the triple (Z,U, p) is the tensor product of (X, 7) and (Y,S) in the class ic. Let Tt, T2 E 7, F E E(Y). Let F’ be any local complement of F. Then, for any wt @ w2 E A, WI @

w2(,u(T1

AF)) = 1 - w1 @zwz(p(TI jtF’))

= 1 - w2(F’)

= ~1 @ w&(Tz~F)), hence, &7t kF’) is a common local complement of p(Tl X F) and p(T2 j
=

c

WI @,c W2(45i,

iEI Wl c3 W2(P(%

=c

Yi)> = ~~&&z(YJ iEI Yi>> = 1,

iEI

hence, 4(F) E W. That is, C$ is a morphism, and h: = C$o p. Finally, let T E U, T E 2’. Then T(i) = p(z,,y2) for some (zi, yi) E X x Y. Put F(i) = (G, yi> f or chosen (si, yi) E X x Y such that ,u(z,, yi) = F(i). Then T = p(F). This proves that (Z,U, ,u) is a tensor product of (X, 7) and (Y,S) in the class K. 0 The tensor product (2, U, 7) from of (X, 7) and (Y,S) with respect to will be denoted by (X @A Y, ‘7 @A S, Theorem 6.5 can be reformulated necessary finite D-tests. We say that completely full (or a-completely fill)

Theorem 6.5 is called the weight tensor product nonempty sets of weight A, and Ay, and it @A), where A = A, x Ay. also for D-algebraic D-test spaces with not a nonempty set of weights A on (X,7) is iff (i) for any index set (any countable index

270

A. 13'UREi‘ENStiI.l and S. PULMANNoV/i

set) I and F E A-‘, we have E’ t 7 iff L(F) = I for all w’ t A, and (ii) in? = d(y) for all w’ E A implies .r = y. For example, if H is a Hilbert space. dim H = cc.then (S(H), B(H)), where S(H) is the unit sphere in H and B(H) is the set of all orthonormal basis in H. is an algebraic test space [ 121 with a completely full system of weights (d,,,: y t S(H)}, where d!,(J) = l(.L!/)I’, .r E S(H). which is not a full system of weights. For these D-test spaces we have the following result on tensor products. THEOREM 6.6. Let (X, 7) trnrl (Y, S) be D-algehruic D-test spaces with nonempt? sets qf weights d.v und Al-, respcctil~ely. Let X be the class qf ull lklgehruic West spuces (I’_ 1/v) such thut there is u himorphism K: S x Ir j I’. 1_ = K( X X Y). untl the set A, := {ti, :~a, l_+: d] t A,.,!, t A)-},

where LJI :$, O(K(S. y)) = d,(z) weights on (V. W), where

d(y),

(.r. y) E .Y x II

is u completely

full

set of

and B be the set of all these himorphisms K’S, 7%len K = (X, 8) i5 a consistent cluss for (X. 7) mnd (Y, S). and there is u tensor product qf (X, I) and (Y, S) in the cluss K. Pro@ It follows the same ideas the proof of Theorem 6.5 to

The tensor product (X. 7) and (Y.S) with

as the proof

of Theorem

6.5. We change

from Theorem 6.7 is said to be a complete respect to A.\- and AI-.

weight

U from

tensor

of

PROYOSITION 6.7. Let (X, 7) he u D-ulgebruic D-test space urid let L = IT(*Y) he the> corresponding D-poset. For culy weight d on (X, 7) there is II unique stuie 2 on L such thut 2(-/r(E)) = d(E)

for uny I? t E(X). Conversely, if’ uny test T in 7 u weight ii on (X. 7) such thut jq.1.) = p(7r(:r)). Proqf’: LJ and z are correctly The following result are very close notions.

shows

defined, that

state

and tensor

is ,finite, ury

state ~1 on

L define

.1‘t ‘Y. the proofs products

are straightforward. and weight

tensor

0 products

THEOREM 6.X. Let P, Q und R be D-posets cd let (X. I), (E-S) un~l (Z,Z4) he the corresponding stardud D-test spuces. LCI Pi urd IJz be two nonernpty sets of weights on (X. 7) and (Y. S). respectively, and let ?‘f = { 2: LV’t P, }, i = I. 2. Let

271

TENSOR PRODUCTS OF D-POSETS AND D-TEST SPACES

(R, 7) be the state tensor product of P and Q in the category of D-posets with respect to is = PI x & and let (V, W, 6) be the weight tensor product of D-algebraic D-test spaces (X, I) and (Y, S) with respect to PI x ‘pz, and let D be the corresponding D-poset. Then D and R are isomorphic. Proof Let rrx be the corresponding mapping from E(X) into n(X) defined just before (4.1); similarly we define R-Y,KZ and TV. Since (V, W, 6) is a weight tensor product, by Proposition 6.7 and the definitions of tensor products, there is a morphism & V + 2 such that #J o S = 6, where r;(z, y) = y(‘irx({z}), ~({y}), (z, y) E X x Y. Let 3: D + R be the morphism be the state

bimorphism

tensor

:=

y = C#J o ?1,oy.

Since

by &TV(E)) ,7ry(F))

by $-irx(E)

there

is a morphism

product,

q7rx_(g>~Y(F))

defined

defined nz(f‘qEkF))

every element

=

Y(TX(Jq

= -irz(~$(E)).

= 7rv(6(EkF)).

Let s: P x Q ---t D Since

(R, y)

is a

$J: R + D such that s = II, o y. Further, x 7ry(F)) implies z o 2 = E = y, whence

in R is a finite

orthogonal

sum of elements

of the

form y(p, q), p E P, q E Q, we conclude z o + = idR. On the other hand, every test in W is finite. Let a E D and let E E E(V) be such that a = 7rv(E). Now E + T for some T E W. By (ii) of the definition of a weight tensor product, there are finite E, and F, E, 6 F, F E (X x Y)’ such that E = S(E,) and T = S(F). Then a = ~(d(&))

= @w({~(G,Y~)))

when E, = {(~,~i))i, shows that

7. Examples

which entails $ o &a) = a (observe $ is surjective, and hence it is an isomorphism.

of tensor

that

2 = $ o g o s).

This 0

products

In the present section, with concrete examples.

we illustrate

the above

developed

theory

of tensor

products

EXAMPLE 7.1. Let L = [0, 11, X = (0, 11, and let any D-test in 7 consist of all finite sequences {lcz} in X such that Ci zi = 1. Then on (X, 7) there is a unique weight w = idx. Therefore, the weight tensor product of (X, 7) with itself is again (X, 7). To prove that it is necessary to repeat the proof of Theorem 5.4. In addition, let f,g, h be generators on [0, 11. Then Lf @ L, and Lh @ Lh are isomorphic, and Lf @p L, and Lh are isomorphic. This is true because all L,‘s are isomorphic with Lid; the mapping 4y: Lid + L,, t H g-l(t), t E [0, 11, is an isomorphism. EXAMPLE 7.2. Let S be a real or complex inner product space. Put X = {z E S: llzll = 1) and let 7 consist of the set of all maximal orthonormal systems in S. Then (X, 7) is a test space, and the following statements are equivalent [3, 71:

272

A. DVLJRE~ENSKIJ and S. PULMANNOVA

(i) (X, 7) is an algebraic test space. (ii) (X,7) possesses at least one weight. (iii) S is complete. If one statement is satisfied, then H(X) is isomorphic to the subspaces in the Hilbert space S, which is a complete OMP.

set of all closed

EXAMPLE 7.3. Let X = E(Ht) \ (0) and Y = &(H2) \ (0) and let D-tests on X and Y be the systems of all finite sequences of effects {Ai} on Hilbert spaces HI and H2 such that C, Ai = I. Denote by 7 and S the systems of all D-tests on X and Y, respectively. Let d,x

= {m,:

m,(A)

= (Aqs),

A E X,

x E H,,IIx:I1 = I}

Ay

= {my:

m,(A)

= (Ay, y)% A E Y.

5 E H2, /l;ylI = 1).

and

Then 0~ and A, are full systems of weights on X and Y, respectively. The weight tensor product of (X, 7) and (Y,S) with respect to AX and Ay exists and is equal to (Z,U, @), where 2 = {A @ B: A E X. B E Y}% i.e., A @ B is an operator and HZ,

in the

l4 = {{A, @ Bi}iEl:

tensor

product

A, E X, B, E Y,

c

HI @ HZ of the

Hilbert

A, cz B, = 11 E I1.V finite

spaces

HI

I}

?EI

and y),

@(A, B) = A @z B. 2 E HI,

This

follows

y E HZ, 11~11= IIyII

=

from the fact that 1 iff (A@Bz@y,~@y)

(42,2)(By.y)

= (CCC,.rz)(Q.

= (C@Ds@y,z@y)

iff (A @ B C, z, @ yi, C, 2, 8 yz) = (C @ D 1, .r, Q3yi, C, zi @ yi) iff A 8 B = C @ B as operators in H1 @ Hz. The last equivalence follows from using polar formulae for Hermitian bilinear forms generated by A, B, C, and D. Hence, {m, 0 my: 5 E = (Ax, rr)(By, y), is a I, Y E 5YH2)), where mr o m,(A E B) := rr~.~(A) m,(B) full system of wights on (Z,U). For H(Z), which by Theorem 6.7 is a state tensor product of &(HI) and E(Hz), we have that it consists of all elements of the form C, A, @ B,, where A, E &(HI), Bi E I(H2) for which there are A,, Q? B, E E(H1) x E(Hz) such that C, A, 18 B, + + C, A, @ B, = II @ 12 (all summations are over finite index sets). We recall that the ordering < on H(2) is defined as follows C, A,@B, I 1, C., Q D, iff there is {A’, c%Bk} such that C, Ai @ B, + Ck AL @ Bk = C, C,, @I D,? and it can be different from the natural ordering on E(H1 @ HZ). EXAMPLE 7.4. Let X = E(H,) \ (0) and Y = &(Hz) \ (0) and let D-tests on X and Y be the systems of all summable effects {A,} on Hilbert spaces Hl and Hz such that C, Ai = I (the convergence is assumed to be, for example, weak). Denote by 7 and S the systems of all D-tests on X and Y, respectively. Let AAy = {m,:

7nr(A)

= (ALz),

A E X.

z E HI, jl.c1I = 1)

TENSOR

PRODUCTS

OF D-POSETS

AND

D-TEST

SPACES

273

and Ay = {m,. * m,(A)

= (AK?/),

A E Y, z E Hz, 1141= 1).

Then A, and Ay are completely full systems of weights on X and Y, respectively. The complete weight tensor product of (X, 7) and (Y, S) with respect to A, and Ay exists and is equal to (Z,U, B), where Z={A@B:

i.e., A 8 B is the operator and Hz,

AEX,BEY},

in the tensor product

U = {{A, @ Bi}i: Ai E X, B, E Y, c

Hi @ Hz of the Hilbert spaces HI Ai @ Bi = II 8 12)

and @(A, B) = A @ B. (Use the same arguments as in Example 7.3; {m, 0 my: CCE is a completely full system of weights on (Z,M).) For 17(Z) we S(Hi), Y E Wf2)) have that it consists of all elements of the form Ci Ai @I Bi, where Ai E E(Hl), Bi E E(H2), for which there are Aj 8 Bj E E(H1) x E(H2) such that xi Ai @ Bi + +CjAj@Bj

= Il@I2.

We recall that the ordering < on n(Z) is defined as follows: xi Ai 8 Bi I Cj Cj 8 Dj iff there is {A’, 8 Bk} such that xi Ai @ Bi + XI, AL 8 Bi = C, Cj @ Dj, and it can be different from the natural ordering on E(H, @ Hz). EXAMPLE 7.5. Let L(H,) be the set of all closed subspaces of a Hilbert space Hi, i = 1,2. Then L(H,) is a complete OMP. Let X = L(Hl) \ (0) and Y = L(H;?) \ (0) and let D-tests on X and Y be all finite orthogonal decompositions of HI and Hz,

respectively, which are in fact tests [12]. Denote by 7 and S the systems on X and Y. Let Ax and Ay have the same meaning as in Example 7.3. Then tensor product of (X, I) and (Y,S) is (Z,U), where Z and U have the as in Example 7.3. In addition, n(Z) is an orthoalgebra; we recall that do not know whether is it an OMP.

of all tests the weight same form the authors

EXAMPLE 7.6. Let P be the Phano plane whose diagram is illustrated in Fig. 1, which is an example of an orthoalgebra. It is known that P 8 P does not exist in the category of orthoalgebras [9], but it exists in the category of D-posets [4]. Let X = {a, b, c, d, e, f, g} be a test space of the Phano plane, where the tests are {a, b, c},

{c, d, e),

{e, f, ~1, {a, g, dl,

{b, g, e),

{c, g, f)

and {b, d, fl.

We assert that the weight tensor product of (X, 7) with itself exists, however the set of all weights on (X,7) consists of a unique weight p with ,u({z}) = l/3, z E X. We claim that the weight tensor product of (X, 7) with itself is any D-algebraic D-test space (Z, U, r), where Z = {z}, U = {T}, where T: { 1, . . . ,9} + Z, and ~(2, y) = z, 2, y E X. We recall that on (Z,U) there is a unique weight w, namely w(z) = l/9. Therefore, in K = (X,B), X consists of the set of all isomorphic D-test spaces with (Z,U). It is worth noting that n(Z) = (20, ~1,. . . ,x9} with 0 = 20 < 21 < . . . < x9 = 1, where the difference 8 is defined as follows: xj 8 xi = xj-i, 0 5 i 5 j 5 9. The

274

A. DVURECENSKIJ

and S. PULMANNOVA

Fig.

1

difference 8 is the unique one for a finite chain 0 = rcg < zt < .. < XC)= 1 to be a D-poset [22]. On 17(Z) there is a unique state p, namely I = i/9, i = 0, 1, . . . ,9. On the other hand, the exact form of the (algebraic) tensor product P @ P is unknown. Little is known about this tensor product. For example, using the same reasoning as in [9] (which has been used to show that P ~3P does not exist in the category of orthoalgebras) we can show that a @ b 1 c @Cd for all c1!b, c, d E L.

8. Concluding

remarks

We studied the tensor product of D-posets and of D-test spaces determined by sets of states or weights. These tensor products may be, in general, different from the algebraic tensor products (not depending on states). However, the statistical information pertinent to individual systems is preserved in the joint system described by a state (weight) tensor product. We found concrete examples of state and weight tensor products in some cases, where the form of the algebraic tensor product is not known yet, in particular, the state tensor product of the sets of all effects in Hilbert spaces is presented. REFERENCES

[II

P. J. Lahti and P. Mittelstaedt: The quantum theory of measurement, in Lecture Springer, Berlin, Heidelberg, New York, London, Budapest 1991. PI P. A. M. Dirac: The Principles of QuantumMechanics, Clarcndon Press, Oxford 1930. PI A. Dvuretenskij: Gleasonk Theorem and ifs Applications, Kluwer Academic Pub]., Dordrecht, London 1993. 141 A. DvureEenskij: Trans. Amer. Math. Sot. (1994) (to appear). A. DvureEenskij and S. Pulmannovi: Inter. I. Thror. Phys. 33 (1994), 819. PI and S. Pulmannova: Rep. Math. Phys. 34 (1994), 151. [61 A. Dvurecenskij Left. Math. P1zy.s. (to appear). [71 A. DvureEenskij and S. Pulmannova: A. DvureEenskij and B. RieEan: Intw. J. Theor. Phys. 33 (1994). 1403. PI D. J. Foulis and M. K. Bennett: Order 19 (lYY3), 271. PI D. J. Foulis and M. K. Bennett: Found. Phys. (lYY4) (to appear). [W P. Busch,

Notes

in

Physics,

Boston,

TENSOR PRODUCTS OF D-POSETS AND D-TEST SPACES

275

D. J. Foulis, R. J. Greechie and G. T. Riittimann: Infer. J. Theor. Phys. 31 (1992) 787. WI D. J. Foulis and C. H. Randall: .I. Math. Phys. 13 (1972) 1667. WI , 1131 J. Hedhkova and S. Pulmannova: Generalized difference posets and orthoalgebras, Preprint MO SAV (1994). . 1141 G. Kalmbach and Z. Riecanova: An axiomatization for abclian relative inverses, PreprintSC!/ Brat&lava (1993).

A. N. Kolmogorov: Grundbegrife der Wahrscheinlichkeitsrechnung, Berlin 1933. F. Kopka and F. Chovanec: Math. Slovaca 44 (1994), 21. WI 1171 R. Mesiar: Fuzzy difference posets and MV-algebras, Preprint STU, Bratislava 1993. M. Navara and P. Ptak: Difference posets and orthoalgebras (submitted). WI 1191 J. von Neumann: Maihematische Grundlagen der Quantenmechanik, Springer, Berlin 1932. PO1 S. Pulmannova: Demonstratio Math. 27 (1994). C. H. Randall and D. J. Foulis: J. Math. Phys. 14 (1973), 1472. WI P21 Z. RieEanova and D. BrSel: Inter. J. Theor. Phys. 33 (1994), 133. [231 V. S. Varadarajan: Geometry of Quantum Theoy, Vol. 1, van Nostrand, Princeton 1968. WI