Vol. 18 (1980) REPORTS ON MATHEMATICAL PHYSICS No. 3.

COMMUTATIVITY IN ORTHOMODULAR POSETS

R. M. GODOWSKI

Institute of Mathematics, Technical University, Warsaw, Poland

(Received January 1 I, 1979)

An abstract characterization of the commutation relation in orthomodular posets is given. This characterization is a generalization of Guz’s result. In particular, if an orthomodular poset P is Boolean, then aCh iff a A b exists in P.

A method of constructing nonregular Boolean orthomodular posets is presented.

1. Introduction

In [4], M. J. Mqczyriski formulated a set of axioms of quantum mechanics, which modify Mackey’s axioms [3]. These axioms imply that with each physical system we can associate an orthomodular poset L such that each observable can be identified with an L-valued measure on the real Bore1 sets. The elements of L are called questions. We say that two questions 2, b are compatible (written aCb), if there exist real Bore1 sets E, F

and there exists an observable x : B(R) --f L such that a = x(E), and b = x(F). We say that L is regular if for every a, b, c E L such that aCb, bCc and aCc, there exists an observable x: B(R) + L and there exist real Bore1 sets E, F, G such that a = x(E), b

= x(F) and c = x(G). V. S. Varadarajan [6] proved that L is regular if L is an orthomodular lattice. In [2], W. Guz gives some abstract characterization of compatibility, in case L is regular.

If an orthomodular poset L is Boolean ([5]) then for every a, b E L such that a A b

exists in L, uCb. Boolean orthomodular posets are algebraic systems which are near to Boolean algebras. For example, every finite Boolean orthomodular poset is a Boolean

algebra. Every Boolean orthomodular poset admits a full set of two-valued states. But there exist Boolean orthomodular posets which are not regular.

2. Basic definitions and properties

Let (L, < ) be a partially ordered set (abbreviated poset) with the least element 0, the greatest element 1 and a one-to-one map a H a’ of L onto L. (L, < ,O, 1,‘) is called an orthomodulur poset if for every u,b E L the following five conditions are satisfied:

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348 R. M. GODOWSKI

(01) a” = a, (02) a < b =- b’ < a’,

(03) if a < b’, then the least upper bound uv b exists in L, (04) UVU’ = 1, (05) a < b + b = uv(uvb’)‘.

When a < b’ we say that a and b are orthogonal and we write a _L b. Observe that if a I b, then al\ b = 0. The converse implication, in general, does not hold. An orthomodular poset satisfying the condition

(OB) u/jb=Oaulb

is called a Boolean orthomodulur poset. Let (L, ,< ,O,l,‘) be an orthomodular poset. A subset L, G L is called an

orthomodular subposet if 0,l E L and for every u,b EL the following conditions hold:

(Sl) UEL, =SU’ELl,

(S2) [u,bEL,;uIb]=az//bEL,.

If moreover (L,, < ,O, 1,‘) is a Boolean algebra, we call it a Boolean subalgebra of L. When for some u,b EL there exists a Boolean subalgebra B G L such that u,b E B, we

say that a commutes with b and write uCb. Observe that if an orthomodular poset L is associated with some physical system, then the conditions: a commutes with b and u,b are compatible are equivalent. From the condition (05) it follows that if a < b, then aCb.

Likewise, if a I b, then uCb. In [l] P. D. Finch proved that if E G L is a set of elements mutually orthogonal, then there exists a Boolean subalgebra B c L such that E c B.

It is a well known fact that in every orthomodular poset L the condition uCb is equivalent to the condition:

3u,,b,,c, EL[u, I b,,u, I c,b, I c,a = a, Vc and b = b, vc].

LEMMA 1. Let L be an orthomodulur poset, u,b E L and assume that the greatest upper bound a A b exists in L. Then the following conditions are equivalent:

(1) uCb,

(2) u/‘/(uAb)‘J-bA(u//b)‘.

Proof: The implication (1) = (2) is obvious. (2) = (1). If a < b, or b d a, then obviously uCb. If a A b = 0, then (a A b)’ = 1 and

from (2) we obtain a I b, thus uCb. In the other cases from the condition (05) it follows that there exists two (possibly equal) Boolean subalgebras B,,B, G L represented on Hasse’s diagrams as follows

k7Ab~~ AL. c

1

w

by (ohb)'

o' 1.; 1. o/'.b wb @

Dt

0 0

COMMUTATIVITY IN ORTHOMODULAR POSETS 349

Because a = [a A (a A b)‘] v (a A b), b = [b A (a A b)‘] v (a A b), we infer that aCb.

THEOREM 1. For any elements a, b of a Boolean orthomodular poset L, thefollowing

conditions are equivalent:

(1) aCb, (2) a p, b exist in L.

Proof: The implication (1) = (2) is obvious. (2) * (1). Observe that [a A (a A b)‘] A [b A (a A b)‘] = a A b A (a A b)’ = 0. Because L

is Boolean, it follows that a ,q (a A b)’ _L b A (a A b)‘. Therefore Lemma 1 implies aCb.

LEMMA 2. Let L be an orthomodular poset and for a,b,c E L let aCb, bCc and

a A bCc. Then aCb A c.

Proof From the assumptions it is immediately evident that in L b A c and a A b A c

exist. Since a A b AC d b AC, from the condition (05) of the orthomodular poset

definition it follows that b A c = (a A b l\c) v [(a A b A c)’ A (b A c)]. As a A bCc, we have

’ A c. Consequently, it follows that [(a A b A c)’ A (b A c)]

!~#$$$$a(‘a~~~ c)] in other words aCb ,q I:. Since aCb, we have (a ~b) A b = a’A b. Therefore b AC

Note that from the assu-mition of the lemma it does not follow that aCc. In fact, if

we assume b = 0, then always aC0, OCc and a ,j OCc.

3. The representation theorem

As a conclusion we get the following theorem:

THEOREM 2. For any elements a, 6, c of an orthomodular poset L the following

conditions are satisfied:

(Cl) aCb =s- bCa, (C2) aCb =S aCb’, (C3) a < b = aCb, (C4) [aCb, bCc and a A bCc] =E- aCb ,q c, (C5) aCb +- a A b exists in L,

(C6) [aCb and aA\b = 0] =z=-al b.

We shall show that the conditions given above characterize the commutation relation in the class of all relations in a given orthomodular poset L. Namely, we have the following theorem.

THEOREM 3. I. Let L be an orthomodular poset and let R be a relation on L such that

for every a, b, c, d E L the following four conditions are satisfied:

(1) aRb =B bRa, (2) aRb 3 aRb’,

350 R. M. GODOWSKI

(3) a < b =s. aRb,

(4) $ aRb, bRc, a A b, b A c exist in P and a A bRc then aRb A c. Then aCb =z. aRb.

II. IfT moreover, the following two conditions are satisfied:

(5) aRb =S a f, b exists in L, (6) [aRbandaAb=O]=>alb,

then aRb =S aCb.

Proof: I. Since aCb, therefore a A (a’ /,/ b) = a A b. From conditions (1) (2) and (3) we have aRa’\/ b, a’~ bRb, a A (a’ v b)Rb. Consequently, it follows from condition (4) that aRb.

II. Since aRb, by virtue of (5) a A b exists in L. From conditions (1) - (3) it is evident that bRa, aR(a A b)‘, b ,j aR(a A b)‘. Taking into account (4) we get bRa A (a A b)‘.

On the other hand we have bR(a A b)’ and a A (ay\ b)’ A bR(a A b)‘. Therefore aA(aAb)‘RbA(a/I\b)‘. But [ar\(aAb)‘]~[b~(a~b)‘] = 0, so a/\(af,b)‘IbA A (a A b)‘. Then f rom Lemma 1 it is evident that aCb.

Remark: Observe that if L is a Boolean orthomodular poset, then the condition (6) clearly holds for every relation R on L and part II of Theorem 3 is self-evident.

4. Regularity of an orthomodular poset

An orthomodular poset is said to be regular (see Guz [2]) if for every a, b, c E L such that aCb, bCc and aCc there exists a Boolean subalgebra B c L such that a,b,cEB

(abbreviated to (a,b,c)C).

THEOREM 4 (W. Guz [2]). Let L be an orthomodular poset, a,b,cEL. 7’hen the following conditions are equivalent:

(1) (a,b,c)C, (2) aCb, bCc, aCc and a A bCc.

COROLLARY. Iffor some elements a, b, c of an orthomodular poset L aCb, bCc, aCc and at least one of the properties a < b, a I b, b < a, a’ I b’ holds, then (a,b,c)C.

EXAMPLE: A simple example of an orthomodular poset L, in which there are a,b,c such that aCb, bCc, aCc and (a,b,c)C does not hold is the family M of even subsets of set X = { 1,2,...,8} with the natural partial order (inclusion) and the orthocomplementa- tionA’:= X\A.Inthisorthomodularposet{1,2,3,4}C{3,4,5,6},{3,4,5,6}C{1,4,5, 7},{1,4,5,7)C{1,2,3,4)but{1,2,3,4}~{3,4,5,6) = (3,4}and{3,4)C{1,4,5,7}does not hold.

By a simple set-theoretical calculus we can prove two theorems stated below.

THEOREM 5. Let M be a family of subsets of some set X. Assume also that the following conditions hold for any A,BE M:

COMMUTATIVITY IN ORTHOMODULAR POSETS 351

(Sl) %XE M, (S2) X\AE M,

(S3) A~B=~)*AuBEM. Then we have:

(1) M with the inclusion as a partial order and with the “natural” orthocomplementa- tion A’ : = X\A is an orthomodular poset. We call it a partialfield of sets.

(2) AIBsAnB=& (3) ACB~A~BE M, (4) Mis Booleano[AAB=$9+AnB=Q)].

THEOREM 6. Let M be a partialfield of subsets of a set X, and let T be an infinite set; Y: = X x T We define a family N of subsets of the set Yin the following way:

ZEN:G(~Z,E M)Z t (Z, x T) isfinite.

Then:

(1) N is a partialfield of sets;

(2) N is a Boolean orthomodular poset; (3) ACB in M -(A x T)C(B x T) in N.

COROLLARY. There exists a Boolean orthomodular poset L and a,b,ce L such that aCb, bCc, aCc and (a,b,c)C does not hold. This poset can be constructed by the method

described in Theorem 6 and applied in Example 1.

REFERENCES

[l] P. D. Finch: Journal of Symbolic Logic 34 (1969), 275-283. [2] W. Guz: 0 podstawach aksjomatycznych klasycznej i kwantowej mechaniki statystycznej, Wyd. Uczel.

Uniw. Gd., Gdalisk, 1977. [3] W. Mackey: ne mathematicalfoundation of quantum mechanics, W. A. Benjamin Inc., New York, 1963. [4] M. J. Mqczydski: Reports Math. Phys. 2 (1971), 135.

[5] M. J. Mqczyriski, T. Traczyk: Bull. Acad. Polon. Sci., Ser. Sci. Math. Astr. et Phys. 21 (1973), 3-8. [6] V. S. Varadarajan: Geometry of quantum theory, Vol. 1, Van Nostrand, Princeton, 1968.