Quasivarieties of orthomodular lattices and Bell inequalities

Quasivarieties of orthomodular lattices and Bell inequalities


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Vol. 37 (1996)




No. 2




Dipartimento di Matematica e Applicazioni, Universita Federico II Complesso Monte S. Angelo, via Cintia 80126 Napoli, Italy

and SYLVIA PULMANNOVA Mathematical

Institute, Slovak Academy of Sciences, Stefanikova 49, 814 73 Bratislava, Slovakia (Received May 25, 1995)

It is shown that the class of orthomodular lattices with at least two elements having a nonzero Boolean quotient, which coincides with the class of orthomodular lattices possessing a state in which Bell inequalities of a special type are satisfied, forms a quasivariety.

1. Algebras of orthomodular lattices We start with recollecting basic notions from the theory of universal algebras [8, 91 and orthomodular lattices [2, 10, 1.51. We will consider orthomodular lattices as algebras of the type {A, V, I> 0, l}, where the operations are subject to the standard axioms. Further, let us denote by OML the class of all orthomodular lattices, by I the class of all one-point orthomodular lattices and by BA the class of all Boolean algebras. Evidently, 1 c BA c OML. Let V = {q,z2,. . .} be a set of variables (sometimes we write also 2, y instead of x~,xz). Let T denote a set of all terms. For t E T, let var(t) denote the set of all variables occurring in the term t. For n E N, let T, = {t E T : var(t) C_ xn}}. For the sake of transparency we sometimes write t(z1,52,. . . ,x,) {51rZ27..., for t E Tn. If t E T,, the formula t[q + tl, 22 + t2,. . . ,zn + tn] will denote the term which comes into existence by substituting every zi in t by t; (i = 1,2,. . . , n). Suppose that L E OML. A mapping U: V + L will be called valuation in L. The mapping TJ can be naturally extended over T to a mapping wr: T + L. If t E T,, and w(zi) = ai (i = 1,2,. .., n), then the element wT(t) will be denoted by tL(al, a2, f * . , a,). So, every term t E T,, determines a mapping tL: T, --+ L. The identities are words of the type s g t, where s, t E T. Here the sign 2 stands for the relation which transforms in a given L E OML to an equality. Quasiidentities are words of the type (sl~ttl&...&s,rt,)~s~t, sl,..., sn,tl ,..., t,,s,tET. The semantic validity will be denoted by the symbol I=. If s, t E T and OML I= s g t, we will write s - t. If OML I= s 5 t, we will write s 5 t. 12611



If crl, 02 are identities or quasiidentities, we say that ~1 is equivalent to 13 if in any L E OML, L I= (~1 u L I= ~22. It is easy to see that every identity is equivalent to an identity of the form s g 0 (recall that in an orthomodular lattice x = y iff (zVy)A(z%yl) = 0). Al so, every identity is equivalent to a quasiidentity, and every quasiidentity is equivalent to a quasiidentity of the form s % 0 + t 2 0. Suppose that 0 denotes a set of quasiidentities in OML. Put Mod(o) = {L E OML : L I= (Y for any cr E 0). By a variety (resp. a quasivariety) in OML we call any subclass of OML of the type Mod(Q), where fi is a set of identities (resp. a set of quasiidentities). A quasivariety which is not a variety is called a proper quasivariety. The following statement recalls a famous result of universal algebras [8]. PROPOSITION 1. Suppose that Q is a subclass of OML. Then Q is a quasivariety if and only if it is closed under the formation of subalgebras, products, ultraproducts, isomorphic algebras and it contains a trivial algebra. A quasivariety Q is a variety if and only if it is closed under the formation of epimorphic images.

2. Quotients of orthomodular lattices Let L be an orthomodular

lattice. Put

com(z, y) = (x A y)

V (&

A y) V (cc A y’)

V (z’

A y/I).

The term com(z, y) is called the (lower) commutator of 2, y (the upper commutator is the orthocomplement com(a,b)l of com(a, b)). Recall that a set P C L is called p-filter in L if P is a filter in L (in the lattice theoretic sense) and a E P implies x v (xl A u) E P for all x E L; p-ideal is defined dually. The following properties of commutators are well known [2, lo]. PROPOSITION 2. Suppose that L E OML.

Put n



, b, E L : A com(ai, b;) 5 z}. i=l

Then PC is a p-filter.

by J, the p-ideal in L generated by all upper commutators in L, i.e., = {u’- : a E PC}. Recall that there is a l-l correspondence between p-ideals (dually, p-filters) and congruences in L (see, e.g., [2, lo]). More precisely, if 0 Denote


is a congruence, the class containing 0 is a p-ideal (dually, the class containing 1 is a p-filter). Conversely, if J is a p-ideal, then the relation defined by [email protected] iff (a V b) A (al V bl) E J is a congruence relation and J is the congruence class containing 0. In what follows, L/Q or L/J (resp. L/P) will denote the quotient corresponding to the congruence 0 or, equivalently, to the corresponding p-ideal J (resp. to the corresponding p-filter P). A quotient L/Q is an orthomodular lattice. We will say that L/O is nonzero if L/O # {0}, and we will say that L/Q is a Boolean quotient if L/Q is a Boolean algebra.



An J is a p-ideal p. 88). An

ideal J in L is called a prime ideal if a Ab E J * a E J or b E J. Equivalently, prime ideal if a E J or cl E J for any a E L. Every prime ideal is a maximal (the converse need not hold, maximal ideals need not be prime, see [lo],

ideal J of L is called a semiprime ideal if a A b E J and a A c E J =+a A (b v c) E J. It is easy to see that a prime ideal is semiprime and a semiprime ideal is a p-ideal. It is also straightforward to show that a proper ideal J of an orthomodular lattice L is a prime ideal iff J is a p-ideal and L/J = (0,1). The notions of a prime (semiprime)

filter can be defined dually.

PROPOSITION 3 [6]. Let L be an orthomodular lattice and let J be a proper ideal of L. The following statements are equivalent:


J rS a p-ideal and L/J

is a Boolean algebra;

(2) (3) (4)

J is a semiprime ideal; J is the intersection of all prime

J I Jc;

ideals that contain it.

Accordingly, an orthomodular lattice L has a nonzero a proper filter. Let L be an orthomodular lattice and G a commutative G is finitely additive if m(a~ b) = m(u) + m(b) whenever a a 5 b’. m is called a valuation’ if, in addition, m(u V b) for all a, b E L. According to [l], a characterization of orthomodular Boolean quotient is the following. PROPOSITION

iff P, is

group. A function m: L + and b are orthogonal, i.e., + m(u A b) = m(u) + m(b)

lattices having a nonzero

4. Let

order different from

(1) L has (2) L has (3) There element of G

Boolean quotient

L be an otihomodular lattice, G a commutative group of two. The following statements are equivalent:

a nonzero Boolean quotient. a proper prime Jilter. is a nonzero valuation on L with values in (0, A} c G, where A is an of order different from two.

For 51. x2, . . . , z,, ~1, . . . , yn E V, let t, E Tzn. be defined as follows: t&l,

X2? . . .,X7x.Yl,Y2,...,Yn)

= i;\ com(Xi,









yn E L


# 0L)UI.

where I is the class of all one-point orthomodular lattices, and the indices indicate to which L belongs the element. Clearly, L E NBQ iff one of the equivalent conditions of Proposition 4 is satisfied. ‘Not to be confused with the function v from Section 1, which is also called a valuation.



Let F2 denote a free orthomodular lattice over the set {z, y}. Recall that F2 has 96 elements and it has been treated in detail in [2], p. 82. PROPOSITION 5.

(i) Every free orthomodular lattice belongs to NBQ. (ii) BA is a proper subset of NBQ. (iii) The class NBQ h a proper quasivariety of OML. Proof: (i) Let F be a free orthomodular lattice over a set X (X # 0). Let P, be the commutator p-filter in F. Then the quotient mapping q : F + F/P, maps F into BA, and F/P is a free Boolean algebra over h(X) (h(X) # 0). (ii) Evidently, BA c NBQ. On the other hand, F2 belongs to NBQ but it does not belong to BA. (iii) From our definition of NBQ, we immediately see that



,..., 2,,y1,y2 ,..., y+O+l=O}).


Therefore NBQ is a quasivariety. To show that NBQ is not a variety, let us consider the orthomodular lattice MO2 (the horizontal sum of two 4-point Boolean algebras), which is a homomorphic image of F2, but it is simple. Remark: In [l l] there was introduced

a quasivariety

NC={L~0ML;Vz,y~L:com(z~y)#0~}~1 of orthomodular lattices with fully nontrivial commutators. It was proved that the orthomodular lattice L(H) of all closed subspaces of a Hilbert space H belongs to NC if and only if dim H is a (finite) odd number ([ll], Theorem 1.4). This enables us to prove the following statement. PROPOSITION 6. Quasivariety NBQ is a proper subquasivariety of NC. Proof: Clearly, NBQ g NC. To prove that the equality does not hold, it suffices to take into account that L(H) for any Hilbert space H has no nonzero Boolean quotient [6].

3. Bell inequalities on orthomodular lattices In contrast to classical mechanics, where experiments can be described in the frame of Boolean algebras and the Kolmogorov axiomatic model of probability, quantum mechanics provides more general structures. J. S. Bell [3] introduced examples of inequalities involving probabilities which are valid in classical probability theory, but are violated by some quantum mechanical experiments. In the logico-algebraic approach to quantum mechanics, classical model based on a Boolean algebra is replaced by a more general model based on an orthomodular a-lattice (also called quantum logic). Bell inequalities in the frame of quantum logics were studied by several authors, see, e.g., [13, 17, 16, 5, 71.





In [14] and [12] the relations between properties of the probability measures (Bell inequalities, in particular) and the Boolean quotients have been studied. The famous Bell inequalities can be considered as “tests” for the existence of the so-called hidden variables, that is, “Kolmogorovianity” of the probabilistic model. Since Bell inequalities involve only finitely many elements, we will consider the couple (L, s), where L is an orthomodular lattice and s is a (finitely additive) state on L. We recall that the state on an orthomodular lattice L is a positive valued finitely additive function on L such that s(l) = 1. The state s on L is called two-valued (dispersion free) if s(u) E (0,1) Vu E L, and the state s is called pure if it is an extremal point in the convex set of all states on L. Observe that if L E I, there is no state on L. The theory of hidden variables on a quantum logic L can be defined as follows (see [4], pp. 172, 174). Let S be a set of physical states on L associated with the considered physical system. Then there exists a Kolmogorovian triple (a, C, P) such that every pure state s E S has a “completion”. That is, there is a function w I-+ (s-w), where (s,w) is a two-valued valuation2 on L for every w E 0; for every a E L, the function w H (s, w) (a) is measurable, and s(a) = J (s, w)(a) P(dw) . f-2

It is clear that (s, w)(u) = 0 VWE 0 =+-s(u) = 0. In other words, s is a superposition (in the sense of Varadarajan [18]) of {(s, w) : w E 0). Proposition 7(iv) below indicates that in the hidden variable theory every pure state satisfies Bell inequalities. Put S(u, b) = s(u) + s(b) - 2s(u A b), a, b E L. The following types of Bell inequalities were studied in [14, 131: Qu,bE L,

s(u) + s(b) - s(u A b) 5 1,


e L,

S(u, b) L S(a, c) + S(b, c),



Vu, b, c, d E L,

0 2 s(u A b) + s(b A c) + s(c A d)- s(u A d) - s(b) - s(c),

‘da, b. c E L.

- s(u A c) - s(b A c) 5 1,

Vu, b, c, d E L,


s(u) + s(b) + s(c) - s(u A b)-


s(u A b) + s(b A c) + s(c A d) - s(u A d)- s(b) - s(c) 2 -1.


Inequalities (1) and (3) are called the inequalities of the Bell-Wigner type, (2) and (4) are called the inequalities of the Clauser-Horne type [13]. In the following theorem [12] we are mainly concerned with the inequalities of type (3) and (4). We note that any of (3) and (4) imply (1) (2) and (2’), and the latter three are satisfied if and only if s is a valuation [14]. 2Equivalently, a two-valued Jauch-Piron state. The state m on L is called Jauch-Piron if m(a) = 0 = m(b) implies m(a V b) = 0, a, b E L. It is easy to check that a two-valued state is a valuation if and only if it is Jauch-Piron.



(i) (ii) (iii) (iv)

7 [12]. The following statements are equivalent for (L, s):

inequalities (3) hold; s(com(a, b)l) = 0 for any a, b E L;

s]J, = 0; s is a superposition of a set of two-valued valuations on L;

(v) there exists a Boolean algebra B, a surjective homomorphism a state s on B such that B o cp = s; (vi) inequalities (4) hold.

cp: L --+ B, and

By the symbol Bell we will denote the class of orthomodular lattices L satisfying the following property: L E I or there is a state s on L such that (L, s) satisfies one of the equivalent conditions of Proposition 7. Proposition 7(v) yields the following COROLLARY 8. The class Bell coincides with the class NBQ, hence it is a proper quasivariety in OML.

Proof: If L E Bell \ I, then by Proposition 7(v), there exists a Boolean quotient B 9 I. Conversely, if L E NBQ \ I, then there is a nonzero Boolean quotient B of L, and any state z on B can be lifted to a state s = SO h on L, where h: L -+ B

is the quotient

mapping. Hence Proposition

7(v) is satisfied.


d’Andrea, A. B. and Ptdmannova, S: Boolean quotients of orthomodular

lattices (to appear).

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