One physical motivation for this approach is a result proven for the first time by Kochen and Specker [] (see also Specker [], Zierler and Schlessinger [], John Bell [], and Mermin []) stating the impossibility to ``complete'' quantum physics by the introduction of noncontextual hidden parameter models. Such a possible ``completion'' had been suggested, though in not very concrete terms, by Einstein, Podolsky and Rosen (EPR) []. These authors speculated that ``elements of physical realitiy'' exist irrespective of whether they are actually measured or not. Moreover, EPR conjectured, the quantum formalism can be ``embedded'' into a larger theoretical framework which would reproduce the quantum theoretical results but would otherwise be classical and deterministic from an algebraic and logical point of view.
A proper formalization of the term ``element of physical reality'' suggested by EPR can be given in terms of two-valued states or valuations, which can take on only one of two values 0 and 1 and which are interpretable as the classical logical truth assignments false and true, respectively. Kochen and Specker's results [] state that for quantum systems representable by Hilbert spaces of dimension higher than two, there does not exist any such valuation f: L® {0,1} on the set of closed linear subspaces L interpretable as quantum mechanical propositions preserving the lattice operations and the orthocomplement, even if these lattice operations are carried out among commuting (orthoghonal) elements.1
The Kochen and Specker result, it is commonly believed, is directed against the noncontextual hidden parameter program envisaged by EPR. Indeed, if one takes into account the entire logic spanned by Hilbert space (of dimension larger than two) and if one considers all states thereon, any truth value assignment to quantum propositions prior to the actual measurement yields a contradiction. One ``fall-back'' option we shall consider here is a restriction to a subset of all possible states, corresponding to specifications of physical preparation procedures. It is then possible to recover consistent truth value assignments and therefore valuations for a subclass of quantum mechanical propositions. In this way one is naturally lead to the notion of quasiboolean algebras.
The aim of the present paper is to put these ideas into relations with related results previously obtained in lattice theory and quantum logic approach to quantum mechanics. In particular, we are interested in semiprime ideals, Boolean quotients, commutators, partial compatibility, joint distributions, Bell inequalities and hidden variables.
In [], the ring-theoretical concept of semiprime ideal is appropriately defined for lattices. It is proved that an ideal I of a lattice L is semiprime iff I is the kernel of some homomorphism of L onto a distributive lattice with 0. The theory of semiprime ideals is developed there without assuming the axiom of choice and it is proved that the Ultrafilter Principle 2. is equivalent to the statement that every semiprime ideal is representable as an intersection of prime ideals.
In [], distributivity of a finitely generated orthomodular lattice is characterized using the concept of a semiprime ideal. A generalization of these results can be found in [], where it is proved that an ideal in an orthomodular lattice is semiprime iff it contains the commutator ideal.
Irreducible orthomodular lattice with Boolean quotients are studied in [], where a nontrivial example of an irreducible orthomodular lattice is found, all proper quotients of which are Boolean algebras.
Some Bell-type inequalities in orthomodular lattices and their relations to subadditivity of states, commutators and Boolean quotients are studied in [], [], [], and a further development of these ideas can be found in [], [], [].
2. Semiprime ideals and quasidistributive lattices In the theory of commutative rings, the following result is known as Krull's Lemma (cf. []): Axiom of Choice is equivalent, in Zermelo-Frankel set theory, to the condition that any proper ideal in a commutative ring with unit is contained in a maximal ideal, which in turn is equivalent to the formally simpler condition that any non-trivial commutative ring with unit contains a maximal ideal. Another variant of Krull's Lemma says that the Boolean Ultrafilter thoerem is equivalent to the condition that every nontrivial commutative ring with unit contains a prime ideal. There are several attempts to find an analogue of Krull's Lemma in the theory of distributive lattices (cf. []). Recall that an ideal I of a commutative ring R with unit is called semiprime whenever an Î I (n is a positive integer) entails a Î I. According to a well-known result by Krull ([]), using well-ordering theorem, every semiprime ideal is the intersection of all prime ideals that contain it (cf. [] for a proof using only the Ultrafilter Principle). In the following definition, an appropriate analogue of the notion of a semiprime ideal for lattices is given ([]).
Recall that an ideal in a lattice L is a subset I of L such that a Î I, b £ a imply b Î I, and a,b Î I imply aÚb Î I. The definition of a filter is dual, that is, A subset F of L is a filter if a Î F, b ³ a imply b Î F and a,b Î F imply aÙb Î F. An ideal (filter) is proper if it is not equal to the whole L, and a proper ideal (filter) is maximal if there is no bigger proper ideal (filter).
An ideal I of a lattice L is called semiprime if for every x,y,z Î L, whenever xÙy Î I and xÙz Î I, then xÙ(yÚz) Î I. Dually, a filter F is semiprime if xÚy Î F and xÚz Î F imply that xÚ(yÙz) Î F. In a distributive lattice, every ideal and every filter is semiprime. Recall that an ideal I is called prime if xÙy Î I implies that either x Î I or y Î I. It is easy to see that every prime ideal is semiprime, and consequently any nonempty intersection of prime ideals or filters is semiprime.
The Ultrafilter Principle (in the form of the Boolean Prime Ideal Theorem) says that every Boolean algebra contains a prime ideal. In [], there is proved that the Ultrafilter Principle is equivalent to the statement that every semiprime ideal in a lattice is representable as an intersection of prime ideals (a dual result holds for filters).
An ideal I is principal if I = (a] = { b Î L:b £ a}. We note that for principal ideals, the notion of a semiprime ideal coincides with the notion of 0-distributivity due to Varlet []. According to [], a lattice with 0 is 0-distributive if xÙy = 0 and xÙz = 0 imply that xÙ(yÚz) = 0. In [], a 0-distributive lattice is called semiprime (i.e., the zero ideal (0] is semiprime). Dually, a lattice with 1 is called dual-semiprime if the unit filter [1) is semiprime. A bounded 3 lattice is called bi-semiprime if it is both semiprime and dual semiprime.
Recall that a binary relation R in a lattice is a congruence if R is an equivalence relation preserving lattice operations, i.e. aRa1 and bRb1 imply aÚbRa1Úb1 and aÙbRa1Ùb1. A mapping h:L1® L2, where Li,i = 1,2 are lattices, is a homomorphism if it preserves the lattice operations, i.e., h(aÚb) = h(a)Úh(b) and h(aÙb) = h(a)Ùh(b). The kernel of a homomorphism is the set { a Î L: h(a) = 0. The kernel of any homomorphism is an ideal, but not every ideal gives rise to a homomorphism, in general. Let R be a congruence on a lattice L. For a Î L, let [`a] denote the equivalence class with respect to R to which a belongs. The set of all equivalence classes, denoted by L/R, is a lattice called a quotient of L. The mapping a® [`a] assigning to every elements a Î L its corresponding equivalence class [`a] in L/R, is a surjective homomorphism (called also the canonical epimorphism).
The main theorem in [] is the following.
Let L be a lattice and I an ideal in L. Then the following conditions are equivalent: