Pavel Pták

Almost Boolean orthomodular posets

Abstract Let C be the class of concrete (=set-representable) orthomodular partially ordered sets. Let C 0 be the class of Boolean OMPs (Boolean algebras). In-between C 0 and C ( C 0 ⊂ C ) there are three classes originating in quantum axiomatics — the class C 1 of concrete Jauch-Piron OMPs ( A ϵ C 1 ⇔ if s ( A ) = s ( B ) = 1 for a state s on A and A, B ϵ A , then s ( C ) = 1 for some C ϵ A with C ⊂ A ∩ B ), the class C 2 of ‘compact-like’ OMPs ( A ϵ C 2 ⇔ A is concrete and for every pair A, B ϵ A we have a finite A -covering of A ∩ B ), and the class C 3 of ‘infimum faithful’ OMPs ( A ϵ C 3 ⇔ if a ∧ b = 0 for a, b ϵ A then a ≤ b ′). We study these classes and show that C 0 ⊂ C 1 ⊂ C 2 ⊂ C 3 ⊂ C . We also exhibit examples establishing that at least three of the latter inclusions are proper. Then we prove a representation theorem — every OMP is an epimorphic image of an OMP from C 3 . Finally, we comment on the interpretation of the results in quantum axiomatics and formulate open questions.

Concrete quantum logics with covering properties

LetL be a concrete (=set-representable) quantum logic. Letn be a natural number (or, more generally, a cardinal). We say thatL admits intrinsic coverings of the ordern, and writeL∈Cn, if for any pairA, B∈L we can find a collection {Ci∶ i∈I}, where cardI<n andCi∈L for anyi∈I, such thatA ∩B=∪i∈lCi. Thus, in a certain sense, ifL∈Cn, then “the rate of noncompatibility” of an arbitrary pairA,B∈L is less than a given numbern. In this paper we first consider general and combinatorial properties of logics ofCn and exhibit typical examples. In particular, for a givenn we construct examples ofL∈Cn+1\Cn. Further, we discuss the relation of the classesCn to other classes of logics important within the quantum theories (e.g., we discover the interesting relation to the class of logics which have an abundance of Jauch-Piron states). We then consider conditions on which a class of concrete logics reduce to Boolean algebras. We conclude with some open questions.

Weak dispersion‐free states and the hidden variables hypothesis

We investigate dispersion‐free states which are additive only on the pairs containing a central element (central‐absolutely compatible). We show that any logic possesses plenty of such states, in fact, as many as a certain Boolean algebra. The latter result matches the hidden variables conjecture.

On compatibility in quantum logics

We offer a variant of the intrinsic definition of compatibility in logics. We shown that any compatible subset can be embedded into a Boolean σ-algebra, we show how the algebra is constructed, and we demonstrate that our definition cannot be weakened unless we put additional assumptions on the logic.

On states on the product of logics

We take up the question of when a state (= σ-additive measure) on the product of logics (=σ-orthomodular posets) depends on at most countably many coordinates. We show that it is always so provided there are no real-measurable cardinals. The manner of dependence is a kind of convex combination. We derive some consequences of the latter statement.

Orthocomplemented Posets with a Symmetric Difference

We endow orthocomplemented posets with a binary operation–an abstract symmetric difference of sets–and we study algebraic properties of this class,

Concrete Quantum Logics

{\cal ODP}

Measures on orthomodular partially ordered sets

. Denoting its elements by ODP, we first investigate on the features related to compatibility in ODPs. We find, among others, that any ODP is orthomodular. This explicitly links

Jauch-Piron property (everywhere!) in the logicoalgebraic foundation of quantum theories

{\cal ODP}

On identities in orthocomplemented difference lattices

with the theory of quantum logics. By analogy with Boolean algebras, we then ask if (when) an ODP is set representable. Though we find that general ODPs do not have to be set representable, many natural ODPs are shown to be. We characterize the set-representable ODPs in terms of two valued morphisms and prove that they form a quasivariety. This quasivariety contains the class of pseudocomplemented ODPs as we show afterwards. At the end we ask whether any orthomodular poset can be converted or, more generally, enlarged to an ODP. By countre-examples we answer these questions to the negative.