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Content available remote PBZ*-lattices : structure theory and subvarieties
EN
We investigate the structure theory of the variety of PBZ*-lattices and some of its proper subvarieties. These lattices with additional structure originate in the foundations of quantum mechanics and can be viewed as a common generalisation of orthomodular lattices and Kleene algebras expanded by an extra unary operation. We lay down the basics of the theories of ideals and of central elements in PBZ*-lattices, we prove some structure theorems, and we explore some connections with the theories of subtractive and binary discriminator varieties.
EN
The present paper is devoted to modelling of a probabi‐ lity measure of logical connectives on a quantum logic via a G‐map, which is a special map on it. We follow the work in which the probability of logical conjunction (AND), dis‐ junction (OR), symmetric difference (XOR) and their nega‐ tions for non‐compatible propositions are studied. Now we study all remaining cases of G‐maps on quantum lo‐ gic, namely a probability measure of projections, of impli‐ cations, and of their negations. We show that unlike clas‐ sical (Boolean) logic, probability measures of projections on a quantum logic are not necessarilly pure projections. We indicate how it is possible to define a probability me‐ asure of implication using a G‐map in the quantum logic, and then we study some properties of this measure which are different from a measure of implication in a Boolean algebra. Finally, we compare the properties of a G‐map with the properties of a probability measure related to logical connectives on a Boolean algebra.
3
Content available remote Comparability groups
EN
A comparability group is a unital group with a compression base and with the general comparability property. The additive group of self-adjoint elements in a von Neumann algebra, and any Dedekind sigma-complete lattice-ordered abelian group with order unit are examples of comparability groups. We develop the basic theory of comparability groups, and show that an archimedean comparability group with the Rickart projection property can be embedded in a partially ordered rational vector space the elements of which admit a rational spectral resolution.
4
Content available remote Implications and equivalences in orthomodular lattices
EN
The present article describes a method for checking the validity of implications or equivalences in the free orthomodular lattice on two generators and in the F(a, b, c1,..., cn), which is the free orthomodular lattice generated by the elements a, 6, ci,... Cn, where the elements ci, i = 1,..., n are central in it. The structure of the previous lattices is described in [3] and [1]. The method presented is based on comparing the elements that are assigned to each expression on both sides of an implication or an equivalence. It gives a necessary condition for the implication or equivalence of arbitrary positive statements (a combination of identities and logical connectives AND and OR) to hold. When the conclusion part is an identity or a conjunction of identities, these conditions become also sufficient.
5
Content available remote Boolean carried homomorphisms in orthomodular lattices
EN
Let L, L1 be orthomodular lattices. Let us say that a surjective homomorphism f : L - L1 is Boolean carried if for any maximal Boolean subalgebra B1 of L1 there is a maximal Boolean subalgebra B of L such that f(B) = B1. In this note we investigate the class HOMC all L's such that all surjective homomorphisms from L to orthomodular lattices are Boolean carried. We prove as a main result that if L possesses at most countably many infinite maximal Boolean subalgebras then L L HOMC- We also relate the class HOMCto the classes previously studied and provide some model-theoretic propertiesHOMC.
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