Milan Matoušek

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,

On identities in orthocomplemented difference lattices

{\cal ODP}

States with values in the Łukasiewicz groupoid

. 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

Orthocomplemented difference lattices in association with generalized rings

{\cal ODP}

Orthocomplemented lattices with a symmetric difference

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.

Orthomodular Posets Related to Z2-Valued States

In this note we continue the investigation of algebraic properties of orthocomplemented (symmetric) difference lattices (ODLs) as initiated and previously studied by the authors. We take up a few identities that we came across in the previous considerations. We first see that some of them characterize, in a somewhat non-trivial manner, the ODLs that are Boolean. In the second part we select an identity peculiar for set-representable ODLs. This identity allows us to present another construction of an ODL that is not set-representable. We then give the construction a more general form and consider algebraic properties of the ‘orthomodular support’.

ORTHOCOMPLEMENTED DIFFERENCE LATTICES WITH FEW GENERATORS

Abstract In this paper we consider certain groupoid-valued measures and their connections with quantum logic states. Let ∗ stand for the Łukasiewicz t-norm on [0, 1]2. Let us consider the operation ⋄ on [0, 1] by setting x ⋄ y = (x⊥ ∗y⊥)⊥ ∗ (x∗y)⊥, where x⊥ = 1−x. Let us call the triple L = ([0, 1], ⋄, 1) the Łukasiewicz groupoid. Let B be a Boolean algebra. Denote by L(B) the set of all L-valued measures (L-valued states). We show as a main result of this paper that the family L(B) consists precisely of the union of classical real states and Z2-valued states. With the help of this result we characterize the L-valued states on orthomodular posets. Since the orthomodular posets are often understood as “quantum logics” in the logico-algebraic foundation of quantum mechanics, our approach based on a fuzzy-logic notion actually select a special class of quantum states. As a matter of separate interest, we construct an orthomodular poset without any L-valued state.

Orthomodular lattices that are \(Z_2\)-rich

Orthocomplemented difference lattices (ODLs) are orthocomplemented lattices endowed with an additional operation of “abstract symmetric difference”. In studying ODLs as universal algebras or instances of quantum logics, several results have been obtained (see the references at the end of this paper where the explicite link with orthomodularity is discussed, too). Since the ODLs are “nearly Boolean”, a natural question arises whether there are “nearly Boolean rings” associated with ODLs. In this paper we find such an association — we introduce some difference ring-like algebras (the DRAs) that allow for a natural one-to-one correspondence with the ODLs. The DRAs are defined by only a few rather plausible axioms. The axioms guarantee, among others, that a DRA is a group and that the association with ODLs agrees, for the subrings of DRAs, with the famous Stone (Boolean ring) correspondence.