Sublattices and Δ-blocks of orthomodular posets. (28th July 2020) - Chajda, Ivan; Länger, Helmut
Abstract: States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation $\mathrel Δ$ and a binary operator $d(x, y)$ that are generalizations of the binary relation $C$ and the commutator $c(x, y)$, respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations $\mathrel Δ$ and $\leftrightarrow$ (the latter was introduced by P. Pták and S. Pulmannová) and the operator $d(x, y)$ . In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of $Δ$ -blocks with respect to Boolean subalgebras and distributive subposets they include.