Sublattices and Δ-blocks of orthomodular posets. (28th July 2020)

• Record Type: Journal Article
• Authors: Chajda, Ivan; Länger, Helmut
• Is Part Of: Journal of logic and computation. Volume 30:Number 7(2020)
• Page Start: 1401
• Page End: 1423
• Publication Date: 2020-07-28
• Subjects: Orthomodular poset -- orthoposet -- orthogonal poset -- Boolean poset -- generalized commutator -- relation Δ -- Δ-block; Logic programming -- Periodicals; Logic, Symbolic and mathematical -- Periodicals; Computational complexity -- Periodicals; 005.115
• DOI: 10.1093/logcom/exaa037
• Languages: English
• ISSNs: 0955-792X

Abstract

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 $\mathrel \Delta$ and a binary operator $d(x, y)$ that are generalizations of the binary relation $\textrm{C}$ 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 $\mathrel \Delta$ 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 $\Delta$ -blocks with respect to Boolean subalgebras and distributive subposets they include.