Sublattices and Δ-blocks of orthomodular posets

Abstract

For orthoposets we introduce a binary relation ∆ and a binary operator d(x, y) which are generalizations of the binary relation C and the commutator c(x, y), respectively, known for orthomodular lattices. We characterize orthomodular posets among orthoposets and orthogonal posets. Moreover, we describe connections between the relations ∆ and ↔ and the operator d(x, y). In details 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 ∆-blocks with respect to Boolean sublattices and distributive subposets they include. AMS Subject Classification: 06A11, 06C15, 03G12