Sublattices and $\Delta$-blocks of orthomodular posets

Abstract

For orthoposets we introduce a binary relation Delta 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 Delta and \leftrightarrow 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 Delta-blocks with respect to Boolean sublattices and distributive subposets they include.