Sheffer operations in complemented posets

Abstract

We show that in every downward directed poset with an antitone involu-tion the so-called Sheffer operation can be introduced satisfying certain identities.However, also conversely, if we have given a Sheffer operation|on a setP, thenPcan be converted into a poset with an antitone involution′, where both′and theorder relation≤are derived by|. Using this, we can characterize orthoposets, i.e.bounded posets with complementation which is an antitone involution by means ofidentities satisfied by this Sheffer operation. Also conversely, if|is a Sheffer oper-ation on a given setPsatisfying these identities, thenPcan be organized into anorthoposet.