On orthomodular posets

Abstract

Let S be a poset with a greatest element 1. We denote order in S by ‘≦’ and, whenever they exist in S, l.u.b and g.l.b by ‘∨’ and ‘∧’ respectively. An orthocomplementation of S is a bijection w: S → S such that x ∨ xω exists for each x in S and (i) xωω = x, (ii) x ≦ y implies yω ≦ xω and (iii) x ∨xω = 1. If a poset S admits an orthocomplementation ω we call the pair (S, ω) an orthoposet.