On Frink Ideals in Orthomodular Posets

Abstract

Let $\mathcal {S}$ denote the class of orthomodular posets in which all maximal Frink ideals are selective. Let $\mathcal {R}$ (resp. HCode $\mathcal {T}$ ) be the class of orthomodular posets defined by the validity of the following implications: $P\in \mathcal {R}$ if the implication a,b ∈ P, $a\wedge b=0\ \Rightarrow \ a\le b^{\prime }$ holds (resp., $P\in \mathcal {T}$ if the implication $a\wedge b=a\wedge b^{\prime }=0\ \Rightarrow \ a=0$ holds). In this note we prove the following slightly surprising result: $\mathcal {R}\subset \mathcal {S}\subset \mathcal {T}$ . Since orthomodular posets are often understood as quantum logics, the result might have certain bearing on quantum axiomatics.