On Frink Ideals in Orthomodular Posets

Abstract

Let S denote the class of orthomodular posets in which all maximal Frink ideals are selective. Let R (resp. T ) be the class of orthomodular posets defined by the validity of the following implications:In this note we prove the following slightly surprising result: R ⊂ S ⊂ T . Since orthomodular posets are often understood as quantum logics, the result might have certain bearing on quantum axiomatics. Notions and ResultsWe study three classes of orthomodular posets introduced in the abstract. Though the classes R and T have naturally appeared in the investigation of orthomodular posets ([5-8, 11]), and they may have certain interpretation in theoretical physics as quantum logics (for instance, if P ∈ R and a, b ∈ P , then a, b are compatible exactly when a ∧ b exists in P , see [8]), the link with purely order-theoretic notion of Frink ideal may appear unexpected. Prior to formulating our results, let us recall the definitions as we shall use them in the sequel. We shall exclusively deal with orthomodular posets (OMPs, see [10]). By an OMP we mean a 5-tuple (P , ≤, , 0, 1) such that the following conditions are fulfilled:(1) (P , ≤) is a partially ordered set with a least element, 0, and a greatest element, 1;

Quantum logics close to Boolean algebras

“…The main result reads as follows. Prior to its formulation, let us note that the proof strategy follows that of [10]; the presence of △ and the pecularities of the classes R, S, and T require some additional checking in places. Proof.…”

On the Set-Representable Orthomodular Posets that are Point-Distinguishing

“…[2,4,10]). In particular, it is worth noticing that the examples of conceptually important orthomodular posets with the property that x is compatible with y exactly when x ∨ y exists (see [13]) could be constructed point-distinguishing. For a potential further research, let us conclude this paragraph with a few observations concerning the natural point-distinguishing representation.…”