Algebraic properties of paraorthomodular posets

TL;DR

This paper explores the algebraic and order-theoretic properties of paraorthomodular posets, providing new representations and characterizations relevant to quantum logic structures.

Contribution

It introduces a smooth algebraic representation of paraorthomodular posets and characterizes their order-theoretic features and completion conditions.

Findings

Paraorthomodular posets can be represented by bounded directoids with involution.

Order-theoretic features are characterized by forbidden configurations.

Conditions for their Dedekind-MacNeille completion to be paraorthomodular are established.

Abstract

Paraorthomodular posets are bounded partially ordered set with an antitone involution induced by quantum structures arising from the logico-algebraic approach to quantum mechanics. The aim of the present work is starting a systematic inquiry into paraorthomodular posets theory both from an algebraic and order-theoretic perspective. On the one hand, we show that paraorthomodular posets are amenable of an algebraic treatment by means of a smooth representation in terms of bounded directoids with antitone involution. On the other, we investigate their order-theoretical features in terms of forbidden configurations. Moreover, sufficient and necessary conditions characterizing bounded posets with an antitone involution whose Dedekind-MacNeille completion is paraorthomodular are provided.