Orthogonality spaces associated with posets

TL;DR

This paper explores the relationship between posets and orthogonality spaces, showing how the structure of the poset influences the logic of its associated orthogonality space, with characterizations for lattices and chains.

Contribution

It introduces a novel association between posets and orthogonality spaces and characterizes finite bounded lattices and chains via the logic of these spaces.

Findings

Finite bounded lattices correspond to orthomodular lattices in the orthogonality space logic.

Posets that are chains correspond to Boolean algebras in the orthogonality space logic.

Provides a new perspective on the structure of posets through orthogonality spaces.

Abstract

An orthogonality space is a set equipped with a symmetric, irreflexive relation called orthogonality. Every orthogonality space has an associated complete ortholattice, called the logic of the orthogonality space. To every poset, we associate an orthogonality space consisting of proper quotients (that means, nonsingleton closed intervals), equipped with a certain orthogonality relation. We prove that a finite bounded poset is a lattice if and only if the logic of its orthogonality space is an orthomodular lattice. We prove that that a poset is a chain if and only if the logic of the associated orthogonality space is a Boolean algebra.