Orthogonality spaces associated with posets
Rings and Algebras
2024-11-20 v1
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.
Cite
@article{arxiv.2206.08113,
title = {Orthogonality spaces associated with posets},
author = {Gejza Jenča},
journal= {arXiv preprint arXiv:2206.08113},
year = {2024}
}