Orthomodular and generalized orthomodular posets

TL;DR

This paper characterizes the smallest non-lattice orthomodular poset, explores generalized orthomodular posets including Boolean posets, and introduces new concepts like compatibility, commutator, and ternary discriminator with illustrative examples.

Contribution

It identifies the minimal orthomodular poset, extends the class to generalized orthomodular posets, and introduces new structural concepts and their relationships.

Findings

The 18-element non-lattice orthomodular poset is unique and minimal.

Generalized orthomodular posets include all Boolean posets.

Introduces compatibility, commutator, and ternary discriminator concepts.

Abstract

We prove that the 18-element non-lattice orthomodular poset depicted in the paper is the smallest one and unique up to isomorphism. Since not every Boolean poset is orthomodular, we consider the class of the so-called generalized orthomodular posets introduced by the first and third author in a previous paper. We show that this class contains all Boolean posets and we study its subclass consisting of horizontal sums of Boolean posets. For this purpose we introduce the concept of a compatibility relation and the so-called commutator of two elements. We show the relationship between these concepts and we introduce the notion of a ternary discriminator for these posets. Numerous examples illuminating these concepts and results are included in the paper.