Free skew Boolean algebras

TL;DR

This paper investigates the structure of free skew Boolean algebras, providing representations, formulas for their size, and characterizations of key elements, with implications for their minimal generating sets.

Contribution

It introduces new structural descriptions and formulas for free skew Boolean algebras, including their representations, atomic and central elements, and properties of infinite free algebras.

Findings

Finite free skew Boolean algebras are represented as products of primitive algebras.

Formulas for calculating the cardinality of finite free skew Boolean algebras.

The center of infinitely generated free skew Boolean algebras is trivial.

Abstract

We study the structure and properties of free skew Boolean algebras. For finite generating sets, these free algebras are finite and we give their representation as a product of primitive algebras and provide formulas for calculating their cardinality. We also characterize atomic elements and central elements and calculate the number of such elements. These results are used to study minimal generating sets of finite skew Boolean algebras. We also prove that the center of the free infinitely generated algebra is trivial and show that all free algebras have intersections.