Free skew Boolean intersection algebras and set partitions

TL;DR

This paper characterizes the atoms of free skew Boolean intersection algebras via set partitions, explores their structure, and connects these findings to combinatorial numbers like Bell and Stirling numbers, including infinite cases.

Contribution

It provides a bijective correspondence between atoms and pointed partitions, and describes the algebraic structure and combinatorial properties of free skew Boolean intersection algebras.

Findings

Atoms correspond to pointed partitions of non-empty subsets.

Atoms decompose into joins consistent with partition containment.

Finite free algebras' structure relates to Bell and Stirling numbers.

Abstract

We show that atoms of the $n$-generated free left-handed skew Boolean intersection algebra are in a bijective correspondence with pointed partitions of non-empty subsets of $\{1,2,\dots, n\}$. Furthermore, under the canonical inclusion into the $k$-generated free algebra, where $k\geq n$, an atom of the $n$-generated free algebra decomposes into an orthogonal join of atoms of the $k$-generated free algebra in an agreement with the containment relation on the respective partitions. As a consequence of these results, we describe the structure of finite free left-handed skew Boolean intersection algebras and express several their combinatorial characteristics in terms of Bell numbers and Stirling numbers of the second kind. We also look at the infinite case. For countably many generators, our constructions lead to the `partition analogue' of the Cantor tree whose boundary is the `partition variant' of the Cantor set.