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Free skew Boolean intersection algebras and set partitions

Ganna Kudryavtseva

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.

Free skew Boolean intersection algebras and set partitions

Abstract

We show that atoms of the -generated free left-handed skew Boolean intersection algebra are in a bijective correspondence with pointed partitions of non-empty subsets of . Furthermore, under the canonical inclusion into the -generated free algebra, where , an atom of the -generated free algebra decomposes into an orthogonal join of atoms of the -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.

Paper Structure

This paper contains 16 sections, 27 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Skew Boolean algebras
  4. Left-handed skew Boolean algebras
  5. Skew Boolean intersection algebras
  6. Some properties of left-handed SBIAs
  7. Elementary elements and normal forms
  8. Set partitions
  9. Elements determined by pointed partitions
  10. The Decomposition Rule
  11. Normal forms
  12. The commutative case
  13. Free algebras
  14. Finite free algebras
  15. Infinine free algebras
  16. ...and 1 more sections

Key Result

Theorem 1

The relations ${\mathcal{L}}$ and ${\mathcal{R}}$ are congruences for any skew lattice $S$. Moreover, $S/{\mathcal{L}}$ is the maximal right-handed image of $S$, $S/{\mathcal{R}}$ is the maximal left-handed image of $S$, and the following diagram is a pullback:

Figures (1)

  • Figure 1: The first four levels of the infinite partition tree

Theorems & Definitions (55)

  • Theorem 1: Leech L3
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 45 more