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Commuting degree for BCK-algebras

C. Matthew Evans

TL;DR

The paper studies the commuting degree cd(A) = |C(A)|/|A|^2 in finite BCK-algebras, i.e., the probability that two randomly chosen elements commute, and builds on sharp order-n bounds to prove that every commuting degree in CD(n) is realized by some non-commutative algebra of order n, with a unique M_n achieving the minimum. It develops constructive tools based on BCK-union and Iséki extension to realize all degrees and shows that every rational in (0,1] occurs as a commuting degree of some finite BCK-algebra. These results extend the understanding of commutativity probabilities in non-classical logics and parallel analogous findings in semigroups.

Abstract

We discuss the following question: given a finite BCK-algebra, what is the probability that two randomly selected elements commute? We call this probability the \textit{commuting degree} of a BCK-algebra. In a previous paper, the author gave sharp upper and lower bounds for the commuting degree of a BCK-algebra with order $n$. We expand on those results in this paper: we show that, for each $n\geq 3$, there is a BCK-algebra of order $n$ realizing each possible commuting degree and that the minimum commuting degree is achieved by a unique BCK-algebra of order $n$ Additionally, we show that every rational number in $(0,1]$ is the commuting degree of some finite BCK-algebra.

Commuting degree for BCK-algebras

TL;DR

The paper studies the commuting degree cd(A) = |C(A)|/|A|^2 in finite BCK-algebras, i.e., the probability that two randomly chosen elements commute, and builds on sharp order-n bounds to prove that every commuting degree in CD(n) is realized by some non-commutative algebra of order n, with a unique M_n achieving the minimum. It develops constructive tools based on BCK-union and Iséki extension to realize all degrees and shows that every rational in (0,1] occurs as a commuting degree of some finite BCK-algebra. These results extend the understanding of commutativity probabilities in non-classical logics and parallel analogous findings in semigroups.

Abstract

We discuss the following question: given a finite BCK-algebra, what is the probability that two randomly selected elements commute? We call this probability the \textit{commuting degree} of a BCK-algebra. In a previous paper, the author gave sharp upper and lower bounds for the commuting degree of a BCK-algebra with order . We expand on those results in this paper: we show that, for each , there is a BCK-algebra of order realizing each possible commuting degree and that the minimum commuting degree is achieved by a unique BCK-algebra of order Additionally, we show that every rational number in is the commuting degree of some finite BCK-algebra.

Paper Structure

This paper contains 3 sections, 7 theorems, 33 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Commuting degrees

Key Result

Lemma 3.1

Let $\mathbf{A}$ be a BCK-algebra with $|\mathbf{A}|=n$ and $\mathop{\mathrm{cd}}\nolimits(\mathbf{A})=\frac{k}{n^2}$. Then

Figures (5)

  • Figure 1: Hasse diagram for $\mathbf{B}_n$
  • Figure 2: Possible Hasse diagrams for $\mathbf{A}$
  • Figure 3: Numerators appearing in $\mathcal{CD}(n)$ for $n=3,4,5,6,7$
  • Figure 4: Obtaining $\mathcal{CD}(6)$ from $\mathcal{CD}(5)$, and $\mathcal{CD}(7)$ from $\mathcal{CD}(6)$.
  • Figure 5: Paths to obtain commuting degree $2/5$

Theorems & Definitions (21)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Lemma 3.1
  • Theorem 3.2
  • proof : Sketch of proof.
  • Lemma 3.3
  • Theorem 3.4
  • proof : Sketch of proof.
  • Theorem 3.5
  • ...and 11 more