Commuting degree for BCK-algebras

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.