Commuting Regularity degree of finite semigroups

A pair (x,y)(x,y) of elements xx and yy of a semigroup SS is said to be a commuting regular pair, if there exists an element z∈Sz∈S such that xy=(yx)z(yx)xy=(yx)z(yx). In a finite semigroup SS, the probability that the pair (x,y)(x,y) of elements of SS is commuting regular will be denoted by dcr(S)dcr(S) and will be called the Commuting Regularity degree of SS. Obviously if SS is a group, then dcr(S)=1dcr(S)=1. However for a semigroup SS, getting an upper bound for dcr(S)dcr(S) will be of interest to study and to identify the different types of non-commutative semigroups. In this paper, we calculate this probability for certain classes of finite semigroups. In this study we managed to present an interesting class of semigroups where the probability is 1212. This helps us to estimate a condition on non-commutative semigroups such that the commuting regularity of (x,y)(x,y) yields the commuting regularity of (y,x)(y,x).