Comparing Numbers of Diagonal Subsemigroups and Congruences for Semigroups

Abstract

Given a semigroup $S$, a diagonal subsemigroup $\ddot{I}$ is defined to be a reflexive and compatible relation on $S$, i.e. a subsemigroup of the direct square $S\times S$ containing the diagonal $\{(s,s):s\in S\}$. When $S$ is finite, we define the DSC coefficient $\ddot{I}‡(S)$ to be the ratio of the number of congruences to the number of diagonal subsemigroups. In a previous work we observed that $\ddot{I}‡(S)=1$ if and only if $S$ is a group. Here we show that for any rational $\hat{I}\pm$ with $0<\hat{I}\pm \le 1$, there exists a semigroup with $\ddot{I}‡(S)=\hat{I}\pm$. We do this by utilizing the Rees matrix construction and adapting the congruence classification of such semigroups to describe their diagonal subsemigroups.