Comparing Numbers of Diagonal Subsemigroups and Congruences for Semigroups

Given a semigroup $S$, a diagonal subsemigroup $\rho$ 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 $\{ (s,s)\colon s\in S\}$. When $S$ is finite, we define the DSC coefficient $\chi(S)$ to be the ratio of the number of congruences to the number of diagonal subsemigroups. In a previous work we observed that $\chi(S) = 1$ if and only if $S$ is a group. Here we show that for any rational $\alpha$ with $0 < \alpha \leq 1$, there exists a semigroup with $\chi(S) = \alpha$. We do this by utilizing the Rees matrix construction and adapting the congruence classification of such semigroups to describe their diagonal subsemigroups.