Comparing Numbers of Diagonal Subsemigroups and Congruences for Semigroups

Given a semigroup $S$, a diagonal subsemigroup $ρ$ 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 $χ(S)$ to be the ratio of the number of congruences to the number of diagonal subsemigroups. In a previous work we observed that $χ(S) = 1$ if and only if $S$ is a group. Here we show that for any rational $α$ with $0 < α\leq 1$, there exists a semigroup with $χ(S) = α$. We do this by utilizing the Rees matrix construction and adapting the congruence classification of such semigroups to describe their diagonal subsemigroups.