Comparing Numbers of Diagonal Subsemigroups and Congruences for Semigroups

TL;DR

The paper defines and analyzes the DSC coefficient $\chi(S)=\frac{|\text{Cong}(S)|}{|\text{Diag}(S)|}$ for finite semigroups, establishing that $\chi(S)=1$ characterizes groups. It proves that any rational value in $(0,1]$ can be realized by some finite semigroup using Rees matrix constructions and a diagonal-subsemigroup classification via linked reflexive triples, extending the classical congruence framework to diagonal subsemigroups when the underlying group is DSC. An explicit formula for $\chi$ on Rees matrix semigroups is derived, and a constructive method (via choices of $G$ and $P$) shows how to realize any target rational value, including a lower bound given by the rectangular band case. The work also discusses implications for Clifford semigroups and outlines open questions about the spectrum of $\chi$ across semigroup families.

Abstract

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.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5