arXiv subjects

Callum Barber

Publications and source records attributed to Callum Barber.

2 recordsLinked to original sources

arXiv2026

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.

math.RA↗

arXiv2025

Semigroup Congruences and Subsemigroups of the Direct Square

We investigate semigroups $S$ which have the property that every subsemigroup of $S\times S$ which contains the diagonal $\{ (s,s)\colon s\in S\}$ is necessarily a congruence on $S$. We call such $S$ a DSC semigroup. It is well known that all finite groups are DSC, and easy to see that every DSC semigroup must be simple. Building on this, we show that for broad classes of semigroups -- including periodic, stable, inverse and several well-known types of simple semigroups -- the only DSC members are groups. However, it turns out that there exist non-group DSC semigroups, which we obtain utilising a construction introduced by Byleen for the purpose of constructing interesting congruence-free semigroups. Such examples can additionally be regular or bisimple.

math.RA↗