On Schur Rings Over Semigroups

TL;DR

This paper extends the concept of Schur rings from groups to semigroups, demonstrating that many properties carry over, providing examples of differences, and fully enumerating Schur rings for small semigroups.

Contribution

It generalizes Schur rings to semigroups, proves key properties, and enumerates all Schur rings for semigroups of orders up to 7.

Findings

Many group Schur ring properties hold for semigroups

Examples show differences between group and semigroup Schur rings

Complete enumeration of Schur rings for semigroups of order 0-7

Abstract

We generalize the idea of a Schur ring of a group to the category of semigroups. Fundamental results of Schur rings over groups are shown to be true for Schur rings over semigroups. Examples where Schur rings differ between the two categories are provided. We prove some results for Schur rings over specific families of semigroups. We consider parallels between semigroup extensions and their Schur rings. We fully enumerate the Schur rings for all semigroups of orders 0-7, and some statistical analysis is performed.