On Schur Rings Over Semigroups

TL;DR

This paper extends Schur rings from groups to semigroups, defining partitions that form subring-like structures within semigroup algebras and developing a broad toolkit of constructions (automorphic, direct product, trivial, ideal extensions via Rees quotients, rosters, cloning) to generate and analyze Schur rings. It introduces the concept of recursive indecomposable imposition (RIISR) to describe Schur rings over nilpotent and related semigroups, and proves correspondence principles for Rees quotients and ideal extensions. A major empirical contribution is the complete enumeration of Schur rings over all semigroups of orders 0–7 using GAP/smallsemi, revealing a strong tendency toward Bell-number counts and highlighting both highly flexible and highly rigid semigroups. The results illuminate the interplay between semigroup structure (nilpotency, indecomposables, idempotents) and the combinatorial richness of Schur rings, with implications for algebraic combinatorics and potential extensions to other algebraic categories.

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.

Figures (2)

• Figure 1: The Five Partitions of Order 3
• Figure 2: The Fifteen Partitions of Order 4

Theorems & Definitions (103)

• Proposition 1.1
• proof
• Proposition 1.2
• proof
• Proposition 1.3
• proof
• Proposition 1.4
• proof
• Proposition 1.5
• proof