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Counting finite semirings

J. Edwards, J. D. Mitchell, P. Ragavan

TL;DR

The paper addresses counting finite semirings up to isomorphism and, for small orders, up to isomorphism and anti-isomorphism. It adopts a double-coset framework: a semiring is determined by a pair of semigroups $(S,+)$ and $(S,×)$ with distributivity, and two semiring structures are isomorphic if their multiplicative actions lie in the same double coset of $\operatorname{Aut}(S,×)\backslash \operatorname{Sym}(S)/\operatorname{Aut}(S, +)$ (and anti-isomorphic via $\operatorname{Aut}^*(S,×)$). The authors implement the computation in GAP using the Smallsemi library, compute automorphism groups via a graph-based method, and obtain representative counts for small $n$ along with several tables of results. They validate their approach against existing data where possible, report a notable discrepancy for $n=5$ due to distributivity-check errors in an alternative tool, and illustrate the substantial computational effort required for enumeration (up to ~2400 CPU hours for certain cases). Overall, the work demonstrates a practical enumeration methodology for semirings and highlights the close relationship between semiring counting and semigroup structure, in contrast with the more constrained counting in rings or groups.

Abstract

In this short note we count the finite semirings up to isomorphism, and up to isomorphism and anti-isomorphism for some small values of $n$; for which we utilise the existing library of small semigroups in the GAP package Smallsemi.

Counting finite semirings

TL;DR

The paper addresses counting finite semirings up to isomorphism and, for small orders, up to isomorphism and anti-isomorphism. It adopts a double-coset framework: a semiring is determined by a pair of semigroups and with distributivity, and two semiring structures are isomorphic if their multiplicative actions lie in the same double coset of (and anti-isomorphic via ). The authors implement the computation in GAP using the Smallsemi library, compute automorphism groups via a graph-based method, and obtain representative counts for small along with several tables of results. They validate their approach against existing data where possible, report a notable discrepancy for due to distributivity-check errors in an alternative tool, and illustrate the substantial computational effort required for enumeration (up to ~2400 CPU hours for certain cases). Overall, the work demonstrates a practical enumeration methodology for semirings and highlights the close relationship between semiring counting and semigroup structure, in contrast with the more constrained counting in rings or groups.

Abstract

In this short note we count the finite semirings up to isomorphism, and up to isomorphism and anti-isomorphism for some small values of ; for which we utilise the existing library of small semigroups in the GAP package Smallsemi.

Paper Structure

This paper contains 2 sections, 2 theorems, 3 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Tables of results

Key Result

Theorem 1.3

Let $(S, +)$ be a commutative semigroup, let $(S, \times)$ be a semigroup, and let $\sigma, \tau\in\operatorname{Sym}(S)$ be such that $(S, +, \times ^ \sigma)$ and $(S, +, \times ^ \tau)$ are semirings. Then $(S, +, \times ^ \sigma)$ and $(S, +, \times^\tau)$ are isomorphic if and only if $\sigma$

Theorems & Definitions (5)

  • Definition 1.1: Semiring
  • Definition 1.2: Semiring isomorphism
  • Theorem 1.3
  • proof
  • Corollary 1.4