Representation Theory of Finite Semigroups over Semirings

TL;DR

This paper develops the representation theory of finite semigroups over arbitrary semirings, including classification of irreducible and minimal representations, with special focus on the boolean semiring and its simple representations.

Contribution

It extends classical representation theory to semirings, providing new classifications and characterizations, especially for the boolean semiring.

Findings

Classified irreducible and minimal representations over semirings.

Characterized simple representations of the boolean semiring.

Established foundational aspects of character theory for semirings.

Abstract

We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the results for a field. Special attention is paid to the boolean semiring, where we also characterize the simple representations and introduce the beginnings of a character theory.