Hyperbolicity of Semigroup Algebras II

TL;DR

This paper completes the classification of finite semigroups whose rational semigroup algebra has a hyperbolic unit group, focusing on the case where the semigroup is semi-simple, building on prior classifications of units in semigroup rings.

Contribution

It provides a complete classification of semi-simple finite semigroups with hyperbolic unit groups in their rational semigroup algebras, extending previous partial results.

Findings

Classification of semi-simple finite semigroups with hyperbolic unit groups

Extension of prior classifications of units in semigroup rings

Identification of algebraic structures leading to hyperbolic unit groups

Abstract

In 1996 Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki-Juriaans-Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra %over a field of characteristic zero, contains a ${\mathbb{Z}}$-order with hyperbolic unit group. In this paper we complete this classification by handling the case in which the semigroup is semi-simple.