Finite complete rewriting systems for regular semigroups

TL;DR

This paper demonstrates that certain regular semigroups with finitely many ideals can be presented by finite complete rewriting systems if their maximal subgroups have such presentations, extending known results in algebraic structures.

Contribution

It proves that finite complete rewriting system presentations are preserved under ideal extensions and for completely 0-simple semigroups with finitely many ideals.

Findings

Regular semigroups with finitely many ideals inherit finite complete rewriting systems from their maximal subgroups.

Ideal extensions by semigroups with finite complete rewriting systems preserve this property.

Completely 0-simple semigroups with finitely many ideals admit finite complete rewriting system presentations.

Abstract

It is proved that, given a (von Neumann) regular semigroup with finitely many left and right ideals, if every maximal subgroup is presentable by a finite complete rewriting system, then so is the semigroup. To achieve this, the following two results are proved: the property of being defined by a finite complete rewriting system is preserved when taking an ideal extension by a semigroup defined by a finite complete rewriting system; a completely 0-simple semigroup with finitely many left and right ideals admits a presentation by a finite complete rewriting system provided all of its maximal subgroups do.