The Abelian Kernel of an Inverse Semigroup

A Ballester-Bolinches,V Pérez-Calabuig

doi:10.3390/math8081219

Abstract

The problem of computing the abelian kernel of a finite semigroup was first solved by Delgado describing an algorithm that decides whether a given element of a finite semigroup S belongs to the abelian kernel. Steinberg extended the result for any variety of abelian groups with decidable membership. In this paper, we used a completely different approach to complete these results by giving an exact description of the abelian kernel of an inverse semigroup. An abelian group that gives this abelian kernel was also constructed.

Highlights

This paper concerns the computability of kernels of finite semigroups with respect to varieties of groups
The kernel of a finite semigroup S with respect to a variety of finite groups F is defined to be the set of all elements that relate to 1 under every relational morphism from S to a group from F
We determine the set of elements of the abelian kernel by means of a normal subgroup of the free abelian group whose quotient is finite. This quotient group is the natural candidate to be the image of a relational morphism which gives the abelian kernel. This result is closely related to profinite topologies and the problem of extending partial automorphisms, which provides extra motivation for undertaking such problems

Summary

Introduction

This paper concerns the computability of kernels of finite semigroups with respect to varieties of groups. Our main result gives a complete description of the abelian kernel of every finite inverse semigroup. The proof our main result depends heavily on the description of the relational morphisms between the members of a class of inverse semigroups and abelian groups (Lemma 6). We determine the set of elements of the abelian kernel by means of a normal subgroup of the free abelian group whose quotient is finite This quotient group is the natural candidate to be the image of a relational morphism which gives the abelian kernel. This result is closely related to profinite topologies and the problem of extending partial automorphisms, which provides extra motivation for undertaking such problems.

Basic Results on Semigroups

Relational Morphisms and Kernels

Minimal Pairs

The Abelian Kernel of an Inverse Semigroup

Conclusions