# Representations of inverse semigroups in complete atomistic inverse meet-semigroups

## TL;DR

This paper develops a new theory for representing inverse semigroups within complete atomistic inverse algebras, extending classical ideas to broader algebraic structures like partial automorphism monoids.

## Contribution

It introduces a generalized representation framework for inverse semigroups in complete atomistic inverse algebras, including partial automorphism monoids of various mathematical structures.

## Key findings

- Established a theory of decompositions for these representations
- Identified the necessity of complete distributivity for classical-like results
- Extended classical inverse semigroup representation concepts to new algebraic contexts

## Abstract

As an appropriate generalisation of the features of the classical (Schein) theory of representations of inverse semigroups in $\mathscr{I}_{X}$, a theory of representations of inverse semigroups by homomorphisms into complete atomistic inverse algebras is developed. This class of inverse algebras includes partial automorphism monoids of entities such as graphs, vector spaces and modules. A workable theory of decompositions is reached; however complete distributivity is required for results approaching those of the classical case.