# A combinatorial approach to the structure of locally inverse semigroups

## TL;DR

This paper develops a graph-based framework for understanding locally inverse semigroups, introducing new graphical representations that generalize existing concepts and facilitate structural analysis.

## Contribution

It introduces a novel presentation and bipartite graph structure for locally inverse semigroups, extending the tools available for their structural study.

## Key findings

- Graphs describe idempotents and inverses
- Characterization of subclasses via graph properties
- Graph-based methods for semigroup analysis

## Abstract

This paper introduces a notion of presentation for locally inverse semigroups and develops a graph structure to describe the elements of locally inverse semigroups given by these presentations. These graphs will have a role similar to the role that Cayley graphs have for group presentations or that Sch\"utzenberger graphs have for inverse monoid presentations. However, our graphs have considerable differences with the latter two, even though locally inverse semigroups generalize both groups and inverse semigroups. For example, the graphs introduced here are not `inverse word graphs'. Instead, they are bipartite graphs with both oriented and non-oriented edges, and with labels on the oriented edges only. A byproduct of the theory developed here is the introduction of a graphical method for dealing with general locally inverse semigroups. These graphs are able to describe, for a locally inverse semigroup given by a presentation, many of the usual concepts used to study the structure of semigroups, such as the idempotents, the inverses of an element, the Green's relations, and the natural partial order. Finally, the paper ends characterizing the semigroups belonging to some usual subclasses of locally inverse semigroups in terms of properties on these graphs.