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

A combinatorial approach to the structure of locally inverse semigroups