Presentations for locally inverse semigroups

Luís Oliveira

Published 2019-03-25Version 1

In this paper we introduce a concept of presentation for locally inverse semigroups and begin studying it. Our main goal is the development of a graph structure to describe the elements of a locally inverse semigroup given by one such presentation. We can look at the graphs introduced here as having a role for locally inverse semigroup presentations similar to the role of Cayley graphs for group presentations or of Sch\"utzenberger graphs of $R$-classes for inverse semigroup presentations. However, although locally inverse semigroups generalize both groups and inverse semigroups, the graphs considered here look very different from both Cayley graphs and Sch\"utzenberger graphs. For example, they are not `inverse word graphs' as the latter two are. Instead, our graphs are bipartite graphs with both oriented and non-oriented edges, and with labels only on the oriented edges. A byproduct of the theory developed here is the introduction of a graphical method for dealing with general locally inverse semigroups. Our graphs are able to characterize and describe, on the locally inverse semigroups given by presentations, 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, we also characterize some usual subclasses of locally inverse semigroups in terms of properties on these graphs.