Transitive representations of inverse semigroups

TL;DR

This paper characterizes which inverse semigroups can be represented as transitive inverse semigroups of partial transformations, extending the classical group theory concept to a broader algebraic context.

Contribution

It provides a characterization of inverse semigroups that are isomorphic to transitive inverse semigroups of partial transformations.

Findings

Identifies conditions for inverse semigroups to be transitive

Extends the concept of transitivity from groups to inverse semigroups

Provides a classification of such inverse semigroups

Abstract

While every group is isomorphic to a transitive group of permutations, the analogous property fails for inverse semigroups: not all inverse semigroups are isomorphic to transitive inverse semigroups of one-to-one partial transformations of a set. We describe those inverse semigroups that are.