Closed inverse subsemigroups of graph inverse semigroups

ABSTRACT

As part of his study of representations of the polycylic monoids, Lawson described all the closed inverse submonoids of a polycyclic monoid P_n and classified them up to conjugacy. We show that Lawson’s description can be extended to closed inverse subsemigroups of graph inverse semigroups. We then apply Schein’s theory of cosets in inverse semigroups to the closed inverse subsemigroups of graph inverse semigroups: we give necessary and sufficient conditions for a closed inverse subsemigroup of a graph inverse semigroup to have finite index, and determine the value of the index when it is finite.

KEYWORDS:

• Coset
• directed graph
• index
• inverse semigroup
• subsemigroup

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 20M30
• Secondary 05C25

Acknowledgement

The authors are grateful to the referee for the close scrutiny given to the paper, and for several scholarly suggestions for improvements. In particular, the referee indicated how our original arguments could be adapted to deal with infinite graphs, and these suggestions have much improved the paper.