Closed inverse subsemigroups of graph inverse semigroups

摘要

As part of his study of representations of the polycylic monoids, M.V. Lawsondescribed all the closed inverse submonoids of a polycyclic monoid $P_n$ andclassified them up to conjugacy. We show that Lawson's description can beextended to closed inverse subsemigroups of graph inverse semigroups. We thenapply B. Schein's theory of cosets in inverse semigroups to the closed inversesubsemigroups of graph inverse semigroups: we give necessary and sufficientconditions for a closed inverse subsemigroup of a graph inverse semigroup tohave finite index, and determine the value of the index when it is finite.