On universal objects in the class of graph inverse semigroups

Abstract

We show that polycyclic monoids are universal objects in the class of graph inverse semigroups. In particular, we prove that a graph inverse semigroup G(E) over a directed graph E embeds into the polycyclic monoid $${\mathscr {P}}_{\lambda }$$ where $$\lambda =|G(E)|$$. We show that each graph inverse semigroup G(E) admits the coarsest inverse semigroup topology $$\tau $$. Moreover, each injective homomorphism from $$(G(E),\tau )$$ to $$({\mathscr {P}}_{|G(E)|},\tau )$$ is a topological embedding.