On universal objects in the class of graph inverse semigroups

TL;DR

This paper demonstrates that polycyclic monoids serve as universal objects for graph inverse semigroups, showing embeddings and topological properties that unify these algebraic structures.

Contribution

It establishes that all graph inverse semigroups can embed into polycyclic monoids and introduces a natural coarsest inverse semigroup topology for these structures.

Findings

Graph inverse semigroups embed into polycyclic monoids.

A coarsest inverse semigroup topology exists for each graph inverse semigroup.

Injective homomorphisms are topological embeddings.

Abstract

In this paper 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 $\mathcal{P}_{\lambda}$ 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 the $(\mathcal{P}_{|G(E)|},\tau)$ is a topological embedding.