The structure of a graph inverse semigroup

Zachary Mesyan, J. D. Mitchell

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)


Given any directed graph E one can construct a graph inverse semigroup G(E), where, roughly speaking, elements correspond to paths in the graph. In this paper we study the semigroup-theoretic structure of G(E). Specifically, we describe the non-Rees congruences on G(E), show that the quotient of G(E) by any Rees congruence is another graph inverse semigroup, and classify the G(E) that have only Rees congruences. We also find the minimum possible degree of a faithful representation by partial transformations of any countable G(E), and we show that a homomorphism of directed graphs can be extended to a homomorphism (that preserves zero) of the corresponding graph inverse semigroups if and only if it is injective.
Original languageEnglish
Number of pages20
JournalSemigroup Forum
VolumeFirst online
Early online date30 Mar 2016
Publication statusPublished - 2016


  • Inverse semigroup
  • Directed graph
  • Congruence


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