Semigroup Forum

, Volume 93, Issue 1, pp 111–130 | Cite as

The structure of a graph inverse semigroup

  • Zachary Mesyan
  • J. D. Mitchell
Research Article


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.


Inverse semigroup Directed graph Congruence 



We would like to thank Benjamin Steinberg for pointing us to relevant literature. We are also grateful to the referee for helpful suggestions about improving the paper and for noticing a gap in the proof of an earlier version of Lemma 9.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ColoradoColorado SpringsUSA
  2. 2.Mathematical InstituteUniversity of St AndrewsSt AndrewsScotland, UK

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