Cayley Posets

Abstract

We introduce Cayley posets as posets arising naturally from pairs \(S<T\) of semigroups, much in the same way that a Cayley graph arises from a (semi)group and a subset. We show that Cayley posets are a common generalization of several known classes of posets, e.g., posets of numerical semigroups (with torsion) and more generally affine semigroups. Furthermore, we give Sabidussi-type characterizations for Cayley posets and for several subclasses in terms of their endomorphism monoid. We show that large classes of posets are Cayley posets, e.g., series–parallel posets and (generalizations of) join-semilattices, but also provide examples of posets which cannot be represented this way. Finally, we characterize (locally finite) auto-equivalent posets (with a finite number of atoms)—a class that generalizes a recently introduced notion for affine semigroups—as those posets coming from a finitely generated submonoid of an abelian group.

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Notes

  1. 1.

    Note that if X is a (semi)group, \(S\subseteq X\) is just a subset, and the \(\cdot \) is right-multiplication, then the resulting relation coincides with the Cayley graph of X with respect to S.

  2. 2.

    Upsets and downsets are sometimes called filters and ideals, respectively.

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Acknowledgements

We are grateful to Ulrich Knauer for his comments on the manuscript. Moreover, we thank an anonymous referee for helpful comments and pointing our attention to constructions of transitive digraphs that are not Cayley graphs of a monoid. The first author was partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades through grant PID2019-105896GB-I00 and by the ULL Research Project MASCA. The second author was partially supported by the French Agence nationale de la recherche through project ANR-17-CE40-0015 and by the Spanish Ministerio de Ciencia, Innovación y Universidades through grant RYC-2017-22701. Both authors were partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades through grant PID2019-104844GB-I00.

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Correspondence to Ignacio García-Marco.

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García-Marco, I., Knauer, K. & Mercui-Voyant, G. Cayley Posets. Mediterr. J. Math. 17, 186 (2020). https://doi.org/10.1007/s00009-020-01634-z

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Keywords

  • Numerical semigroup
  • Cayley poset
  • Monoid
  • Semigroup

Mathematics Subject Classification

  • 06A11
  • 06A07
  • 20M99