Cayley posets

TL;DR

This paper introduces Cayley posets, a new class of posets derived from pairs of semigroups, unifying several known classes and providing characterizations and examples of their structure and limitations.

Contribution

It defines Cayley posets from semigroup pairs, generalizes known poset classes, and characterizes their automorphisms and representability.

Findings

Cayley posets unify various known poset classes.

Series-parallel posets are Cayley posets.

Some posets cannot be represented as Cayley posets.

Abstract

We introduce Cayley posets as posets arising naturally from pairs $S<T$ of semigroups, much in the same way that 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, with a finite number of atoms) auto-equivalent posets - a class that generalizes a recently introduced notion for numerical semigroups - as those posets coming from a finitely generated submonoid of an abelian group.