Cayley Posets

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.