  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.
