In mathematics, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S.
If S is a group and it is ordered as a semigroup, one obtains the notion of ordered group, and similarly if S is a monoid it may be called ordered monoid. The terms posemigroup, pogroup and pomonoid are also in use. Additive semigroup of natural numbers (N,+) and additive group of integers (Z,+) endowed with natural order are examples of a posemigroup and pogroup. On the other hand, (N∪{0},+) with the natural order is a pomonoid. Clearly, every semigroup can be treated as a posemigroup endowed with the trivial (discrete) partial order: '='. The class of all semigroups may therefore be viewed as a subclass of the class of all posemigroups (indeed one may then prefer to denote a posemigroup by a triple (S,•,≤)).
One can attribute two types of morphisms (in the sense of category theory) to posemigroups, namely the posemigroup homomorphisms which are 'order preserving' (equivalently monotone) semigroup homomorphisms and the posemigroup order-embeddings that are (besides being semigroup homomorphisms) both order preserving and reflecting.
Category-theoretic interpretation
A pomonoid (M, •, 1, ≤)can be considered as a skeletal thin monoidal category, with an object of for each element of M, a unique morphism from m to n if and only if m ≤ n, the tensor product is given by •, and the unit is given by 1.