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.
An ordered monoid and an ordered group are, respectively, a group or a monoid that are endowed with a partial order that makes them ordered semigroups. The terms posemigroup, pogroup and pomonoid are sometimes used, where "po" is an abbreviation for "partially ordered".
Every semigroup can be considered as a posemigroup endowed with the trivial (discrete) partial order "=".
A pomonoid (M, •, 1, ≤) can be considered as a monoidal category that is both skeletal and thin, 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 being given by •, and the unit by 1.
- T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, ISBN 1-85233-905-5, chap. 11.