Inmathematics, anordered semigroup is asemigroup (S,•) together with apartial order ≤ that iscompatible with the semigroup operation, meaning thatx ≤y implies z•x ≤ z•y and x•z ≤ y•z for allx,y,z inS.
Anordered monoid and anordered group are, respectively, amonoid or agroup that are endowed with a partial order that makes them ordered semigroups. The termsposemigroup,pogroup andpomonoid are sometimes used, where "po" is an abbreviation for "partially ordered".
Thepositive integers, thenonnegative integers and theintegers form respectively a posemigroup, a pomonoid, and a pogroup under addition and the natural ordering.
Every semigroup can be considered as a posemigroup endowed with the trivial (discrete) partial order "=".
Amorphism orhomomorphism of posemigroups is asemigroup homomorphism thatpreserves the order (equivalently, that ismonotonically increasing).
A pomonoid(M, •, 1, ≤) can be considered as amonoidal category that is bothskeletal andthin, with an object of for each element ofM, a unique morphism fromm ton if and only ifm ≤n, the tensor product being given by•, and the unit by1.
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