Ordered semigroup
Template:Short description 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 monoid or a group 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".
The positive integers, the nonnegative integers and the integers 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 "=".
A morphism or homomorphism of posemigroups is a semigroup homomorphism that preserves the order (equivalently, that is monotonically increasing).
Category-theoretic interpretation
A pomonoid (M, •, 1, ≤)Script error: No such module "Check for unknown parameters". can be considered as a monoidal category that is both skeletal and thin, with an object of for each element of MScript error: No such module "Check for unknown parameters"., a unique morphism from mScript error: No such module "Check for unknown parameters". to nScript error: No such module "Check for unknown parameters". if and only if m ≤ nScript error: No such module "Check for unknown parameters"., the tensor product being given by •Script error: No such module "Check for unknown parameters"., and the unit by 1Script error: No such module "Check for unknown parameters"..
References
- T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, Template:Isbn, chap. 11.