# Ordered semigroup

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".

The positive integers, the nonnegative integers and the integers form respectively a posemigroup, a pomonoid, and a pogroup under addition and natural ordering.

Every semigroup can be considered as a posemigroup endowed with the trivial (discrete) partial order "=".

A morphisms or homomorphism of posemigroups is a homomorphism of semigroups, that *preserves* the order (equivalently, that is monotonically increasing).

## Category-theoretic interpretation

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.

## References

- T.S. Blyth,
*Lattices and Ordered Algebraic Structures*, Springer, 2005, ISBN 1-85233-905-5, chap. 11.