---
tag: HA79YFZAQR
kind: concept
name: Ordered semigroup
msc: ["06F05"]
specializes: [0XMEM406JM]
sources:
  - url: https://en.wikipedia.org/wiki/Ordered_semigroup
same_as:
  wikidata: Q7100714
  wikipedia: Ordered semigroup
---

## Summary

An ordered semigroup is a [[semigroup|0XMEM406JM]] with a [[partial order|0QV2JS9RJ3]] that is preserved by multiplication on both sides.
It is the most general of the ordered algebraic structures with one associative operation.
Ordered monoids and [[ordered groups|KRDR04RQ7N]] are special cases.
Ordered semigroups appear in the theory of formal languages, of semirings and of residuated structures.

## Explanations

### L1 human

The natural numbers under addition are an ordered semigroup: if $a \le b$, then $a + c \le b + c$.
The positive integers under multiplication are another: if $a \le b$, then $ac \le bc$.
In general, an ordered semigroup is any associative operation together with a way to compare elements that the operation respects.