Ordered semigroup

An ordered semigroup is a semigroup $S$ equipped with a partial ordering for which $y\mapsto xy$ and $y\mapsto yx$ are monotonic for every $x\in S$.

Examples

• $({\mathbb{N}}_1,+)$ forms a posemigroup (a (partially-)ordered semigroup).

• $(\mathbb{N},+)$ forms a pomonoid.

• $(\mathbb{Z},+)$ forms a pogroup.

• Noted by the Emacs community's Tecosaur, Unix-style file paths form a posemigroup with path concatenation. TODO: Elaborate.