commutative monoid



A commutative monoid is a monoid where the multiplication satisfies the commutative law:

xy=yx.x y = y x.

Alternatively, just as a monoid can be seen as a category with one object, a commutative monoid can be seen as a bicategory with one object and one morphism (or equivalently, a monoidal category with one object).

Commutative monoids with homomorphisms between them form a category of commutative monoids.


  • An abelian group is a commutative monoid that is also a group.

  • The natural numbers (together with 0) form a commutative monoid under addition.

  • Every bounded semilattice is an idempotent commutative monoid, and every idempotent commutative monoid yields a semilattice, (see that entry).

Revised on March 20, 2015 13:45:01 by Noam Zeilberger (