nLab commutative monoid

Context

Algebra

higher algebra

universal algebra

Contents

Definition

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

$x y = y x.$

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.

Examples

• 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 April 21, 2017 03:24:18 by Urs Schreiber (92.218.150.85)