# 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 monoidal category with one object and one morphism.

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 28, 2014 06:01:02 by Urs Schreiber (88.128.80.161)