Commutative monoid

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article defines a monoid property, viz a property that can be evaluated for any monoid. Recall that a monoid is a set with an associative binary operation, having a neutral element (viz multiplicative identity)

Definition

A monoid in which all elements commute is called a commutative monoid. That is, a commutative monoid satisfies $ab=ba$ for all $a,b$ in the monoid.

Related notions

Weaker than

For a monoid with all elements invertible, i.e. a group, the related notion is an abelian group.

Examples

• Any abelian group.

• The additive monoid of natural numbers.

• The multiplicative monoid of non-zero integers.