Commutative monoids

Abbreviation: CMon

Definition

A commutative monoid is a monoid M = (M,·,e) such that · is commutative:   x·y = y·x.

Definition

A commutative monoid is a structure M = (M,·,e), where · is an infix binary operation, called the monoid product, and e is a constant (nullary operation), called the identity element, such that
· is commutative:   x·y = y·x,
· is associative:   (x·y)·z = x·(y·z),
e is an identity for ·:   e·x = x.

Morphisms

Let M and N be commutative monoids. A morphism from M to N is a function h : M→N that is a homomorphism: h(x·y) = h(x)·h(y)  and  h(e) = e.

Some results

Examples

(N, + ,0), the natural numbers, with addition and zero. The finitely generated free commutative monoids are direct products of this one.

Properties

Finite members

Subclasses

Superclasses