A commutative monoid is a monoid M = (M,·,e) such that · is commutative: x·y = y·x.
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.
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.
(N, + ,0), the natural numbers, with addition and zero. The finitely generated free commutative monoids are direct products of this one.
|Congruence extension property|
|Definable principal congruences|
|Equationally definable principal congruences||no|
|Strong amalgamation property|
|Epimorphisms are surjective|