nLab
abelian monoid

An abelian monoid is a monoid where the multiplication satisfies the commutative law:

$$xy=yx.$$x y = y x.

Alternatively, just as a monoid is a category with one object, an abelian monoid is a monoidal category with one object and one morphism.

An abelian group is an abelian monoid that is also a group.