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A commutative monoid is a monoid where the multiplication satisfies the commutative law:
Alternatively, just as a monoid can be seen as a category with one object, a commutative monoid can be seen as a bicategory with one object and one morphism (or equivalently, a monoidal category with one object).
Commutative monoids with homomorphisms between them form a category of commutative monoids.
More generally, the concept makes sense internal to any symmetric monoidal category. See at commutative monoid in a symmetric monoidal category for details.
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).
Examples of commutative monoids in a symmetric monoidal category:
A commutative monoid in the symmetric monoidal category of vector spaces is a commutative algebra;
A commutative monoid in the symmetric monoidal category of chain complexes of vector spaces is a differential graded-commutative algebra;
A commutative monoid in the symmetric monoidal category of chain complexes of super vector spaces is a differential graded-commutative superalgebra.
Last revised on July 27, 2018 at 05:12:49. See the history of this page for a list of all contributions to it.