nLab commutative monoid

Definition

In Sets

A commutative monoid is a monoid where the multiplication satisfies the commutative law:

$$x y = y x.$$

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.

Every commutative monoid has the canonical structure of a module over the commutative rig $\mathbb{N}$. That is, CMon = $\mathbb{N}$-Mod.

In any symmetric monoidal category

More generally, the concept makes sense internal to any symmetric monoidal category (or even any braided monoidal category). See at commutative monoid in a symmetric monoidal category for details.

As a 0-truncated object

A commutative monoid is a 0-truncated commutative H-monoid?. It is also a 0-truncated $E_2$-space.

Examples

• 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:

1. A commutative monoid in the symmetric monoidal category of vector spaces is a commutative algebra;

2. A commutative monoid in the symmetric monoidal category of chain complexes of vector spaces is a differential graded-commutative algebra;

3. A commutative monoid in the symmetric monoidal category of chain complexes of super vector spaces is a differential graded-commutative superalgebra.

Results on commutative monoids in Set

1. If a commutative monoid is finitely generated it is finitely presented.

2. Finitely generated commutative monoids have decidable word problems, the isomorphism problem for them is decidable, and indeed the first-order theory of finitely generated commutative monoids is decidable (see KharlampovichSapir).

3. If a finitely generated commutative monoid is cancellative ($a + b = a' + b \Rightarrow a = a'$) then it embeds in a finitely generated abelian group.

4. If a finitely generated commutative monoid is cancellative and torsion-free ($a + a + \cdots + a = 0 \Rightarrow a = 0$) then it embeds in a finitely generated free abelian group (or more concretely, $\mathbb{Z}^n$). Conversely any submonoid of $\mathbb{Z}^n$ is cancellative and torsion-free (but not necessarily finitely generated). A commutative monoid is called an affine monoid if it is isomorphic to a finitely generated submonoid of $\mathbb{Z}^n$, and there is an extensive theory of these, connected to toric varieties (see BrunsGubeladze).

5. If a finitely generated commutative monoid is cancellative and nonnegative ($a + b = 0 \Rightarrow a,b = 0$), it embeds in a finitely generated free commutative monoid, or more concretely, $\mathbb{N}^n$ (Thm. 3.11, RosalesGarcía-Sánchez). Conversely, any finitely generated submonoid of $\mathbb{N}^n$ is cancellative and nonnegative (but not necessarily finitely generated).

Related concepts

• commutative magma

• CMon

• group completion

• category of monoids

• super commutative monoid

• cocommutative comonoid

• braided monoidal groupoid

• symmetric monoidal groupoid

• k-tuply monoidal n-groupoid

• symmetric monoidal category

• model structure on commutative monoids in a symmetric monoidal model category

• commutative ∞-monoid

• E-infinity algebra

• G-commutative monoid

• periodic table

• commutative monoidal setoid

References

The word problem for commutative monoids is discussed here:

• Olga G. Kharlampovich and Mark V. Sapir. Algorithmic problems in varieties, International Journal of Algebra and Computation 5.04n05 (1995), 379-602. (pdf)

For affine monoids and other finitely generated commutative monoids see:

• Winfried Bruns and Joseph Gubeladze, Polytopes, Rings, and K-theory, Springer, Berlin, 2009. (pdf of preliminary incomplete version)

• José Carlos Rosales and Pedro A. García-Sánchez, Finitely Generated Commutative Monoids, Nova Publishers, 1999.

For a treatment of free commutative monoids in homotopy type theory/univalent foundations:

• Vikraman Choudhury, Marcelo Fiore, Free Commutative Monoids in Homotopy Type Theory (arXiv:2110.05412)