nLab
commutative monoid

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Definition

In Sets

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

xy=yx.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. See at commutative monoid in a symmetric monoidal category for details.

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

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.

Last revised on May 2, 2021 at 06:37:00. See the history of this page for a list of all contributions to it.