submonoid in nLab

Contents

Contents

Definition

A submonoid of a monoid MM with unit 11 is a subset NN of MM containing 11 which is also a monoid with respect to the inherited multiplication.

Examples

• Given a group GG, any subgroup is a submonoid.
• Given a category CC and a subcategory DD, an object XX in both, the monoid Hom D(X,X)\Hom_D(X,X) is a submonoid of Hom C(X,X)\Hom_C(X,X).
• A commutative and cancellative? monoid MM is a submonoid of its Grothendieck group G(M)G(M).

See also

• GCD ring