Submonoids

A submonoid of a monoid M is a subset that is itself a monoid under the same operation. If M = (M, ·, e) with associative · and identity e, a subset N ⊆ M is a submonoid if e ∈ N and for all a, b ∈ N we have a · b ∈ N. Then N with the restricted operation is a monoid.

Basic properties include that the intersection of any family of submonoids of M is again a submonoid, and the identity element of M lies in every submonoid. The concept is compatible with homomorphisms: the image of a monoid homomorphism is a submonoid, and the preimage of a submonoid under a homomorphism is itself a submonoid.

The submonoid generated by a subset S ⊆ M, denoted ⟨S⟩, is the smallest submonoid containing S. It consists of all finite products of elements of S (and the identity e). If S is empty, ⟨S⟩ = {e}. When M is commutative, ⟨S⟩ often has a particularly tractable structure.

Examples: In the additive monoid (N, +) with identity 0, submonoids are the sets nN = {0, n, 2n, ...} for n ≥ 0. In the free monoid on an alphabet, A*, submonoids containing the empty word ε correspond to sublanguages closed under concatenation.

Submonoids generalize subgroups by dropping the requirement of inverses. If M is a group, a submonoid of M need not be a subgroup unless every element of the submonoid has its inverse in the submonoid. Submonoids play a role in various constructions, including semigroup theory and the study of affine semigroups in algebraic geometry.