Definition:Generated Algebraic Substructure

Definition

Let $\struct {A, \circ}$ be an algebraic structure.

Let $G \subseteq A$ be any subset of $A$.

The algebraic substructure generated by $G$ is the smallest substructure of $\struct {A, \circ}$ which contains $G$.

It is written $\gen G$.

Also known as

Some sources refer to such a $G$ as a set of generators of $B$, but this terminology is misleading, as it can be interpreted to mean that each of the elements of $G$ is itself a generator of $\gen G$ independently of the other elements.

Also see

The concept of a generator is usually defined in the context of particular types of structure:

• Definition:Generator of Subsemigroup
• Definition:Generator of Submonoid
• Definition:Generator of Subgroup

• Definition:Generator of Algebraic Structure