In the context of groups we also have the notion `generated by'.
Definition
Let D be a subset of a group G. The set of all products
g1·g2 ··· gn
where n is a positive integer and gi an element or inverse of an element of D, is a subgroup of G, called the subgroup generated by D.
If G
= <D>, then we say that G is generated by D.
A group is called finitely generated if the group is
generated
by a finite set. We call a group cyclic if it can
be
generated by a single element.
The subgroup of a group G
generated by
a set D equals the submonoid generated by
D 
D-1,
where D-1 =
{d-1 | d
D}.