  Since its introduction by Symons, the semigroup of maps with restricted range
has been studied in the context of transformations on a set, or of linear maps
on a vector space. Sets and vector spaces being particular examples of
independence algebras, a natural question that arises is whether by taking the
semigroup $T(\mathcal{A},\mathcal{B})$ of all endomorphisms of an independence
algebra $\mathcal{A}$ whose image lie in a subalgebra $\mathcal{B}$, one can
obtain corresponding results as in the cases of sets and vector spaces. In this
paper, we put under a common framework the research from Sanwong, Sommanee,
Sullivan, Mendes-Gon\c{c}alves and all their predecessors. We describe Green's
relations as well as the ideals of $T(\mathcal{A},\mathcal{B})$ following their
lead. We then take a new direction, completely describing all of the extended
Green's relations on $T(\mathcal{A},\mathcal{B})$. We make no restriction on
the dimension of our algebras as the results in the finite and infinite
dimensional cases generally take the same form.
