The semigroup of endomorphisms with restricted range of an independence algebra

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.