The semigroup of endomorphisms with restricted range of an independence algebra
Ambroise Grau · Semigroup Forum · 2022
Abstract 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 $$T(\mathscr {A}, \mathscr {B}) $$ T ( A , B ) of all endomorphisms of an independence algebra $$\mathscr {A} $$ A whose image lie in a subalgebra $$\mathscr {B} $$ 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-Gonçalves and all their predecessors. We describe Green’s relations as well as the ideals of $$T(\mathscr {A}, \mathscr {B}) $$ T ( A , B ) following their lead. We then take a new direction, completely describing all of the extended Green’s relations on $$T(\mathscr {A}, \mathscr {B}) $$ T ( A , 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.