Generators of transformation semigroups with restricted range

Onto a non-empty set X, denote the set of all partial transformations by 〖PT〗_X and the set of all (full) transformations by T_X. Then, 〖PT〗_X and T_X are semigroups with function composition and these semigroups are called partial transformations semigroup and (full) transformations semigroup, respectively. Let Y be a non-empty subset of X and Xα=im(α)={xα∶x∈X} for any α∈P_X. Then, the sets T(X,Y)={α∈T_X ∶Xα⊆Y}, PT(X,Y)={α∈PT_X ∶Xα⊆Y} and F(X,Y)={α∈T(X,Y) ∶Xα⊆Yα}) are also semigroups with function composition. In this paper, the studies which study the subsemigroups T(X,Y), PT(X,Y) and F(X,Y) were compiled, and especially some results about the generator sets of these subsemigroups were informed.