Transformation semigroups with restricted range

Let ∅≠X be any set. Then, the semigroup T(X,Y) defined as T(X,Y)={α∈T_X ∶Xα⊆Y}, where Y is a nonempty subset of X, is a subsemigroup of T_X, the semigroup of transformations on set X with the composition of transformations. The elements of this semigroup are called transformations with restricted range. Similarly, the set T_((X,Y))={α∈T_X∶Y_α=Y} is also a subsemigroup of T_X. If we specifically consider the finite sets of X, hence the finite sets ∅≠Y⊆X, without loss of generality, when |X|=n,|Y|=m (n,m∈Z^+ ) we can take X=X_n={1,…,n } and Y=X_m={1,…,m}. In this case, we will write T_((n,m) ) instead of T_((X,Y) ). In this thesis, we will compile some studies that have obtained some important algebraic results of T(X,Y). Afterwards, we will compile some studies that contain important results related to the generating set and rank of T_((n,m) ) and present them to the information of the researchers. Key words: Transformation semigroups, Green's equivalences, generating set, rank.