Let T (X) be the full transformation semigroup on a nonempty set X. For ∅ Z ⊆ Y ⊆ X, let T (X, Y, Z) = {α ∈ T (X) : Yα ⊆ Z}. It is not difficult to see that it is a generalized form of three well-known semigroups. This paper obtains an isomorphism theorem of T (X, Y, Z). In addition, when X is finite and Z ⊂ Y ⊂ X, the rank of the semigroup T (X, Y, Z) is calculated