Isomorphisms of semigroups of transformations

If M is a centered operand over a semigroup S , the suboperands of M containing zero are characterized in terms of S -homomorphisms of M . Some properties of centered operands over a semigroup with zero are studied. A Δ -centralizer C of a set M and the semigroup S ( C , Δ ) of transformations of M over C are introduced, where Δ is a subset of M . When Δ = M , M is a faithful and irreducible centered operand over S ( C , Δ ) . Theorems concerning the isomorphisms of semigroups of transformations of sets M i over Δ i -centralizers C i , i = 1 , 2 are obtained, and the following theorem in ring theory is deduced: Let L i , i = 1 , 2 be the rings of linear transformations of vector spaces ( M i , D i ) not necessarily finite dimensional. Then f is an isomorphism of L 1 → L 2 if and only if there exists a 1 − 1 semilinear transformation h of M 1 onto M 2 such that f T = h T h − 1 for all T ∈ L 1 .