International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 3, Pages 487-501
doi:10.1155/S0161171283000447

Isomorphisms of semigroups of transformations

Abstract

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 Mi over Δi-centralizers Ci, i=1,2 are obtained, and the following theorem in ring theory is deduced: Let Li, i=1,2 be the rings of linear transformations of vector spaces (Mi,Di) not necessarily finite dimensional. Then f is an isomorphism of L1→L2 if and only if there exists a 1−1 semilinear transformation h of M1 onto M2 such that fT=hTh−1 for all T∈L1.