Relative rank of the finite full transformation semigroup with restricted range

Abstract

In this paper, we determine the relative rank of the semigroup T(X; Y ) of all transformations on a nite set X with restricted range Y modulo the semigroup of all extensions of the bijections on Y , modulo the idempotent order-preserving transformations in T(X; Y ), and modulo the semigroup of all order-preserving transformations in T(X; Y ).