Magnifiers in Some Generalization of the Full Transformation Semigroups

Abstract

An element a of a semigroup S is called a left [right] magnifier if there exists a proper subset M of S such that a M = S ( M a = S ) . Let T ( X ) denote the semigroup of all transformations on a nonempty set X under the composition of functions, P = { X i ∣ i ∈ Λ } be a partition, and ρ be an equivalence relation on the set X. In this paper, we focus on the properties of magnifiers of the set T ρ ( X , P ) = { f ∈ T ( X ) ∣ ∀ ( x , y ) ∈ ρ , ( x f , y f ) ∈ ρ and X i f ⊆ X i for all i ∈ Λ } , which is a subsemigroup of T ( X ) , and provide the necessary and sufficient conditions for elements in T ρ ( X , P ) to be left or right magnifiers.