Left and Right Magnifying Elements in Generalized Semigroups of Transformations by Using Partitions of a Set (European Journal of Pure and Applied Mathematics, 2018)
An element a of a semigroup S is called left [right] magnifying if there exists a proper subset M of S such that S = aM [S = Ma]. Let X be a nonempty set and T(X) be the semigroup of all transformation from X into itself under the composition of functions. For a partition P = {X_α | α ∈ I} of the set X, let T(X,P) = {f ∈ T(X) | (X_α)f ⊆ X_α for all α ∈ I}. Then T(X,P) is a subsemigroup of T(X) and if P = X, T(X,P) = T(X). Our aim in this paper is to give necessary and suﬃcient conditions for elements in T(X,P) to be left or right magnifying. Moreover, we apply those conditions to give necessary and suﬃcient conditions for elements in some generalized linear transformation semigroups.