Left and Right Magnifying Elements in Generalized Semigroups of Transformations by Using Partitions of a Set

Ronnason Chinram, Pattarawan Petchkaew, Samruam Baupradist

Abstract

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 | αI} of the set X, let T(X,P) = {fT(X) | (X_α)fX_α for all αI}. 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.

Keywords

functions, transformation semigroups, partitions, left magnifying elements, right magnifying elements

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