Magnifying Elements in the Generalized Semigroups of Transformations Preserving an Equivalence Relation

Abstract

An element a of a semigroup S is called a left (right) magnifying element if there exists a proper subset M of S such that aM = S (Ma = S). Let T(X) and P(X) denote the semigroup of the full and partial transformations on a nonempty set X, respectively. For an equivalence relation E and a partition P = {X; | i E A} on the set X, let TE(X) = {a e T(X)(x,y) E E implies (xa, ya) € E}, PE(X) = {a e P(X)(x,y) € E implies (2a, ya) € E}, T(X,P) = {a ET(X) Xịa CX, for all i E A}, and P(X,P) = {a e P(X) Xịa CX; for all i E A} Then TE(X), PE(X), T(X, P) and P(X, P) are semigroups under the composition of functions, as well. The main purpose of this thesis is to provide the properties of magnifying elements in the semigroups TE(X), PE(X), TE(X,P) = TE(X) n T(X,P) and PE(X,P) = PE(X) n P(X,P). Futhermore, the necessary and sufficient conditions for elements in these semigroups to be a left or right magnifying element are established.