Magnifying elements of transformation semigroups

Abstract

An element a of a semigroup S is a left (right) magnifying element if aM = S (Ma = S) for some proper subset M of S. It is a strong left (right) magnifying element if aT = S (Ta = S) for a proper subsemigroup T of S. The notion of a magnifying element was introduced by E.S. Ljapin [7] and that of a strong magnifying element by K. Tolo [ll]. For a list of other papers dealing with magnifying elements, see the first paragraph of [1]. V.M. Klimov has shown that any Bear-Levi semigroup consists of nonstrong left magnifying elements and we will see that there is an abundance of semigroups with the property that all magnifying elements are strong. It is noted in [1], however, that no one has yet produced an example of a semigroup which contains both strong and nonstrong left (right) magnifying elemcnts. In Section 2, we gather together several elementary results about magnifying elements in abstract semigroups which will be of use to us in our subsequent discussion. In Section 3, we investigate magnifying elements for quite general transformation semigroups. Among other things, we characterize those transformation semigroups with identities which have magnifying elements and we characterize those magnifying elements. The results are then applied in Section 4 to such transformation semigroups as the semigroup of all linear transformations of a vector space and S(X) , the semigroup of all continuous selfmaps of the topological space X. We are able to show, for example, that if X is any local dendrite with finite branch number, then the subsemigroup of all left magnifying elements of S(X) is isomorphic to the dual of the semigroup of all monomorphisms which map S(X) properly into S(X) and the subsemigroup of all right magnifying elements is a homomorphic image of that semigroup.