Regularity in Semigroups of Quasi-onto Transformations

Abstract

Let X be an in nite set with cardinal p and let T(X) denote the semigroup (under composition) of transformations on X. For each in nite cardinal q such that q ≤ p, let AE(X; q) = {α ∈ T(X) : d(α) < q} when d(α) = |X\imα|. Then AE(X, q) is a semigroup of quasi-onto tranformations on X. In this thesis, we characterize left regular elements, right regular elements, completely regular elements, unit regular elements and intra-regular elements of AE(X,q).