Regularity of some transformation semigroups with restricted range

Abstract

Let X be a nonempty set and T(X) the full transformation semigroup on X. We denote by AM(X) the set of almost one-to-one transformations on X and AE(X) the set of almost onto transformations on X. Also, we de ne OM(X) and OE(X) to be the complement of the set AM(X) and AE(X) in T(X), respectively. It is known that AM(X) and AE(X) are regular semigroups under certain conditions, but OM(X) and OE(X) are not regular semigroups. In this thesis, some generali- sations of T(X) and its subsemigroups are introduced. Let Y be a nonempty subset of X. We de ne T(X; Y ) to be the set of transformations on X whose range is a subset of Y . Likewise, we have a generalisation of AM(X), AE(X), OM(X) and OE(X), namely, AM(X; Y ) = AM(X)\T(X; Y ), AE(X; Y ) = AE(X)\T(X; Y ), OM(X; Y ) = OM(X) \ T(X; Y ) and OE(X; Y ) = OE(X) \ T(X; Y ). Our thesis is devoted to the study of regularity of these semigroups, and their parallels linear transformation semigroups.