Some Research About Type-A Semigroups

Abstract: In this thesis, by using of the commutativity of idempotents, the idempotent-connection, the compatibility of the natural partial order with respect to the multiplication and the closure in a type A semigroup as well as the localization theory of type A semigroups, we investigate the following four basic questions:1. characterization of the Vagner-Preston representation of a type-A semigroup;2. characterization of the Schutzenberger representation of a type-A semigroup;3. characterization of right ω-coset representation on a closed *- subsemi-group of a type A semigroup;4. characterization of localization of a type-A semigroup on its subsemi-groups(in particular, on its idempotent semilattice).The thesis is divided into three chapters.In the first chapter, we give some basic concepts of the Green's*-equivalences and their important properties related to the idempotents such as abundance, the commutativity of idempotents and the idempotent-connection, and the natural partial order on abundant semigroups and, and its compatibility with respect to the multiplication on the semigroups, so on, which are useful for the thesis.In the second chapter, we discuss three kinds of representations for type-A semigroups: the Vagner-Preston representation, the Schutzenberger representation and the representation on ω-cosets of closed *—subsemigroups. The...