Some Contributions To Ordered Semigroups

In this thesis, some classes of ordered semigroups are studied. This thesis consists of three Chapters, and each Chapter is an integrated papers.Chapter 1 is devoted to integrally closed strong Dubreil-Jacotin semigroups. After providing some known results on strong Dubreil-Jacotin semigroups, we introduce the definition of integrally closed strong Dubreil-Jacotin semigroups and obtain characterizations of such ordered semigroups. Then we establish the structure of such ordered semigroups in terms of the cartesian ordered semidirect product and the lexicographic ordered semidirect product. These results extend main results of Blyth in [1].In Chapter 2, we research a class of ordered abundant semigroups, so-called, SNF-ordered IC quasi-adequate semigroups. After obtaining some properties of such ordered semigroups, we establish their structure, which generalizes results of Blyth in [2].In Chapter 3, we study naturally ordered quasi-adequate semigroups with greatest idempotents. We provide properties of such semigroups generalizing some results of naturally ordered regular semigroups with greatest idempotents. Then we specialize a construction of Guo in [17] to obtain the structure theorem of amenably and naturally ordered quasi-adequate semigroups with greatest idempotents.