Some Studies On Semigroups With Some Kind Of Transversals

This thesis consists of six chapters. We mainly investigate regular semigroups with orthodox transversals and abundant semigroups with adequate transversals and quasi-adequate transversals respectively.The first chapter is introduction and preliminaries.In Chapter 2, the concept of left simplistic orthodox transversals is introduced. An example is given to demonstrate that left simplistic orthodox transversals are proper generalizations of quasi-ideal orthodox transversals. Regular semigroups with left sim-plistic orthodox transversals are explored and a structure theorem for this class of semigroups is also established.In Chapter 3, the concept of S-orthodox transversals is introduced. An example is given to demonstrate that S-orthodox transversals are proper generalizations of left simplistic orthodox transversals. A structure theorem for regular semigroups with S-orthodox transversals is established. As applications of this structure theorem, the structures of regular semigroups with quasi-ideal orthodox transversals and inverse transversals respectively are acquired.In Chapter 4, relationship between the adequate transversals of an abundant semi-group is explored. It is proved that if an abundant semigroup S has a multiplicative adequate transversal, then every quasi-ideal adequate transversal of S is multiplica-tive. Weakly multiplicative adequate transversals are introduced and explored. The connection of an adequate transversal between the properties of being multiplicative, weakly multiplicative and a quasi-ideal is obtained. The product of quasi-ideal ade-quate transversals of an abundant semigroup is investigated.In Chapter 5, abundant semigroups with S-adequate transversals are studied. The structures of abundant semigroups with quasi-ideal S-adequate transversals and S-adequate transversals respectively are established by means of two methods In Chapter 6, abundant semigroups with multiplicative quasi-adequate trans sals are studied. Some properties of abundant semigroups with multiplicative q(?) adequate transversals are characterized. A new structure theorem for this clas semigroups is also established.