On The Structures Of Cryptogroups And Orthogroups

This Ph.D dissertation consists of three chapters. We study properties and structures of cryptogroups and orthogroups.Some definitions and properties for completely regular semigroups, cryptogroups, orthogroups and inverse transversals are proposed in Chapter 1. And then, we fix some notations which are used later.Chapter 2 is composed of two sections. By the virtue of inverse transversals, we give constructions of cryptogroups and orthogroups with inverse transversals respectively. Cryptogroups with inverse transversals are constructed in Section 1: One construction is given by means of a band with semilattice transversals and a Clifford semigroup in the first part; the other one is present by means of a band with semilattice transversals and a family of homomorphisms of completely simple semigroups in the second part. Orthogroups with inverse transversals are constructed by means of bands with semilattice transversals and Clifford semigroups in Section 2.Chapter 3 is devoted to construct cryptogroups and orthogroups. We give a construction theorem of cryptogroups in terms of a band and a family of homomorphisms between completely simple semigroups in Section 1, which is a generalization of the strong semilattice structure theorem of normal cryptogroups, and in Section 2, orthogroups are constructed by means of bands and Clifford semigroups.