Structure theory of generalized regular semigroups

In this thesis, we study the structure theory of generalized regular semigroups, including quasiregular semigroups, abundant semigroups and rpp semigroups.; The first part of the thesis is divided into two chapters. In Chapter 1, we obtain a structure theorem for left Clifford semigroups by using left quasi-direct product of left regular bands and Clifford semigroups. The concept of left quasi-direct product developed in this chapter is a new technique in studying structure theory of semigroups. In Chapter 2, we provide a method of construction for generalized orthogroups and as a consequence, the structure theorem of Petrich for orthogroups follows as an immediate corollary of our theorem on generalized orthogroups.; The second part of the thesis is composed of Chapters 3 to 6. The main purpose of these chapters is to study abundant semigroups. In Chapter 3, we show that an L* -inverse semigroup can be described as a left cohort product of a type A semigroup Gamma and a left regular band B with respect to a left cohort mapping. This result generalizes the structure theorem of Yamada for left inverse semigroups. In Chapter 4, we establish the structure theorem for quasi*-inverse semigroups by using the sandwich cohort product of semigroups. In particular, we prove that a semigroup is a quasi*-inverse semigroup if and only if it is a spined product of an L* -inverse semigroup and an R* -inverse semigroup. In Chapter 5 we show that a superabundant semigroup can be represented by a semilattice of normalized Rees matrix semigroups over some cancellative monoids. This last result extends the well known result of Petrich on completely regular semigroups.; In the third part of the thesis, we concentrate on the structure of rpp semigroups. In Chapter 7, a structure theorem for right C-rpp semigroups is given. We show in Chapter 8 that a semigroup S is an rpp semigroup with left central idempotents if and only if S is a strong semilattice of left cancellative right stripes.