Inductive * - Half Ring And Related Semi-ring

In this thesis, we define the notions of μ-semirings, A-semirings, *-μ-semirings and *-λ-semirings which extend the notions of inductive *-semirings. we also study the basic properties of these semirings and discuss the interrelation between them. Also, we discuss whether these semirings are closed for the construction of formal power series via a fixed point-theoretical viewpoint.This dissertation can be divided into five parts. The first chapter is an introduction to the history of semiring theory. we emphasize that the thesis is mainly concern with several important semirings arising in theoretical computer science. The second chapter, which serves as the basis of our discussion, gives the basic notions and results of semirings, posets and fixed points. The third chapter introduces the notion of inductive *-semirings and weak inductive *-semirings. We give some basic properties of them and discuss the formal power series semirings. In chapter 4, we define the notions of μ-semirings and λ-semirings and also study the primary properties of them. Moreover, we discuss the interrelation between them and the formal power series semirings of them. In chapter 5, we define the notions of *-μ-semirings as well as *-λ-semirings and also study the primary properties of them. We discuss the interrelation between *-μ-semirings and *-λ-semirings, and their relation with those semirings introduced above. In the end, we study the formal power series semirings of *-λ-semirings as well as weak inductive *-semirings.