Semi-ring. Partial Order Partial Order Expansion

Ordered algebric theory is a very important branch of algebra. Many experts and scholars have investigated it thoroughly and painstakingly. In this paper, we mainly study extensions of partial order and finitely totally order for partially ordered semirings, and obtain some new results.This dissertation can be divided into three parts. The first chapter, we introduce the history of semigroups, semirings, partially ordered semigroups, partially ordered semirings. we emphasize that the thesis is mainly concerned with several important papers arising in extensions of partial order for ordered algebraic theory. Also, we introduce the basic notions and results of partially ordered semirings, pseudoorder, pseudochain, semi-lattices, lattices, isotonic homomorphism and homomorphism of them. The second chapter, we introduce the notions of semi-pseudoorderσand semi-pseudochain modσon the partially ordered semirings and give some basic properties of them. we also give a constructive method of extensions of partial order for partially ordered semirings. Besides, we get sufficient condition of extensions of partial order for partially ordered semirings by semi-pseudochain modσ, and obtain some interesting results. In chapter 3. we give the notion of extensions of finitely totally order for partially ordered semirings, Besides, we discuss sufficient and necessary conditions on extensions of finitely totally order for partially ordered semirings.