Half Ring Green-relations And Minimal Intended Ideal

In this thesis, we discussed the connection of the Green's relation between Semirings and it's multiplicative semigroups, presented a characteristic of (L|·) -class which reduced to only one element of the semiring. Additionally we studied the structure of the Green's-L relation, and give a sufficient condition that guarantee the equation L = (L|·) is tenable, at the same time we investigate the semirings which satisfies the equation above. At the last part, based on the conclusions we have already studied, we discussed the Minimal Quasi-ideals of semirings assisted with the Green relations in semirings.This dissertation has been divided into five parts. In the first chapter, we introduced the background information, such as the history and expectation of semiring theory, and the most concernment of our article about and the primary conclusions we get. In the second chapter, we give the basic notions and results of semirings, which serves as the basis of our discussion. The third chapter is mainly about the Green's relation in semirings, and some definitions and properties are given. All the concept introduced in this chapter are the foundation stone of our future investigation. The fourth and fifth chapters are of the most signification, for these two chapters contain all the important results we studied. In the fourth chapter, we characterized the (L|·) -class which reduced to only one element, and give a more precise description about the structure of the L -class of semiring. In the last chapter, Using the results we have get above, we discussed the Minimal Quasi-ideals of semrings, and build a very strong connection between Green's-H-relation and Minimal quasi-ideal.