The Full H-ideals Of An Additively Inverse Semirings In Which Addition Is Commutative And The Left Ideals Of Some Ordered Semigroups

Ideals have an important position in the algebra. One side, they are one of the research tools of the algebra structure, and have a close relations with congruence and Green's relation. Another side, ideals are also an algebra system, having themselves special property. Based on above, firstly, this thesis studys the property of full h -ideals of additively inverse semirings in which addition is commutative, presents the characterization of additively inverse semirings in which addition is commutative assiste with full h-ideals, then obtains the corresponding ring congruence. Secondly, we discuss the property, the relations and the structures of ideals on the nil ordered semigroups and the related conclusion between Green's relation, congruence and ideals.There are four chapters in this paper. In chapter I, we mainly introduce the background informations, research restults of semirings, ordered semigroups and ideals, additionly, the primary restults we get and the meaning of this article. The chapter II plays the basis role in this paper, we import the concept and lemma about ordered, semigroups, semirings, ordered semigroups, congruence, ideals, Green's-relation. In chapter III, we investigate the property of h -ideals and its generation. Next we get the algebra structure of full h -ideals sets, depict the structure of additively inverse semirings in which addition is commutative because full h -ideals and rings congruence can be shown on additively inverse semirings in which addition is commutative each other. In the last chapter, when ideal elements and left ideals combine with prime, semiprime and weakly prime, we give the new characters and the relations between them on the specific ordered semigroups, by the way, extend the construction of intra-regular and illustrate some ordered semigroups' decompositions. Finally, under the assistance of principal left ideals, there are a series of interesting results about L -trivial ordered semigroups, divisibility ordering, completely congruence and Rees congruence on nil ordered semigroups.