On Ideals Of Ordered Semigroups

In this thesis, some important ideals of ordered semigroups have been discussed,at the same time, a few new results have been gained. we divide the thesis into three chapters.The first chapter is introduction. In this chapter,we have expounded that the research of ordered algebras in science is important and necessary, we also introduced the subjects connecting with our work.In chapter two , we first use the concept of m—system (n—system), some characterizations of weakly prime ideals of ordered semigroups are obtained by m- system and those of weakly semiprime ideals by n-system,some interesting properties of prime ideals, weakly prime ideals of ordered semigroups are given. At last,the concept of strong reversible ordered semigroups are introduced, some characterization of (semi) primary ordered semigroups and (semi) primary ordered semigroups of the class of strong reversible ordered semigroups are given, and some characterization of strong reversible ordered semigroups whose all ideals are prime ideals are given too. In chapter three, submaximal ideals and subminimal ideals of ordered semigroups are introduced, and their fundamental properties are studied. The existance of submaximal (subminimal) ideal is proved. We obtained that every ideal of ordered semigroups can be decomposed as intersection (union) of submaximal (subminimal) ideals. Specially, in the semigroup with descending (ascending) chain condition of ideals, every ideal can be decomposed as finite intersection (union) of submaximal (subminimal) ideals, at last, based on the bi-ideals and bi-ideal elements, two important conditions on regular poe-semigroups are given.