On Some Issues Of The Ordered Semigroups

The theory of congruences and various ideals play an increasing important role in the evo-lution of the algebraic theory of ordered semigroups. Most of time, studying these congruences and ideals, we obtained the inner structures of the ordered semigroups.In this paper, the concepts of inverse strongly regular congruences. C-ideals and prime bi-ideals of ordered semigroups are introduced, and some theorems are given to characterize the structures of the ordered semigroups. Moreover, we introduce the concepts of f-prime ideals and f-prime radicals and extend the related results of semirings and semigroups theory to ordered semigroups.In Chapter1, we introduce the background and some basic concepts.In Chapter2, on the basic of the theory of congruences on ordered semigroups and the concept of pseudoorders, we introduce the concepts of inverse quasi-chains and inverse strongly regular congruences on ordered semigroups, and give some theorems to characterize regular congruences and inverse strongly regular congruences.In Chapter3, we introduce the concept of C-ideals of ordered semigroups and study some elementary properties of C-ideals. Furthermore, we give the conditions of the existence of the greatest C-ideals of an ordered semigroup.In Chapter4, we introduce the concepts of prime、strongly prime、semiprime、irreducible、 strongly irreducible bi-ideals of ordered semigroups. Moreover, the ordered semigroups in which each bi-ideal is strongly prime and the ordered semigroups in which each bi-ideal is strongly irreducible are characterized respectively.In Chapter5, we introduce the concepts of f-prime ideals、f-semiprime ideals and f-prime radicals and extend the theory of f-primary decomposition in semirings to ordered semi-groups.