ON SIMPLE LEFT, RIGHT AND TWO-SIDED IDEALS OF AN ORDERED SEMIGROUP HAVING A KERNEL

Abstract

Abstract. The intersection of all two-sided ideals of an ordered semi-group, if it is non-empty, is called the kernel of the ordered semigroup. Aleft ideal Lof an ordered semigroup (S,·,≤) having a kernel Iis said to besimple if I is properly contained in Land for any left ideal L ′ of (S,·,≤),I is properly contained in L ′ and L ′ is contained in Limply L ′ = L. Thenotions of simple right and two-sided ideals are defined similarly. In thispaper, the author characterize when an ordered semigroup having a ker-nel is the class sum of its simple left, right and two-sided ideals. Further,the structure of simple two-sided ideals will be discussed. 1. IntroductionThe intersection of all two-sided ideals of a semigroup, if it is non-empty, iscalled the kernel [5] of the semigroup.Let S be a semigrouphavinga kernelI. A left ideal L of S is said to be simpleif I ⊂ L, (The symbol ⊂ stands for proper inclusion of sets.) and for any leftideal L ′ of S, if I ⊂ L ′ ⊆ L, then L ′ = L. The notions of simple right and two-sided ideals are defined similarly. These concepts were introduced and studiedby Schwarz [6]. Indeed, the author characterized when a semigroup having akernel is the class sum of its simple left, right or two-sided ideals. Further, thestructure of simple two-sided ideals of some semigroups was investigated.The purpose of this paper is to extend Schwarz’s results to ordered semi-groups. Section 2 recalls some certain definitions and results used throughoutthis paper. In Section 3, we introduce the concepts of simple left, right and two-sided ideals of an ordered semigroup having a kernel. Necessary and sufficientconditions for an ordered semigroup having a kernel to be the class sum of itssimple left, right or two-sided ideals are given (Theorems 3.9-3.11). Further,we obtain important corollaries using annihilators of an ordered semigroup.Sections 4-5 investigate the structure of simple two-sided ideals of an orderedsemigroup having a kernel.