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

The intersection of all two-sided ideals of an ordered semigroup, if it is non-empty, is called the kernel of the ordered semigroup. A left ideal L of an ordered semigroup ($S,{\cdot},{\leq}$) having a kernel I is said to be simple if I is properly contained in L and for any left ideal L' of ($S,{\cdot},{\leq}$), I is properly contained in L' and L' is contained in L imply L' = L. The notions of simple right and two-sided ideals are defined similarly. In this paper, the author characterize when an ordered semigroup having a kernel is the class sum of its simple left, right and two-sided ideals. Further, the structure of simple two-sided ideals will be discussed.