Bulletin of the
Korean Mathematical Society BKMS

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 $(S,\cdot,\le)$ having a kernel $I$ is said to be simple if $I$ is properly contained in $L$ and for any left ideal $L'$ of $(S,\cdot,\le)$, $I$ 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.

Keywords: semigroup, simple ordered semigroup, kernel, simple ideal, ideal, annihilator, radical, $I$-potent

MSC numbers: Primary 06F05