On the Ordered $n$-Prime Ideals in Ordered $\Gamma$-Semigroups
Commun. Korean Math. Soc. 2008 Vol. 23, No. 1, 19-27
Printed March 1, 2008
Manoj Siripitukdet and Aiyared Iampan
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
Abstract : The motivation mainly comes from the conditions of the (ordered) ideals to be prime or semiprime that are of importance and interest in (ordered) semigroups and in (ordered) $\Gamma$-semigroups. In 1981, Sen \cite{2} has introduced the concept of the $\Gamma$-semigroups. We can see that any semigroup can be considered as a $\Gamma$-semigroup. The concept of ordered ideal extensions in \pos s was introduced in 2007 by Siripitukdet and Iampan \cite{6}. Our purpose in this paper is to introduce the concepts of the \on s and the \ons s in \pos s and to characterize the relationship between the \on s and the \oi~extensions in \pos s.
Keywords : (ordered) semigroup, (ordered) $\Gamma$-semigroup, (ordered) ideal, \oi~extension, (ordered) prime ideal, (ordered) semiprime ideal, \on~and \ons
MSC numbers : 20M99, 06F99, 06B10
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