On left weakly prime, left weakly semiprime ideals in ordered semigroups (Q1033693)

In this paper the author gives some results based on the following definitions: Let \((S,\cdot ,\leq)\) be an ordered semigroup. \(S\) is called \textit{left semiregular} if for every \(a\in S\) there exist \(x,y\in S\) such that \(a\leq xaya\) [\textit{S. K. Lee} and \textit{Y. I. Kwon}, ``On left, right weakly prime ideals on po-semigroups'', Commun. Korean Math. Soc. 11, No. 2, 315--321 (1996; Zbl 0943.06008)] (that is, \(a\in (SaSa]\) for every \(a\in S\), equivalently \(A\subseteq (SASA]\) for every \(A\subseteq S\)). A subset \(T\) of \(S\) is called \textit{left weakly prime} if for any left ideals \(A,B\) of \(S\) such that \(AB\subseteq T\), we have \(A\subseteq T\) or \(B\subseteq T\). A subset \(T\) of \(S\) is called \textit{left weakly semiprime} if for every left ideal \(A\) of \(S\) such that \(A^2\subseteq T\), we have \(A\subseteq T\). A subset \(T\) of \(S\) is called \textit{left strongly prime} if for any left ideals \(A,B\) of \(S\) such that \(AB\cap BA\subseteq T\), we have \(A\subseteq T\) or \(B\subseteq T\). A left ideal \(L\) of \(S\) is called \textit{irreducible} (resp. \textit{strongly irreducible}) if for any left ideals \(A,B\) of \(S\) such that \(A\cap B=L\) (resp. \(A\cap B\subseteq L\)), we have \(A=L\) or \(B=L\) (resp. \(A\subseteq L\) of \(B\subseteq L\)). The author accepts and uses that the intersection of two left ideals is a left ideal referring to [\textit{N. Kehayopulu}, Math. Jap. 35, No. 6, 1051--1056 (1990; Zbl 0717.06006)] while no such result is given there. The intersection of two left ideals is not a left ideal, in general, unless their intersection is nonempty.