On ordered quasi-Gamma-ideals of regular ordered Gamma-semigroups. (Q895944)

Summary: We introduce the notion of ordered quasi-\(\Gamma\)-ideals of regular ordered \(\Gamma\)-semigroups and study the basic properties of ordered quasi-\(\Gamma\)-ideals of ordered \(\Gamma\)-semigroups. We also characterize regular ordered \(\Gamma\)-semigroups in terms of their ordered quasi-\(\Gamma\)-ideals, ordered right \(\Gamma\)-ideals, and left \(\Gamma\)-ideals. Finally, we have shown that (i) a partially ordered \(\Gamma\)-semigroup \(S\) is regular if and only if for every ordered bi-\(\Gamma\)-ideal \(B\), every ordered \(\Gamma\)-ideal \(I\), and every ordered quasi-\(\Gamma\)-ideal \(Q\), we have \(B\cap I\cap Q\subseteq(B\Gamma I\Gamma Q]\) and (ii) a partially ordered \(\Gamma\)-semigroup \(S\) is regular if and only if for every ordered quasi-\(\Gamma\)-ideal \(Q\), every ordered left \(\Gamma\)-ideal \(L\), and every ordered right-\(\Gamma\)-ideal \(R\), we have that \(R\cap Q\cap L\subseteq(R\Gamma Q\Gamma L]\).