On complete ideals in ordered semigroups (Q2844297)

According to this paper, a left (resp. right) ideal \(A\) of an ordered semigroup \((S,\cdot,\leq)\) is said to be \textit{complete} if \((SA]=A\) (resp. \((AS]=A\)) and an ideal \(A\) of \(S\) is said to be \textit{complete} if \((SA]=(AS]=A\). This is the usual concept of \((m,n)\)-regularity: If \((S,\cdot,\leq)\) is an ordered semigroup and \(m\), \(n\) non-negative integers, then \(S\) is called \textit{\((m,n)\)-regular} if for every \(a\in S\) there exists \(x\in S\) such that \(a\leq a^mxa^n\) (\(a^0x=xa^0=a)\). The author first proves that the union of two complete left ideals of \(S\) is a complete left ideal of \(S\). Then he proves that if \(L_i\) is a complete left ideal of \(S_i\) for every \(i\in I\), then the direct product \(\prod\limits_{i \in I} {{L_i}} \) is a complete left ideal of the ordered semigroup \(\prod\limits_{i \in I} {{S_i}} \) and that the direct product \(\prod\limits_{i \in I} {{S_i}} \) of the ordered semigroups \(\{S_i \mid i\in I\}\) is \((m,n)\)-regular if and only if every \(S_i\) is so. Finally he shows that every left ideal of \(\prod\limits_{i \in I} {{S_i}} \) is complete if and only if \(\prod\limits_{i \in I} {{S_i}} \) is \((0,1)\)-regular.