Semigroups with strong and nonstrong magnifying elements (Q1815357)

An element \(a\) of a semigroup \(S\) is a left (right) magnifying element if \(aM=S\) (\(Ma=S\)) for some proper subset \(M\) of \(S\). It is a strong left (right) magnifying element if \(aT=S\) (\(Ta=S\)) for some proper subsemigroup \(T\) of \(S\). In [Semigroup Forum 48, No. 1, 119-126 (1994; Zbl 0805.20050)], the reviewer reiterated the observation of \textit{F. Catino} and \textit{F. Migliorini} [ibid. 44, No. 3, 314-319 (1992; Zbl 0746.20035)] that no one to that time had produced a semigroup which contains both strong and nonstrong left magnifying elements. Well, now someone has. Let \(LM(S)\) and \(\overline{LM}(S)\) denote the collections of left magnifying elements and strong left magnifying elements, respectively, of a semigroup \(S\). The author proves that if \(S\) and \(T\) are two semigroups such that (1) Neither \(S\) nor \(T\) has a left identity and (2) \(LM(S)\neq\emptyset\), \(\overline{LM}(S)=\emptyset\), and \(\text{LM}(T)\neq\emptyset\), then \(S\times T\) contains both strong and nonstrong left magnifying elements. He applies this result to the bicyclic semigroup \(\mathcal T\) and the Baer-Levi semigroup \(\mathcal B\) to show that \({\mathcal T}\times{\mathcal B}\) contains both strong and nonstrong left magnifying elements.