×

Semigroups with strong and nonstrong magnifying elements. (English) Zbl 0859.20050

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 F. Catino and 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.

MSC:

20M10 General structure theory for semigroups
20M12 Ideal theory for semigroups
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Catino, F. and F. Migliorini,Magnifying elements n semigroups, Semigroup Forum44 (1992), 314–319. · Zbl 0746.20035 · doi:10.1007/BF02574350
[2] Ljapin, E. S., ”Semigroups,” Amer. Math. Soc., Providence, R.I. (1963).
[3] Magill, K. D. Jr.,Magnifying elements of transformation semigroups, Semigroup Forum48 (1994), 119–126. · Zbl 0805.20050 · doi:10.1007/BF02573659
[4] Tolo, K.,Factorizable semigroups, Pacific J. Math.,32 (1969), 523–535. · Zbl 0188.05401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.