Semigroups with good and bad magnifiers. (Q1408759)

An element \(a\) of a semigroup \(S\) is a left (right) magnifier if there exists a proper subset \(M\) of \(S\) such that \(aM=S\) (\(Ma=S\)) and \(M\) is said to be associated with the element \(a\). If \(M\) is a minimal subset which is also a subsemigroup of \(S\), then \(a\) is said to be a good magnifier. Otherwise it is a bad magnifier. The authors show how to obtain every semigroup with a good magnifier \(e\) and a minimal semigroup associated with \(e\) from a semigroup \(S\) with a left identity \(e\) and a particular type of endomorphism of \(S\). A number of additional results about magnifiers are also obtained. They also settle a problem which had been open for quite some time. It was not known if a semigroup could contain both good and bad magnifiers. The authors show that the answer is, indeed, yes.