Semigroups with good and bad magnifiers

An element a of a semigroup S is called a (left) magnifier if there exists a proper subset M of S such that aM=S. If there is a minimal subset M with this property that is a right ideal of S, then the magnifier a is called very good. If such a minimal subset is a subsemigroup of S, then a is called good. Otherwise, it is called bad. It is well known that if a semigroup S has a very good magnifier, then all magnifiers in S are very good. A long-standing open problem is whether there exist semigroups having both good and bad magnifiers. In this paper we answer this question in positive and prove several results concerning such semigroups.