Magnifying elements of transformation semigroups (Q1313483)

An element \(a\) of a semigroup \(S\) is called left (right) magnifying if \(aM = S\) (\(Ma = S\)) for a proper subset \(M\) of \(S\). If \(M\) is a proper semigroup then \(a\) is a strong left (right) magnifying element. It is well known that any left (right) magnifying element in a semigroup \(S\) with a left (right) identity is strong. An element \(a\) of a semigroup \(S\) with an identity \(e\) is left magnifying if and only if there exists \(b \in S\) with \(ab = e\) and \(ba \neq e\). Other characterization conditions for the existence of a left magnifying element in a semigroup \(S\) with an identity are given. As a consequence we obtain that the monoid of linear transformations of a vector space \(V\) has a left magnifying element if and only if \(V\) has infinite dimension. An analogous characterization for the monoid of continuous maps of a topological space is given.