A property of transformation semigroups

A semigroup S (with zero) is called right resp. left (0-) quasiresiduated with respect to its natural partial order a parts per thousand currency sign (S) if for any a,baS (a,b not equal 0) there exists xaS (x not equal 0) resp. yaS (y not equal 0) such that axa parts per thousand currency sign (S) b resp. yaa parts per thousand currency sign (S) b. It is shown that the most important semigroups of mappings are-at least for finite sets and finite-dimensional vector spaces-left and/or right (0-) quasiresiduated.