A semigroup S (with zero) is called right resp. left (0-) quasiresiduated with respect to its natural partial order ≤S if for any a,b∈S (a,b≠0) there exists x∈S (x≠0) resp. y∈S (y≠0) such that ax≤Sb resp. ya≤Sb. 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.