Algebras of functions on mappings

Topological algebras of functions on mappings are defined and investigated. It is proved that each algebra satisfying certain conditions (which are necessary and sufficient) is topologically (and isometrically with respect to its semi-norms) isomorphic to a subalgebra of an algebra of functions on some mapping. It is interesting to note that a completely regular space does not define a topology of its algebra of continuous functions uniquely if this space contains an infinite compact subspace, while a mapping do this, of course, amongst topologies of the definite kind. It is possible to define a new conception (connected with mappings) of a completeness of algebras and to prove some usual properties of complete algebras.