On -Algebra and -Polynomial Relations

The note deals with categories whose objects are functions from sets to -algebras and morphisms are -homomorphisms of -algebras making appropriate diagrams commutative. In the theory of universal -algebras, such categories satisfying certain additional axioms are called -relations. Those -relations that determine universal -algebras are said to be compact. In this note, we construct functors between compact -relations. These functors arise from -homomorphisms between universal -algebras which are determined by compact -relations. In the case when a functor is defined by an isomorphism of the universal -algebras, we show that this functor is an isomorphism of compact -relations. Moreover, we consider -relations which are called -polynomial relations associated with -polynomial pairs. It is shown that every -algebra is the universal -algebra generated by a -polynomial pair. As a consequence, we obtain that every compact -relation is isomorphic to a -polynomial relation.