Representations of superassociative algebras by commutative functions with different types

For fixed integers [Formula: see text] and [Formula: see text], we study the faithful representation of superassociative algebras by two kinds of functions of the arity [Formula: see text], the so called [Formula: see text]-commutative and [Formula: see text]-commutative. The automorphism on the algebra of all [Formula: see text]-commutative functions is determined. In general, an isomorphism from a superassociative system which consists of a family of nonempty sets and a family of operations satisfying the axiom of superassociativity to a system of full multiplace functions is discussed. Particularly, we also provide necessary and sufficient conditions under which a superassociative system and a system of [Formula: see text]-commutative functions are isomorphic. With these results, a close connection with the theory of terms and tree languages is described.