Representations of superassociative algebras by commutative functions with different types

For fixed integers 1 <= i < j <= n and sigma is an element of S-n, we study the faithful representation of superassociative algebras by two kinds of functions of the arity n, the so called (i, j)-commutative and sigma-commutative. The automorphism on the algebra of all (i, j) 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 sigma-commutative functions are isomorphic. With these results, a close connection with the theory of terms and tree languages is described.