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A hypersubstitution is a map which takes n-ary operation symbols to n-ary terms. Any such map can be uniquely extended to a map defined on the set W-tau(X) of all terms of type tau, and any two such extensions can be composed in a natural way. Thus, the set Hyp(tau) of all hypersubstitutions of type tau forms a monoid. In this paper, we characterize Green's relation R on the monoid Hyp(tau) for the type tau = (n, n). In this case, the monoid of all hypersubstitutions is isomorphic with the monoid of all Clone endomorphisms. The results can be applied to mutually derived varieties
Hypersubstitutions were introduced in [3] as a way of making precise the concepts of hyperidentity and M- hyperidentity. The monoid of hypersubstitutions has been widely studied by many authors. Knowledge of the monoid of hypersubstitutions can be applied to the concept of M-hyperidentities. In this paper, we show that the order of hypersubstitutions of type tau = (3) is 1, 2, 3 or infinite