Green's Relation ℛ on the Monoid of Clone Endomorphisms

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 Wτ(X) of all terms of type τ, and any two such extensions can be composed in a natural way. Thus, the set Hyp (τ) of all hypersubstitutions of type τ forms a monoid. In this paper, we characterize Green's relation ℛ on the monoid Hyp (τ) for the type τ=(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.