POWER CLONES AND NON-DETERMINISTIC HYPERSUBSTITUTIONS

Hypersubstitutions map operation symbols to terms of the corresponding arity. Any hypersubstitution can be extended to a mapping defined on the set Wτ(X) of all terms of type τ. If σ : {fi | i ∈ I} → Wτ(X) is a hypersubstitution and [Formula: see text] its canonical extension, then the set [Formula: see text] is a tree transformation where the original language and the image language are of the same type. Tree transformations of the type Tσ can be produced by tree transducers. Here Tσ is the graph of the function [Formula: see text]. Since the set of all hypersubstitutions of type r forms a semigroup with respect to the multiplication [Formula: see text], semigroup properties influence the properties of tree transformations of the form Tσ. For instance, if σ is idempotent, the relation Tσ is transitive (see [1]). Non-deterministic tree transducers produce tree transformations which are not graphs of some functions. If such tree transformations have the form Tσ, then σ is no longer a function. Therefore, there is some interest to study non-deterministic hypersubstitutions. That means, there are operation symbols which have not only one term of the corresponding arity as image, but a set of such terms. To define the extensions of non-deterministic hypersubstitutions, we have to extent the superposition operations for terms to a superposition defined on sets of terms. Let [Formula: see text] be the power set of the set of all n-ary terms of type τ. Then we define a superposition operation [Formula: see text] and get a heterogeneous algebra [Formula: see text] (ℕ+ is the set of all positive natural numbers), which is called the power clone of type τ. We prove that the algebra [Formula: see text] satisfies the well-known clone axioms (C1), (C2), (C3), where (C1) is the superassociative law (see e.g. [5], [4]). It turns out that the extensions of non-deterministic hypersubstitutions are precisely those endomorphisms of the heterogeneous algebra [Formula: see text] which preserve unions of families of sets. As a consequence, to study tree transformations of the form Tσ, where σ is a non-deterministic hypersubstitution, one can use the structural properties of non-deterministic hypersubstitutions. Sets of terms of type τ are tree languages in the sense of [3] and the operations [Formula: see text] are operations on tree languages. In [3] also another kind of superposition of tree languages is introduced which generalizes the usual complex product of subsets of the universe of a semigroup. We show that the extensions of non-deterministic hypersubstitutions are not endomorphisms with respect to this kind of superposition.