On some Sufficient Condition for the Equality of Multi-clone and Super-clone

Abstract

Clones are studied most actively in the theory of functional systems [1]. Clones are sets of operations that are closed with respect to superposition, and they contain all projection operators. Recently interest in generalizations of clones, namely, hyperclones, multiclones and superclones has been raised [2]. Multi-clone is a set of multi-operations which are closed with respect to superposition, and it contains all complete, empty and projection operations. A super-clone is obtained from a multi-clone by adding the closure condition with respect to solvability of the simplest equation. It is known that super-clones are closely related to clones. Complete Galois connection between them was established [3]. Condition of the equality of multi-clone and super-clone is obtained in this paper. Let A be an arbitrary finite set, and B(A) be the set of all subsets of A including ∅. A mapping from A into A is described as an n-ary operation on A (the case n = 0 is possible). The set of all n-ary operations on A is described as P A, and the set of all operations on A is described as PA = ∪