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ON VARIETIES GENERATED BY BOOLEAN CLONES

Clones are collections of functions defined on the same set A which are closed under superposition and contain all projections. One of the applications of clones is the composition of switching circuits. The combination of switching circuits to logical networks can be regarded as a technical realization of the superposition of Boolean functions, i.e. functions defined on the set {0, 1}. In [5] A. I. Mal’cev described clones as carriers of algebras of the type (2, 1, 1, 1, 0). W. Taylor ([6]) showed that the identities of such clone algebras (for short also clones) correspond to so-called hyperidentities. Hyperidentities are special formulas in a second order language. To the differences between clones of Boolean functions and clones of functions defined on sets with more than two elements belongs the following fact ([2]): Every clone of Boolean functions is (up to isomorphisms) uniquely determined by the sets of its hyperidentities. In this sense, Boolean clones can be “separated” by hyperidentities. This is not longer true for clones of functions defined on sets with more than two elements. There are non-isomorphic clones of functions defined on the same set A with |A| > 2 satisfying the same hyperidentities. Using these results we characterize that part of a variety generated by a clone of Boolean functions which consists only of clones of Boolean functions and describe all homomorphic images of clones of Boolean functions.