Classification of P<inf>k2</inf>

Abstract

The set of functions of Pk2 (mapping the set {0,1,...,k-1}n into {0,1}, n = 1,2,...) is divided into equivalence classes so that two functions are in the same class if their membership in the maximal subclones of Pk2 coincides. This also leads to a natural classification of the set of bases (i.e. irredundant complete subsets) of Pk2. We determine all nonempty classes of functions of Pk2 and show that their number is 13B(k) - 11B(k - 1), where B(k) is the number of equivalence relations on the set of k elements (Bell's number). The maximal number of elements in a base of Pk2 is proved to be k + 2. Computational results for the numbers of classes of bases are also presented for k=3 and k=4. © 1989.