CLASSIFICATION AND ENUMERATION OF BASES IN Pk(2)
Dietlinde Lau, Masahiro Miyakawa · Asian-European Journal of Mathematics · 2008
Let k ≥ 2, Ek := {0, 1,…, k – 1}, and let Pk denote the set of all k-valued logical functions, i.e., of maps [Formula: see text] for n = 1, 2, … We denote by Pk(2) the set of all functions of Pk whose range contains no more than two elements. The set Pk(2) is a closed set (or class) with respect to the usual superposition operations and all maximal closed subsets of Pk(2) are known. For each f ∈ Pk(2) we determine its characteristic vectors with respect to the maximal classes and give an explicit formula for the total number of the characteristic vectors in terms of the numbers of the equivalence relations on Ek. Then we show that P3(2) has exactly 75 characteristic vectors and 33,678 classes of bases, and show that the cardinality of an arbitrary basis for P3(2) is 3, 4 or 5.