Classifications and Enumeration of Bases in P_{3}(2)
Dietlinde Lau, Masahiro Miyakawa · Proceedings/Proceedings - International Symposium on Multiple-Valued Logic · 2007
Let Ek:= {0,1,. .. , k - 1}, k ges 2, and let Pkbe the set of all k-valued logical functions, i.e., maps f : Eknrarr Ekfor n = 1,2,.... Denote by Pk(2) the set of all functions of Pkwhose range contains no more than two elements. The set Pk(2) is a class (i.e., a closed set with respect to the usual superposition operations) and all maximal subclasses of Pk(2) are known. In this paper we study the characteristic vectors for each f isin Pk(2) 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 equivalence classes of bases, and show that every basis of P3(2) consists of either 3, 4 or 5 functions.