Classifications and Enumeration of Bases in P_{3}(2)

Let E_k := {0,1,. .. , k - 1}, k ges 2, and let P_k be the set of all k-valued logical functions, i.e., maps f : E_k ⁿ rarr E_k for n = 1,2,.... Denote by P_k (2) the set of all functions of P_k whose range contains no more than two elements. The set P_k(2) is a class (i.e., a closed set with respect to the usual superposition operations) and all maximal subclasses of P_k(2) are known. In this paper we study the characteristic vectors for each f isin P_k(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 E_k- 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.