Classification of three-valued logical functions preserving 0

(0, 1,2}. The set of three-valued logical functions (i.e., f: E” --t E for n=l,2,... ) is denoted by P3. A subset F of P3 is said to be closed if it contains all superpositions (i.e., compositions or substitutions) of its members (cf. [l, 1 l-131). For closed sets F and H such that FC H (proper inclusion), F is an H-maxima/ set if FC G c H for no closed set G. A subset F of H is complete in H if H is the least closed set containing F. We assume throughout that H has finitely many H-maximal sets and that each proper closed subset of H extends to a maximal one (or, equivalently, it is finitely generated, i.e., there is a finite F complete in H). Clearly a subset of H is complete in H if and only if it is not contained in any H-maximal set (completeness condition, cf. [l]). Completeness (also called functional completeness or primality) is directly related to universal algebra and to logical circuit design. A complete set Fin His called an H-base if no proper subset of F is complete in H. A subset F of His called pivotal in H, if for each function f EF there exists an H-maximal set A4 such that f $ M>F\{f}. F rom these definitions it follows that an H-base is a complete and pivotal set in H. The rank of a set is the number of its elements. Let HI,..., H, be all the H-maximal sets. For f;gE H put f =g if either both A g E Hi or both f,g $ Hi for all i = 1, . . . , m. The relation = is an equivalence rela-