A CRITERION OF FUNCTIONAL COMPLETENESS FOR B
Виктор Константинович Финн · 2015
The study of the class of the functions B corresponding to D. A. Bochvar’s three-valued logic [1] is subject of the present paper. In [2] V. I. Shestakov noticed that B can be embedded in the class of the functions corresponding to Lukasiewicz logic L3. In [3], [4] the author examined normal forms for the functions belonging to B and the axiomatized algebra corresponding to B. We make use here of some results and symbolism from [3]. Following A. V. Kuznecov we consider the closure operation [ ] determined on the subsets of the set B. Let K ⊆ B. Then we call [K] the closure of the set K. [K] comprises all superpositions ([5]) of functions belonging to K. The set of functions K is called closed if [K] = K; the set of functions K ⊆ B is called pre-complete in B provided that [K] 6= B and for any function F ∈ B such that F 6∈ K, [K ∪ {F}] = B; a set K is said to be functionally complete in B if [K] = B. Let 0, 1, 2 be the logical values of the logic B3 (0 =falsehood). By ∼ x1, x1 · ∩x2, x1 · ∪x2 we denote the functions called: internal negation, internal conjunction, and internal disjunction, respectively [1,3]. Their truth-tables are as follows: