On Yablonskii theory concerning functional completeness of $k$-valued logic.
Kanzo Hino · Notre Dame Journal of Formal Logic · 1977
In his paper, S. B. Yablonskii [l] proved a theorem concerning the functional completeness in ^-valued logic (see [l], p. 64).The theorem asserts that the system of functions consisting of constant k -2, ~x9 and x γ D x 2 is functionally complete in this logic.His proof is incomplete.In this paper, we shall give a simple proof of this theorem.Let P k be the set of all functions that are defined on the set {θ, 1, . .., k -1} and take their values on the same set.First, we shall give a lemma needed for the proof of the theorem.Lemma The system consisting of functions 0, 1, . .., k -1, max(i b x 2 ) 9 π\ (x l9 x 2 ) and j,-(x)(0 «£ i ^ k -1) defined by (k-l,ifx= i,