On the structural completeness of some pure implicational propositional calculi

1. The notion of the structural completeness of systems was introduced by W. A. Pogorzelski in [4]. The main result of this article is a theorem on structural completeness of Hubert's implicational propositional calculus. We also prove that Lewis's implicational cal? culi1 S5 and S3 are not structurally complete. 2. Let piqis,p1,p2, ... be an infinite sequence of various propositional variables. Let S stand for the least of the sets containing all propositional variables and closed with regard to connective ->. The variables ol, ?, y represent elements of the set S, and A, X, Y those of the set 2s. By At we denote the set of all propositional variables, by Ve the extension of the function e, transforming the set At into S, to the endo morphism of the algebra , and thus: Ve(p) = e (p), Ve(tx -> ?) = Vc(ol) -> -> Fe(?), for any a, ? e S, p e At. Moreover, we assume the symbols and definitions employed by W. A. Pogorzelski in [4], Let us now repeat Pogorzelski's definition: A system with primitive rules R and axioms A is structurally complete (in symbolic notation: e rnx = S = n A A (.Ve(n) F?(a) e X) , e: At -> 5