Abstract
LetN. be the set of all natural numbers (except zero), and letD * n = {k ∈N ∶k|n} ∪ {0} wherek¦n if and only ifn=k.x f or somex∈N. Then, an ordered setD * n = 〈D * n , ⩽ n , wherex⩽ ny iffx¦y for anyx, y∈D *n , can easily be seen to be a pseudo-boolean algebra.
In [5], V.A. Jankov has proved that the class of algebras {D * n ∶n∈B}, whereB =,{k ∈N∶ ⌉\(\mathop \exists \limits_{n \in N} \) (n > 1 ≧n 2 k)is finitely axiomatizable.
The present paper aims at showing that the class of all algebras {D * n ∶n∈B} is also finitely axiomatizable.
First, we prove that an intermediate logic defined as follows:
finitely approximatizable. Then, defining, after Kripke, a model as a non-empty ordered setH = 〈K, ⩽〉, and making use of the set of formulas true in this model, we show that any finite strongly compact pseudo-boolean algebra ℬ is identical with. the set of formulas true in the Kripke modelH B = 〈P(ℬ), ⊂〉 (whereP(ℬ) stands for the family of all prime filters in the algebra ℬ). Furthermore, the concept of a structure of divisors is defined, and the structure is shown to beH D * n = 〈P (D * n ), ⊂〉for anyn∈N. Finally, it is proved that for any strongly compact pseudo-boolean algebraU satisfying the axiomp 3∨ [p 3→(p1→p2)∨(p2→p1)] there is a structure of divisorsD n * such that it is possible to define a strong homomorphism froomiH D * n ontoH D U .
Exploiting, among others, this property, it turns out to be relatively easy to show that\(LD = \mathop \cap \limits_{n \in N} E(\mathfrak{D}_n^* )\).
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I am indebted to Professors T. Prucnal and A. Wroński for their valuable remarks concerning the paper.
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Janioka-Żuk, I. Finite axiomatization for some intermediate logics. Stud Logica 39, 415–423 (1980). https://doi.org/10.1007/BF00713551
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DOI: https://doi.org/10.1007/BF00713551