Finite axiomatization for some intermediate logics

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:

$$LD = Cn(INT \cup \{ p_3 \vee [p_3 \to (p_1 \to p_2 ) \vee (p_2 \to p_1 )]\} )$$

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$LD = \mathop \cap \limits_{n \in N} E(\mathfrak{D}_n^* )$.