ISSN:
1572-8730
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Philosophy
Notes:
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 $$LD = \mathop \cap \limits_{n \in N} E(\mathfrak{D}_n^* )$$ .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00713551
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