Let*N.* be the set of all natural numbers (except zero), and let*D*_{n}^{*}
= {*k* ∈*N* ∶*k*|*n*} ∪ {0} where*k¦n* if and only if*n=k.x* f or some*x∈N.* Then, an ordered set*D*_{n}^{*}
= 〈*D*_{n}^{*}
, ⩽_{n}, where*x⩽*_{n}y iff*x¦y* for any*x, 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*}, where*B* =,{*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 set*H* = 〈*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 model*H*_{B} = 〈*P*(ℬ), ⊂〉 (where*P*(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 be*H**D*_{n}^{*}
= 〈*P* (*D*_{n}^{*}
), ⊂〉for any*n∈N*. Finally, it is proved that for any strongly compact pseudo-boolean algebra*U* satisfying the axiom*p*_{3}∨ [*p*_{3}→(p_{1}→p_{2})∨(p_{2}→p_{1})] there is a structure of divisors*D*_{*}^{n}
such that it is possible to define a strong homomorphism froomi*H**D*_{n}^{*}
onto*H**D*_{U}.

Exploiting, among others, this property, it turns out to be relatively easy to show that
$$LD = \mathop \cap \limits_{n \in N} E(\mathfrak{D}_n^* )$$
.