The decidability of some classes of Stone algebras

Iterating the triple construction applied consecutively to n Boolean algebras, we introduce two finitely axiomatizable subclasses SA(n)(i) and SA(n)(s) of the class SA n of all Stone algebras of degree n with all the structure homomorphisms in their P-product representation injective or surjective, respectively. Then the class of all Post algebras of degree n is definitionally equivalent to the intersection SA(n)(i) boolean AND SA(n)(s). We show that for each n >= 2 the class SA(n)(i) is hereditarily undecidable while SA(n)(s) is decidable. As a consequence we obtain several (un) decidability results for various axiomatic classes of Stone algebras: among them the decidability of the class of all Stone algebras of degree n which are dually pseudocomplemented and form a dual Stone algebra under the operation of dual pseudocomplement, and undecidability of the class of all Stone algebras with Boolean dense set. On the other hand, the class of all finite members in SA(n) is decidable.