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Undecidability of theories of Boolean algebras with selected ideals. (English. Russian original) Zbl 0628.03004
Algebra Logic 25, 206-219 (1986); translation from Algebra Logika 25, No. 3, 326-346 (1986).
Main result: Let (A,I) be a countable Boolean algebra with a selected ideal I. Then the theory of (A,I) is decidable iff A is superatomic. To prove necessity, it is shown how to obtain in an atomless B, for any $$k\in \omega$$ a formula $$\psi_ k(z)$$, in the language of Boolean algebras with a selected ideal, and for any (infinite) set $$A\subseteq \omega$$ an ideal $$I_ A$$ of B, such that $$(B,I_ A)\vDash \exists z \psi_ k(z)$$ iff $$k\in A.$$
As to sufficiency, it is shown that the theory of (A,I), when A is superatomic, is (recursively) axiomatizable. The main tool is the construction of three mappings $$r_ 1,r_ 2,r_ 3$$ from A into $$\omega\cup \{\infty \}$$ with the properties:
i) For $$x\in A$$, $$x\in I$$ iff $$r_ 3(x)\leq 1$$; ii) for $$x\in A$$, there are x’, x” such that $$x=x'\vee x''$$, $$x'\wedge x''=0$$ and $$r_ 2(x')=r_ 2(x'')=0$$; iii) each statement $$r_ j(\ell_ A)=n_ j''$$ $$(j=1,2,3)$$ is equivalent to a r.e. set of formulas.
It is shown that if (A,I) and (A’,I’) are both atomic and satisfy i) and ii), then they are elementary equivalent iff $$r_ j(\ell_ A)=r_ j(\ell_{A'})$$ for $$j=1,2,3$$.
Reviewer: A.Ursini

##### MSC:
 03B25 Decidability of theories and sets of sentences
Full Text:
##### References:
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