×

zbMATH — the first resource for mathematics

Strong constructibility of Boolean algebras of elementary characteristic (1,1,0). (English. Russian original) Zbl 0824.03018
Algebra Logic 32, No. 6, 334-341 (1993); translation from Algebra Logika 32, No. 6, 618-630 (1993).
The paper contains a step in the solution of the problem formulated by S. P. Odintsov [“Restricted theories of constructive Boolean algebras in the lower layer”, Inst. Mat., Novosibirsk, Prepr. No. 12 (1986)]: does there exist a constructive but not strongly constructivizable Boolean algebra of the Ershov-Tarski characteristic (1,1,0) admitting a constructivization with a recursive set of atoms? The result is the following Theorem. If a constructive Boolean algebra \(\langle B, \nu\rangle\) of the Ershov-Tarski characteristic (1,1,0) has a recursive set of atoms and is effectively presented as a direct sum of constructive Boolean algebras \(\langle B_ i, \nu_ i\rangle\) with the first characteristic 0, then \(B\) is strongly constructivizable. To solve this problem completely, now it would be good to find some effective representation for any constructive Boolean algebra with recursive set of atoms.

MSC:
03D45 Theory of numerations, effectively presented structures
06E99 Boolean algebras (Boolean rings)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Yu. L. Ershov, ”Constructive models,” in:Selected Problems in Algebra and Logic, III [in Russian], Nauka, Novosibirsk (1973).
[2] N. G. Khisamiev, ”Strongly constructive models of decidable theories,”Izv. Akad. Nauk KazSSR, Ser. Fiz.-Mat., 83–84 (1974).
[3] Yu. L. Ershov,Decidability Problems and Constructive Models [in Russian], Nauka, Moscow (1980).
[4] S. S. Goncharov,Countable Boolean Algebras [in Russian], Nauka, Novosibirsk (1988). · Zbl 0667.03024
[5] S. S. Goncharov, ”Some properties of constructivizations of Boolean algebras,”Sib. Mat. Zh.,16, No. 2, 264–278 (1975).
[6] S. S. Goncharov, ”Restricted theories of constructive Boolean algebras,”Sib. Mat. Zh.,17, No. 4, 797–812 (1976). · Zbl 0361.02066
[7] S. P. Odintsov,Restrictions in the Theory of Constructive Boolean Algebras from the Lower Layer, Institute of Mathematics, Novosibirsk, Preprint No. 12 (1986).
[8] J. B. Remmel, ”Recursive isomorphism types of recursive Boolean algebras,”J. Symb. Logic,46, No. 3, 572–594 (1981). · Zbl 0543.03031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.