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Universal recursively enumerable Boolean algebras. (English. Russian original) Zbl 0552.03026
Sib. Math. J. 24, 852-858 (1983); translation from Sib. Mat. Zh. 24, No. 6(142), 36-43 (1983).
Let \({\mathfrak K}\) be a class of r.e. Boolean algebras. A r.e. Boolean algebra \({\mathfrak A}_{\nu}\) is called universal in the class \({\mathfrak K}\), if \({\mathfrak A}_{\nu}\in {\mathfrak K}\) and for each r.e. Boolean algebra \({\mathfrak B}_{\mu}\in {\mathfrak K}\) there exists a r.e. Boolean algebra \({\mathfrak C}_{\pi}\) such that \({\mathfrak B}_{\mu}\times {\mathfrak C}_{\pi}\) and \({\mathfrak A}_{\nu}\) are recursively isomorphic. Let B denote the class of atomless r.e. Boolean algebras, A be the class of atomic r.e. Boolean algebras and \(A_ r\) be the class of atomic recursive Boolean algebras. The author studies the question of existence of universal r.e. Boolean algebras for the classes B, A, \(A_ r\) and proves the following theorems: Theorem 1. There is no universal r.e. Boolean algebra in the class B of atomless r.e. Boolean algebras. Theorem 2. For each atomic r.e. Boolean algebra (\({\mathfrak A},\mu)\) there exists a recursive atomic Boolean algebra which is not a direct summand in (\({\mathfrak A},\mu)\). Corollary 1. The class \(A_ r\) has no universal recursive Boolean algebra. Corollary 2. The class A has no universal r.e. Boolean algebra.
MSC:
03D45 Theory of numerations, effectively presented structures
06E99 Boolean algebras (Boolean rings)
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References:
[1] Yu. L. Ershov, Theory of Numerations [in Russian], 3, Novosibirsk Univ. Press (1974).
[2] R. Sikorski, Boolean Algebras [Russian translation], Mir, Moscow (1970).
[3] H. Rogers, The Theory of Recursive Functions and Effective Computability [Russian translation], Mir, Moscow (1972).
[4] M. G. Peretyat’kin, ?Strongly constructive models and numerations of the Boolean algebra of recursive sets,? Algebra Logika,10, No. 5, 535-557 (1971).
[5] S. S. Goncharov, ?Some properties of constructivizations of Boolean algebras,? Sib. Mat. Zh.,16, No. 2, 264-278 (1975).
[6] S. S. Goncharov, ?Nonautoequivalent constructivizations of atomic Boolean algebras,? Mat. Zametki,19, No. 6, 853-858 (1976).
[7] A. T. Nurtazin, ?Computable classes and algebraic criteria of autostability,? Candidate’s Dissertation, Mathematics Institute, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk (1974).
[8] W. Hanf, ?Characterization of B?2?,? Preprint, University of Hawaii, Honolulu (1979).
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