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On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem Löwenheim theorems and compactness of related quantifiers

Published online by Cambridge University Press:  12 March 2014

Matatyahu Rubin
Affiliation:
Hebrew University, Jerusalem, Israel University of Colorado, Boulder, Colorado 80309
Saharon Shelah
Affiliation:
Ben Gurion University, Beer Sheva, Israel University of Colorado, Boulder, Colorado 80309

Abstract

Theorem 1. (◊ℵ1,) If B is an infinite Boolean algebra (BA), then there is B1, such that ∣ Aut (B1) ≤∣B1∣ = ℵ1 and 〈B1, Aut (B1)〉 ≡ 〈B, Aut(B)〉.

Theorem 2. (◊ℵ1) There is a countably compact logic stronger than first-order logic even on finite models.

This partially answers a question of H. Friedman. These theorems appear in §§1 and 2.

Theorem 3. (a) (◊ℵ1) If B is an atomic ℵ-saturated infinite BA, Ψ Є Lω1ω and 〈B, Aut (B)〉 ⊨Ψ then there is B1, Such that ∣Aut(B1)∣ ≤ ∣B1∣ =ℵ1, andB1, Aut(B1)〉⊨Ψ. In particular if B is 1-homogeneous so is B1. (b) (a) holds for B = P(ω) even if we assume only CH.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1980

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References

[F]Friedman, H., One hundred and two problems in mathematical logic, this Journal, vol. 40 (1975), pp. 113129.Google Scholar
[K]Keisler, H. J., Models with tree structures, Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics, vol. XXV, American Mathematical Society, Providence, R. I., 1974, pp. 331348.Google Scholar
[M]McKenzie, R., On elementary types of symmetric groups, Algebra Universalis, vol. 1 (1971), pp. 1320.CrossRefGoogle Scholar
[R1]Rubin, M., On the automorphism groups of homogeneous and saturated Boolean algebras, Algebra Unhersalis, vol. 9 (1979), pp. 5486.CrossRefGoogle Scholar
[R2]Rubin, M., On the automorphism groups of countable Boolean algebras, Israel Journal of Mathematics (to appear).Google Scholar
[S1]Shelah, S., Models with second order properties. I, Boolean algebras with no definable automorphisms, Annals of Mathematical Logic, vol. 14 (1978), pp. 5772.CrossRefGoogle Scholar
[S2]Shelah, S., First order theory of permutation groups, Israel Journal of Mathematics, vol. 14 (1973), pp. 149169; errata, Israel Journal of Mathematics, vol. 15 (1973), pp. 437–447.CrossRefGoogle Scholar
[S3]Shelah, S., Models with second order properties IV (preprint).Google Scholar
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On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem Löwenheim theorems and compactness of related quantifiers
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