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Infinite joins and meets

  • Roman Sikorski
Conference paper
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 25)

Abstract

Let A 1,..., A n be elements of a Boolean algebra U.

Keywords

Boolean Algebra Borel Subset Stone Space Complete Boolean Algebra Infinite Cardinal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literatur

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    This fundamental representation theorem for Boolean σ-algebras was found independently by Loomis [1] and Sikorski [4]. It was published by Loomis in August 1947 and it was presented (with proof) by Sikorski at the Congress of the Polish Mathematical Society in Kraków on May 1947, but published one year later because of the printing difficulties in Poland after the second world war. The proof presented here is that of Sikorski [4]. A similar proof was found independently by P. R. Halmos. See also Aumann [1]. A new proof based on metamathematical ideas was given by Tarski [12].Google Scholar
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  74. 1.
    Conditions (r5) and (r’5) are due to Chang [1]. See also Scott [3]. Condition (r4) is due to Pierce [2]. Condition (r1) is a modification (Sikorski [25]) of a sufficient condition given by Smith [1]. The whole theorem 29.3 was published by Sikorski [25].Google Scholar
  75. 1.
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  76. 1.
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    This theorem is a particular case of a more general topological theorem proved by Gleason [1]. For a discussion of the connection between theorem 33.1 and the Gleason theorem and for application of the Gleason theorem and other related questions, see Halmos [8], Isbell and Semadeni [1], Rainwater [1], Semadeni [4, 5].Google Scholar
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    Matthes [1]. The proof given below is a slight modification of a proof (not published) communicated by K. Matthes to the author.Google Scholar
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    Gaifman [1, 3] and Hales [1, 2]. The construction of quoted here is that of Hales [1, 2].Google Scholar
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    Pierce [3]. The above proof of 35.5 differs from that of Pierce.Google Scholar
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    σ-extensions were fiirst examined by Sikorski [13]. Later Kerstan [1] proved the existence of free (J, M, m)-extensions for any m. Another proof of the existence of free (J, M, m)-extensions was given by Sikorski [32]. Independently the same proof of the existence of free m-extensions was found by Yaqub [1] who also examined the case where J and M are empty. See also Day [1] and Day and Yaqub [1]. The exposition in this section is a slight modification of that in Sikorski [32].Google Scholar
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    Recently Yaqub [1] proved that (for empty J and M) if m ≥ 2χ 0 and if {h 0, 𝔉m} is the maximal (J, M, m)-extension, then 𝔄 is superatomic. Day [1] proved the converse statement. Thus {h 0, 𝔉m} is a maximal (J, M, m) extension of 𝔄 for empty J and M and m ≥ 2χ 0 if and only if 𝔄 is superatomic. See Day and Yaqub [1].Google Scholar
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    Example C) is due to M. Katětov (not published).Google Scholar
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    Sikorski [11, 14].Google Scholar
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    Sikorski [11].Google Scholar
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    See e.g. Halmos [1], p. 154 —158.Google Scholar
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    Sikorski [11]. The equivalence of (i) and (ii) was also proved simultaneously and independently by Sherman [1].Google Scholar
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    For definition and fundamental properties, see e.g. Kuratowski [3], §§ 34 and 35.Google Scholar
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    Example A) was given by Sikorski [14].Google Scholar
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    (m, 0)-products were examined by Sikorski [13] (in the case where m = χ0) and Sikorski [32] (in the case of any m ≥ χ0).Google Scholar
  121. 1.
    m-products were investigated by Sikorski [13] (in the case of m = χ0) and Sikorski [32] (in the case of any m ≥ χ0).Google Scholar
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    Theorem 38.13 was proved by Christensen and Pierce [1]. The proof given above is due to Sikorski and Traczyk [2].Google Scholar
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    Steinhaus [1].Google Scholar
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    Example C) was given by Sikorrski [15]. A part of the proof is a slight modification of an argument in Helson [2].Google Scholar

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© Springer-Verlag Berlin Heidelberg 1960

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  • Roman Sikorski

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