Infinite joins and meets

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


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


Boolean Algebra Borel Subset Stone Space Complete Boolean Algebra Infinite Cardinal 
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  1. 1.
    For investigation of this Boolean algebra, see Sierpinski [3, 5].Google Scholar
  2. 1.
    See e.g. Rasiowa and Sikorski [1, 7]. See also § 40. To perform the substitution α(t), a change of bound variables is sometimes necessary.Google Scholar
  3. 1.
    For a study of infinite distributivity in Boolean algebras, see Chin and Tarski [1], Christensen and Pierce [1], Enomoto [1], Kerstan [1], Kowalsky [1], Matthes [1, 2], Pierce [3, 5, 6], Scott [1], Sikorski [25, 27], Sikorski and Traczyk [2], Smith [1], Smith and Tarski [1]. See also § 20, 24, 25, 29, 34, 35, 36, 38.Google Scholar
  4. 2.
    Pierce [3], Smith and Tarski [1].Google Scholar
  5. 1.
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  6. 2.
    See Tarski [3], Birkhoff [2] and Macneille [1].Google Scholar
  7. 1.
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  8. 2.
    See e.g. Kuratowski [3], p. 54–56.Google Scholar
  9. 3.
    Marczewski [3].Google Scholar
  10. 1.
    Marczewski [3, 10].Google Scholar
  11. 2.
    Smith and Tarski [1].Google Scholar
  12. 1.
    Pierce [3].Google Scholar
  13. 1.
    Smith ard Tarski [1]. See also Pierce [3].Google Scholar
  14. 2.
    Pierce [3].Google Scholar
  15. 2.
    Scott [1].Google Scholar
  16. 1.
    Sikorski [12].Google Scholar
  17. 2.
    This follows from a more general theorem on separability of Cartesian products. See Marczewski [9].Google Scholar
  18. 3.
    Tarski [3].Google Scholar
  19. 1.
    σ-measures on Boolean algebras have, roughly speaking, the same properties as σ-measures on fields of sets. We shall often use this fact without any reference. For details see e.g. Aumann [3].Google Scholar
  20. 1.
    We quote here [examples C), D), E)] only the simplest cases of the overcompleteness of quotient algebras. For other interesting theorems of this kind, see Smith and Tarski [1].Google Scholar
  21. 1.
    This result is due to Birkhoff and Ulam. See Birkhoff [3]. See also Von Neumann [2].Google Scholar
  22. 2.
    Banach [3]. See also Kuratowski [3], p. 49.Google Scholar
  23. 3.
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  24. 1.
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  25. 2.
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  26. 3.
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  27. 1.
    This remark is due to Birkhoff and Ulam. See Birkhoff [3].Google Scholar
  28. 2.
    See e.g. Marczewski [1] or Marczewski and Sikorski [2].Google Scholar
  29. 3.
    For another proof of the non-existence of any σ-measure on 𝔄1, see Horn and Tarski [1]. The non-existence of any strictly positive σ-measure on 𝔄1 and the nonisomorphism of 𝔄0 and 𝔄1 follow also from § 29 C) and D).Google Scholar
  30. 2.
    The existence of a mapping φ with this property was proved by SierpiŃski [2, 6]. See also Marczewski [11], Oxtoby and Ulam [1].Google Scholar
  31. 3.
    See Sikorski [10].Google Scholar
  32. 1.
    This lemma is a generalization of an argument in the proof of F. bernstein’s theorem on the existence of totally imperfect sets. See e.g. Kuratowski [3], p. 422.Google Scholar
  33. 2.
    See Sikorski [10].Google Scholar
  34. 3.
    Pierce [3]. A part of this theorem was proved independently by Smith and Tarski [1]. See also Sikorski [21].Google Scholar
  35. 1.
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  36. 1.
    For a detailed exposition of properties of extremally disconnected spaces, see Gillman and Jerison [1].Google Scholar
  37. 1.
    Semadeni [1].Google Scholar
  38. 1.
    Sikorski [1] and Tarski [8]. The hypothesis thatis σ-complete is essential. See Kinoshita [1] and Hanf [1]. The set-theoretical meaning of this theorem is explained on p. 193.Google Scholar
  39. 1.
    Sikorski [13].Google Scholar
  40. 1.
    Sikorski [13].Google Scholar
  41. 2.
    Banach [3]. See also Kuratowski [3], p. 49.Google Scholar
  42. 3.
    For an investigation of separable Boolean algebras, see Horn and Tarski [1].Google Scholar
  43. 4.
    Horn and Tarski [1].Google Scholar
  44. 1.
    Horn and Tarski [1], Tarski [6].Google Scholar
  45. 2.
    See Semadeni [2, 3].Google Scholar
  46. 1.
    This proof of theorem 23.4 is a slight modification of the proof of a similar theorem in Halmos [1], p. 27.Google Scholar
  47. 2.
    This fact was first observed by Tarski [3]. See also Marczewski [4], Sikorski [4].Google Scholar
  48. 3.
    Pauc [2], Horn and Tarski [1], Pierce [2], Sikorski [4], Tarski [3].Google Scholar
  49. 1.
    See Sikorski [12].Google Scholar
  50. 1.
    Smith and Tarski [1].Google Scholar
  51. 1.
    Theorem 24.10 was found by Sikorski. See Rasiowa and Sikorski [1] and Rasiowa [2]. Another algebraic proof of 24.10 was found by A. Tarski.Google Scholar
  52. 1.
    Čech [1]. See also Sikorski [4].Google Scholar
  53. 2.
    Bruns [1].Google Scholar
  54. 1.
    Theorem 25.1 is due to A. Lindenbaum and A. Tarski (see Tarski [2]), theorem 25.2 is due to Tarski [2]. See also Horn and Tarski [1].Google Scholar
  55. 1.
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  56. 2.
    Pierce [4].Google Scholar
  57. 1.
    Pierce [4]. Earlier Ginsburg [2] proved that for every infinite cardinal n there is a complete homogeneous algebra of cardinality 2η.Google Scholar
  58. 2.
    Pierce [4] where the theorem is formulated more generally ; card A is supposed to be any cardinal-valued mapping which has the monotonicity property. The proof is the same. See also Pierce [9].Google Scholar
  59. 1.
    Pierce [4]Google Scholar
  60. 2.
    Dwinger [2].Google Scholar
  61. 1.
    See Sikorski [4], Smith and Tarski [1].Google Scholar
  62. 2.
    Tarski [3]. See also Smith and Tarski [1]. They prove other interesting theorems stating a higher completeness of ideals.Google Scholar
  63. 1.
    This theorem is due to Ulam [1]. Under a more restrictive hypothesis on the cardinality of X, it is a particular case of a general theorem stating that every σ-finite σ-measure on the field of all subsets of X is concentrated on an enumerable set. See Banach and Kuratowski [1], Banach [1], Ulam [1]. For the case X=20 see also Marczewski [1], Sierpinski [1]. Further generalizations were given by Marczewski and Sikorski [1], Mazur [1], Sierpinski [2] Chapter V, Tarski [3].Google Scholar
  64. 1.
    The solution is due to Tarski [18] who obtained it as a corollary to a metamathematical theorem of Hanf [2, 3]. The first proof of the Tarski theorem was metamathematical. A mathematical proof was given by Keisler [1]. For a full exposition see Keisler and Tarski [1]. The papers quoted concern also other related problems in the theory of Boolean algebras. See also Erdös and Tarski [2] and Hanf [4].Google Scholar
  65. 2.
    Scott [4] proved that the existence of cardinals that are non-v-perfect implies the existence of sets that are not constructive in the sense of Gödel [1]. This result also shows that practically it would be impossible to define a non-σperfect cardinal.Google Scholar
  66. 1.
    Marczewski and Sikorski [1]. Under a hypothesis on the cardinal X, it is a particular case of a general theorem stating that every finite σ-measure on the field of all Borel subsets of X is concentrated on a separable subset. See Marczewski and Sikorski [1]. For a generalization to non-metrizable spaces, see Katetov [2].Google Scholar
  67. 2.
    This statement is a particular case of a general theorem due to Montgomery [1]. See also Kuratowski [2].Google Scholar
  68. 1.
    Theorems 28.1–28.3 were proved by Sikorski [4].Google Scholar
  69. 1.
    Tarski [3].Google Scholar
  70. 2.
    The same construction is used in Sikorski [21] for the problem of axiomatization of the notion of σ-fields of sets. See also Horn and Tarski [1], Sikorski [4], Smith and Tarski [1].Google Scholar
  71. 1.
    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
  72. 2.
    Čech [1]. See also Sikorski [4].Google Scholar
  73. 1.
    Recently Karp [1] proved that for every m > χ χ0 there is a non-m-representable m-algebra. More precisely, for every m ≥ χ0 there exists a complete m-representable algebra which is not m+-representable, m+ being the smallest cardinal greater than m.Google Scholar
  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.
    The proof of this implication is a slight modification of A. Bialynicki-Birula’s proof (not published) of Chang’s [1] representation theorem.Google Scholar
  76. 1.
    Example B) is a particular case of general theorem proved by von Neumann and Stone [1]. The paper quoted contains a whole discussion of this question. See also Tarski [3].Google Scholar
  77. 1.
    For a discussion of the connection between m-distributivity, weak m-distributivity and m-representability, see Sikorski [27].Google Scholar
  78. 2.
    See Sikorski [25] where the theorem is proved for m = nn.Google Scholar
  79. 1.
    This theorem was explicitly formulated and proved by Horn and Tarski [1] but it was used earlier by Banach and Kuratowski [1]. See also Von Neumann [2].Google Scholar
  80. 1.
    See Von Neumann [2].Google Scholar
  81. 3.
    This theorem is due to Banach and Kurratowski [1].Google Scholar
  82. 1.
    This theorem is due to Kelley [2] and J. Oxtoby.Google Scholar
  83. 2.
    Rieger [5]. The above proof differs from that of Rieger.Google Scholar
  84. 1.
    This result was recently obtained by Gaifman [1, 3] and Hales [1, 2]. This result has the following interesting consequence. Replacing everywhere in the definition of free Boolean m-algebra the words “m-algebra” by “complete algebra” and “m-generates” by “completely generates” we get the analogous definition of free complete Boolean algebra A with a set A of n free complete generators. The free complete Boolean algebra with n generators exists if n is finite (viz. it coincides then with A0, n) but does not exist if n is infinite. For it can be proved (by the same method as a similar statement in § 35 G)) that the smallest m-subalgebra containing ⊂ would then be a free Boolean m-algebra with n generators. Hence if n is infinite, we get A m for every cardinal m which is impossible.Google Scholar
  85. 1.
    In the case m = n = χ0, theorems 31.4 — 31.6 were proved by Sikorski [14]. The case of n > χ0 was examined first by Rieger [5]. The proof quoted here was given by Sikorski [17].Google Scholar
  86. 1.
    This also follows from a general remark on free algebras, due to Pierce [8].Google Scholar
  87. 1.
    Theorems 32.1— 32.2 and 32.4 — 32.6 are slight modifications of analogous theorems proved for m = χ0 by Sikorski [6, 8, 18].Google Scholar
  88. 1.
    See e.g. Kuratowski [3], p. 337.Google Scholar
  89. 1.
    See e.g. Kuratowski [3], p. 358.Google Scholar
  90. 2.
    Sikorski [6].Google Scholar
  91. 1.
    Traczyk [1].Google Scholar
  92. 2.
    Sikorski [5].Google Scholar
  93. 1.
    This part of the proof can be replaced by verification that the mapping f which coincides with h>o on A0 and assumes the value B at Ao satisfies condition § 12 (4).Google Scholar
  94. 1.
    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
  95. 2.
    Lemmas 33.2 — 33.4 were proved by Sikorski [13].Google Scholar
  96. 1.
    Sikorski [32], Sikorski and Traczyk [2].Google Scholar
  97. 1.
    Matthes [1]. The proof given below is a slight modification of a proof (not published) communicated by K. Matthes to the author.Google Scholar
  98. 1.
    Example A) was given by Sikorski [14].Google Scholar
  99. 1.
    Dubins [1].Google Scholar
  100. 1.
    For an examination of completions of Boolean algebras, see Glivenko [1], Macneille [1], Sikorski [13] and Stone [5]. See also Dilworth [3], Gleason [1], Rainwater [1], Semadeni [2].Google Scholar
  101. 2.
    Sikorski [13].Google Scholar
  102. 1.
    See Macneille [1].Google Scholar
  103. 1.
    This result is due to S. Jaskowski (not published). See Tarski [3], p. 199.Google Scholar
  104. 2.
    This method of construction of completions of Boolean algebras was applied by Macneille [1].Google Scholar
  105. 1.
    For the case m = χ0, see Sikorski [13].Google Scholar
  106. 1.
    Gaifman [1, 3] and Hales [1, 2]. The construction of quoted here is that of Hales [1, 2].Google Scholar
  107. 1.
    Pierce [3]. The above proof of 35.5 differs from that of Pierce.Google Scholar
  108. 1.
    Traczyk [5].Google Scholar
  109. 2.
    Pierce [3].Google Scholar
  110. 4.
    Traczyk [5].Google Scholar
  111. 1.
    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
  112. 1.
    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
  113. 2.
    Example C) is due to M. Katětov (not published).Google Scholar
  114. 1.
    Sikorski [11, 14].Google Scholar
  115. 2.
    Sikorski [11].Google Scholar
  116. 1.
    See e.g. Halmos [1], p. 154 —158.Google Scholar
  117. 2.
    Sikorski [11]. The equivalence of (i) and (ii) was also proved simultaneously and independently by Sherman [1].Google Scholar
  118. 1.
    For definition and fundamental properties, see e.g. Kuratowski [3], §§ 34 and 35.Google Scholar
  119. 2.
    Example A) was given by Sikorski [14].Google Scholar
  120. 3.
    (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
  122. 1.
    Theorem 38.13 was proved by Christensen and Pierce [1]. The proof given above is due to Sikorski and Traczyk [2].Google Scholar
  123. 1.
    For detailed proofs of these statements see Sikorski [13].Google Scholar
  124. 2.
    See e.g. Halmos [1], p. 38.Google Scholar
  125. 1.
    See e.g. Halmos [1], p. 38.Google Scholar
  126. 1.
    Steinhaus [1].Google Scholar
  127. 2.
    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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  • Roman Sikorski

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