Boolean Algebras pp 54-190 | Cite as

# Infinite joins and meets

Conference paper

## 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
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## Literatur

- 1.For investigation of this Boolean algebra, see Sierpinski [3, 5].Google Scholar
- 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 - 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
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- 1.
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- 2.This follows from a more general theorem on separability of Cartesian products. See Marczewski [9].Google Scholar
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- 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
- 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
- 1.
- 2.
- 3.
- 1.
- 2.
- 3.
- 1.This remark is due to Birkhoff and Ulam. See Birkhoff [3].Google Scholar
- 2.
- 3.For another proof of the non-existence of any σ-measure on 𝔄
_{1}, see Horn and Tarski [1]. The non-existence of any strictly positive σ-measure on 𝔄_{1}and the nonisomorphism of 𝔄_{0}and 𝔄_{1}follow also from § 29 C) and D).Google Scholar - 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
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- 3.
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- 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
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- 1.Pierce [4]. Earlier Ginsburg [2] proved that for every infinite cardinal n there is a complete homogeneous algebra of cardinality 2
^{η}.Google Scholar - 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 - 1.Pierce [4]Google Scholar
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- 2.Tarski [3]. See also Smith and Tarski [1]. They prove other interesting theorems stating a higher completeness of ideals.Google Scholar
- 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 - 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
- 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
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*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 - 2.This statement is a particular case of a general theorem due to Montgomery [1]. See also Kuratowski [2].Google Scholar
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- 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
- 2.
- 1.Recently Karp [1] proved that for every m > χ χ
_{0}there is a non-m-representable m-algebra. More precisely, for every m ≥ χ_{0}there exists a complete m-representable algebra which is not m^{+}-representable, m^{+}being the smallest cardinal greater than m.Google Scholar - 1.Conditions (r
_{5}) and (r’_{5}) are due to Chang [1]. See also Scott [3]. Condition (r_{4}) is due to Pierce [2]. Condition (r_{1}) 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 - 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
- 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
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- 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 A
_{0}, 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 - 1.In the case m = n = χ
_{0}, theorems 31.4 — 31.6 were proved by Sikorski [14]. The case of n > χ_{0}was examined first by Rieger [5]. The proof quoted here was given by Sikorski [17].Google Scholar - 1.This also follows from a general remark on free algebras, due to Pierce [8].Google Scholar
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*f*which coincides with h>o on A_{0}and assumes the value*B*at*A*o satisfies condition § 12 (4).Google Scholar - 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
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- 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
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- 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
- 1.Recently Yaqub [1] proved that (for empty J and M) if m ≥ 2
^{χ}_{0}and if {*h*_{0}, 𝔉_{m}} is the maximal (J, M, m)-extension, then 𝔄 is superatomic. Day [1] proved the converse statement. Thus {*h*_{0}, 𝔉_{m}} is a maximal (J, M, m) extension of 𝔄 for empty J and M and m ≥ 2^{χ}_{0}if and only if 𝔄 is superatomic. See Day and Yaqub [1].Google Scholar - 2.Example C) is due to M. Katětov (not published).Google Scholar
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_{0}) and Sikorski [32] (in the case of any m ≥ χ_{0}).Google Scholar - 1.m-products were investigated by Sikorski [13] (in the case of m = χ
_{0}) and Sikorski [32] (in the case of any m ≥ χ_{0}).Google Scholar - 1.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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