Abstract
A Boolean algebra is a non-empty set 𝔄 in which two binary operations ∪, ∩ and one unary operation — are defined which have, roughly speaking, the same properties as the set-theoretical union, intersection and complementation of subsets of a fixed space. Since the elements of 𝔄 have many of the properties of sets, we shall denote them by capital letters A, B, . . . used generally to denote sets. For arbitrary elements A, B∈ 𝔄, A ∪ B and A∩ B are elements in 𝔄, uniquely determined by A and B and called respectively the join and the meet of A and B. For each element A ∈ 𝔄, — A is an element in 𝔄, uniquely determined by A and called the complement of A. The operations ∪, ∩, — are characterized by a set of axioms assuring that these operations have properties ana-logoues to those of union, intersection and complementation of sets respectively. Many equivalent sets of axioms characterizing ∪, ∩, — are known1.
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References
See Bennett [1], Bernstein [1, 2, 4, 6, 7], Birkhoff and Birkhoff [1], Braithwaite [1], Byrne [1, 2, 3], Croisot [1], Diamond [1, 2], Frink [1], Grau [1, 2], Hammer [1], Hoberman and McKinsey [1], Huntington [1, 2], Kalicki [1], Miller [1], Montague and Tarski [1], Newman [1], Sheffer [1], Sholander [1, 2], Stabler [1], Stamm [1], Stone [2, 3], Tarski [2, 5, 8, 12], Whiteman [1, 2]. See also Rudeanu [2].
The set of axioms (A1—(A5) is not the simplest one. It can be shortened since some axioms are consequences of other ones. For instance, one of the axioms (A4) can be omitted (see e. g. Birkhoff [2], p. 133). Many papers quoted in footnote1contain much shorter sets of axioms. However, if the set of axioms is short, then it is more difficult to deduce from it various important properties of Boolean operations. To omit these algebraic difficulties we start from the convenient axioms (A1)-(A5).Observe that every set of axioms for Boolean algebras has to contain at least three variables A, B, C. This follows from the fact that there exists an algebra which is not Boolean but every one of its subalgebras generated by two elements is Boolean. See Diamond and McKinsey [1].
See Kuratowski [3], p. 38 and Stone [6].
See Birkhoff [2], p. 176.
For details, see Rasiowa and Sikorski [9].
For a detailed study and classification of ideals, see Stone [7]. See also Maeda [1], Mori [1], Pospišil [3], Tarski [3, 6].
For examination of a notion more general than that of homomorphism, see Halmos [4], Wright [2, 3, 4].
This fundamental theorem is due to Stone [5]. See also Tarski [1], Ulam [2].
See Horn and Tarski [1], Łoš and Marczewski [1], Tarski [11].
This fundamental representation theorem is due to Stone [1, 4, 5, 6, 10]. The representation theorem 8.2 and theorem 6.1 on the existence of maximal ideals and filters are investigated in many papers. See Aumann [2], Dilworth [1], Dunford and Schwartz [1] (p. 41), Dunford and Stone [1], Engelking and Kuratowski [1], Enomoto [1], Frink [1], Iseki [1], Kakutani [1], Livenson [1], Mori [1], Nolin [1], Stabler [2], Tarski [1]. Another representation theorem for Boolean algebras was given by Haimo [2].
This theorem is due to Urysohn. See e.g. Alexandroff and Hopf [1], p. 88.
Mostowski [1]. See also Mazurkiewicz and Sierpiński [1].
Alexandroff and Urysohn [1].
For a similar construction of the Stone space of the field of all sets A∪ B where A ∈ 𝔄 and B is finite, see Marczewski [12, 13]. See also Semadeni [3], p. 79.
Discussed in the paper Stone [7].
For a generalization of the notion of atom, see Pierce [5].
See e.g. Kuratowski [4], p. 58.
Stone [6].
Sikorski [6].
An example of such a space was given by Katětov [1] by means of the β-compactification technique. Two similar examples of linearly ordered compact spaces with this property were given independently and simultaneously by Jónsson [1] and Rieger [6].
For an investigation of this notion, see Day [1].
Pelczyński and Semadeni [1], Rudin [1].
This remark is due to C. Goffman.
Alexandroff and Urysohn [1]. See also Semadini [1], p. 20.
Pelczyński and Semadini [1]. For a characterization of scattered compact spaces (i.e. of superatomic Boolean algebras) in terms of measure theory, see Rudin [2], Pelczyński and Semadeni [1].
Sikorski [14].
Kuratowski and Posament [l].
Sikorski [14].
Sikorski [32].
Büchi [1], For another proof see Sikorski [31].
The notion of independence (and of m-independence — see § 37) is a slight generalization of the notion of set-theoretical independence of sets introduced by Fichtenholz and Kantorovitch [1] Hausdorff [1]. Marczewski [5, 6, 8, 14, 16] examined this notion from the point of view of applications to measure theory. See also Kappos [5]. The theorems mentioned in § 13 were proved by Sikorski [11, 13, 14].
Sikorski [3]. Another equivalent definition was given by Kappos [2, 5] and Ridder [1].
This theorem is a particular case of a general theorem (due to Hewitt [2]) on Cartesian products of topological spaces. The case m = χ0of this theorem was independently found by Marczewski [9].
This theorem is a particular case of more general theorems proved by Hausdorff [1] and Tarski [6]. Form = χ0 see also Fichtenholz and Kantorovitch [1].
Gillman and Jerison [1], Novak [1], Pospišil [5, 6]. See also Tarski [6], Hausdorff [1].
Sikorski [6]. Theorems 15.1–15.2 were proved by Sikorski [6, 18]. The first theorem of this kind was that of von Neumann [1] on induced measure-preserving isomorphisms.
Sikorski [6, 18].
Sierpiński [4].
Sanin [1].
The mapping φ is called the characteristic function of. The notion of characteristic functions was examined by Marczewski [2, 10] in the case where Tis the set of positive integers. For the generalization to uncountable sets T, see Stone [11]. Marczewski’s method of characteristic functions is used several times in this book.
Halmos [10]. For a generalization, see Semadeni [5].
Dwinger [2].
Theorem 17.1 is due to Stone [2, 5]. There also exist other binary operations with respect to which every Boolean algebra is a ring and which define uniquely the Boolean operations. See Rudeanu [2].
For the investigation of the symmetric difference, see e.g. Helson [1], Marczewski [7].
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Sikorski, R. (1960). Finite joins and meets. In: Boolean Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-01507-0_2
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