Abstract
Each Boolean algebra is isomorphic to an algebra of sets. This fact was already mentioned in the preceding chapters. Below, we give an exact formulation and a complete proof of the celebrated Stone Theorem and discuss the problems that arise in connection with this theorem.
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As q, we can take a maximal ideal that contains Cv.
The mapping, denoted by Φ on that page, is the identity mapping in our case (X 2 ≡ &O(,D2)) HERE the letter Φ is the symbol of an ARBITRARY homomorphism.
We need to replace e by e\φ -1(E), where E is the greatest clopen subset of φ (e) (such a subset always exists in an extremally disconnected compact space).
The projection & related to ξassigns to each q ∈ the element of ξ containing q.
K. Kuratowski [2, Theorem 1, p. 194].
D. A. Vladimirov and P. Zenf [1].
K. Kuratowski [2, pp. 194–195].
Z. T. Dikanova [1].
For instance, see H. Schubert [1]. Semadeni’s book [1] contains an exposition of the basics of category theory which is oriented to functional analysis.
Each set is a class but not vice versa.
Together with j, we distinguish P by the condition j ∈ M or (P, Q).
Choosing j and P, we prefer to select the main “name” for the subobject and forget about other names.
In particular, a retraction f can be identical on B. This case is very important (for instance, in topology).
The term “Boolean product” is also employed.
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© 2002 Springer Science+Business Media Dordrecht
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Vladimirov, D.A. (2002). Representation of Boolean Algebras. In: Boolean Algebras in Analysis. Mathematics and Its Applications, vol 540. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0936-1_4
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DOI: https://doi.org/10.1007/978-94-017-0936-1_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5961-1
Online ISBN: 978-94-017-0936-1
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