Abstract
A subset X 0 of a Boolean algebra X is called a subalgebra of X if X0 contains 0 and 1 and is closed under the main Boolean operations ∨, ∧, and C;i.e.,
whenever x, y ∈ X0. The set X0, furnished with the induced order, is a Boolean algebra with the same zero and unity as in X. By duality we easily show that for a subset X0 to be a subalgebra it is sufficient that X0 be closed under the operations ∨ and C or under the operations ∧ and C. Finally, the induction demonstrates that each subalgebra contains the suprema and infima of its finite subsets.
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© 2002 Springer Science+Business Media Dordrecht
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Vladimirov, D.A. (2002). The Basic Apparatus. In: Boolean Algebras in Analysis. Mathematics and Its Applications, vol 540. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0936-1_2
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DOI: https://doi.org/10.1007/978-94-017-0936-1_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5961-1
Online ISBN: 978-94-017-0936-1
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