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Every superatomic subalgebra of an interval algebra is embeddable in an ordinal algebra


Authors: Uri Abraham and Robert Bonnet
Journal: Proc. Amer. Math. Soc. 115 (1992), 585-592
MSC: Primary 06E05; Secondary 03G05, 06A05, 54F05
DOI: https://doi.org/10.1090/S0002-9939-1992-1074745-2
MathSciNet review: 1074745
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Abstract: Let us recall that a Boolean algebra is superatomic if every subalgebra is atomic. So by the definition, every subalgebra of a superatomic algebra is superatomic. An obvious example of a superatomic algebra is the interval algebra generated by a well-ordered chain. In this work, we show that every superatomic subalgebra of an interval algebra is embeddable in an ordinal algebra, that is by definition, an interval algebra generated by a well-ordered chain. As a corollary, if $ B$ is an infinite superatomic subalgebra of an interval algebra, then $ B$ and the set $ \operatorname{At}(B)$ of atoms of $ B$ have the same cardinality.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1074745-2
Keywords: Boolean algebras, interval algebras, superatomic Boolean algebras
Article copyright: © Copyright 1992 American Mathematical Society

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