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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Every superatomic subalgebra of an interval algebra is embeddable in an ordinal algebra
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by Uri Abraham and Robert Bonnet PDF
Proc. Amer. Math. Soc. 115 (1992), 585-592 Request permission

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
  • © Copyright 1992 American Mathematical Society
  • 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