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 is an infinite superatomic subalgebra of an interval algebra, then and the set of atoms of 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