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.

**[1]**R. Bonnet, M. Rubin, and H. Si-Kaddour,*On Boolean algebras with well-founded set of generators*, Trans. Amer. Math. Soc., submitted.**[2]**R. Bonnet and S. Shelah,*On**spaces. An uncountable compact**space, different of*,*which is homeomorphic to each of its uncountable closed subspace*, Israel J. Math., (second version, 1991, submitted).**[3]**George W. Day,*Superatomic Boolean algebras*, Pacific J. Math.**23**(1967), 479–489. MR**0221993****[4]**Paul R. Halmos,*Lectures on Boolean algebras*, Van Nostrand Mathematical Studies, No. 1, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. MR**0167440****[5]**Sabine Koppelberg,*Handbook of Boolean algebras. Vol. 1*, North-Holland Publishing Co., Amsterdam, 1989. Edited by J. Donald Monk and Robert Bonnet. MR**991565****[6]**R. D. Mayer and R. S. Pierce,*Boolean algebras with ordered bases*, Pacific J. Math.**10**(1960), 925–942. MR**0130842****[7]**Judy Roitman,*Superatomic Boolean algebras*, Handbook of Boolean algebras, Vol. 3, North-Holland, Amsterdam, 1989, pp. 719–740. MR**991608****[8]**J. Roseinstein,*Linear ordering*, Academic Press, NY, 1982.**[9]**M. Rubin and S. Shelah,*On the cardinality of superatomic subalgebra of an interval algebra*, 1988 (unpublished).**[10]**Roman Sikorski,*Boolean algebras*, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Heft 25, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR**0126393**

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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