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), 585592
MSC:
Primary 06E05; Secondary 03G05, 06A05, 54F05
MathSciNet review:
1074745
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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 wellordered 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 wellordered 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. SiKaddour, On Boolean algebras with wellfounded 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
(36 #5045)
 [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 (29 #4713)
 [5]
Sabine
Koppelberg, Handbook of Boolean algebras. Vol. 1,
NorthHolland Publishing Co., Amsterdam, 1989. Edited by J. Donald Monk and
Robert Bonnet. MR
991565 (90k:06002)
 [6]
R.
D. Mayer and R.
S. Pierce, Boolean algebras with ordered bases, Pacific J.
Math. 10 (1960), 925–942. MR 0130842
(24 #A696)
 [7]
Judy
Roitman, Superatomic Boolean algebras, Handbook of Boolean
algebras, Vol.\ 3, NorthHolland, 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M. Rubin and S. Shelah, On the cardinality of superatomic subalgebra of an interval algebra, 1988 (unpublished).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, SpringerVerlag,
BerlinGöttingenHeidelberg, 1960. MR 0126393
(23 #A3689)
 [1]
 R. Bonnet, M. Rubin, and H. SiKaddour, On Boolean algebras with wellfounded 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]
 G. W. Day, Superatomic Boolean algebras, Pacific J. Math. 23 (1967), 479489. MR 0221993 (36:5045)
 [4]
 P. Halmos, Lectures on Boolean algebras, Van Nostrand Math. Studies, vol. 1, Van Nostrand Co., NY, 1963. MR 0167440 (29:4713)
 [5]
 S. Koppelberg, Special classes of Boolean algebras, Handbook on Boolean algebras, vol. 1, part I, Chapter 6 (J. D. Monk, ed.), NorthHolland, Amsterdam, 1989, pp. 239284. MR 991565 (90k:06002)
 [6]
 D. Mayer and R. S. Pierce, Boolean algebras with ordered basis, Pacific J. Math. 10 (1960), 925942. MR 0130842 (24:A696)
 [7]
 J. Roitman, Superatomic Boolean algebras, Handbook on Boolean algebras, vol. 3, Part II, Chapter 19 (J. D. Monk, ed.), NorthHolland, Amsterdam, 1989, pp. 719740. 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]
 R. Sikorski, Boolean algebras, Ergeb. Math. Grenzgeb (3), vol. 25, SpringerVerlag, Berlin, 1964. MR 0126393 (23:A3689)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
06E05,
03G05,
06A05,
54F05
Retrieve articles in all journals
with MSC:
06E05,
03G05,
06A05,
54F05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199210747452
PII:
S 00029939(1992)10747452
Keywords:
Boolean algebras,
interval algebras,
superatomic Boolean algebras
Article copyright:
© Copyright 1992
American Mathematical Society
