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$ 2\sp {2\sp \omega}$ nonisomorphic short ordered commutative domains whose quotient fields are long


Author: Krzysztof Ciesielski
Journal: Proc. Amer. Math. Soc. 113 (1991), 217-227
MSC: Primary 03E05; Secondary 03E75, 06F25, 12J15
DOI: https://doi.org/10.1090/S0002-9939-1991-1079888-4
MathSciNet review: 1079888
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Abstract: A linearly ordered set is short if it does not contain any monotonic sequence of length $ {\omega _1}$, and it is long if it contains a monotonic sequence of length $ \alpha $ for every ordinal $ \alpha < {({2^\omega })^ + }$. We prove that there exists a family $ {\mathbf{F}}$ of power $ {2^{{2^\omega }}}$ of long ordered fields of size $ {2^\omega }$ that are pairwise nonisomorphic (as fields) and such that every field $ F \in {\mathbf{F}}$ has $ {2^{{2^\omega }}}$ nonisomorphic short subdomains whose field of quotients is $ F$. The generalization of this result for higher cardinals is also discussed. This generalizes the author's result of [Ci].


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1079888-4
Keywords: Nonisomorphic, ordered commutative domains, quotient fields
Article copyright: © Copyright 1991 American Mathematical Society

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