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Proceedings of the American Mathematical Society

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

 

 

Comfort types of ultrafilters


Author: Salvador García-Ferreira
Journal: Proc. Amer. Math. Soc. 120 (1994), 1251-1260
MSC: Primary 54A25; Secondary 03E35, 54A35, 54D80
DOI: https://doi.org/10.1090/S0002-9939-1994-1170543-1
MathSciNet review: 1170543
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Abstract: Let $ \alpha $ be an infinite cardinal. Comfort pre-order on $ \beta (\alpha )\backslash \alpha $ is defined as follows: for $ p,q \in \beta (\alpha )\backslash \alpha ,\;p{ \leqslant _{\text{c}}}q$ if every $ q$-compact space is $ p$-compact. For $ p \in U(\alpha )$, we let $ {T_{\operatorname{RK} }}(p)$ be the type of $ p$ and $ {T_{\text{c}}}(p) = \{ q \in U(\alpha ):q{ \leqslant _{\text{c}}}p{ \leqslant _{\text{c}}}q\} $. Since $ {T_{\text{c}}}(p)$ is a union of types, it is natural to define $ {c_p} = \vert\{ {T_{\operatorname{RK} }}(q):q \in {T_{\text{c}}}(p)\} \vert$. It is evident that $ \omega \leqslant {c_p} \leqslant {2^\alpha }$ for $ p \in U(\alpha )$. We show that if $ p \in U(\alpha )$ then $ \vert{T_{\operatorname{RK} }}(p)\vert = {\vert^\alpha }\alpha /p\vert$, and we use this equality to prove that $ {c_p} = {2^\alpha }$ whenever $ p$ is decomposable. We also note that if $ p$ is countably incomplete then $ {2^\omega } \leqslant {c_p} \leqslant {2^\alpha }$; if $ p$ is RK-minimal (selective) and $ \omega < \alpha $ then $ {c_p} = \omega $ and $ {c_p} = {2^{{\aleph _n}}}$ for each $ p \in U({\aleph _n})$ and for $ n < \omega $. Finally, we prove that, if $ \alpha $ is a strong limit and $ p \in U(\alpha )$ is indecomposable, then $ {\beta _p}(\alpha )$ is a $ p$-compact, noninitially $ \alpha $-compact space, where $ {\beta _p}(\alpha )$ is the $ p$-compactification of $ \alpha $.


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DOI: https://doi.org/10.1090/S0002-9939-1994-1170543-1
Keywords: Comfort pre-order, Rudin-Keisler order, type, Comfort type, $ p$-compact space, RK-minimal, countably incomplete, $ \alpha $-complete, decomposable, indecomposable, tensor product of ultrafilters, ultraproducts
Article copyright: © Copyright 1994 American Mathematical Society