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Gaps in $({\mathcal P}(\omega ),\subset ^*)$ and $(\omega ^{\omega },\le ^*)$


Author: Zoran Spasojevic
Journal: Proc. Amer. Math. Soc. 124 (1996), 3857-3865
MSC (1991): Primary 03E05
DOI: https://doi.org/10.1090/S0002-9939-96-03378-3
MathSciNet review: 1327045
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Abstract: For a partial order $(P,\le _P)$, let $\Gamma (P,\le _P)$ denote the statement that for every $\le _P$-increasing $\omega _1$-sequence $a\subseteq P$ there is a $\le _P$-decreasing $\omega _1$-sequence $b\subseteq P$ on top of $a$ such that $(a,b)$ is an $(\omega _1,\omega _1)$-gap in $P$. The main result of this paper is that $\mathfrak t>\omega _1\leftrightarrow \Gamma(\mathcal P(\omega ),\subset ^*)\leftrightarrow \Gamma (\omega ^\omega ,\le ^*)$. It is also shown, as a corollary, that $\Gamma (\omega ^\omega ,\le ^*)\to \mathfrak b>\omega _1$ but $\mathfrak b>\omega _1\not \to\Gamma (\omega ^\omega ,\le ^*)$.


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Additional Information

Zoran Spasojevic
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: Institute of Mathematics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel
Email: zoran@math.huji.ac.il

DOI: https://doi.org/10.1090/S0002-9939-96-03378-3
Received by editor(s): September 6, 1994
Received by editor(s) in revised form: March 27, 1995
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1996 American Mathematical Society