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Ultrafilters on \(\omega\) and atoms in the lattice of uniformities. II. (English) Zbl 0657.54023
The authors continue their investigations on the interaction between ultrafilters and uniformities (on a countable set) [see part I reviewed above]. Under CH they study proximally discrete atoms in the lattice of uniformities on \(\omega\). Various ultrafilters on \(\omega\) are constructed so that atoms in the lattice of uniformities refining the corresponding ultrafilter-uniformities have special properties. Methods from finite combinatorics are used. The following results are typical. Let \(1\leq s<\omega\) and let \({\mathcal G}\) be an ultrafilter on \(\omega\). Then there is an ultrafilter \({\mathcal F}>{\mathcal G}\) on \(\omega\) such that there are precisely s distinct atoms refining \({\mathcal U}_{{\mathcal F}}\). All these atoms are zero-dimensional. (Here \(>\) denotes the Rudin-Keisler order.) Let \({\mathcal G}\) be an ultrafilter on \(\omega\). Then there exists an ultrafilter \({\mathcal F}>{\mathcal G}\) on \(\omega\) such that there are \(2^ c\) distinct atoms refining \({\mathcal U}_{{\mathcal F}}\). There exists a uniformity N, generated by an ultrafilter \({\mathcal F}\) and a partition \({\mathcal R}\) into finite sets on \(\omega\), such that (i) \({\mathcal N}\) is an atom in the lattice of zero-dimensional uniformities on \(\omega\), (ii) \({\mathcal N}\) is not an atom in the lattice of all uniformities on \(\omega\) ; in fact, there are at least \(2^{\omega}\) pairwise uniformly non- homeomorphic atoms finer than \({\mathcal N}\), and these atoms are non-zero- dimensional.
All these non-zero-dimensional atoms have the same distal modification, viz. \({\mathcal N}\). Thus d\({\mathcal A}_ 1\wedge d{\mathcal A}_ 2\neq d({\mathcal A}_ 1\wedge {\mathcal A}_ 2)\) for any couple \({\mathcal A}_ 1,{\mathcal A}_ 2\) of these atoms. (For any uniformity \({\mathcal U}\), \(d{\mathcal U}\) denotes the distal modification of \({\mathcal U}\), i.e. the uniformity whose basis consists of all covers of finite order in \({\mathcal U}.)\)
Reviewer: H.-P.Künzi

MSC:
54E15 Uniform structures and generalizations
03E50 Continuum hypothesis and Martin’s axiom
05C55 Generalized Ramsey theory
05C65 Hypergraphs
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