arXiv subjects

Andrzej Starosolski

Publications and source records attributed to Andrzej Starosolski.

7 recordsLinked to original sources

arXiv2020

The Rudin-Kisler ordering of P-points under $\mathfrak{b} = \mathfrak{c}$

M. E. Rudin proved under CH that for each P-point there exists another P-point strictly RK-greater (M. E. Rudin, Partial orders on the types of $β\mathbb{N}$ , Trans. Amer. Math. Soc., 155 (1971), 353-362). Assuming $\mathfrak{p}=\mathfrak{c}$ A. Blass showed the same, and proved that each RK-increasing $ω$-sequence of P-points is upper bounded by a P-point, and that there is an order embedding of the real line into the class of P-points with respect to the RK-(pre)ordering (A. Blass, Rudin - Keisler ordering on P-points, Trans. Amer. Math. Soc., 179 (1973), 145-166). In the present paper the results cited above are proved under a (weaker) assumption $\mathfrak{b}=\mathfrak{c}$. A. Blass also asked in (A. Blass, Rudin - Keisler ordering on P-points, Trans. Amer. Math. Soc., 179 (1973), 145-166) what ordinals can be embedded in the set of P-points and pointed out, that such an ordinal may not be greater then $\mathfrak{c}^+$. In the present paper the question is answered showing (under $\mathfrak{b} = \mathfrak{c}$) that there is an order embedding of $\mathfrak{c}^+$ into P-points.

math.LO↗

arXiv2019

Continuous extension of maps between sequential cascades

The contour of a family of filters along a filter is a set-theoretic lower limit. Topologicity and regularity of convergences can be characterized with the aid of the contour operation. Contour inversion is studied, in particular, for iterated contours of sequential cascades. A related problem of continuous extension of maps between maximal elements of sequential cascades to full subcascades is solved in full generality.

math.GN↗

arXiv2012

Ordinal ultrafilters versus P-hierarchy

An earlier paper, entitled "P-hierarchy on $βω$", investigated the relations between ordinal ultrafilters and the so-called P-hierarchy. This study is continued in the present paper and focuses on the aspects of characterization of classes of finite index, existence, generic existence and the Rudin-Keisler-order.

math.LO↗

arXiv2012

Rudin-Kisler ordering on the P-hierarchy

M. E. Rudin in "Partial orders on the types of $β{N}$", (Trans. Amer. Math. Soc., 155, 1971, 353-362) proved (under CH) that for each P-point $u$ there is a P-point $v$ such that $v>_{RK}u$. A. Blass in "Rudin - Kisler ordering on P-points" (Trans. Amer. Math. Soc. 179, 1973, 145-166) improved that theorem assuming MA in the place of CH, in that paper he also proved that under MA each RK-increasing sequence of P-points is upper bounded by a P-point. We improve Blass results simultaneously in 3 directions - we prove it for each class of index $\geq 2$ of P-hierarchy (P-points coincidence with a class $P_2$ of P-hierarchy), assuming $\mathfrak{b}=\mathfrak{c}$ in the place of MA and we show that there are at least $\mathfrak{b}$ many Rudin-Kisler incomparable such upper bounds.

math.LO↗

arXiv2012

Cascades, Order and Ultrafilters

We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. Using that relation we prove (ZFC) that the class of strict $J_{ω^ω}$-ultrafilters, introduced by J. E. Baumgartner in \textit{Ultrafilters on $ω$}, is empty. We translate the result to the language of $<_\infty$-sequences under an ultrafilter, investigated by C. Laflamme in \textit{A few special ordinal ultrafilters}, to show that if there is an arbitrary long finite $<_\infty$-sequence under $u$ than $u$ is at least strict $J_{ω^{ω+1}}$- ultrafilter.

math.LO↗

arXiv2012

Thin ultrafilters, P-hierarchu and MArtin Axiom

Under MA we prove that for the ideal $\cal I$ of thin sets on $ω$ and for any ordinal $γ\leq ω_1$ there is an ${\cal I}$-ultrafilter (in the sense of Baumgartner), which belongs to the class ${\cal P}_γ$ of P-hierarchy of ultrafilters. Since the class of ${\cal P}_2$ ultrafilters coincides with a class of P-points, out result generalize theorem of Flašková, which states that there are ${\cal I}$-ultrafilters which are not P-points. It is also related to theorem which states that under CH for any tall P-ideal $\cal I$ on $ω$ there is an ${\cal I}$-ultrafilter, however the ideal of thin sets is not P-ideal.

math.LO↗

arXiv2011

How high can Baumgartner's {\cal I}-ultrafilters lie in the P-hierarchy?

Under CH we prove that for any tall ideal $\cal I$ on $ω$ and for any ordinal $γ\leq ω_1$ there is an ${\cal I}$-ultrafilter (in the sense of Baumgartner), which belongs to the class ${\cal P}_γ$ of P-hierarchy of ultrafilters. Since the class of ${\cal P}_2$ ultrafilters coincides with a class of P-points, out result generalize theorem of Flašková, which states that there are ${\cal I}$-ultrafilters which are not P-points.

math.LO↗