<p>This version provides details to fill a gap in a previous version of this article. It is shown that if various cardinal invariants of the continuum related to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_830_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathfrak {d}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">d</mi> </math></EquationSource> </InlineEquation> are equal to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_830_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\aleph _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℵ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> then there is a non-trivial automorphism of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_830_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathscr {P}}({\mathbb {N}})/[{\mathbb {N}}]^{<\aleph _0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msup> <mrow> <mo stretchy="false">[</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mo><</mo> <msub> <mi>ℵ</mi> <mn>0</mn> </msub> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. Some of these results extend to automorphisms of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_830_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathscr {P}}(\kappa )/[\kappa ]^{<\kappa }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mrow> <mo stretchy="false">(</mo> <mi>κ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msup> <mrow> <mo stretchy="false">[</mo> <mi>κ</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mo><</mo> <mi>κ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_830_Article_IEq7.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation> is inaccessible.</p>

Revised version of “Non-trivial automorphisms of ${\mathscr {P}}({\mathbb {N}})/[{\mathbb {N}}]^{<\aleph _0}$ from variants of small dominating number”

期刊论文
发表日期：

2025年5月30日

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Saharon Shelah,
Juris Steprāns

摘要

This version provides details to fill a gap in a previous version of this article. It is shown that if various cardinal invariants of the continuum related to ${\mathfrak {d}}$ $d$ are equal to $\aleph _1$ $ℵ_{1}$ then there is a non-trivial automorphism of ${\mathscr {P}}({\mathbb {N}})/[{\mathbb {N}}]^{<\aleph _0}$ $P (N) / {[N]}^{< ℵ_{0}}$ . Some of these results extend to automorphisms of ${\mathscr {P}}(\kappa )/[\kappa ]^{<\kappa }$ $P (κ) / {[κ]}^{< κ}$ if $\kappa $ $κ$ is inaccessible.

Revised version of “Non-trivial automorphisms of \({\mathscr {P}}({\mathbb {N}})/[{\mathbb {N}}]^{<\aleph _0}\) from variants of small dominating number”

摘要