Description:
Form 322 (\(KW(WO,\infty)\)) is true in \(\cal N1\)
Content:
In this note we prove
Form 322 (\(KW(WO,\infty)\)) is true in \(\cal N1\).
(Brunner [1982a], (Proposition 4.5) also has a proof of this result.)
Assume that \(X\) is a well ordered set in \(\cal N1\) and that \(\forall y\in X\), \(|y|\ge 2\). Let
\(E\) be a (finite) support of a well ordering of \(X\) and let fix\((E)\) be the group of permutations of \(A\) (the set of atoms) which
fix \(E\) pointwise. Then for all \(\phi\in \hbox{fix}(E)\), \(\phi\) fixes \(X\) pointwise. Choose an atom \(a \in A-E\). We will show
that for every \(y\in X\), \(E\cup \{a\}\) is a support of a non-empty proper subset \(y'\) of \(y\). This will suffice since it follows that
the function \(y\mapsto y'\) is in \(\cal N1\).
Choose an element \(y\in X\). If every element of \(y\) has support \(E\) then any non-empty proper subset of \(y\) has support \(E\) and
therefore has support \(E\cup \{a\}\). So we may assume that \(y\) has some element \(t\) such that \(E\) is not a support of \(t\). Choose
a finite non-empty subset\(F\) of \(A\) such that \(E\cap F=\emptyset\) and \(E\cup F\) is a support of \(t\) and such that \(F\) is minimal
in the sense that for no proper subset \(F'\) of \(F\) is \(E\cup F'\) a support of \(t\). We may assume without loss of generality that
\(a\in F\). (If \(a\not\in F\) then we replace \(t\) by \(\psi(t)\) and \(F\) by \(\psi(F)\) where \(\psi\) is a permutation of \(A\) chosen
so that \(\psi\in\hbox{fix}(E)\) and \(a\in \psi(F)\).) Let \(y' = \{\phi(t) : \phi\in\hbox{fix}(E\cup\{a\})\}\). Clearly \(y'\) has support
\(E\cup\{a\}\), \(y'\subseteq y\) and \(y'\ne\emptyset\) (since\(t\in y'\)). It only remains to show that \(y'\ne y\).
If \(\beta\) and \(\gamma\) are in fix\((E)\) and \(\beta(F) \ne \gamma(F)\) then \(\beta(t) \ne \gamma(t)\).
Assume \(b\in \beta(F) - \gamma(F)\). Since \(F\) is minimal, there is no proper subset \(B\) of \(\beta(F)\) for which \(E\cup B\) is a
support of \(\beta(t)\). Hence there is a \(\psi\in \hbox{fix}((E\cup\beta(F))-\{b\})\) such that \(\psi(\beta(t)) \ne \beta(t)\). We can
find such a \(\psi\) for which \(\psi(b)\notin \gamma(F)\). (If \(\psi(b)\in\gamma(F)\) then we choose \(d\in A -(\gamma(F)\cup \beta(F)\cup E)\).
Since\(\psi(\beta)\notin\beta(F)\) the cycle \(\rho =(d,\psi(b))\) of length two fixes \(E\cup \beta(F)\) pointwise so that
\(\rho(\beta(t)) = \beta(t)\). If we let \(\psi' = \rho\circ\psi\) then \(\psi'\) fixes \((E \cup \beta(F))-\{b\}\) pointwise. Further,
since \(\rho(\beta(t))=\beta(t)\) and \(\psi(\beta(t))\ne \beta(t)\), we conclude that \(\psi'(\beta(t)) = \rho(\psi(\beta(t))) \ne \beta(t)\).
We can therefore replace \(\psi\) by \(\psi'\).)
Let \(\sigma\) be the cycle \((b,\psi(b))\). Since \(\sigma\) agrees with \(\psi\) on a support of \(\beta(t)\),
\(\sigma(\beta(t)) = \psi(\beta(t))\ne \beta(t)\). Since \(\sigma \) fixes \(\gamma(F)\) pointwise, \(\sigma(\gamma(t)) = \gamma(t)\).
Therefore \(\beta(t) \ne\gamma(t)\). This completes the proof of the lemma.
Choose an \(\eta\in\hbox{fix}(E)\) so that \(\eta(F)\cap F = \emptyset\). Then for every \(\phi\in \hbox{fix}(E\cup\{a\})\),
\(\eta(F)\ne \phi(F)\). It follows from the lemma that for all such \(\phi\), \(\eta(t)\ne \phi(t)\). Therefore \(\eta(t)\in (y-y')\).
Howard-Rubin number:
66
Type:
Proof
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