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Description: Definitions from Shannon [1990]

Content:

Definitions from Shannon [1990]

Definition:

Assume that \((P,\le )\) is a quasi-order (reflexive and transitive).
  1. \(x\) and \(y\) in \(P\) are compatible if \(\exists z\in P\) such that \(z\le x\) and \(z \le y\).
  2. Incompatible means not compatible.
  3. \(I\subseteq P\) is an antichain if each pair of elements of \(I\) are incompatible.
  4. \(D\subseteq P\) is dense if \(\forall x\in P\), \(\exists y\in D\) such that \(y\le x\).
  5. \(F\subseteq P\) is a filter if
    1. \(\forall x, y\in F\, \exists z\in F\) such that \(z\le x\) and \(z\le y\) and
    2. \(\forall x\in F\forall y\in P\), \(x\le y\) implies \(y\in F\).<\li>
  6. If \({\cal D}\subseteq {\cal P}(P)\), a filter \(F\) is \({\cal D}\) generic if \(F\cap D\neq\emptyset\) for all \(D\) in \({\cal D}\).  \(F\) is \(P\) generic if \(F\cap D\neq\emptyset\) for all dense subsets \(D\) of \(P\).
  7. A chain \(C\) of \((P,\le)\) is a covering chain iff for all \(x\in C\), if \(x\) is not maximal in \(C\), then \(\exists y\in C\) such that \(y\) covers \(x\) in \(P\) (i.e., \(y > x\) and there is no \(z \in  P\) such that \(y > z > x\)).
  8. If \(Q \subseteq  P\) and \(x\in P\), the depth of \(x\) (relative to \(Q\))  is  the order type of the smallest well ordered (under  the  quasi-order) covering chain \(C\) which begins at \(x\) and ends at an  element  of \(Q\).  (The depth of \(x\) is only defined if such a chain exists.)
  9. Two elements \(x\) and \(y\) of \(P\) are \(Q\) strongly incompatible if \(x\) and \(y\) are incompatible and the depth of \(x\) is different  from  the depth of y.
  10. \(I\subseteq  P\) is a \(Q\) strong antichain if any two elements are \(Q\) strongly incompatible.

Shannon [1990] proves that the following are equivalent for each ordinal \(\alpha\):

  1. \(C(\aleph_{\alpha },<\aleph_{0})\)
  2. If \((P,\le)\) is a quasi-order such that \(P \neq\emptyset\) and \(P\) is a union of \(\aleph_{\alpha}\) finite sets and every antichain is finite, then  there  is  a well orderable, \(P\) generic filter.
  3. For some ordinal \(\beta\ge\alpha\), for every quasi-order \((P,\le )\) such that \(P\neq\emptyset\)  and such that \(P\) is the union of \(\aleph_{\alpha}\) finite sets and such that every antichain is finite then for  every family \({\cal D}\) of \(\aleph_{\beta}\) dense sets there is a well orderable \({\cal D}\) generic filter.

Howard-Rubin number: 47

Type: Definitions

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