, then the diagonal intersection, denoted by is defined to be That is, an ordinal
is used to avoid restricting the range of the intersection.
For κ an uncountable regular cardinal, in the Boolean algebra P(κ)/INS where INS is the nonstationary ideal (the ideal dual to the club filter), the diagonal intersection of a κ-sized family of subsets of κ does not depend on the enumeration.
That is to say, if one enumeration gives the diagonal intersection X1 and another gives X2, then there is a club C so that X1 ∩ C = X2 ∩ C. A set Y is a lower bound of F in P(κ)/INS only when for any S ∈ F there is a club C so that Y ∩ C ⊆ S. The diagonal intersection ΔF of F plays the role of greatest lower bound of F, meaning that Y is a lower bound of F if and only if there is a club C so that Y ∩ C ⊆ ΔF.
This article incorporates material from diagonal intersection on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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