Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory. If \displaystyle\delta is an ordinal number and \displaystyle\langle X_\alpha \mid \alpha is a sequence of subsets of \displaystyle\delta , then the diagonal intersection, denoted by : \displaystyle\Delta_{\alpha is defined to be : \displaystyle\{\beta That is, an ordinal \displaystyle\beta is in the diagonal intersection \displaystyle\Delta_{\alpha if and only if it is contained in the first \displaystyle\beta members of the sequence. This is the same as : \displaystyle\bigcap_{\alpha where the closed interval from 0 to \displaystyle\alpha is used to avoid restricting the range of the intersection.

Diagonal intersection is a term used in mathematics, especially in set theory. If \displaystyle\delta is an ordinal number and \displaystyle\langle X_\alpha \mid \alpha is a sequence of subsets of \displaystyle\delta , then the diagonal intersection, denoted by \displaystyle\Delta_{\alpha is defined to be \displaystyle\{\beta That is, an ordinal \displaystyle\beta is in the diagonal intersection \displaystyle\Delta_{\alpha if and only if it is contained in the first \displaystyle\beta members of the sequence. This is the same as \displaystyle\bigcap_{\alpha where the closed interval from 0 to \displaystyle\alpha is used to avoid restricting the range of the intersection. ==Relationship to the Nonstationary Ideal== For κ an uncountable regular cardinal, in the Boolean algebra P(κ)/I NS where I NS 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 X 1 and another gives X 2 , then there is a club C so that X 1 ∩ C = X 2 ∩ C. A set Y is a lower bound of F in P(κ)/I NS 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 makes the algebra P(κ)/I NS a κ + -complete Boolean algebra, when equipped with diagonal intersections. ==See also== Club set Fodor's lemma ==References== Thomas Jech, Set Theory, The Third Millennium Edition, Springer-Verlag Berlin Heidelberg New York, 2003, page 92, 93. Akihiro Kanamori, The Higher Infinite, Second Edition, Springer-Verlag Berlin Heidelberg, 2009, page 2.

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