Diagonal intersection is a term used in mathematics, especially in set theory.

If <math>\displaystyle\delta</math> is an ordinal number and <math>\displaystyle\langle X_\alpha \mid \alpha<\delta\rangle</math>

is a sequence of subsets of <math>\displaystyle\delta</math>, then the diagonal intersection, denoted by

:<math>\displaystyle\Delta_{\alpha<\delta} X_\alpha,</math>

is defined to be

:<math>\displaystyle\{\beta<\delta\mid\beta\in \bigcap_{\alpha<\beta} X_\alpha\}.</math>

That is, an ordinal <math>\displaystyle\beta</math> is in the diagonal intersection <math>\displaystyle\Delta_{\alpha<\delta} X_\alpha</math> if and only if it is contained in the first <math>\displaystyle\beta</math> members of the sequence. This is the same as

:<math>\displaystyle\bigcap_{\alpha < \delta} ( [0, \alpha] \cup X_\alpha ),</math>

where the closed interval from 0 to <math>\displaystyle\alpha</math> 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<sub>NS</sub> where I<sub>NS</sub> 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<sub>1</sub> and another gives X<sub>2</sub>, then there is a club C so that X<sub>1</sub> ∩ C = X<sub>2</sub> ∩ C.

A set Y is a lower bound of F in P(κ)/I<sub>NS</sub> 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<sub>NS</sub> a κ<sup>+</sup>-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.