Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory. If [math]\displaystyle{ \displaystyle\delta }[/math] is an ordinal number and [math]\displaystyle{ \displaystyle\langle X_\alpha \mid \alpha\lt \delta\rangle }[/math] is a sequence of subsets of [math]\displaystyle{ \displaystyle\delta }[/math], then the diagonal intersection, denoted by

[math]\displaystyle{ \displaystyle\Delta_{\alpha\lt \delta} X_\alpha, }[/math]

is defined to be

[math]\displaystyle{ \displaystyle\{\beta\lt \delta\mid\beta\in \bigcap_{\alpha\lt \beta} X_\alpha\}. }[/math]

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

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

where the closed interval from 0 to [math]\displaystyle{ \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_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 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)/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.