Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory. If $\delta$ is an ordinal number and $⟨X_{\alpha }\mid \alpha <\delta ⟩$ is a sequence of subsets of $\delta$, then the diagonal intersection, denoted by

${\mathrm{\Delta }}_{\alpha <\delta }{X}_{\alpha },$

is defined to be

$\{\beta <\delta \mid \beta \in \bigcap _{\alpha <\beta }{X}_{\alpha }\}.$

That is, an ordinal $\beta$ is in the diagonal intersection ${\mathrm{\Delta }}_{\alpha <\delta }{X}_{\alpha }$ if and only if it is contained in the first $\beta$ members of the sequence. This is the same as

$\bigcap _{\alpha <\delta }([0,\alpha ]\cup X_{\alpha }),$

where the closed interval from 0 to $\alpha$ 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 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.