Diagonal intersection

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

If is an ordinal number and is a sequence of subsets of , then the diagonal intersection, denoted by

is defined to be

That is, an ordinal is in the diagonal intersection if and only if it is contained in the first members of the sequence. This is the same as

where the closed interval from 0 to 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.