Diagonal Intersection

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

If $\displaystyle\delta$ is an ordinal number and $\displaystyle\langle X_\alpha \mid \alpha<\delta\rangle$
is a sequence of subsets of $\displaystyle\delta$, then the ''diagonal intersection'', denoted by

:$\displaystyle\Delta_ X_\alpha,$

is defined to be

:$\displaystyle\.$

That is, an ordinal $\displaystyle\beta$ is in the diagonal intersection $\displaystyle\Delta_ X_\alpha$ if and only if it is contained in the first $\displaystyle\beta$ members of the sequence. This is the same as

:\displaystyle\bigcap_ ( , \alpha\cup X_\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 ''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 Δ''F'' 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'' ⊆ Δ''F''.

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.

Ordinal numbers
Set theory

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