View source for Diagonal intersection

'''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 number|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 Δ''F'' of ''F'' plays the role of [[infimum|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<sub>NS</sub>'' a κ<sup>+</sup>-complete Boolean algebra, when equipped with diagonal intersections.

==See also==

* [[Club set]]
* [[Fodor's lemma]]

==References==
{{reflist}}
{{refbegin}}
* [[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.
{{refend}}
{{PlanetMath attribution|id=3233|title=diagonal intersection}}

{{Mathematical logic}}
{{Set theory}}

[[Category:Ordinal numbers]]
[[Category:Set theory]]


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