Diagonal intersection
Diagonal intersection is a term used in mathematics, especially in set theory.
If 
is defined to be
That is, an ordinal 
![\displaystyle \bigcap _{\alpha <\delta }([0,\alpha ]\cup X_{\alpha }),](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6316e249339d97a2d13066c9d9f1d66a7b5259f)
where the closed interval from 0 to 
01Relationship to the Nonstationary Ideal
For κ an uncountable regular cardinal, in the Boolean algebra P(κ)/INS where INS 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)/INS 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)/INS a κ+-complete Boolean algebra, when equipped with diagonal intersections.
Sources and credits
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