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


 * $$\displaystyle\Delta_{\alpha<\delta} X_\alpha,$$

is defined to be


 * $$\displaystyle\{\beta<\delta\mid\beta\in \bigcap_{\alpha<\beta} X_\alpha\}.$$

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


 * $$\displaystyle\bigcap_{\alpha < \delta} ( [0, \alpha] \cup X_\alpha ),$$

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)/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.