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

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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 X1C = X2C.

A set Y is a lower bound of F in P(κ)/INS only when for any SF there is a club C so that YCS. 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 YC ⊆ ΔF.

This makes the algebra P(κ)/INS a κ+-complete Boolean algebra, when equipped with diagonal intersections.

See also

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References

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This article incorporates material from diagonal intersection on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.