Diagonal intersection (original) (raw)

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

If δ {\displaystyle \displaystyle \delta } {\displaystyle \displaystyle \delta } is an ordinal number and ⟨ X α ∣ α < δ ⟩ {\displaystyle \displaystyle \langle X_{\alpha }\mid \alpha <\delta \rangle } {\displaystyle \displaystyle \langle X_{\alpha }\mid \alpha <\delta \rangle }is a sequence of subsets of δ {\displaystyle \displaystyle \delta } {\displaystyle \displaystyle \delta }, then the diagonal intersection, denoted by

Δ α < δ X α , {\displaystyle \displaystyle \Delta _{\alpha <\delta }X_{\alpha },} {\displaystyle \displaystyle \Delta _{\alpha <\delta }X_{\alpha },}

is defined to be

{ β < δ ∣ β ∈ ⋂ α < β X α } . {\displaystyle \displaystyle \{\beta <\delta \mid \beta \in \bigcap _{\alpha <\beta }X_{\alpha }\}.} {\displaystyle \displaystyle \{\beta <\delta \mid \beta \in \bigcap _{\alpha <\beta }X_{\alpha }\}.}

That is, an ordinal β {\displaystyle \displaystyle \beta } {\displaystyle \displaystyle \beta } is in the diagonal intersection Δ α < δ X α {\displaystyle \displaystyle \Delta _{\alpha <\delta }X_{\alpha }} {\displaystyle \displaystyle \Delta _{\alpha <\delta }X_{\alpha }} if and only if it is contained in the first β {\displaystyle \displaystyle \beta } {\displaystyle \displaystyle \beta } members of the sequence. This is the same as

⋂ α < δ ( [ 0 , α ] ∪ X α ) , {\displaystyle \displaystyle \bigcap _{\alpha <\delta }([0,\alpha ]\cup X_{\alpha }),} {\displaystyle \displaystyle \bigcap _{\alpha <\delta }([0,\alpha ]\cup X_{\alpha }),}

where the closed interval from 0 to α {\displaystyle \displaystyle \alpha } {\displaystyle \displaystyle \alpha } 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 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 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)/INS a κ+-complete Boolean algebra, when equipped with diagonal intersections.

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