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Frink ideal

In mathematics, a Frink ideal, introduced by Orrin Frink, is a certain kind of subset of a partially ordered set.

Basic definitions

LU(A) is the set of all common lower bounds of the set of all common upper bounds of the subset A of a partially ordered set.

A subset I of a partially ordered set (P, ≤) is a Frink ideal, if the following condition holds:

For every finite subset S of I, we have LU(S) \subseteq I.

A subset I of a partially ordered set (P, ≤) is a normal ideal or a cut if LU(I) \subseteq I.

Remarks

  1. Every Frink ideal I is a lower set.
  2. A subset I of a lattice (P, ≤) is a Frink ideal if and only if it is a lower set that is closed under finite joins (suprema).
  3. Every normal ideal is a Frink ideal.

References

Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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