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 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 is a Frink ideal, if the following condition holds:
For every finite subset S of I, we have LU I.
A subset I of a partially ordered set is a normal ideal or a cut if LU I.

Remarks

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

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