Frink ideal

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 P, S  ⊆  I implies that LU(S)  ⊆  I.

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

Related notions