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) [math]\displaystyle{ \subseteq }[/math] I.

A subset I of a partially ordered set (P, ≤) is a normal ideal or a cut if LU(I) [math]\displaystyle{ \subseteq }[/math] 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

• pseudoideal
• Doyle pseudoideal

References

• Frink, Orrin (1954). "Ideals in Partially Ordered Sets". American Mathematical Monthly 61 (4): 223–234. doi:10.2307/2306387. 
• Niederle, Josef (2006). "Ideals in ordered sets". Rendiconti del Circolo Matematico di Palermo 55: 287–295. doi:10.1007/bf02874708.