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

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...

.

Basic definitions

LU(A) is the set of all lower bounds of the set of all upper bound

Upper bound

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...

s of the subset A of a partially ordered set

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...

.

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
   If and only if
   In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
   
    it is a lower set that is closed under finite joins (suprema).
3. Every normal ideal is a Frink ideal.