Jump to content

Frink ideal

From Wikipedia, the free encyclopedia

This is an old revision of this page, as edited by 2001:861:3ac0:4fa0:5422:2f4e:bc6c:1682 (talk) at 08:54, 16 October 2019 (→‎Basic definitions). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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 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

References

  • Frink, Orrin (1954). "Ideals in Partially Ordered Sets". American Mathematical Monthly. 61: 223–234. doi:10.2307/2306387. MR 0061575.
  • Niederle, Josef (2006). "Ideals in ordered sets". Rendiconti del Circolo Matematico di Palermo. 55: 287–295. doi:10.1007/bf02874708.