Frink ideal: Difference between revisions

Content deleted Content added

Inline

Revision as of 08:21, 4 May 2014

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

LU(A) is the set of all lower bounds of the set of all 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 $\subseteq$ I implies that LU(S) $\subseteq$ I.

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

Every Frink ideal I is a lower set.
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).
Every normal ideal is a Frink ideal.

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.

@@ Line 20: / Line 20: @@
 ==References==
-*{{cite journal|author=Frink, Orrin|title=Ideals in Partially Ordered Sets|journal=[[American Mathematical Monthly]]|volume=61|year=1954|pages=223–234|mr=61575}}
+*{{cite journal|author=Frink, Orrin|title=Ideals in Partially Ordered Sets|journal=[[American Mathematical Monthly]]|volume=61|year=1954|pages=223–234|mr=61575|doi=10.2307/2306387}}
-*{{cite journal|author=Niederle, Josef|title=Ideals in ordered sets|journal=[[Rendiconti del Circolo Matematico di Palermo]]|volume=55|year=2006|pages=287–295}}
+*{{cite journal|author=Niederle, Josef|title=Ideals in ordered sets|journal=[[Rendiconti del Circolo Matematico di Palermo]]|volume=55|year=2006|pages=287–295|doi=10.1007/bf02874708}}
 [[Category:Order theory]]