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-In mathematics, a '''Frink ideal''', introduced by [[Orrin Frink]], is a certain kind of subset of a [[partially ordered set]].
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-LU(''A'') is the set of all [[lower bound]]s of the set of all [[upper bound]]s of the subset ''A'' of a [[partially ordered set]].
-A  subset ''I'' of a partially ordered set (''P'',&nbsp;≤) is a '''Frink ideal''', if the following condition holds:
-For every finite subset ''S'' of ''P'', ''S''&nbsp;<math>\subseteq</math>&nbsp;''I''  implies that LU(''S'')&nbsp;<math>\subseteq</math>&nbsp;''I''.
-A  subset ''I'' of a partially ordered set (''P'',≤) is a '''normal ideal''' or a '''cut''' if LU(''I'')&nbsp;<math>\subseteq</math>&nbsp;''I''.
-== Remarks==
-#Every Frink ideal ''I'' is a [[lower set]].
-#A subset ''I'' of a lattice (''P'',&nbsp;≤) 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.
-==Related notions==
-*[[pseudoideal]]
-*[[Doyle pseudoideal]]
-==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=Niederle, Josef|title=Ideals in ordered sets|journal=[[Rendiconti del Circolo Matematico di Palermo]]|volume=55|year=2006|pages=287–295}}
-[[Category:Order theory]]

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