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2014-12-22T14:04:03Z

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	~~In mathematics, a '''Frink ideal''', introduced by~~ [~~[Orrin Frink~~]~~], is a certain kind of subset of a [[partially ordered set]].~~		Just do not [http://www.natwestconstructions.com/brand/coach.asp?Topic=31 www.natwestconstructions.com/brand/coach.asp?Topic=31] getten they<br><br>

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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'', ≤)~~ is ~~a '''Frink ideal''', if~~ the ~~following condition holds~~:

	~~For every finite subset ''S'' of ''P'', ''S'' <math>\subseteq<~~/~~math> ''I'' implies that LU~~(~~''S''~~)~~ ~~<~~math~~>~~\subseteq~~<~~/math~~>~~ ''I'~~'.

	~~A subset ''~~I'~~' of~~ a ~~partially ordered set (''P'',≤~~) ~~is a '''normal ideal''' or a '''cut''' if LU(''~~I~~'') <math>\subseteq<~~/~~math> ''I''~~.

	== ~~Remarks~~==
	~~#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~~.

	==~~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]~~]

en>777sms at 04:09, 14 April 2012

2012-04-14T04:09:52Z

New page

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 [[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'', ≤) is a '''Frink ideal''', if the following condition holds:

For every finite subset ''S'' of ''P'', ''S'' <math>\subseteq</math> ''I'' implies that LU(''S'') <math>\subseteq</math> ''I''.

A subset ''I'' of a partially ordered set (''P'',≤) is a '''normal ideal''' or a '''cut''' if LU(''I'') <math>\subseteq</math> ''I''.

== Remarks==
#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.

==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]]

Reinhardt domain - Revision history

en>Workbit at 14:04, 22 December 2014

en>777sms at 04:09, 14 April 2012