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In the theory of [[partially ordered set]]s, a '''pseudoideal''' is a subset characterized by a bounding operator LU.
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== 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 '''Doyle pseudoideal''', if the following condition holds:
 
For every finite subset ''S'' of ''P'' which has a [[supremum]] in ''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 '''pseudoideal''', if the following condition holds:
 
For every subset ''S'' of ''P'' having at most two elements which has a [[supremum]] in ''P'', ''S''<math>\subseteq</math> ''I''  implies that LU(''S'') <math>\subseteq</math> ''I''.
 
== Remarks==
#Every [[Frink ideal]] ''I'' is a Doyle pseudoideal.
#A subset ''I'' of a lattice (''P'',≤) is a Doyle pseudoideal [[if and only if]] it is a lower set that is closed under finite joins ([[suprema]]).
 
==Related notions==
*[[Frink ideal]]
 
==References==
*Abian, A., Amin, W. A. (1990) "Existence of prime ideals and ultrafilters in partially ordered sets", Czechoslovak Math. J., 40: 159–163.
*Doyle, W.(1950) "An arithmetical theorem for partially ordered sets", [[Bulletin of the American  Mathematical Society]], 56: 366.
*Niederle, J. (2006) "Ideals in ordered sets", [[Rendiconti del Circolo Matematico di Palermo]] 55: 287–295.
 
[[Category:Order theory]]

Revision as of 07:34, 20 February 2014

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