# Pseudoideal

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In the theory of [partially ordered set](/source/partially_ordered_set)s, a '''pseudoideal''' is a subset characterized by a bounding operator LU.
== Basic definitions ==
LU(''A'') is the set of all [lower bound](/source/lower_bound)s of the set of all [upper bound](/source/upper_bound)s of the subset ''A'' of a [partially ordered set](/source/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'' that has a [supremum](/source/supremum) in ''P'', if <math>S\subseteq I</math> then <math>\operatorname{LU}(S)\subseteq I</math>.

A  subset ''I'' of a partially ordered set (''P'',&nbsp;≤) is a '''pseudoideal''', if the following condition holds:

For every subset ''S'' of ''P'' having at most two elements that has a [supremum](/source/supremum) in ''P'', if ''S'' <math>\subseteq</math> ''I''  then LU(''S'') <math>\subseteq</math> ''I''.

== Remarks==
#Every [Frink ideal](/source/Frink_ideal) ''I'' is a Doyle pseudoideal.
#A subset ''I'' of a lattice (''P'',&nbsp;≤) is a  Doyle pseudoideal [if and only if](/source/if_and_only_if) it is a lower set that is closed under finite joins ([suprema](/source/suprema)).

==Related notions==
*[Frink ideal](/source/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](/source/Bulletin_of_the_American_Mathematical_Society), 56: 366.
*Niederle, J. (2006) "Ideals in ordered sets", [Rendiconti del Circolo Matematico di Palermo](/source/Rendiconti_del_Circolo_Matematico_di_Palermo) 55: 287–295.

Category:Order theory

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