Pseudoideal

In the theory of partially ordered sets, a pseudoideal is a subset characterized by a bounding operator LU.

Basic definitions

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 Doyle pseudoideal, if the following condition holds:

For every finite subset S of P that has a supremum in P, S $\subseteq$ I implies that LU(S) $\subseteq$ 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 that has a supremum in P, S $\subseteq$ I implies that LU(S) $\subseteq$ 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.

This article is issued from Wikipedia - version of the Wednesday, April 09, 2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.