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${\displaystyle \subseteq }$ I implies that LU(S) ${\displaystyle \subseteq }$ 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${\displaystyle \subseteq }$ I implies that LU(S) ${\displaystyle \subseteq }$ I.

Remarks

1. Every Frink ideal I is a Doyle pseudoideal.
2. 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).