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