Pseudoideal


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

Basic definitions

LU 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 is a Doyle pseudoideal, if the following condition holds:
For every finite subset S of P that has a supremum in P, if then.
A subset I of a partially ordered set 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, if S I then LU I.

Remarks

  1. Every Frink ideal I is a Doyle pseudoideal.
  2. A subset I of a lattice is a Doyle pseudoideal if and only if it is a lower set that is closed under finite joins.

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