title: Pseudoideal
text: # 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, if $S\subseteq I$ then $\operatorname {LU} (S)\subseteq I$.
  
  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, if S $\subseteq$ I then 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
  