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- Pseudoideal

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

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*

*\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*

*\subseteq*

- 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).

- 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.