In the theory of partially ordered sets, a pseudoideal is a subset characterized by a bounding operator LU.
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 I implies that LU(S) 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 I implies that LU(S) I.
- 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.