Pseudoideal

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 which has a supremum in P, S$\subseteq$ I implies that 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 which has a supremum in P, S$\subseteq$ I implies that LU(S) $\subseteq$ I.

Remarks

1. Every Frink ideal I is a Doyle pseudoideal.
2. 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

References

• Abian, A., Amin, W. A., Existence of prime ideals and ultrafilters in partially ordered sets. Czechoslovak Math. J., 40 (1990), 159–163.
• Doyle, W., An arithmetical theorem for partially ordered sets. Bull. Amer. Math. Soc., 56 (1950), 366.
• Niederle, J., Ideals in ordered sets. Rendiconti Circ. Math. Palermo, 55 (2006), 287–295.