Frink ideal
In mathematics, a Frink ideal, introduced by Orrin Frink, is a certain kind of subset of a partially ordered set.
Basic definitions
LU(A) is the set of all common lower bounds of the set of all common upper bounds of the subset A of a partially ordered set.
A subset I of a partially ordered set (P, ≤) is a Frink ideal, if the following condition holds:
For every finite subset S of P, S I implies that LU(S) I.
A subset I of a partially ordered set (P,≤) is a normal ideal or a cut if LU(I) I.
Remarks
- Every Frink ideal I is a lower set.
- A subset I of a lattice (P, ≤) is a Frink ideal if and only if it is a lower set that is closed under finite joins (suprema).
- Every normal ideal is a Frink ideal.
Related notions
References
- Frink, Orrin (1954). "Ideals in Partially Ordered Sets". American Mathematical Monthly 61: 223–234. doi:10.2307/2306387. MR 61575.
- Niederle, Josef (2006). "Ideals in ordered sets". Rendiconti del Circolo Matematico di Palermo 55: 287–295. doi:10.1007/bf02874708.
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