Consider a poset $P$ and suppose that every finite subset admits a supremum. Call an ideal $I$ of $P$ *minimal infinite* if it is infinite and every ideal properly contained in $I$ is finite. I am interested in the following "compactness property": Every infinite ideal of $P$ contains a minimal infinite ideal.


Question: Has this property studied before? Are there posets arising in "mathematical nature" having this property?



A compactness property of posets
(I will write "semilattice" for "Posets in which every finite subset has a supremum".)


This is a partial answer only. I am not aware of a name for this property, but there are many examples in "mathematical nature" with this property:


1. (All finite semilattices. Not a good example.)
2. The natural numbers
3. Let V be a vector space over an infinite field K. Then the subspace lattice of V has your property. The minimal infinite ideals are exactly the principal ideals $(U]$, where $U$ is any 2-dimensional subspaces.
4. If P is any infinite semilattice with your property, and Q is any semilattice, then the vertical sum P+Q (if p in P, q in Q, then p is below q) has your property, too. (For example, all well-orders have this property.)


