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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?

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(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.)

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  • $\begingroup$ Many thanks for this answer. I like in particular the 3rd example, because it is in spirit related to the example I have in mind: A recent result of C. M. Ringel [Minimal infinite submodule-closed subcategories, available from math.uni-bielefeld.de/sfb701/preprints/] says that a specific poset attached to the representations of a finite dimensional algebra has this property. $\endgroup$
    – Henning Krause
    Commented May 29, 2011 at 22:02

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