# Set of upper bounds is finite for any finite subset

Is there a term to describe a preordered set $P$ in which any finite subset $S \subset P$ has at most finitely many minimal upper bounds? The preordered sets I'm studying generally aren't join-semilattices. Apologies if this is well-known.

The principle example I'm looking at is the set of vertices in a directed graph, where we take $v \leq w$ to mean that there is a directed path from $w$ to $v$.

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I don't know, as I am not an order theorist. However, the terms 'compact' and 'algebraic' spring to mind; perhaps there is a paper out there which contains those terms as well as what you seek. Gerhard "Somewhat Ignorant About Order Theory" Paseman, 2013.04.08 –  Gerhard Paseman Apr 8 '13 at 19:59
I think Gerhard is on the right track: this is a kind of weak compactness property. I would try Continuous Lattices and Domains by Gierz, Hofmann, Keimel, Lawson, Mislove, and Scott, but it's probably obscure even there. –  François G. Dorais Apr 15 '13 at 14:46