Does anyone have a reference for this notion of "general position" in a poset:
A set $S$ in a poset $(X,\le)$ is said to be in general position if for any $A,B\subseteq S$, $\{x:\forall y\in B,y\le x\}\subseteq \{x:\forall y\in A,y\le x\}$ implies $A\subseteq B$.
The condition can also be written $\bigvee A\le\bigvee B\to A\subseteq B$ in a complete lattice or other poset where the least upper bounds exist. The converse implication is trivial. The terminology is based on projective geometry, where the points are atoms of the poset, lines are joins of two atoms, etc. Here any one or two atoms is in general position, and three atoms are in general position iff they are not collinear (that is, $a\le b\vee c$). This also extends to sets of different kinds of objects, for example a point and a line are in general position iff the point does not lie on the line (i.e. $a\le \ell$). A single poset element is in general position iff it is not the poset zero.
Does this definition look similar to anything from projective geometry or order theory? I'm interested to know what is the usual approach here so I don't get too far afield. In particular, I'm not certain whether to replace "for all $A,B$" with "for all finite $A,B$" in the definition.