What do you call a poset with this property? For any elements $a,b,c,d$ such that $\{a,b\}\le\{c,d\}$, there is an element x such that $\{a,b\}\le x\le\{c,d\}$. (Equivalently, for any finite sets $A\le B$, there is an element x such that $A\le x\le B$.) 


For example, any upper or lower semilattice has this property. Also, overlooking the fact that it's not a set, the class of all cardinal numbers has this property in ZF.



Does this property of a partially ordered set have a name?
In the world of partially ordered abelian groups, this is the *interpolation property*.
These groups are called *partially ordered abelian groups with interpolation*, or simply *interpolation groups*. Intuitively, I think about them as "almost as nice as lattice ordered abelian groups".


A simple example of a non-lattice ordered interpolation group is the set of all
polynomial functions $\mathbb R\to\mathbb R$.


Probably the most important subclass is the class of *dimension groups*. As proved by Effros, Handelman and Shen [in 1980](http://www.jstor.org/discover/10.2307/2374244?uid=3739024&uid=2129&uid=2&uid=70&uid=4&sid=21102296591717), dimension groups classify
the [approximate finite dimensional $C^\*$-algebras](http://en.wikipedia.org/wiki/Approximately_finite_dimensional_C%2A-algebra) (via the $K\_0$ functor).


I recommend this [book by Goodearl](http://books.google.sk/books/about/Partially_Ordered_Abelian_Groups_With_In.html?id=dXUsGysPqRUC&redir_esc=y) -- very readable, he is an excellent writer.


