TITLE: Does this property of a partially ordered set have a name?
QUESTION [9 upvotes]: 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.

REPLY [8 votes]: 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, dimension groups classify
the approximate finite dimensional $C^*$-algebras (via the $K_0$ functor).
I recommend this book by Goodearl -- very readable, he is an excellent writer.