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 $\{a,b\}\le x\le\{c,d\}$. 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.