# Does this property of a partially ordered set have a name?

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.

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Trees also have this property. – Ramiro de la Vega May 9 '13 at 22:16
That´s sometimes called a well-joined (well-met for the dual notion) partial order. – Ramiro de la Vega May 9 '13 at 22:50
In analogy with the LUB property, you could call it the DUB property (directed upper bound property), since you are saying that every bounded-above finite set has a directed collection of upper bounds (and similarly for lower bounds). – Joel David Hamkins May 9 '13 at 23:33
What does it mean that $\lbrace a,b\rbrace\leq\lbrace c,d\rbrace$? Both $a$ and $b$ are smaller than both $c$ and $d$? – Asaf Karagila May 10 '13 at 0:03
For the last example in the post, see mathoverflow.net/questions/130768/… – Andrés E. Caicedo May 17 '13 at 17:05

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).