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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… – Andrés E. Caicedo May 17 '13 at 17:05
up vote 8 down vote accepted

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.

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