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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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    $\begingroup$ Trees also have this property. $\endgroup$
    – Ramiro de la Vega
    Commented May 9, 2013 at 22:16
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    $\begingroup$ That´s sometimes called a well-joined (well-met for the dual notion) partial order. $\endgroup$
    – Ramiro de la Vega
    Commented May 9, 2013 at 22:50
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    $\begingroup$ 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). $\endgroup$
    – Joel David Hamkins
    Commented May 9, 2013 at 23:33
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    $\begingroup$ What does it mean that $\lbrace a,b\rbrace\leq\lbrace c,d\rbrace$? Both $a$ and $b$ are smaller than both $c$ and $d$? $\endgroup$
    – Asaf Karagila ♦
    Commented May 10, 2013 at 0:03
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    $\begingroup$ For the last example in the post, see mathoverflow.net/questions/130768/… $\endgroup$
    – Andrés E. Caicedo
    Commented May 17, 2013 at 17:05

1 Answer 1

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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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