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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.

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.

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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user33772
user33772

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