Is it true that in a partial order the intersection of two upper cones is either disjoint or again an upper cone? Suppose $(P, \leq)$ is a partially ordered set. 
For $x \in P$, define $U_x := \{ y \in P \ | \ y \geq x\}$.
Is it true that for any $x,y \in P$, either $U_x \cap U_y = \emptyset$ or $U_x \cap U_y = U_z$ for some $z \in P$ ?

No. Consider the set $P=\{\{a\},\{b\},\{a,b,c\},\{a,b,d\}\}$ partially ordered by inclusion. $U_{\{a\}}\cap U_{\{b\}}=\{\{a,b,c\},\{a,b,d\}\}$, which is neither empty nor of the form $U_z$ for some $z\in P$.
