Let $(L,\land,\lor)$ be a complete distributive lattice. Given $x\neq y \in L$, is there a finite set ${\cal I}$ of closed intervals in $L$ such that

- no member of ${\cal I}$ contains both $x$ and $y$, and
- $\bigcup {\cal I} = L$

?

(A *closed interval* in $L$ is a subset of the form $[a, b] = \{x\in L: a\leq x\leq b\}$ where $a\leq b \in L$.)

completelydistributive? (i.e. do arbitrary joins distribute over arbitrary meets?) $\endgroup$ – Ilya Bogdanov Sep 29 '17 at 9:19