Is there a Boolean lattice $(X,\le)$ and a nonempty collection $(a\_{ij})\_{i\in I,j\in J}\subseteq X$ such that
$$\bigvee\_{i\in I}\bigwedge\_{j\in J}{a\_{ij}}\ne \bigwedge\_{f:I\to J}\bigvee\_{i\in I}a\_{if(i)}\quad ?$$


where $\bigvee$ stands for $\sup$ and $\bigwedge\_{f:I\to J}\bigvee\_{i\in I}a\_{if(i)}$ is $\inf\{\bigvee\_{i\in I}a\_{if(i)}\mid f:I\to J\text{ is any function} \}$.



Existence of more distributive Boolean lattices
Yes as soon as $X$ is not atomic. See Handbook of Boolean algebras, Theorem 14.5 (page 215 in Volume 1)


