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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 $\inf\{\bigvee_{i\in I}a_{if(i)}\mid f:I\to J\text{ is any function} \}$.

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

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