In modal logic, we have
$\Box(p\wedge q)\leftrightarrow\Box p\wedge\Box q$
but only
$\Box p\vee\Box q\rightarrow\Box(p\vee q)$,
where $\Box$ is the operator of necessity.
Do the same relations hold in infinitary modal logic, i.e. do we have
$\Box\bigwedge_{n\in\mathbb{N}}p_n\leftrightarrow\bigwedge_{n\in\mathbb{N}}\Box p_n$
but only
$\bigvee_{n\in\mathbb{N}}\Box p_n\rightarrow\Box\bigvee_{n\in\mathbb{N}}p_n$?