For Boolean algebras with operators, we know that if a variety/universal class of BAOs is closed under taking MacNeille completions, then it is also closed under taking canonical extensions. Universal classes correspond to algebra classes defined by formulas universally quantified, so they are in the Pi_1 hierarchy. If we consider Pi_2 statements (i.e. statements of the form forall......exists......) and their corresponding inductive classes (closed under taking union of chains or directed limits), would the situation be the same or different?

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