> I don't see why you say that. Are you saying that any poset with this chain of adjunctions must be a Boolean algebra? That's imaginable, but I don't know such a theorem.

I think so.

Here's my sketch of the proof: in a Heyting algebra with the left adjoint \$$\setminus\$$ to \$$\vee\$$, first show \$$\top \leq \top \setminus\ (\top \setminus\ a) \to a\$$ from the adjunction laws. After that, along with \$$\top \leq a \to b \to a\$$ and \$$\top \leq (a \to b \to c) \to (a \to b) \to a \to c\$$ and modus ponens \$$\top \leq a \to b \Longrightarrow \top \leq a \Longrightarrow \top \leq b\$$ you have enough to show all of classical propositional logic. Then, using \$$\top \leq a \to b \Longleftrightarrow a \leq b\$$ you can use classical propositional logic to recover the traditional axioms for a Boolean algebra.