Question on Heyting algebras

Question

Does $a \Rightarrow b = 1$ iff $a≤b$ hold for any complete Heyting algebra? If not, please provide a counterexample.

Answer 1

Yes; in fact, the Heyting algebra need not be complete.

If $a \leq b$, then $1 \land a = a \leq b$, so $1 \leq a \Rightarrow b$. Therefore, $1 = a \Rightarrow b$.

If $a \Rightarrow b = 1$, then $1 \leq a \Rightarrow b$, and therefore $a = 1 \land a \leq b$.