A new characterization of complete Heyting and co-Heyting algebras

We give a new order-theoretic characterization of a complete Heyting and co-Heyting algebra $C$. This result provides an unexpected relationship with the field of Nash equilibria, being based on the so-called Veinott ordering relation on subcomplete sublattices of $C$, which is crucially used in Topkis’ theorem for studying the order-theoretic stucture of Nash equilibria of supermodular games.