Interpretations into Heyting algebras (Q1820182)

A variety \({\mathcal V}\) is interpretable in a variety \({\mathcal W}\) if for each \({\mathcal V}\)-operation \(F_ t(x_ 1,...,x_ n)\) there is a \({\mathcal W}\)- term \(f_ t(x_ 1,...,x_ n)\) such that if \((A;G_ s)\) is in \({\mathcal W}\), then \((A;f^ A_ t)\) is in \({\mathcal V}\). In this paper we prove that there are only two interpretations from the variety of bounded distributive lattices into the variety of Heyting algebras, namely, the identity interpretation and its dual. As a corollary there is only one interpretation from the variety of Heyting algebras into itself. The method used is by successively showing that the result holds for larger and larger subvarieties of Heyting algebras.