Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 120 / 1 / b / Solution

A Heyting algebra is a bounded lattice $H$ equipped with an operation $a\Rightarrow b$ satisfyingThus, for fixed $a$, the map $x\mapsto x\wedge a$ is left adjoint to $y\mapsto a\Rightarrow y$. A left adjoint preserves joins, soThis is one distributive law for lattices; the other follows from it and the absorption laws. Hence every Heyting algebra is a distributive lattice. This argument is the distributivity of a Heyting algebra.