Heyting algebra

Remarks

Clause 3 says that $a \to b$ is the largest element $c$ with $c \wedge a \le b$, so $\to$ is determined by the lattice.

Example: let $X$ be a topological space and let $H$ be the set of its open sets, with $\wedge = \cap$, $\vee = \cup$, $0 = \emptyset$, $1 = X$, and $U \to V$ the interior of $(X \setminus U) \cup V$. For an open set $W$, the inclusion $W \subseteq (X \setminus U) \cup V$ holds if and only if $W \cap U \subseteq V$. Since $W$ is open, the first inclusion holds if and only if $W$ lies in the interior; this is clause 3.

Non-example: the five-element lattice $M_3$ has a least element, a greatest element and three pairwise incomparable elements between them. It is a bounded lattice but carries no operation $\to$ satisfying clause 3. Indeed, every Heyting algebra is a distributive lattice, and $M_3$ is not distributive.

A Boolean algebra is a Heyting algebra with $a \to b = \neg a \vee b$, and a Heyting algebra is Boolean if and only if $a \vee \neg a = 1$ for all $a$. Every finite distributive lattice is a Heyting algebra.