Exercise 3, § 3

Let $\top$ and $\perp$ be two internal laws on a set $E$ such that $\perp$ is right distributive with respect to $\top$.

(a) If the law $\top$ has an identity element $e$, $x \perp e$ is idempotent under the law $\top$ for all $x \in E$; if further there exists $y$ such that $x \perp y$ is cancellable under the law $\top$, then $x \perp e = e$.

(b) If the law $\perp$ has an identity element $u$ and there exists $z \in E$ which is cancellable for both laws $\perp$ and $\top$, $u$ is cancellable under the law $\top$.