Exercise 3, § 1

Let $\top$ be a law of composition not everywhere defined on E.

(a) For the law $(X, Y) \mapsto X \top Y$ (Exercise 1) between subsets of E to be associative, it is necessary and sufficient that, for all $x \in E, y \in E, z \in E$, if one of the two sides of the formula $(x \top y) \top z = x \top (y \top z)$ is defined, so is the other and is equal to it (use Exercise 1 to show that the condition is sufficient).

(b) If this condition holds, show that Theorem 1 of no. 3 can be generalized as follows: if one of the two sides of formula (3) is defined, the other is defined and is equal to it.