Exercise 2, § 1

Let $\top$ be a law of composition not everywhere defined on a set E. Let $E'$ be the subset of $\mathscr{P}(E)$ consisting of the sets $\{x\}$ where $x$ runs through E and the empty subset $\varnothing$ of E; show that $E'$ is a stable subset of $\mathscr{P}(E)$ under the law $(X, Y) \mapsto X \top Y$ (Exercise 1); deduce that, if $\overline{E}$ denotes the set obtained by adjoining (Set Theory, II, § 4, no. 8) to E an element $\omega$, the law $\top$ can be extended to $\overline{E} \times \overline{E}$, so that the law $\top$ is identical with the law induced on E by this extended law.