Exercise 2, § 2

Let $\tau$ be a law everywhere defined on E. (a) For $\tau$ to be associative, it is necessary and sufficient that every left translation $\gamma_x$ be permutable with every right translation $\delta_y$ in the set of mappings of E into E (with the law $f \circ g$). (b) Suppose that E has an identity element. For $\tau$ to be associative and commutative, it is necessary and sufficient that the mapping $(x, y) \mapsto x \tau y$ be a morphism of the magma $E \times E$ into the magma E.