Exercise 10, § 3

Let E be a commutative monoid whose law is written additively and whose identity element is denoted by 0. Let $\top$ be an internal law on E which is distributive with respect to addition and such that $0 \top x = x \top 0 = 0$ for all $x \in E$. Let S be a subset of E which is stable under addition and under each of the external laws derived from $\top$; let $\overline{E}$ denote the monoid of differences of E with respect to S under addition and $\varepsilon$ the canonical homomorphism of E into $\overline{E}$. Then there exists on $\overline{E}$ one and only one law $\overline{\top}$ which is distributive with respect to addition and such that $\varepsilon$ is a homomorphism for the laws $\top$ and $\overline{\top}$. If the law $\top$ is associative (resp. commutative), so is the law $\overline{\top}$.