Definition 7

Let $E$ be a commutative monoid, $S$ a subset of $E$ and $S'$ the submonoid of $E$ generated by $S$. The quotient monoid $(E \times S')/R$, where the equivalence relation $R$ is as described in Lemma 1, is denoted by $E_S$ and is called the monoid of fractions† of $E$ associated with $S$ (or with denominators in $S$).

For $a \in E$ and $p \in S'$ the class of $(a, p)$ modulo $R$ is in general denoted by $a/p$ and called the fraction with numerator $a$ and denominator $p$. Then by definition $(a/p).(a'/|p') = aa'/|pp'$. The fractions $a/p$ and $a'/|p'$ are equal if and only if there exists $s$ in $S'$ with $spa' = sp'a$; if so, there exist $\sigma$ and $\sigma'$ in $S'$ with $a\sigma = a'\sigma'$ and $p\sigma = p'\sigma'$. In particular, $a/p = sa/sp$ for $a \in E$ and $s, p$ in $S'$. The identity element of $E_S$ is the fraction $e/e$.

Let $a/e = \varepsilon(a)$ for all $a \in E$. The above shows that $\varepsilon$ is a homomorphism of $E$ into $E_S$, called canonical. For all $p \in S'$, $(p/e).(e/p) = e/e$ and hence $e/p$ is the inverse of $\varepsilon(p) = p/e$; every element of $\varepsilon(S')$ is therefore invertible. Then $a/p = (a/e).(e/p)$, whence

for $a \in E$ and $p \in S'$, the monoid $E_S$ is therefore generated by $\varepsilon(E) \cup \varepsilon(S)^*$.