Quotient group of a monoid
Definition
Let \(S\) be a cancellative monoid. It is defined the relation \(\sim\) on \(S \times S\) as follows:
\[ (a, b) \sim (c, d) \Longleftrightarrow a + d = b + c. \]
It can be proven that the relation \(\sim\) is a congruence. It is defined the quotient group, denoted by \(Q(S)\), as \(Q(S) = (S \times S)/ \sim\).
Examples
\(\circ\) Let \(S = (\mathbb{Z}\setminus \{0\}, \cdot)\). Then,
\[ (a,b) \sim (c,d) \Longleftrightarrow ad = bc. \]
It is well known that the relation defined above on \(\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})\) generates the quotient field of \(\mathbb{Z}\), which is \(D(\mathbb{Z}) = \mathbb{Q}\). Since \((0,n) \sim (0,1)\) for all \(n \in \mathbb{Z}\setminus \{0\}\), it is deduced that \(Q(S) = (S \times S) / \sim \hspace{0.1cm} = D(\mathbb{Z}) \setminus \{[(0,1)]\}\).