Cancellative monoid
Definition
Let \(A\) be a monoid. It is said that \(A\) is a cancellative monoid if for any \(a, b, c \in A\) such that \(a + c = b + c\), it holds \(a = b\).
Examples
\(\circ\) Every group is a cancellative monoid since every element has an inverse.
\(\circ\) Let \(R\) be an unitary ring and let us consider \(A = (M_2(R), \cdot)\), where \(M_2(R)\) is the set of \(2 \times 2\) matrix with coefficients in \(R\) and \(\cdot\) denotes the usual matrix product. It is well known that \(A\) is a monoid. We have that
\[ \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \]
where it is deduced that \((M_2(R), \cdot)\) is not a cancellative monoid.