Weakly cancellative monoid
Definition
Let \(A\) be a monoid. It is said that \(A\) is weakly cancellative if given \(x, y \in A\) such that \(x + a = y + a\) for all \(a \in A \setminus \{0\}\), it holds \(x = y\).
Examples
\(\circ\) If \(A\) is a cancellative monoid, then it is a weakly cancellative monoid.
\(\circ\) Let \(R\) be an unitary ring and let us consider \(A = (M_2(R), \cdot)\), where \(M_2(R)\) denotes the set of \(2 \times 2\) matrix with coefficients in \(R\) and \(\cdot\) denotes the usual matrix product. Let us suppose that there exist \(A,B \in M_2(R)\) such that for all \(C \in M_2(R) \setminus \{Id\}\), it holds \(AC = BC\). Let us consider
\[ C_{1,1} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}. \]
If \(AC = BC\), then \(a_{1,1}c_{1,1} + a_{1,2}c_{2,1} = b_{1,1}c_{1,1} + b_{1,2}c_{2,1}\), and consequently \(a_{1,1} = b_{1,1}\). With an analogous argument it is obtained that \(a_{i,j} = b_{i,j}\) for \(i,j \in \{1,2\}\), concluding that \(A = B\). Therefore, \((M_2(R), \cdot)\) is a weakly cancellative monoid.