Commuting elements

Remarks

Example: in the set of $2 \times 2$ real matrices with matrix multiplication, every matrix $A$ commutes with $2A$, because $A(2A) = 2A^2 = (2A)A$.

Non-example: the matrices $$A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$$ do not commute, because $AB = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ while $BA = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$.

The relation "commutes with" is reflexive and symmetric but not transitive in general. In the example, $A$ commutes with the identity matrix $I$ and $I$ commutes with $B$, but $A$ and $B$ do not commute. The operation $\ast$ is commutative if and only if every two elements of $S$ commute.