Talk:Condition on Congruence Relations for Cancellable Monoid to be Group/Counterexample

In answer to the question:

"Is all this necessary, considering we have Group is Cancellable Monoid?"

What we are doing here is demonstrating that there exists a (commutative) monoid $\struct {Z_3, \times_3}$ which is specifically not cancellable, and such that every non-trivial congruence relation on $\struct {Z_3, \times_3}$ is defined by a normal subgroup of $\struct {Z_3, \times_3}$.

The fact that $\struct {Z_3, \times_3}$ is not a group should of course be highlighted as soon as we have determined that $\struct {Z_3, \times_3}$ is not cancellable, that is, half way down the page.

The rest of the page is spent in demonstrating that $\struct {Z_3, \times_3}$ has that property of its congruence relations.

Is that the question you were asking? --prime mover (talk) 12:16, 17 July 2022 (UTC)