Converse to a proposition on divisors in commutative monoids

Question

Let $(M,*,1)$ be a commutative monoid. Define the binary relation $R$ on $M$ by $aRb$ iff there exists an $x$ in $M$ such that $a*x=b$. $R$ is the "divides" relation. Since $M$ is a commutative monoid, clearly $R$ is both reflexive and transitive. I read in a text that if $M$ is both cancellative and pure (pure meaning the only invertible element is $1$), then $R$ is antisymmetric. Is the converse true? That is, given a commutative monoid where $R$ is antisymmetric, is $M$ also cancellative and pure? In fact, is there a counterexample where $M$ is neither cancellative or pure?

Answer 1

No. Consider the multiplicative monoid $\{0,1\}$ . The division relation is anti-symmetric. But the monoid is not cancellative $0.1=0.0$ but $0\ne 1$.

On the other hand,antisymmetric division relation implies "pure". Indeed every invertible element divides 1 and 1 divides every element.