Given a commutative monoid $(M,0,\oplus)$. Then we can define an ordering on $M$ by
$a\geq b :\Leftrightarrow \exists c: a=b\oplus c$.
The relation is then transitive and reflexive. The claim is now the following:
Assume that $M$ is idempotent, i.e. $m\oplus m = m$ for any $m\in M$. Then the relation $\geq$ defined above is antisymmetric, i.e. if $a\geq b$ and $b\geq a$ then $b=a$.
Can anyone help with this?
If $a=b\oplus c$, then $$ a\oplus b=b\oplus (b\oplus c)=(b\oplus b)\oplus a=b\oplus c=a.$$ In other words, $$ a\ge b\implies a\oplus b=a.$$ So if $a\ge b$ and $b\ge a$, we have $$a=a\oplus b=b\oplus a = b.$$