Non-cancellative commutative monoids in which no element absorbs another

Question

A commutative monoid $M$ is cancellative if $a + c = b + c$ implies $a = b$ for all $a,b,c \in M$. Let's call $M$ positive if $a + b = a$ implies $b = 0$ for all $a,b \in M$ (i.e. no element absorbs another non-trivially). The easiest way of getting non-cancellative monoids is to find ones that are non-positive (idempotent monoids, etc.), but there are monoids that are positive but non-cancellative. I was wondering whether there are nice classes of examples I could draw from.

Answer 1

If I am not wrong, the following commutative monoid is positive but not cancellative. Let $\sim$ be the congruence on $(\Bbb N^3, +)$ generated by the relation $(1, 1, 0) = (1, 0, 1)$. Actually, one has $$ (a, b, c) \sim (a',b',c') $$ if and only if $$a = a', \quad b + c = b' + c'\quad \text{and} \quad a \geqslant |b-b'| $$ Let $M = \Bbb N^3/{\sim}$. I claim that $M$ is positive. Indeed, if $$ (a,b,c) \sim (a,b,c) + (x,y,z) $$ then $a = a+x$, whence $x=0$ and $b+c = b +y + c + z$, whence $y=z=0$. However, $$ (1, 0, 0) + (0, 1, 0) \sim (1, 0, 0) + (0, 0, 1) $$ but $$ (0, 1, 0) \not\sim (0, 0, 1) $$ and thus $M$ is not cancellative.