I'm looking for examples of infinite non-cancellative commutative semigroups. The only example I have readily at hand is the monoid of homeomorphism classes of closed surfaces. Can anyone provide other examples?
Ideally, I'm interested in examples that do not have an absorbing element and that don't consist entirely of idempotents.
The case with non-idempotents and withouth absorbing elements is more interesting. We can "mix" some of the previous examples.
Take any infinite commutative, non-cancelative semigroup $K$, and any semigroup $L$ without an absorbing element. Then the product $L\times K$ has the desired properties.
For example, $X=\mathbb{Z}_2\times\mathbb{Z}$ with operation $(x,k)(y,l)=(xy,\operatorname{min}(k,l))$.