Infinite right- and left-cancellable semigroup which is not a group?

Question

I was given this question and the proof.

Let $G$ be a nonempty set and let $*$ be an associative binary operation on $G$. Assume that both the left and right cancellation laws hold in $(G,*)$.

Assume moreover that $G$ is finite. Show that $(G,*)$ is a group.

My question is why does the set have to be finite? What would happen if its the opposite?

Answer 1

If $G$ is infinite, the conclusion is no longer true. You can find such examples in the literature, e.g., the positive (equally non-negative) integers form a cancellative semigroup under addition.