Monoid / semigroup with cancellation property $ ab=ac\Rightarrow b = c,$ $\ ba = ca\Rightarrow b = c$

Question

An important group lemma is the left and right cancellation property:

$$ \forall a b c: a * b = a * c \implies b = c $$ $$ \forall a b c: b * a = c * a \implies b = c $$

This lemma does not only hold for groups, but also some monoids, for example the free monoid $\Sigma^*$ over an alphabet $\Sigma$.

It doesn't hold for all monoids, as monoids with absorbing elements are counterexamples:

$$ <\mathbb{Z}, \times> : 0 \times 1 = 0 \times 2 \nRightarrow 1 = 2 $$

What are Monoids with the cancellation property called?

Answer 1

These are called "cancellative monoids", or "cancellative semigroups", respectively.