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