Cancellative semigroup

Symbol-free definition
A cancellative semigroup (also called cancellation semigroup) is a semigroup where every element is cancellative.

Definition with symbols
A semigroup $$(S,*)$$ (a set $$S$$ with associative binary operation $$*$$) if the following hold for all $$a,b,c \in S$$:


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