 Cancellation law

A binary operation has left cancellation when $ax=ay$ implies $x=y$, and right cancellation when $xa=ya$ implies $x=y$. The common factor must be in the same position in the two expressions. A group has both laws: multiplying by the inverse element on the appropriate side removes that factor.

Cancellation does not itself require inverse elements. The nonnegative integers under addition satisfy $a+x=a+y\Rightarrow x=y$, but a positive integer has no additive inverse in that set. Conversely, matrix multiplication need not allow cancellation: a zero matrix multiplied by different matrices gives the same zero matrix. The finite cancellative semigroup theorem explains why adding finiteness and the associative property changes the conclusion.