= Cancellation law
{title2=$ax=ay\Rightarrow x=y$}
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.
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