Cancellativity

Cancellativity is a property of semigroups and monoids describing when equation solving with the product is unique. A semigroup S with operation * is left cancellative if a*b = a*c implies b=c for all a,b,c in S. It is right cancellative if b*a = c*a implies b=c for all a,b,c. If both conditions hold, S is cancellative. Groups are cancellative because ab=ac implies a^{-1}ab=a^{-1}ac, hence b=c.

Examples and non-examples illustrate the idea. The natural numbers under addition (N, +) form a cancellative monoid, as does the positive integers under multiplication (N>0, ×) since multiplication by a nonzero number is injective. A monoid containing a zero, such as (N, ×) including 0, is not cancellative because 0*b = 0*c for all b,c. In finite semigroups, cancellativity implies the structure is a group.

Two common extensions of cancellativity are left and right cancellativity. If a structure is cancellative, left (resp. right) cancellation means left (resp. right) translations by any element are injective maps.

Connections to groups and fractions. A cancellative monoid can often be embedded into a group of fractions: under suitable Ore conditions (both left and right) one can form a localization that embeds the monoid into a group. In the commutative case, cancellation allows the construction of the Grothendieck group, the universal group completion of the monoid. For a commutative cancellative monoid M, M embeds into its Grothendieck group G(M); the classic example is N under addition embedding into Z.

Applications of cancellativity appear in solving equations in semigroups, in factorization theory, and in constructions of localized or fractional objects within algebra.