rightcancellation

Right cancellation is a property of certain algebraic structures such as semigroups, monoids, or groups. A structure is right-cancellative if for all elements a, b, and c, ba = ca implies b = c. Equivalently, for each a in the structure, the right multiplication map R_a: S → S defined by R_a(x) = x a is injective.

Right cancellation is distinct from left cancellation, which requires ab = ac to imply b = c for all a, b, c. If a structure has both properties, it is called cancellative. In groups, both left and right cancellation hold because multiplication by any element is invertible; specifically, ba = ca implies b = c by multiplying on the right by a^{-1}.

Examples and non-examples help distinguish the concept. Every group is right-cancellative (and left-cancellative). A simple non-group example is the left-zero semigroup on a nonempty set S, defined by xy = x for all x, y in S. In this semigroup, ba = b and ca = c, so ba = ca implies b = c, giving right cancellation, but left cancellation fails since ab = a for all b.

A standard failure case is a semigroup with a zero element, where 0 acts as a left zero: for some a = 0, we have b0 = c0 for all b, c, which can violate right cancellation. Thus right cancellation can hold independently of left cancellation in general.

Right cancellation is equivalently the injectivity of right multiplication by any element, a useful property in the study of semigroups and their potential embedding into groups.