Rightcancellative

Rightcancellative is a property of semigroups and monoids describing a one-sided cancellation law. A semigroup S is right cancellative if, for all a, b, c in S, ac = bc implies a = b. Equivalently, for every c in S, the right translation map R_c: S → S defined by R_c(x) = x c is injective.

Right cancellativity is a one-sided version of the cancellative property. A semigroup that is both left and right cancellative is called cancellative; a monoid or group that is cancellative satisfies the usual cancellation laws on both sides. Right cancellativity alone does not guarantee left cancellation, and various one-sided cancellative semigroups exist.

Examples illustrate the concept. The left-zero semigroup, defined by the operation x*y = x for all x, y in a set S, is right cancellative but not left cancellative: ac = a and bc = b for all a, b, c, so ac = bc implies a = b, but ca = cb holds for all a, b since ca = cb = c. The free semigroup (or free monoid) on any set is cancellative, hence right cancellative. The positive integers under multiplication are cancellative as well. Non-examples include the two-element semigroup {0,1} with multiplication where 0 is absorbing; then 0*c = 0 and 1*c may equal 0 for some c, so ac = bc can occur with a ≠ b, violating right cancellativity.

See also: left cancellative, cancellative semigroup, group.