rightcancellability

Rightcancellability is a property studied in semigroups and monoids. A semigroup or monoid S is right cancellative if, for all elements b, c, a in S, the equality ba = ca implies b = c. In other words, right multiplication by any element a is an injective map: the function R_a: S → S defined by R_a(x) = x a is injective for every a in S.

Equivalently, S is right cancellative exactly when, for every a in S, the equation x a = y a has the unique solution x = y for given y. Left cancellativity is defined dually by the condition ab = ac implying b = c.

If a semigroup is cancellative (both left and right cancellative), it satisfies both cancelation laws: ab = ac implies b = c and ba = ca implies b = c for all a, b, c. Groups are canonical examples of cancellative structures, hence they are both left and right cancellative.

Examples and contrasts help intuition. A group is right cancellative by virtue of the existence of inverses. A left zero semigroup with operation a b = a for all a, b is right cancellative but not left cancellative, since ba = b and ca = c imply b = c for right cancellation, but ab = ac holds for all b, c, failing left cancellation.

Notes. Right cancellativity is preserved by taking subsemigroups but need not be preserved under homomorphic images. In contrast, cancellativity together with additional conditions (like the Ore condition) relates to embedding a semigroup into a group of fractions.