Rightcancelable

Rightcancelable, also written right-cancelable or right-cancellative, is a term used in algebra to describe a cancellation property of a binary operation. Given a set S with a binary operation ∗, an element a ∈ S is right-cancelable if for all x, y ∈ S, x ∗ a = y ∗ a implies x = y. If every element of S is right-cancelable, the operation is called right-cancellative; equivalently, the right multiplication map R_a: S → S, defined by R_a(x) = x ∗ a, is injective for all a ∈ S.

Some authors define the cancellation property at the level of the operation rather than individual elements. A semigroup (S, ∗) is right-cancellative if x ∗ a = y ∗ a for all a ∈ S implies x = y. This property is closely related to the absence of right zero divisors and has consequences for solving equations of the form x ∗ a = b.

Examples help illustrate the concept. In the semigroup of all functions from a set to itself under composition, a function f is right-cancelable exactly when f is surjective: if g ∘ f = h ∘ f, then g and h agree on the image of f; surjectivity of f ensures equality everywhere. In the additive semigroup of natural numbers (or, more generally, in any cancellative group), every element is right-cancelable since x + a = y + a implies x = y. In the full multiplicative semigroup of real numbers, right cancellation fails for a = 0, so it is not right-cancellative; restricting to nonzero elements restores the property.

Groups are cancellative: in a group, x ∗ a = y ∗ a implies x = y and a ∗ x = a ∗ y implies x = y for all elements, so all elements are right- and left-cancelable.

See also: cancellation law, left-cancelable, cancellative monoid, surjective.