rightcancellable

Rightcancellable is a term used in algebra to describe a property of an element in a semigroup or monoid. An element a is rightcancellable if right multiplication by a is injective: for all x, y in the structure, x a = y a implies x = y. Equivalently, the right-translation map R_a: S → S, defined by R_a(x) = x a, is one-to-one.

In a cancellative semigroup, every element is both left- and right-cancellable. In a group, all elements are cancellable on both sides because inverse elements allow cancellation. By contrast, rightcancellability can fail in the presence of zero divisors or idempotents. For instance, in rings with zero divisors, an element like 0 or other nonzero divisors may fail the property when multiplication produces the same result from different factors.

Examples: In the natural numbers under multiplication, any a > 0 is rightcancellable, since x a = y a with a ≠ 0 implies x = y. In a semigroup with a zero element, a = 0 is typically not rightcancellable because x 0 = y 0 for all x, y. In the modular ring Z/6Z with multiplication, the element 2 is not right-cancellable, since 1·2 ≡ 4·2 (mod 6) while 1 ≠ 4.

In function semigroups, a function f is right-cancellable if, for all X, Y, X ∘ f = Y ∘ f implies X = Y; this holds when f is surjective. The concept is sometimes called right-cancellative or right-cancellable, and in category theory, the analogous notion on morphisms is cancellable on the right.