Leftcancellable

Leftcancellable, also written as left-cancellable or left-cancelable, is a property of an element in an algebraic structure such as a semigroup, monoid, or category. An element a is leftcancellable if, for all elements x and y in the same structure, a x = a y implies x = y. Equivalently, left multiplication by a defines an injective map from the structure to itself.

In a monoid or semigroup, left-cancellation is often described in terms of left translations. The map L_a: S → S given by L_a(x) = a x is injective when a is leftcancellable. This concept is part of the broader cancellation laws; the dual notion is right-cancellation, where x a = y a implies x = y.

In category theory, a morphism f is leftcancellable if, for any pair of morphisms g, h with a common domain, f ∘ g = f ∘ h implies g = h. Such morphisms are precisely the monomorphisms in the categorical sense.

Examples and observations:

- In any group, every element is leftcancellable, since left-multiplication by a has an inverse, namely multiplication by a^{-1}.

- In the natural numbers under multiplication, every nonzero a is leftcancellable because ax = ay implies x = y.

- In the semigroup of n-by-n matrices over a field, left-cancellation holds for a if and only if a is left-invertible (in particular, for invertible matrices with nonzero determinant). If a is singular, left-cancellation can fail: a x = a y may hold for distinct x ≠ y when a has a nontrivial kernel.

Leftcancellable elements are central to the study of cancellative semigroups and monoids. They help characterize injective behavior of left multiplication and connect to broader notions such as monomorphisms in category theory.