In mathematics, the notion of cancellativity (or cancellability) is a generalization of the notion of invertibility. An element a in a magma (M, ∗) has the left cancellation property (or is left-cancellative) if for all b and c in M, a ∗ b = a ∗ c always implies that b = c. An element a in a magma (M, ∗) has the right cancellation property (or is right-cancellative) if for all b and c in M, b ∗ a = c ∗ a always implies that b = c. An element a in a magma (M, ∗) has the two-sided cancellation property (or is cancellative) if it is both left- and right-cancellative. A magma (M, ∗) has the left cancellation property (or is left-cancellative) if all a in the magma are left cancellative, and similar definitions apply for the right cancellative or two-sided cancellative properties. A left-invertible element is left-cancellative, and analogously for right and two-sided. If a−1 is the inverse of a, then a ∗ b = a ∗ c implies a−1 ∗ a ∗ b = a−1 ∗ a ∗ c which implies b = c. For example, every quasigroup, and thus every group, is cancellative. To say that an element a in a magma (M, ∗) is left-cancellative, is to say that the function g : x ↦ a ∗ x is injective. That the function g is injective implies that given some equality of the form a ∗ x = b, where the only unknown is x, there is only one possible value of x satisfying the equality. More precisely, we are able to define some function f, the inverse of g, such that for all x f(g(x)) = f(a ∗ x) = x. Put another way, for all x and y in M, if a * x = a * y, then x = y. The positive (equally non-negative) integers form a cancellative semigroup under addition. The non-negative integers form a cancellative monoid under addition. In fact, any free semigroup or monoid obeys the cancellative law, and in general, any semigroup or monoid embedding into a group (as the above examples clearly do) will obey the cancellative law. In a different vein, (a subsemigroup of) the multiplicative semigroup of elements of a ring that are not zero divisors (which is just the set of all nonzero elements if the ring in question is a domain, like the integers) has the cancellation property.

À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Graph Chatbot

Chattez avec Graph Search

Posez n’importe quelle question sur les cours, conférences, exercices, recherches, actualités, etc. de l’EPFL ou essayez les exemples de questions ci-dessous.

AVERTISSEMENT : Le chatbot Graph n'est pas programmé pour fournir des réponses explicites ou catégoriques à vos questions. Il transforme plutôt vos questions en demandes API qui sont distribuées aux différents services informatiques officiellement administrés par l'EPFL. Son but est uniquement de collecter et de recommander des références pertinentes à des contenus que vous pouvez explorer pour vous aider à répondre à vos questions.