Inversne

Inversne is a term used in abstract algebra to denote a generalized inverse operation defined within a binary algebraic structure. It extends the usual notion of an inverse by allowing partial existence and non-uniqueness, applicable in structures where a true inverse for every element does not exist. Formally, let (S, ∘, e) be an algebraic structure with a designated identity e and a partially defined operation ∘. An element y in S is called an inversne of x in S if x ∘ y = e and y ∘ x = e whenever these products are defined. The set of all elements x that have an inversne is called the domain of invertibility, and the mapping x ↦ inv(x) (when defined) may or may not be unique.

In many settings inversne reduces to the standard inverse: in a group, every element has a unique inverse and inv(x) = x^{-1}. In more general semigroups or rings with partial inverses, inversne may exist only for a subset of elements and can be nonunique.

Applications of inversne appear in the study of partial symmetries, reversible computing models, and reconstruction problems where full invertibility is too strong a requirement. It provides a framework for discussing when and how information can be recovered from a transformed object when the transformation is not globally invertible.

See also: inverse, pseudoinverse, involution, dual, retract.