Inverse element

In mathematics, the concept of an inverse element generalises the concepts of opposite (−x) and reciprocal (1/x) of numbers.

Given an operation denoted here ∗, and an identity element denoted e, if x ∗ y = e, one says that x is a left inverse of y, and that y is a right inverse of x. (An identity element is an element such that x * e = x and e * y = y for all x and y for which the left-hand sides are defined.[1])

When the operation ∗ is associative, if an element x has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the inverse element or simply the inverse. Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse.

Inverses are commonly used in groups—where every element is invertible, and rings—where invertible elements are also called units. They are also commonly used for operations that are not defined for all possible operands, such as inverse matrices and inverse functions. This has been generalized to category theory, where, by definition, an isomorphism is an invertible morphism.

The word ‘inverse’ is derived from Latin: inversus that means ‘turned upside down’, ‘overturned’. This may take its origin from the case of fractions, where the (multiplicative) inverse is obtained by exchanging the numerator and the denominator (the inverse of {\displaystyle {\tfrac {x}{y}}}{\displaystyle {\tfrac {x}{y}}} is {\displaystyle {\tfrac {y}{x}}}{\displaystyle {\tfrac {y}{x}}}).

In this article, the operations are associative and have identity elements, except when otherwise stated and in section § Generalizations.

Contents
1 Definitions and basic properties
1.1 Associativity
1.2 Identity elements
1.3 Left and right inverses
1.4 Inverses
2 In groups
3 In monoids
4 In rings
5 Matrices
6 Functions, homomorphisms and morphisms
7 Generalizations
7.1 In a unital magma
7.2 In a semigroup
7.3 U-semigroups
7.4 Semirings
7.5 Examples
7.5.1 Galois connections
7.5.2 Generalized inverses of matrices
8 See also

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