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Concept
Inverse element
Summary
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.)
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. In a ring, an invertible element, also called a unit, is an element that is invertible under multiplication (this is not ambiguous, as every element is invertible under addition).
Inverses are commonly used in groupswhere every element is invertible, and ringswhere 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 , where, by definition, an isomorphism is an invertible morphism.
The word 'inverse' is derived from 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 is ).
The concepts of inverse element and invertible element are commonly defined for binary operations that are everywhere defined (that is, the operation is defined for any two elements of its domain). However, these concepts are commonly used with partial operations, that is operations that are not defined everywhere. Common examples are matrix multiplication, function composition and composition of morphisms in a .
Official source
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