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 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 x y {\displaystyle {\tfrac {x}{y}}} is y x {\displaystyle {\tfrac {y}{x}}} ).
Definitions and basic properties 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 also 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 category. It follows that the common definitions of associativity and identity element must be extended to partial operations; this is the object of the first subsections. In this section, X is a set (possibly a proper class) on which a partial operation (possibly total) is defined, which is denoted with ∗ . {\displaystyle *.}
Associativity A partial operation is associative if
x ∗ ( y ∗ z ) = ( x ∗ y ) ∗ z {\displaystyle x*(y*z)=(x*y)*z}
for every x, y, z in X for which one of the members of the equality is defined; the equality means that the other member of the equality must also be defined. Examples of non-total associative operations are multiplication of matrices of arbitrary size, and function composition.
Identity elements Let ∗ {\displaystyle *} be a possibly partial associative operation on a set X. An identity element, or simply an identity is an element e such that
x ∗ e = x and e ∗ y = y {\displaystyle x*e=x\quad {\text{and}}\quad e*y=y}
for every x and y for which the left-hand sides of the equalities are defined. If e and f are two identity elements such that e ∗ f {\displaystyle e*f} is defined, then e = f . {\displaystyle e=f.} (This results immediately from the definition, by e = e ∗ f = f . {\displaystyle e=e*f=f.} ) It follows that a total operation has at most one identity element, and if e and f are different identities, then e ∗ f {\displaystyle e*f} is not defined. For example, in the case of matrix multiplication, there is one n×n identity matrix for every positive integer n, and two identity matrices of different size cannot be multiplied together. Similarly, identity functions are identity elements for function composition, and the composition of the identity functions of two different sets are not defined.
Left and right inverses If x ∗ y = e , {\displaystyle x*y=e,}
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