U-semigroup

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

Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is a right inverse of . (An identity element is an element such that and for all and for which the left-hand sides are defined.)

When the operation is associative, if an element 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 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 category theory, where, by definition, an isomorphism is an invertible morphism.

The word 'inverse' is derived from la, 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 $\tfrac x y$ is $\tfrac y x$).

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 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, is a set (possibly a proper class) on which a partial operation (possibly total) is defined, which is denoted with $*.$

Associativity

A partial operation is associative if
:$x*(y*z)=(x*y)*z$
for every in 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 $*$ be a possibly partial associative operation on a set .

An '' identity element'', or simply an ''identity'' is an element such that
:$x*e=x \quad\text\quad e*y=y$
for every and for which the left-hand sides of the equalities are defined.

If and are two identity elements such that $e*f$ is defined, then $e=f.$ (This results immediately from the definition, by $e=e*f=f.$)

It follows that a total operation has at most one identity element, and if and are different identities, then $e*f$ is not defined.

For example, in the case of matrix multiplication, there is one identity matrix for every positive integer , 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 $x*y=e,$
where is an identity element, one says that is a ''left inverse'' of , and is a ''right inverse'' of .

Left and right inverses do not always exist, even when the operation is total and associative. For example, addition is a total associative operation on nonnegative integers, which has as additive identity, and is the only element that has an additive inverse. This lack of inverses is the main motivation for extending the natural numbers into the integers.

An element can have several left inverses and several right inverses, even when the operation is total and associative. For example, consider the functions from the integers to the integers. The ''doubling function'' $x\mapsto 2x$ has infinitely many left inverses under function composition, which are the functions that divide by two the even numbers, and give any value to odd numbers. Similarly, every function that maps to either $2n$ or $2n+1$ is a right inverse of the function $n\mapsto \left\lfloor \frac n2\right\rfloor,$ the floor function that maps to $\frac n2$ or $\frac2,$ depending whether is even or odd.

More generally, a function has a left inverse for function composition if and only if it is injective, and it has a right inverse if and only if it is surjective.

In category theory, right inverses are also called sections, and left inverses are called retractions.

Inverses

An element is ''invertible'' under an operation if it has a left inverse and a right inverse.

In the common case where the operation is associative, the left and right inverse of an element are equal and unique. Indeed, if and are respectively a left inverse and a right inverse of , then
:$l=l*(x*r)=(l*x)*r=r.$

''The inverse'' of an invertible element is its unique left or right inverse.

If the operation is denoted as an addition, the inverse, or additive inverse, of an element is denoted $-x.$ Otherwise, the inverse of is generally denoted $x^,$ or, in the case of a commutative multiplication $\frac 1x.$ When there may be a confusion between several operations, the symbol of the operation may be added before the exponent, such as in $x^.$ The notation $f^$ is not commonly used for function composition, since $\frac 1f$ can be used for the multiplicative inverse.

If and are invertible, and $x*y$ is defined, then $x*y$ is invertible, and its inverse is $y^x^.$

An invertible homomorphism is called an isomorphism. In
category theory, an invertible morphism is also called an isomorphism.

In groups

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