Editing Inverse element (section)

==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 of a function|domain]]). However, these concepts are also commonly used with [[partial operation]]s, that is operations that are not defined everywhere. Common examples are [[matrix multiplication]], [[function composition]] and composition of [[morphism]]s in a [[category (mathematics)|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, {{mvar|X}} is a [[set (mathematics)|set]] (possibly a [[proper class]]) on which a partial operation (possibly total) is defined, which is denoted with <math>*.</math> 

===Associativity===
A partial operation is [[associative]] if 
:<math>x*(y*z)=(x*y)*z</math>
for every {{math|''x'', ''y'', ''z''}} in {{mvar|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 [[matrix multiplication|multiplication of matrices]] of arbitrary size, and [[function composition]].

===Identity elements===
Let <math>*</math> be a possibly [[partial operation|partial]] associative operation on a set {{mvar|X}}.

An ''[[identity element]]'', or simply an ''identity'' is an element {{mvar|e}} such that 
:<math>x*e=x \quad\text{and}\quad e*y=y</math>
for every {{mvar|x}} and {{mvar|y}} for which the left-hand sides of the equalities are defined.

If {{mvar|e}} and {{mvar|f}} are two identity elements such that <math>e*f</math> is defined, then <math>e=f.</math> (This results immediately from the definition, by <math>e=e*f=f.</math>)

It follows that a total operation has at most one identity element, and if {{mvar|e}} and {{mvar|f}} are different identities, then <math>e*f</math> is not defined. 

For example, in the case of [[matrix multiplication]], there is one {{math|''n''×''n''}} [[identity matrix]] for every positive integer {{mvar|n}}, and two identity matrices of different size cannot be multiplied together.

Similarly, [[identity function]]s 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 <math>x*y=e,</math>
where {{mvar|e}} is an identity element, one says that {{mvar|x}} is a ''left inverse'' of {{mvar|y}}, and {{mvar|y}} is a ''right inverse'' of {{mvar|x}}.

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 integer]]s, which has {{math|0}} as [[additive identity]], and {{math|0}} is the only element that has an [[additive inverse]]. This lack of inverses is the main motivation for extending the [[natural number]]s 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 [[function (mathematics)|functions]] from the integers to the integers. The ''doubling function'' <math>x\mapsto 2x</math> 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 {{mvar|n}} to either <math>2n</math> or <math>2n+1</math> is a right inverse of the function <math display=inline>n\mapsto \left\lfloor \frac n2\right\rfloor,</math> the [[floor function]] that maps {{mvar|n}} to <math display=inline>\frac n2</math> or <math display=inline>\frac{n-1}2,</math> depending whether {{mvar|n}} 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 [[section (category theory)|sections]], and left inverses are called [[retraction (category theory)|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 {{mvar|l}} and {{mvar|r}} are respectively a left inverse and a right inverse of {{mvar|x}}, then 
:<math>l=l*(x*r)=(l*x)*r=r.</math>

''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 {{mvar|x}} is denoted <math>-x.</math> Otherwise, the inverse of {{mvar|x}} is generally denoted <math>x^{-1},</math> or, in the case of a [[commutative]] multiplication <math display =inline>\frac 1x.</math>  When there may be a confusion between several operations, the symbol of the operation may be added before the exponent, such as in <math>x^{* -1}.</math> The notation <math>f^{\circ -1}</math> is not commonly used for [[function composition]], since <math display =inline>\frac 1f</math> can be used for the [[multiplicative inverse]].

If {{mvar|x}} and {{mvar|y}} are invertible, and <math>x*y</math> is defined, then <math>x*y</math> is invertible, and its inverse is <math>y^{-1}x^{-1}.</math> 

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