Editing Inverse element (section)

===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]].