Editing Inverse element (section)

===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]].