Editing Inverse element (section)

==Functions, homomorphisms and morphisms==

[[function composition|Composition]] is a [[partial operation]] that generalizes to [[homomorphism]]s of [[algebraic structure]]s and [[morphism]]s of [[category (mathematics)|categories]] into operations that are also called ''composition'', and share many properties with function composition. 

In all the case, composition is [[associative]].

If <math>f\colon X\to Y</math> and <math>g\colon Y'\to Z,</math> the composition <math>g\circ f</math> is defined if and only if <math>Y'=Y</math> or, in the function and homomorphism cases, <math>Y\subset Y'.</math> In the function and homomorphism cases, this means that the [[codomain]] of <math>f</math> equals or is included in the [[domain of a function|domain]] of {{mvar|g}}. In the morphism case, this means that the [[codomain]] of <math>f</math> equals the [[domain of a function|domain]] of {{mvar|g}}.

There is an ''identity'' <math>\operatorname{id}_X \colon X \to X</math> for every object {{mvar|X}} ([[set (mathematics)|set]], algebraic structure or [[object (category theory)|object]]), which is called also an [[identity function]] in the function case.

A function is invertible if and only if it is a [[bijection]]. An invertible homomorphism or morphism is called an [[isomorphism]]. An homomorphism of algebraic structures is an isomorphism if and only if it is a bijection. The inverse of a bijection is called an [[inverse function]]. In the other cases, one talks of ''inverse isomorphisms''.

A function has a left inverse or a right inverse if and only it is [[injective]] or [[surjective]], respectively. An homomorphism of algebraic structures that has a left inverse or a right inverse is respectively injective or surjective, but the converse is not true in some algebraic structures. For example, the converse is true for [[vector space]]s but not for [[module (mathematics)|modules]] over a ring: a homomorphism of modules that has a left inverse of a right inverse is called respectively a [[split epimorphism]] or a [[split monomorphism]]. This terminology is also used for morphisms in any category.