if and only if or, in the function and homomorphism cases, In the function and homomorphism cases, this means that the codomain of equals or is included in the domain of . In the morphism case, this means that the codomain of equals the domain of . There is an identity for every object (set, algebraic structure or 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 spaces but not for 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. Generalizations In a unital magma Let be a unital magma, that is, a set with a binary operation and an identity element . If, for , we have , then is called a left inverse of and is called a right inverse of . If an element is both a left inverse and a right inverse of , then is called a two-sided inverse, or simply an inverse, of . An element with a two-sided inverse in is called invertible in . An element with an inverse element only on one side is left invertible or right invertible. Elements of a unital magma may have multiple left, right or two-sided inverses. For example, in the magma given by the Cayley table the elements 2 and 3 each have two two-sided inverses. A unital magma in which all elements are invertible need not be a loop. For example, in the magma given by the Cayley table every element has a unique two-sided inverse (namely itself), but is not a loop because the Cayley table is not a Latin