In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

In category theory, an automorphism is an endomorphism ( i. e. a morphism from an object to itself ) which is also an isomorphism ( in the categorical sense of the word ).

* Inverses: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism.

* A group automorphism is a group isomorphism from a group to itself.

In this example it is not sufficient for a morphism to be bijective to be an isomorphism.

In a category with exponentials, using the isomorphism ( in computer science, this is called currying ), the Ackermann function may be defined via primitive recursion over higher-order functionals as follows:

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

It is essentially unique ( up to isomorphism ).

The isomorphism is defined by where for all we have

In fact, when A is a commutative unital C *- algebra, the Gelfand representation is then an isometric *- isomorphism between A and C ( Δ ( A )).

* automorphism if f is both an endomorphism and an isomorphism.

* f is an isomorphism.

Sometimes two quite different constructions yield " the same " result ; this is expressed by a natural isomorphism between the two functors.

The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that η < sub > X </ sub > is an isomorphism for every object X in C.

Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative " up to " an isomorphism.

In category theory, currying can be found in the universal property of an exponential object, which gives rise to the following adjunction in cartesian closed categories: There is a natural isomorphism between the morphisms from a binary product and the morphisms to an exponential object.

One can show that this map is an isomorphism, establishing the equivalence of the two definitions.

If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra over F there is a simply connected Lie group G with as Lie algebra, unique up to isomorphism.

which is natural in the variables N and F. The counit of this adjunction is simply the universal cone from lim F to F. If the index category J is connected ( and nonempty ) then the unit of the adjunction is an isomorphism so that lim is a left inverse of Δ.

This is stronger than simply requiring a continuous group isomorphism — the inverse must also be continuous.

In informal contexts, mathematicians often use the word modulo ( or simply " mod ") for similar purposes, as in " modulo isomorphism ".

Two functors F and G are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from F to G.

The subspaces that give such an isomorphism are called Lagrangian subspaces or simply Lagrangians.

This identification is usually called the Curry – Howard isomorphism, which was originally formulated for intuitionistic logic and simply typed lambda calculus.

Typed lambda calculi are closely related to mathematical logic and proof theory via the Curry – Howard isomorphism and they can be considered as the internal language of classes of categories, e. g. the simply typed lambda calculus is the language of Cartesian closed categories ( CCCs ).

To translate this to a subgraph isomorphism problem, simply let H be the complete graph K < sub > k </ sub >; then the answer to the subgraph isomorphism problem for G and H is equal to the answer to the clique problem for G and k. Since the clique problem is NP-complete, this polynomial-time many-one reduction shows that subgraph isomorphism is also NP-complete.

Similarly, if Peggy knew in advance that Victor would ask to see the isomorphism then she could simply generate an isomorphic graph H ( in which she also does not know a Hamiltonian Cycle ).

Every domain D can be embedded in an ω-complete domain completion ( called the ω-completion or simply the completion of D ) that is, in a precise sense, the smallest ω-complete domain containing D. The isolated elements of completion are precisely the isolated elements of D, but in general completion contains limit points that are not found in D. completion is uniquely determined up to isomorphism.

The Curry – Howard isomorphism associates a term in the simply typed lambda calculus with each natural-deduction proof in intuitionistic propositional logic.

In mathematics, the statement that " Property P characterizes object X " means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as " Property Q characterises Y up to isomorphism ".

In fact, it is an isomorphism, and its inverse is simply

The word " noncanonically " prevents one from concluding that supermanifolds are simply glorified vector bundles ; although the functor Π maps surjectively onto the isomorphism classes of supermanifolds, it is not an equivalence of categories.

An isomorphism between linear orders is simply a strictly increasing bijection.

A bijective *- homomorphism π is called a C *- isomorphism, in which case A and B are said to be isomorphic.

Given two groups (< var > G </ var >, *) and (< var > H </ var >, ), a group isomorphism from (< var > G </ var >, *) to (< var > H </ var >, ) is a bijective group homomorphism from < var > G </ var > to < var > H </ var >.

Spelled out, this means that a group isomorphism is a bijective function such that for all < var > u </ var > and < var > v </ var > in < var > G </ var > it holds that

* An isomorphism is a bijective homomorphism.

For instance, since a function is bijective if and only if it is both injective and surjective, a module homomorphism is an isomorphism if and only if it is both a monomorphism and an epimorphism.

In abstract algebra, an isomorphism is a bijective homomorphism.

If one object consists of a set X with a binary relation R and the other object consists of a set Y with a binary relation S then an isomorphism from X to Y is a bijective function such that:

For example, R is an ordering ≤ and S an ordering, then an isomorphism from X to Y is a bijective function such that

There is an isomorphism from X to Y if the bijective function happens to produce results, that sets up a correspondence between the operator and the operator.

For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.

In graph theory, an isomorphism between two graphs G and H is a bijective map f from the vertices of G to the vertices of H that preserves the " edge structure " in the sense that there is an edge from vertex u to vertex v in G if and only if there is an edge from ƒ ( u ) to ƒ ( v ) in H. See graph isomorphism.

It is also obvious that the map is both injective and surjective ; meaning that is a bijective homomorphism, i. e. an isomorphism.

A K-algebra isomorphism is a bijective K-algebra morphism.

A global isometry, isometric isomorphism or congruence mapping is a bijective isometry.

If F is bijective then F is said to be an isomorphism between A and B.

Since this mapping is clearly surjective, it is bijective and thus an algebra isomorphism of A and B.

Because of this, there is a natural bijective correspondence between the isomorphism classes of the left R-modules and the left M < sub > n </ sub >( R )- modules, and between the isomorphism classes of the left ideals of R and M < sub > n </ sub >( R ).

A functor F: C → D yields an isomorphism of categories if and only if it is bijective on objects and on morphism sets.