Isomorphism Structures F''(''u F''(''v Definition Theory Edge

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In mathematics, an isomorphism is a kind of interesting mapping between objects. Douglas Hofstadter provides an informal definition:

The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures. ( Gödel, Escher, Bach, p. 49)

Formally, an isomorphism is a bijective map f such that both f and its inverse f^-1 are homomorphisms, i.e. structure-preserving mappings.

If there exists an isomorphism between two structures, we call the two structures isomorphic. Isomorphic structures are "the same" at a certain level of abstraction; ignoring the specific identities of the elements in the underlying sets and the names of the underlying relations, the two structures are identical.

For example, if one object consists of a set X with an ordering <= and the other object consists of a set Y with an ordering [=, then an isomorphism from X to Y is a bijective function f : X -> Y such that

f(u) [= f(v) iff u <= v.

Such an isomorphism is called an order isomorphism.

Or, if on these sets, the unknown binary operations * and @ are defined, respectively, then an isomorphism from X to Y is a bijective function f : X -> Y such that

f(u) @ f(v) = f(u * v)

for all u, v in X. When the objects in question are groups, such an isomorphism is called a group isomorphism. Similarly, if the objects are fields, it is called a field isomorphism.

In universal algebra, one can give a general definition of isomorphism that covers these and many other cases. The definition of isomorphism given in category theory is even more general.

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 iff there is an edge from f(u) to f(v) in H.

Isomorphism classAn isomorphism class is a collection of mathematical objects isomorphic with a certain mathematical object. A mathematical object usually consists of a set and some mathematical relations and operations defined over this set. Isomorphism classes are often, Homomorphism, MorphismIn mathematics, a morphism is an abstraction of a function or mapping between two spaces. The word can mean different things depending on the type of space in question. In set theory, for example, morphisms are just functions, in group theory they are gro, AutomorphismIn mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an obj

In sociologySociology is the study of social rules and processes that bind and separate people not only as individuals, but as members of associations, groups, and institutions. A typical textbook definition of sociology calls it the study of the social lives of huma, isomorphism is to a kind of "copying" or "imitation", especially of the practices of one organizationAlternative meaning: Organisation (band). An organization (also organisation in many Commonwealth countries) is a formal group of people with one or more shared goals. This topic is a broad one. According to management science, most human organizations fa by another.

Abstract algebraAbstract algebra Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. The term "abstract algebra" is used to distinguish the field from " elementary algebra" or "high school algebr Algebra Category theory

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