# Isomorphism

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This article describes mathematical isomorphism. For the sociological term, see isomorphism (sociology).

In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich.

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 some level of abstraction; ignoring the specific identities of the elements in the underlying sets, and focusing just on the structures themselves, the two structures are identical. Here are some everyday examples of isomorphic structures.

• A solid cube made of wood and a solid cube made of lead are both solid cubes; although their matter differs, their geometric structures are isomorphic.
• A standard deck of 52 playing cards with green backs and a standard deck of 52 playing cards with brown backs; although the colours on the backs of each deck differ, the decks are structurally isomorphic — if we wish to play cards, it doesn't matter which deck we choose to use.
• The Clock Tower in London (that contains Big Ben) and a wristwatch; although the clocks vary greatly in size, their mechanisms of reckoning time are isomorphic.
• A six-sided die and a bag from which a number 1 through 6 is chosen; although the method of obtaining a number is different, their random number generating abilities are isomorphic. This is an example of functional isomorphism, without the presumption of geometric isomorphism.

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 $\sqsubseteq$ then an isomorphism from X to Y is a bijective function f : X → Y such that

$f(u)\sqsubseteq f(v)$ iff uv.

Such an isomorphism is called an order isomorphism.

Or, if on these sets, the unknown binary operations $\star$ and $\Diamond$ are defined, respectively, then an isomorphism from X to Y is a bijective function f : X → Y such that

$f(u)\Diamond f(v)=f(u\star v)$

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.

In linear algebra, an isomorphism can also be defined as a linear map between two vector spaces that is one-to-one and onto.