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Isomorphism - Definition | | Isomorphism - Definition: Encyclopedia II - Isomorphism - Definition | | 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, ...
See also:Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications | | | Isomorphism, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications, Isomorphism - Definition, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Purpose, Isomorphism - Two abstract examples, automorphism, homomorphism, epimorphism, isomorphism class, monomorphism, morphism | | |
| | | Isomorphism: Encyclopedia II - Isomorphism - Definition
Isomorphism - Definition
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 structural level of abstraction; if ignoring the specific identities of the elements in the underlying sets and focusing just on the structures themselves, then the two structures are identical.
Other related archivesAbstract algebra, Algebra, Analysis, Big Ben, Category theory, Douglas Hofstadter, Eilhard Mitscherlich, Greek, Group isomorphism, Gödel, Escher, Bach, Legendre transform, algebra, algebraic, automorphism, bijection, bijective, binary operations, category theory, codomain, differential equations, domain, epimorphism, fields, graph theory, group, groups, homomorphism, homomorphisms, iff, inverse, isomorphism (sociology), isomorphism class, linear algebra, linear map, logarithm, mapping, mathematics, monomorphism, morphism, one-to-one, onto, order isomorphism, range, real numbers, sets, universal algebra, vector spaces, vertex
 Adapted from the Wikipedia article "Definition", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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