isomorphismˌaɪ səˈmɔr fɪz əm
isomorphism
English Definitions:
isomorphism, isomorphy (noun)
(biology) similarity or identity of form or shape or structure
isomorphism (Noun)
Similarity of form
isomorphism (Noun)
the similarity in form of organisms of different ancestry
isomorphism (Noun)
A bijection f such that both f and its inverse f are homomorphisms, that is, structure-preserving mappings.
isomorphism (Noun)
the similarity in the crystal structures of similar chemical compounds
isomorphism (Noun)
the similarity in the structure or processes of different organizations
isomorphism (Noun)
a one-to-one correspondence between all the elements of two sets, e.g. the instances of two classes, or the records in two datasets
Isomorphism
In abstract algebra, an isomorphism is a bijective homomorphism. Two mathematical structures are said to be isomorphic if there is an isomorphism between them. In essence, two objects are isomorphic if they are indistinguishable given only a selection of their features, and the isomorphism is the mapping of the set elements and the selected operations between the objects. A named isomorphism indicates which features are selected for this purpose. Thus, for example, two objects may be group isomorphic without being ring isomorphic, since the latter isomorphism selects the additional structure of the multiplicative operator. In category theory, an isomorphism is a morphism f: X → Y in a category for which there exists an "inverse" f −1: Y → X, with the property that both f −1f = idX and f f −1 = idY.
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism.An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a universal property), or if the isomorphism is much more natural (in some sense) than other isomorphisms. For example, for every prime number p, all fields with p elements are canonically isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique. The term isomorphism is mainly used for algebraic structures. In this case, mappings are called homomorphisms, and a homomorphism is an isomorphism if and only if it is bijective.
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