Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism (or more generally a morphism) that admits an inverse. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.

For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective.

In topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called diffeomorphisms.

A canonical isomorphism is a canonical map that is an isomorphism. Two objects are said to be canonically isomorphic if there is a canonical isomorphism between them. For example, the canonical map from a finite-dimensional vector space V to its second dual space is a canonical isomorphism; on the other hand, V is isomorphic to its dual space but not canonically in general.

Isomorphism (sociology)

In sociology, an isomorphism is a similarity of the processes or structure of one organization to those of another, be it the result of imitation or independent development under similar constraints. There are three main types of institutional isomorphism: normative, coercive and mimetic. The development that these three types of isomorphism promote can also create isomorphic paradoxes that hinder such development. Specifically, these isomorphic paradoxes are related to an organization's remit, resources, accountability, and professionalization.

The concept of institutional isomorphism was primarily developed by Paul DiMaggio and Walter Powell. The concept appears in their classical paper The iron cage revisited: institutional isomorphism and collective rationality in organizational fields from 1983.

Isomorphism (biology)

In biology, an isomorphism is a similarity of form or structure between organisms, generally between organisms with independent ancestries, e.g. after convergent evolution. Two organisms exhibiting isomorphism are referred to as isomorphs.

The separate evolution of camera eyes in vertebrates and cephalopods (and insects, and many more, in as many as fifty separate instances), is an example of isomorphism. So is the evolution of wings in insects, pterosaurs, birds, and bats.

References