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(l) A mapping of one entity into another having
the same elemental structure, whereby the behaviors of the two
entities are identically describable. (John Warfield) (2) A
formal correspondence of general principles or even of special
laws. (Bertalanffy) (3) A set of principles may be transferred
from one field to another without need to duplicate the effort.
(Weinberg) (4) a one-to-one correspondence between the elements
of two sets such that the result of an operation on elements of
one set corresponds to the result of the analogous operation on
their images in the other set.
A one-to-one mapping representing a pattern in the domain of the mapping by another pattern in its range without loss of information. The product of applying an isomorphism is called an isomorphic system that is a model of another by virtue of an isomorphism offers no simplifications. This may be useful, e.g., when one seeks an electronic substitute for a cognitive process such as the automatic pilot for an airplane. The more common simpler models, from a street map to a simulation of economic processes call for homomorphisms not isomorphisms. (Krippendorff)