The word is derived from Ancient Greek ἴσος (isos) 'equal' and μορφή (morphe) 'form, 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).
[citation needed] An automorphism is an isomorphism from a structure to itself.
In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration.
For example: Category theory, which can be viewed as a formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different aspects of the basic idea.
be the multiplicative group of positive real numbers, and let
This facility makes it possible to multiply real numbers using a ruler and a table of logarithms, or using a slide rule with a logarithmic scale.
Even though these two groups "look" different in that the sets contain different elements, they are indeed isomorphic: their structures are exactly the same.
More generally, the direct product of two cyclic groups
If one object consists of a set X with a binary relation R and the other object consists of a set Y with a binary relation S then an isomorphism from X to Y is a bijective function
Some are more specifically studied; for example: Just as the automorphisms of an algebraic structure form a group, the isomorphisms between two algebras sharing a common structure form a heap.
Letting a particular isomorphism identify the two structures turns this heap into a group.
In mathematical analysis, the Laplace transform is an isomorphism mapping hard differential equations into easier algebraic equations.
from P to Q that preserves the order structure in the sense that for any elements
As an example, the set {1,2,3,6} of whole numbers ordered by the is-a-factor-of relation is isomorphic to the set {O, A, B, AB} of blood types ordered by the can-donate-to relation.
In mathematical analysis, an isomorphism between two Hilbert spaces is a bijection preserving addition, scalar multiplication, and inner product.
In early theories of logical atomism, the formal relationship between facts and true propositions was theorized by Bertrand Russell and Ludwig Wittgenstein to be isomorphic.
An example of this line of thinking can be found in Russell's Introduction to Mathematical Philosophy.
In cybernetics, the good regulator theorem or Conant–Ashby theorem is stated as "Every good regulator of a system must be a model of that system".
In algebraic categories (specifically, categories of varieties in the sense of universal algebra), an isomorphism is the same as a homomorphism which is bijective on underlying sets.
[2] Examples of isomorphism classes are plentiful in mathematics.
However, there are circumstances in which the isomorphism class of an object conceals vital information about it.
are equal; they are merely different representations—the first an intensional one (in set builder notation), and the second extensional (by explicit enumeration)—of the same subset of the integers.
[note 1] On this view and in this sense, these two sets are not equal because one cannot consider them identical: one can choose an isomorphism between them, but that is a weaker claim than identity and valid only in the context of the chosen isomorphism.
Also, integers and even numbers are isomorphic as ordered sets and abelian groups (for addition), but cannot be considered equal sets, since one is a proper subset of the other.
This is generally the case with solutions of universal properties.
For example, the rational numbers are formally defined as equivalence classes of pairs of integers, although nobody thinks of a rational number as a set (equivalence class).
The universal property of the rational numbers is essentially that they form a field that contains the integers and does not contain any proper subfield.
This allows identifying these two fields, since every property of one of them can be transferred to the other through the isomorphism.
There is thus a unique isomorphism from this subfield of the reals to the rational numbers defined by equivalence classes.