Dictionary Definition
isomorphism n : (biology) similarity or identity
of form or shape or structure [syn: isomorphy]
User Contributed Dictionary
English
Noun
 the similarity in form of organisms of different ancestry
 A bijection f such that both f and its inverse f −1 are homomorphisms, that is, structurepreserving mappings.
 the similarity in the crystal structures of similar chemical compounds
 the similarity in the structure or processes of different organizations
 a onetoone correspondence between all the elements of two sets, e.g. the instances of two classes, or the records in two datasets
Related terms
Translations
 Croatian: izomorfizam
 Czech: izomorfismus
 French: isomorphisme
 Portuguese: isomorfismo
Extensive Definition
In abstract
algebra, an isomorphism (Greek:
ison "equal", and morphe "shape") is a bijective map f such that both
f and its inverse
f −1 are homomorphisms, i.e.,
structurepreserving mappings.
In the more general setting of 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
ff −1=idY.
Informally, an isomorphism is a kind of mapping
between objects, which shows a relationship between two properties
or operations. If there exists an isomorphism between two
structures, we call the two structures isomorphic. In a certain
sense, isomorphic structures are structurally identical, if you
choose to ignore finergrained differences that may arise from how
they are defined.
Purpose
Isomorphisms are studied in mathematics in order to extend insights from one phenomenon to others: if two objects are isomorphic, then any property which is preserved by an isomorphism and which is true of one of the objects is also true of the other. If an isomorphism can be found from a relatively unknown part of mathematics into some well studied division of mathematics, where many theorems are already proved, and many methods are already available to find answers, then the function can be used to map whole problems out of unfamiliar territory over to "solid ground" where the problem is easier to understand and work with.Physical analogies
Here are some everyday examples of isomorphic
structures:
 A standard deck of 52 playing cards with the four suits hearts, diamonds, spades, and clubs and a standard deck of 52 playing cards with four suits of triangles, circles, squares, and pentagons; although the suits of each deck differ, the decks are structurally isomorphic — if we wish to play cards, it doesn't matter which deck we choose to use.
 The Clock Tower in London (that contains Big Ben) and a wristwatch; although the clocks vary greatly in size, their mechanisms of reckoning time are isomorphic.
 A sixsided die and a bag from which a number 1 through 6 is chosen; although the method of obtaining a number is different, their random number generating abilities are isomorphic. This is an example of functional isomorphism, without the presumption of geometric isomorphism.
 There is a game which is isomorphic to tictactoe, but on the surface appears completely different. Players take it in turn to say a number between one and nine. Numbers may not be repeated. Both players aim to say three numbers which add up to 15. Plotting these numbers on a 3×3 magic square will reveal the exact correspondence with the game of tictactoe, given that three numbers will be arranged in a straight line if and only if they add up to 15.
Practical example
The following are examples of isomorphisms from
ordinary algebra.
Consider the logarithm function: For any
fixed base b, the logarithm function logb maps from the positive
real
numbers \mathbb^+ onto the real numbers \mathbb;
formally:
 \log_b : \mathbb^+ \to \mathbb \!
This mapping is onetoone
and onto,
that is, it is a bijection from the domain
to the codomain of the
logarithm function.
In addition to being an isomorphism of sets, the
logarithm function also preserves certain operations. Specifically,
consider the group
(\mathbb^+,\times) of positive real numbers under ordinary
multiplication. The logarithm function obeys the following
identity:
 \log_b(x \times y) = \log_b(x) + \log_b(y) \!
But the real numbers under addition also form a
group. So the logarithm function is in fact a group isomorphism
from the group (\mathbb^+,\times) to the group (\mathbb,+).
Logarithms can therefore be used to simplify
multiplication of real numbers. By working with logarithms,
multiplication of positive real numbers is replaced by addition of
logs. This way it is possible to multiply real numbers using a
ruler and a table
of logarithms, or using a slide rule
with a logarithmic scale.
Consider the group Z6, the numbers from 0 to 5
with addition modulo
6. Also consider the group Z2 × Z3, the ordered pairs
where the x coordinates can be 0 or 1, and the y coordinates can be
0, 1, or 2, where addition in the xcoordinate is modulo 2 and
addition in the ycoordinate is modulo 3.
These structures are isomorphic under addition,
if you identify them using the following scheme:

 (0,0) > 0
 (1,1) > 1
 (0,2) > 2
 (1,0) > 3
 (0,1) > 4
 (1,2) > 5
or in general (a,b) > ( 3a + 4 b ) mod
6.
For example note that (1,1) + (1,0) = (0,1) which
translates in the other system as 1 + 3 = 4.
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 Zn and Zm is cyclic if and only if n and m are coprime.
Abstract examples
A relationpreserving isomorphism
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 f : X → Y such that
 f(u) S f(v) if and only if u R v.
S is reflexive,
irreflexive,
symmetric,
antisymmetric,
asymmetric,
transitive,
total, , a
partial
order, total order,
strict
weak order,
total preorder (weak order), an equivalence
relation, or a relation with any other special properties, if
and only if R is.
For example, R is an ordering ≤
and S an ordering \sqsubseteq, then an isomorphism from X to Y is a
bijective function f : X → Y
such that
 f(u) \sqsubseteq f(v) if and only if u ≤ v.
If X = Y we have a relationpreserving automorphism.
An operationpreserving isomorphism
Suppose that on these sets X and Y, there are two
binary
operations \star and \Diamond which happen to constitute the
groups
(X,\star) and (Y,\Diamond). Note that the operators operate on
elements from the domain
and range,
respectively, of the "onetoone" and "onto" function f. There is
an isomorphism from X to Y if the bijective function f :
X → Y happens to produce results, that sets up a
correspondence between the operator \star and the operator
\Diamond.
 f(u) \Diamond f(v) = f(u \star v)
Applications
In abstract algebra, two basic isomorphisms are defined: Group isomorphism, an isomorphism between groups
 Ring isomorphism, an isomorphism between rings. (Note that isomorphisms between fields are actually ring isomorphisms)
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 Legendre
transform is an isomorphism mapping hard differential
equations into easier algebraic equations.
In category
theory, Iet the category
C consist of two classes,
one of objects and the other of morphisms. Then a general
definition of isomorphism that covers the previous and many other
cases is: an isomorphism is a morphism f : a → b that has an
inverse, i.e. there exists a morphism g : b → a with fg = 1b and gf
= 1a. For example, a bijective linear map is
an isomorphism between vector
spaces, and a bijective continuous
function whose inverse is also continuous is an isomorphism
between topological
spaces, called a homeomorphism.
In graph
theory, an isomorphism between two graphs G and H is a bijective map f from the
vertices of G to the vertices of H that preserves the "edge
structure" in the sense that there is an edge from vertex
u to vertex v in G if and
only if there is an edge from f(u) to f(v) in H. See graph
isomorphism.
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.
In cybernetics the Good
Regulator or ConantAshby theorem is stated "Every Good
Regulator of a system must be a model of that system". Whether
regulated or selfregulating an isomorphism is required between
regulator part and the processing part of the system.