Isomorphism

In abstract algebra, an isomorphism (Greek: ison "equal", and morphe "shape") is a bijective map f such that both f and its inverse f&minus;1 are homomorphisms, i.e., structure-preserving 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&minus;1:Y→X, with the property that both f&minus;1f=idX and ff&minus;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 finer-grained 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 &mdash; 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 six-sided 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 tic-tac-toe, 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 tic-tac-toe, 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 $$\mathbb{R}^+$$ onto the real numbers $$\mathbb{R}$$; formally:


 * $$\log_b : \mathbb{R}^+ \to \mathbb{R} \!$$

This mapping is one-to-one 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 $$(\mathbb{R}^+,\times)$$ 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 $$(\mathbb{R}^+,\times)$$ to the group $$(\mathbb{R},+)$$. 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 &times; 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 x-coordinate is modulo 2 and addition in the y-coordinate 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.  

A relation-preserving 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.

Such an isomorphism is called an order isomorphism or (less commonly) an isotone isomorphism.

If X = Y we have a relation-preserving automorphism.

An operation-preserving 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 "one-to-one" 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 $$\Diamond$$.


 * $$f(u) \Diamond f(v) = f(u \star v)$$

for all u, v in X.

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 Conant-Ashby theorem is stated "Every Good Regulator of a system must be a model of that system". Whether regulated or self-regulating an isomorphism is required between regulator part and the processing part of the system.