Isomorphism

Related Topics:
Isomorphism

For the term in sociology, see isomorphism (sociology).

In mathematics, an isomorphism (Greek:isos "equal", and morphe "shape") is a bijective map f such that both f and its inverse f⁻¹ are homomorphisms, i.e. structure-preserving mappings.

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 sets are structurally identical, if you choose to ignore finer-grained differences that may arise from how they are defined.

According to Douglas Hofstadter:

"The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures." (Gödel, Escher, Bach, p. 49)

Purpose

Isomorphisms are frequently used by mathematicians to save themselves work. If a good 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 solid cube made of wood and a solid cube made of lead are both solid cubes; although their matter differs, their geometric structures are isomorphic.
A standard deck of 52 playing cards with green backs and a standard deck of 52 playing cards with brown backs; although the colours on the backs 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 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.

Practical example

The following are examples of isomorphisms from ordinary algebra.

Consider the logarithm function: For any fixed base b, the logarithm function log_b 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 $(\mathbb{R},+)$ .

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 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 sets "look" different, they are indeed isomorphic. More generally, the direct product of two cyclic groups Z_n and Z_m is cyclic if and only if n and m are coprime.

Abstract examples

A relation-preserving isomorphism

For example, if one object consists of a set X with an ordering ≤ and the other object consists of a set Y with 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.

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)

In Analysis, the Legendre transform maps hard differential equations into easier algebraic equations.

In universal algebra, one can provide a general definition of isomorphism that covers these and many other cases. For a more general definition, see category theory.

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 linear algebra, an isomorphism can also be defined as a linear map between two vector spaces that is one-to-one and onto.

External links

Abstract algebra | Algebra | Category theory

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