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Encyclopedia > Isomorphism

In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. More specifically, an isomorphism is an information-preserving transformation between one group to another. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... Eilhard Mitscherlich Eilhard Mitscherlich (1794-1863), is most known for his discovery of the principle of isomorphism. ...

Contents


Definition

Douglas Hofstadter provides an informal definition: This article needs to be cleaned up to conform to a higher standard of quality. ...

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)

Formally, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e. structure-preserving mappings. GEB cover Gödel, Escher, Bach: an Eternal Golden Braid (commonly GEB) is a Pulitzer Prize-winning book by Douglas Hofstadter, published in 1979 by Basic Books. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In abstract algebra, a homomorphism is a structure-preserving map. ...


Suppose we have two vector spaces (which can be thought of as groups of objects), V and W. We can assert there is an isomorphism (denoted T) from V to W if the following three properties are satisfied (suppose v is an element of V and w is an element of W): The fundamental concept in linear algebra is that of a vector space or linear space. ...


1) T is a linear transformation ( T(v1 + v2) = T(v1) + T(v2) and k * T(v1) = T(k*v1) )


2) If T(v1) = T(v2), then v1 = v2 (one-to-one)


3) For each w in W there is a v in V such that T(v) = w (onto)


For example, suppose V = odd numbers = {1, 3, 5, 7, ...} and W = even numbers = {2, 4, 6, 8, ...}, and T is the operation such that T(n) = 2n (for example, T(1) = 2 and T(2) = 4). Then it can be easily proved that T is an isomorphism from V onto W. First, T is linear (naturally). Second, suppose T(a)= T(b), where a and b are both odd numbers. Then 2a = 2b, and dividing by 2, a = b. Third, define w to be any arbitrary even number in W. Then w/2 would be an odd number such that T(w/2) = 2*w/2 = w.


Note that the transformation U such that U(n) = n^2 - 4n + 3 would not be an isomorphism because U(1) = U(3).


If there exists an isomorphism between two structures, we call the two structures isomorphic. Isomorphic structures are "the same" at a structural level of abstraction; if ignoring the specific identities of the elements in the underlying sets and focusing just on the structures themselves, then the two structures are identical. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...


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.

The Clock Tower, colloquially known as Big Ben Big Ben is the colloquial name of the Clock Tower of the Palace of Westminster in London, and an informal name for the Great Bell of Westminster, the largest bell in the tower and part of the Great Clock of Westminster. ...

Practical example

The following is an example of an isomorphism from ordinary algebra. Algebra is a branch of mathematics which studies structure and quantity. ...


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: Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

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 mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... In mathematics, the domain of a function is the set of all input values to the function. ... A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a 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: In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

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},+).


Two 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) iff uv.

Such an isomorphism is called an order isomorphism. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets. ...


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 correspondance between the operator star and the operator Diamond. In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, the range of a function is the set of all output values produced by that function. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

f(u) Diamond f(v) = f(u star v)

for all u, v in X.


Applications

Group isomorphism is where the objects in question are groups. Similarly, if the objects are fields, it is called a field isomorphism. In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...


In Analysis, the Legendre transform maps hard differential equations into easier algebraic equations. Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ... In mathematics, two differentiable functions f and g are said to be Legendre transforms of each other if their first derivatives are inverse functions of each other: f and g are then said to be related by a Legendre transformation. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Algebra is a branch of mathematics which studies structure and quantity. ...


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. Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...


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 iff there is an edge from f(u) to f(v) in H. A diagram of a graph with 6 vertices and 7 edges. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...


In linear algebra, an isomorphism can also be defined as a linear map between two vector spaces that is one-to-one and onto. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... In mathematics, a linear transformation (also called linear operator <<wrong! operators are LTs on the same vector space or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... The fundamental concept in linear algebra is that of a vector space or linear space. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...


See also


  Results from FactBites:
 
Isomorphism theorem (312 words)
In mathematics, the isomorphism theorems are 3 theorems that apply broadly in the realm of universal algebra.
First we state the isomorphism theorems for groups, where they take a simpler form and state important properties of factor groups (also called quotient groups).
The isomorphism theorems are also valid for modules over a fixed ring R (and therefore also for vector spaces over a fixed field).
Isomorphism - Wikipedia, the free encyclopedia (747 words)
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations.
Isomorphic structures are "the same" at some level of abstraction; ignoring the specific identities of the elements in the underlying sets, and focusing just on the structures themselves, the two structures are identical.
In linear algebra, an isomorphism can also be defined as a linear map between two vector spaces that is one-to-one and onto.
  More results at FactBites »

 
 

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