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. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector 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 ⁻¹:Y→X, with the property that both f ⁻¹f=id_X and ff ⁻¹=id_Y. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

Informally, an isomorphism is a kind of mapping between objects, which shows a relationship between two properties or operations. In Mathematics and related technical fields the term map or mapping is often a Synonym for function. 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.

Contents 1 Purpose 2 Physical analogies 3 Practical example 4 Abstract examples 4.1 A relation-preserving isomorphism 4.2 An operation-preserving isomorphism 5 Applications 6 See also 7 External links

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. The Clock Tower is the world's largest four-faced chiming Clock.
• 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. In Recreational mathematics, a magic square of order n is an arrangement of n ² numbers usually distinct Integers in a square, such

Practical example

The following are examples of isomorphisms from ordinary algebra. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity.

• Consider the logarithm function: For any fixed base b, the logarithm function log_b maps from the positive real numbers onto the real numbers ; formally:

  

  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, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the codomain, or target, of a function f: X → Y is the set In addition to being an isomorphism of sets, the logarithm function also preserves certain operations. Specifically, consider the group of positive real numbers under ordinary multiplication. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element The logarithm function obeys the following identity:

  

  But the real numbers under addition also form a group. So the logarithm function is in fact a group isomorphism from the group to the group .

  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. A ruler, or rule, is an instrument used in Geometry, Technical drawing and engineering/building to measure distances and/or to rule straight Before Calculators were cheap and plentiful people would use mathematical tables &mdashlists of numbers showing the results of calculation with varying arguments&mdash to simplify The slide rule, also known as a slipstick, is a mechanical Analog computer.

• Consider the group Z₆, the numbers from 0 to 5 with addition modulo 6. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers 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 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 Z_n and Z_m is cyclic if and only if n and m are coprime. In Mathematics, one can often define a direct product of objectsalready known giving a new one In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than

Abstract examples

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. In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of ↔

S is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomoushttp://en.wikipedia.org../../../../articles/b/i/n/Binary_relation.html#Relations_over_a_set, 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. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. In Mathematics, a Binary relation R over a set X is symmetric if it holds for all a and b in X that In Mathematics, a Binary relation R on a set X is antisymmetric if for all a and b in X, if Asymmetric often means simply not symmetric In this sense an asymmetric relation is a Binary relation which is not a Symmetric relation. In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b In Mathematics, a Binary relation R over a set X is total if it holds for all a and b in X that In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In Mathematics, especially Order theory, a strict weak ordering is a Binary relation S that is a strict partial order In Mathematics, especially Order theory, a strict weak ordering is a Binary relation S that is a strict partial order In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent"

For example, R is an ordering ≤ and S an ordering , then an isomorphism from X to Y is a bijective function f : X → Y such that

if and only if u ≤ v. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying

Such an isomorphism is called an order isomorphism or (less commonly) an isotone isomorphism. In the mathematical field of Order theory an order isomorphism is a special kind of Monotone function that constitutes a suitable notion of Isomorphism

If X = Y we have a relation-preserving automorphism. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

An operation-preserving isomorphism

Suppose that on these sets X and Y, there are two binary operations and which happen to constitute the groups (X,) and (Y,). In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Note that the operators operate on elements from the domain and range, respectively, of the "one-to-one" and "onto" function f. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the range of a function is the set of all "output" values produced by that function 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 and the operator . In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

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. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules 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 In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real (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. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Abstract algebra, a heap (sometimes also called a groud) is a mathematical generalisation of a group. 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. Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, it is often desirable to express a functional relationship f(x\ as a different function whose argument is the derivative of f   rather A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity.

In category theory, Iet the category C consist of two classes, one of objects and the other of morphisms. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and 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 = 1_b and gf = 1_a. 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 Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Topological equivalence redirects here see also Topological equivalence (dynamical systems.

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. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property For other uses see Vertex. In Graph theory, a vertex (plural vertices) or node is the fundamental unit out ↔ See graph isomorphism. In Graph theory, an isomorphism of graphs G and H is a Bijection between the vertex sets of G and H

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. Logical atomism is a philosophical belief that originated in the early 20th century with the development of Analytic philosophy. Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian

In cybernetics the Good Regulator or Conant-Ashby theorem is stated "Every Good Regulator of a system must be a model of that system". Cybernetics is the interdisciplinary study of the Structure of Complex systems especially Communication processes control mechanisms and Feedback The Good Regulator is a theorem due to Roger C Conant and W Ross Ashby that is central to Cybernetics. Whether regulated or self-regulating an isomorphism is required between regulator part and the processing part of the system.

See also

• Epimorphism
• Heap (mathematics)
• Isomorphism class
• Monomorphism
• Isometry

External links

• Isomorphism on PlanetMath
• Eric W. Weisstein, Isomorphism at MathWorld. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which In Abstract algebra, a heap (sometimes also called a groud) is a mathematical generalisation of a group. An isomorphism class is a collection of Mathematical objects Isomorphic with a certain mathematical object In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. For the Mechanical engineering and Architecture usage see Isometric projection. PlanetMath is a free, collaborative online Mathematics Encyclopedia. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W

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