In abstract algebra, an isomorphism (Greek: ἴσος isos "equal", and μορφή morphe "shape") is a kind of mapping between objects that 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 more minute definitional differences are ignored. More formally, an isomorphism is a bijective map f such that both f and its inverse f ⁻¹ are structure-preserving maps between two algebraic structures, i.e. they are both homomorphisms.

In category theory, an isomorphism is a morphism in a category for which there exists an "inverse" with the property that both and

Purpose

Isomorphisms are studied in mathematics in order to extend insights from one phenomenon to others: if two objects are isomorphic, then any property that is preserved by an isomorphism and that 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.

==Practical examples== 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 R⁺ onto the real numbers R; formally:

  :$\backslash log\_b\; :\; \backslash bold\{R\}^+\; \backslash to\; \backslash bold\{R\}.$

  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 (R⁺,×) of positive real numbers under ordinary multiplication. The logarithm function obeys the following identity:

  :$\backslash log\_b(xy)\; =\; \backslash log\_b(x)\; +\; \backslash log\_b(y).\; \backslash !$

  But the real numbers under addition also form a group. So the logarithm function is in fact a group isomorphism from the group (R⁺,×) to the group (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 integers 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 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_m and Z_n is isomorphic to Z_mn if and only if m and n are coprime.

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 such that :$\backslash operatorname\{S\}(f(u),f(v))\; \backslash iff\; \backslash operatorname\{R\}(u,v)$

S is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, 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 $\backslash scriptstyle\; \backslash sqsubseteq$, then an isomorphism from X to Y is a bijective function such that :$f(u)\; \backslash sqsubseteq\; f(v)\; \backslash iff\; u\; \backslash le\; v\; .$ Such an isomorphism is called an order isomorphism or (less commonly) an isotone isomorphism.

If we have a relation-preserving automorphism.

An operation-preserving isomorphism

Suppose that on these sets X and Y, there are two binary operations $\backslash scriptstyle\backslash star$ and $\backslash scriptstyle\backslash Diamond$ that happen to constitute the groups (X,$\backslash scriptstyle\backslash star$) and (Y,$\backslash scriptstyle\backslash Diamond$). Note that the operators operate on elements from the domain and range, respectively, of the "one-to-one" and "onto" function ƒ. There is an isomorphism from X to Y if the bijective function happens to produce results, that sets up a correspondence between the operator $\backslash scriptstyle\backslash star$ and the operator $\backslash scriptstyle\backslash Diamond$.

:$f(u)\; \backslash Diamond\; f(v)\; =\; f(u\; \backslash star\; v)$ 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 Laplace 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 that has an inverse, i.e. there exists a morphism with and . 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 ƒ(u) to ƒ(v) in H. See graph isomorphism.

In mathematical analysis, an isomorphism between two Hilbert spaces is a bijection preserving addition, scalar multiplication, and inner product.

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. An example of this line of thinking can be found in Russell's Introduction to Mathematical Philosophy.

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.

Relation with equality

In certain areas of mathematics, notably category theory, it is valuable to distinguish between equality on the one hand and isomorphism on the other. Equality is when two objects are "literally the same", while isomorphism is when two objects "can be made to correspond via an isomorphism". For example, the sets : and $B\; =\; \backslash \{-1,\; 0,\; 1\backslash \}\; \backslash ,$

are equal – they are two different presentations (one in set builder notation, one by an enumeration) of the same subset of the integers. By contrast, the sets {A,B,C} and {1,2,3} are not equal – the first has elements that are letters, while the second has elements that are numbers. These are isomorphic as sets, since finite sets are determined up to isomorphism by their cardinality (number of elements) and these both have three elements, but there are many choices of isomorphism – one isomorphism is :$\backslash text\{A\}\; \backslash mapsto\; 1,\; \backslash text\{B\}\; \backslash mapsto\; 2,\; \backslash text\{C\}\; \backslash mapsto\; 3,$ while another is $\backslash text\{A\}\; \backslash mapsto\; 3,\; \backslash text\{B\}\; \backslash mapsto\; 2,\; \backslash text\{C\}\; \backslash mapsto\; 1,$

and no one isomorphism is better than any other. Thus one cannot identify these two sets: one can choose an isomorphism between them, but any statement that identifies these two sets depends on the choice of isomorphism.

A motivating example is the distinction between a finite-dimensional vector space V and its dual space } of linear maps from V to its field of scalars K. These spaces have the same dimension, and thus are isomorphic as abstract vector spaces (since algebraically, vector spaces are classified by dimension, just as sets are classified by cardinality), but there is no "natural" choice of isomorphism $\backslash scriptstyle\; V\; \backslash ,\; \backslash overset\{\backslash sim\}\{\backslash to\}\; \backslash ,\; V^*$. If one chooses a basis for V, then this yields an isomorphism: For all , :$v\; \backslash \; \backslash overset\{\backslash sim\}\{\backslash mapsto\}\; \backslash \; \backslash phi\_v\; \backslash in\; V^*\; \backslash quad\; \backslash text\{such\; that\}\; \backslash quad\; \backslash phi\_v(u)\; =\; v^\{\backslash top\}\; u$.

This corresponds to transforming a column vector (element of V) to a row vector (element of V*) by transpose, but a different choice of basis gives a different isomorphism: the isomorphism "depends on the choice of basis". More subtly, there is a map from a vector space V to its double dual } that does not depend on the choice of basis: For all :$v\; \backslash \; \backslash overset\{\backslash sim\}\{\backslash mapsto\}\; \backslash \; x\_v\; \backslash in\; V^\{**\}\; \backslash quad\; \backslash text\{such\; that\}\; \backslash quad\; x\_v(\backslash phi)\; =\; \backslash phi(v)$.

This leads to a third notion, that of a natural isomorphism: while V and V** are different sets, there is a "natural" choice of isomorphism between them. This intuitive notion of "an isomorphism that does not depend on an arbitrary choice" is formalized in the notion of a natural transformation; briefly, that one may consistently identify, or more generally map from, a vector space to its double dual, $\backslash scriptstyle\; V\; \backslash ,\; \backslash overset\{\backslash sim\}\{\backslash to\}\; \backslash ,\; V^\{**\}$, for any vector space in a consistent way. Formalizing this intuition is a motivation for the development of category theory.

If one wishes to draw a distinction between an arbitrary isomorphism (one that depends on a choice) and a natural isomorphism (one that can be done consistently), one may write ≈ for an unnatural isomorphism and ≅ for a natural isomorphism, as in and This convention is not universally followed, and authors who wish to distinguish between unnatural isomorphisms and natural isomorphisms will generally explicitly state the distinction.

Generally, saying that two objects are equal is reserved for when there is a notion of a larger (ambient) space that these objects live in. Most often, one speaks of equality of two subsets of a given set (as in the integer set example above), but not of two objects abstractly presented. For example, the 2-dimensional unit sphere in 3-dimensional space :$S^2\; :=\; \backslash \{\; (x,y,z)\; \backslash in\; \backslash mathbf\{R\}^3\; |\; x^2\; +\; y^2\; +\; z^2\; =\; 1\backslash \}$ and the Riemann sphere $\backslash widehat\{\backslash mathbf\{C\}\}$

which can be presented as the one-point compactification of the complex plane } or as the complex projective line (a quotient space) :$\backslash mathbf\{P\}\_\{\backslash mathbf\{C\}\}^1\; :=\; (\backslash mathbf\{C\}^2\backslash setminus\; \backslash \{(0,0)\backslash \})\; /\; (\backslash mathbf\{C\}^*)$

are three different descriptions for a mathematical object, all of which are isomorphic, but not equal because they are not all subsets of a single space: the first is a subset of R³, the second is ² plus an additional point, and the third is a subquotient of C²

See also

Epimorphism

Heap (mathematics)

Isometry

Isomorphism class

Monomorphism

Morphism

Bisimulation

Notes

References

External links

Category:Morphisms Category:Abstract algebra Category:Algebra Category:Category theory

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