Everipedia is now IQ.wiki - Join the IQ Brainlist and our Discord for early access to editing on the new platform and to participate in the beta testing.

Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.

For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective.

In topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called diffeomorphisms.

A canonical isomorphism is a canonical map that is an isomorphism. Two objects are said to be canonically isomorphic if there is a canonical isomorphism between them. For example, the canonical map from a finite-dimensional vector space V to its second dual space is a canonical isomorphism; on the other hand, V is isomorphic to its dual space but not canonically in general.

Isomorphisms are formalized using category theory. A morphism f : XY in a category is an isomorphism if it admits a two-sided inverse, meaning that there is another morphism g : YX in that category such that gf = 1X and fg = 1Y, where 1X and 1Y are the identity morphisms of X and Y respectively.[4]

Examples

Logarithm and exponential

Letbe themultiplicative groupofpositive real numbers, and letbe the additive group of real numbers.
Thelogarithm functionsatisfiesfor all, so it is agroup homomorphism. Theexponential functionsatisfiesfor all, so it too is a homomorphism.
The identitiesandshow thatandareinversesof each other. Sinceis a homomorphism that has an inverse that is also a homomorphism,is an isomorphism of groups.
Becauseis an isomorphism, it translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using arulerand atable of logarithms, or using aslide rulewith a logarithmic scale.

Integers modulo 6

Consider the group, the integers from 0 to 5 with additionmodulo 6. Also consider the group, 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, under 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 + 4b) mod 6.

For example, (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, thedirect productof twocyclic groupsandis isomorphic toif and only if m and n arecoprime, per theChinese remainder theorem.

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 ƒ: XY such that:[5]

S is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, well-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 anordering≤ and S an ordering, then an isomorphism from X to Y is a bijective functionƒ: XYsuch that

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

If X = Y, then this is a relation-preserving automorphism.

Isomorphism vs. bijective morphism

In a concrete category (that is, a category whose objects are sets (perhaps with extra structure) and whose morphisms are structure-preserving functions), such as the category of topological spaces or categories of algebraic objects like groups, rings, and modules, an isomorphism must be bijective on the underlying sets. In algebraic categories (specifically, categories of varieties in the sense of universal algebra), an isomorphism is the same as a homomorphism which is bijective on underlying sets. However, there are concrete categories in which bijective morphisms are not necessarily isomorphisms (such as the category of topological spaces).

Applications

In abstract algebra, two basic isomorphisms are defined:

• Group isomorphism, an isomorphism between groups

• Ring isomorphism, an isomorphism between rings. (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, let 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 that an isomorphism is a morphism ƒ: ab that has an inverse morphism g: ba, i.e. ƒg = 1b and = 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 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 the regulator and processing parts 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.[6] Equality is when two objects are exactly the same, and everything that's true about one object is true about the other, while an isomorphism implies everything that's true about a designated part of one object's structure is true about the other's. For example, the sets

and

are equal; they are merely different representations—the first an intensional one (in set builder notation), and the second extensional (by explicit 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

while another is

and no one isomorphism is intrinsically better than any other.[1][2] On this view and in this sense, these two sets are not equal because one cannot consider them identical: one can choose an isomorphism between them, but that is a weaker claim than identity—and valid only in the context of the chosen isomorphism.

Sometimes the isomorphisms can seem obvious and compelling, but are still not equalities. As a simple example, the genealogical relationships among Joe, John, and Bobby Kennedy are, in a real sense, the same as those among the American football quarterbacks in the Manning family: Archie, Peyton, and Eli. The father-son pairings and the elder-brother-younger-brother pairings correspond perfectly. That similarity between the two family structures illustrates the origin of the word isomorphism (Greek iso-, "same," and -morph, "form" or "shape"). But because the Kennedys are not the same people as the Mannings, the two genealogical structures are merely isomorphic and not equal.

Another example is more formal and more directly illustrates the motivation for distinguishing equality from isomorphism: the distinction between afinite-dimensional vector spaceV and itsdual spaceV* = { φ: V → K} 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. If one chooses a basis for V, then this yields an isomorphism: For allu. vV,
.

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 V** = { x: V* → K} that does not depend on the choice of basis: For all vV and φ ∈ V*,

.
This leads to a third notion, that of anatural 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 anatural transformation; briefly, that one may consistently identify, or more generally map from, a finite-dimensional vector space to its double dual,, for any vector space in a consistent way. Formalizing this intuition is a motivation for the development of category theory.

However, there is a case where the distinction between natural isomorphism and equality is usually not made. That is for the objects that may be characterized by a universal property. In fact, there is a unique isomorphism, necessarily natural, between two objects sharing the same universal property. A typical example is the set of real numbers, which may be defined through infinite decimal expansion, infinite binary expansion, Cauchy sequences, Dedekind cuts and many other ways. Formally these constructions define different objects, which all are solutions of the same universal property. As these objects have exactly the same properties, one may forget the method of construction and considering them as equal. This is what everybody does when talking of "the set of the real numbers". The same occurs with quotient spaces: they are commonly constructed as sets of equivalence classes. However, talking of set of sets may be counterintuitive, and quotient spaces are commonly considered as a pair of a set of undetermined objects, often called "points", and a surjective map onto this set.

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 VV* and VV**. 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

which can be presented as the one-point compactification of the complex plane C ∪ {∞} or as the complex projective line (a quotient space)

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 CR2[3] plus an additional point, and the third is a subquotient of C2

In the context of category theory, objects are usually at most isomorphic—indeed, a motivation for the development of category theory was showing that different constructions in homology theory yielded equivalent (isomorphic) groups. Given maps between two objects X and Y, however, one asks if they are equal or not (they are both elements of the set Hom(XY), hence equality is the proper relationship), particularly in commutative diagrams.

• Bisimulation

• Heap (mathematics)

• Isometry

• Isomorphism class

• Isomorphism theorem

• Universal property

References

[1]
Citation Linkopenlibrary.orgA, B, C have a conventional order, namely alphabetical order, and similarly 1, 2, 3 have the order from the integers, and thus one particular isomorphism is "natural", namely . More formally, as sets these are isomorphic, but not naturally isomorphic (there are multiple choices of isomorphism), while as ordered sets they are naturally isomorphic (there is a unique isomorphism, given above), since finite total orders are uniquely determined up to unique isomorphism by cardinality. This intuition can be formalized by saying that any two finite totally ordered sets of the same cardinality have a natural isomorphism, the one that sends the least element of the first to the least element of the second, the least element of what remains in the first to the least element of what remains in the second, and so forth, but in general, pairs of sets of a given finite cardinality are not naturally isomorphic because there is more than one choice of map—except if the cardinality is 0 or 1, where there is a unique choice.
Sep 27, 2019, 10:07 AM
[2]
Citation Linkopenlibrary.orgIn fact, there are precisely different isomorphisms between two sets with three elements. This is equal to the number of automorphisms of a given three-element set (which in turn is equal to the order of the symmetric group on three letters), and more generally one has that the set of isomorphisms between two objects, denoted is a torsor for the automorphism group of A, and also a torsor for the automorphism group of B. In fact, automorphisms of an object are a key reason to be concerned with the distinction between isomorphism and equality, as demonstrated in the effect of change of basis on the identification of a vector space with its dual or with its double dual, as elaborated in the sequel.
Sep 27, 2019, 10:07 AM
[3]
Citation Linkopenlibrary.orgBeing precise, the identification of the complex numbers with the real plane, depends on a choice of one can just as easily choose , which yields a different identification—formally, complex conjugation is an automorphism—but in practice one often assumes that one has made such an identification.
Sep 27, 2019, 10:07 AM
[4]
Citation Linkbooks.google.comAwodey, Steve (2006). "Isomorphisms". Category theory. Oxford University Press. p. 11. ISBN 9780198568612.
Sep 27, 2019, 10:07 AM
[5]
Citation Linkbooks.google.comVinberg, Ėrnest Borisovich (2003). A Course in Algebra. American Mathematical Society. p. 3. ISBN 9780821834138.
Sep 27, 2019, 10:07 AM
[6]
Citation Linkopenlibrary.orgMazur, Barry (12 June 2007), When is one thing equal to some other thing? (PDF)
Sep 27, 2019, 10:07 AM
[7]
Sep 27, 2019, 10:07 AM
[8]
Sep 27, 2019, 10:07 AM
[9]
Sep 27, 2019, 10:07 AM
[10]
Sep 27, 2019, 10:07 AM
[11]
Sep 27, 2019, 10:07 AM
[12]
Citation Linkwww.math.harvard.eduWhen is one thing equal to some other thing?
Sep 27, 2019, 10:07 AM
[13]