Antihomomorphism
Encyclopedia
In mathematics
, an antihomomorphism is a type of function
defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an antihomomorphism which has an inverse as an antihomomorphism; this coincides with it being a bijection
from an object to itself.
Formally, an antihomomorphism between X and Y is a homomorphism
, where 
equals Y as a set, but has multiplication reversed: denoting the multiplication on Y as 
and the multiplication on 
as 
, we have 
. The object 
is called the opposite object to Y. (Respectively, opposite group, opposite algebra, etc.)
This definition is equivalent to a homomorphism
(reversing the operation before or after applying the map is equivalent). Formally, sending X to 
and acting as the identity on maps is a functor
(indeed, an involution).
, an antihomomorphism is a map between two groups that reverses the order of multiplication. So if φ : X → Y is a group antihomomorphism,
for all x, y in X.
The map that sends x to x−1 is an example of a group antiautomorphism. Another important example is the transpose
operation in linear algebra
which takes row vectors to column vectors. Any vector-matrix equation may be transposed to an equivalent equation where the order of the factors is reversed.
In ring theory
, an antihomomorphism is a map between two rings that preserves addition, but reverses the order of multiplication. So φ : X → Y is a ring antihomomorphism if and only if:
for all x, y in X.
For algebras over a field
K, φ must be a K-linear map of the underlying vector space
. If the underlying field has an involution, one can instead ask φ to be conjugate-linear, as in conjugate transpose, below.
; these are also called s.
A ring with an involutive antiautomorphism is called a *-ring, and these form an important class of examples.
and an antiautomorphism is the same thing as an automorphism
.
The composition
of two antihomomorphisms is always a homomorphism, since reversing the order twice preserves order. The composition of an antihomomorphism with a homomorphism gives another antihomomorphism.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an antihomomorphism is a type of function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an antihomomorphism which has an inverse as an antihomomorphism; this coincides with it being a bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
from an object to itself.
Definition
Informally, an antihomomorphism is map that switches the order of multiplication.Formally, an antihomomorphism between X and Y is a homomorphism
, where equals Y as a set, but has multiplication reversed: denoting the multiplication on Y as and the multiplication on as , we have . The object is called the opposite object to Y. (Respectively, opposite group, opposite algebra, etc.)This definition is equivalent to a homomorphism
(reversing the operation before or after applying the map is equivalent). Formally, sending X to and acting as the identity on maps is a functorFunctor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
(indeed, an involution).
Examples
In group theoryGroup theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, an antihomomorphism is a map between two groups that reverses the order of multiplication. So if φ : X → Y is a group antihomomorphism,
- φ(xy) = φ(y)φ(x)
for all x, y in X.
The map that sends x to x−1 is an example of a group antiautomorphism. Another important example is the transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
operation in linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
which takes row vectors to column vectors. Any vector-matrix equation may be transposed to an equivalent equation where the order of the factors is reversed.
In ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, an antihomomorphism is a map between two rings that preserves addition, but reverses the order of multiplication. So φ : X → Y is a ring antihomomorphism if and only if:
- φ(1) = 1
- φ(x+y) = φ(x)+φ(y)
- φ(xy) = φ(y)φ(x)
for all x, y in X.
For algebras over a field
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
K, φ must be a K-linear map of the underlying vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
. If the underlying field has an involution, one can instead ask φ to be conjugate-linear, as in conjugate transpose, below.
Involutions
It is frequently the case that antiautomorphisms are involutions, i.e. the square of the antiautomorphism is the identity mapIdentity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
; these are also called s.
- The map that sends x to its inverseInverse elementIn abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
x−1 is an involutive antiautomorphism in any group.
A ring with an involutive antiautomorphism is called a *-ring, and these form an important class of examples.
Properties
If the target Y is commutative, then an antihomomorphism is the same thing as a homomorphismHomomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
and an antiautomorphism is the same thing as an automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
.
The composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
of two antihomomorphisms is always a homomorphism, since reversing the order twice preserves order. The composition of an antihomomorphism with a homomorphism gives another antihomomorphism.