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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
antihomomorphism, i.e. an
antiisomorphism In category theory, a branch of mathematics, an antiisomorphism (or anti-isomorphism) between structured sets ''A'' and ''B'' is an isomorphism from ''A'' to the opposite of ''B'' (or equivalently from the opposite of ''A'' to ''B''). If ther ...
, from a set to itself. From bijectivity it follows that antiautomorphisms have inverses, and that the inverse of an antiautomorphism is also an antiautomorphism.


Definition

Informally, an antihomomorphism is a map that switches the order of multiplication. Formally, an antihomomorphism between structures X and Y is a homomorphism \phi\colon X \to Y^, where Y^ equals Y as a set, but has its multiplication reversed to that defined on Y. Denoting the (generally non-
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
) multiplication on Y by \cdot, the multiplication on Y^, denoted by *, is defined by x*y := y \cdot x. The object Y^ is called the opposite object to Y (respectively,
opposite group In group theory, a branch of mathematics, an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action. Monoids, groups, rings, and algebras can be viewed as ca ...
,
opposite algebra In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring w ...
,
opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
etc.). This definition is equivalent to that of a homomorphism \phi\colon X^ \to Y (reversing the operation before or after applying the map is equivalent). Formally, sending X to X^ and acting as the identity on maps is a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
(indeed, an
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
).


Examples

In
group theory In 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 ...
, an antihomomorphism is a map between two groups that reverses the order of multiplication. So if 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 In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
operation in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, which takes
row vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
s to
column vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
s. Any vector-matrix equation may be transposed to an equivalent equation where the order of the factors is reversed. With matrices, an example of an antiautomorphism is given by the transpose map. Since inversion and transposing both give antiautomorphisms, their composition is an automorphism. This involution is often called the contragredient map, and it provides an example of an outer automorphism of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
, where ''F'' is a field, except when and , or and (i.e., for the groups , , and ). In
ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, an antihomomorphism is a map between two rings that preserves addition, but reverses the order of multiplication. So 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 ''K'', ''φ'' must be a ''K''-
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
of the underlying
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. 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
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
s, i.e. the square of the antiautomorphism is the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
; these are also called s. For example, in any group the map that sends ''x'' to its
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
''x''−1 is an involutive antiautomorphism. A ring with an involutive antiautomorphism is called a *-ring, and these form an important class of examples.


Properties

If the source ''X'' or the target ''Y'' is commutative, then an antihomomorphism is the same thing as a
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
. 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.


See also

*
Semigroup with involution In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, consider ...


References

*{{MathWorld, title=Antihomomorphism, urlname=Antihomomorphism Morphisms