Кері операцияны өзгертетін гомоморфизм

Homomorphism reversing the order of something

In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an invertible antihomomorphism, i. e. an antiisomorphism, from a set to itself. From bijectivity it follows that antiautomorphisms have inverses, and that the inverse of an antiautomorphism is also an antiautomorphism.

Informally, an antihomomorphism is a map that switches the order of multiplication. Formally, an antihomomorphism between structures and is a homomorphism , where equals as a set, but has its multiplication reversed to that defined on Denoting the (generally non commutative) multiplication on by , the multiplication on , denoted by , is defined by The object is called the opposite object to (respectively, opposite group, opposite algebra, opposite category etc.). This definition is equivalent to that of a homomorphism (reversing the operation before or after applying the map is equivalent). Formally, sending to and acting as the identity on maps is a functor (indeed, an involution).

It is frequently the case that antiautomorphisms are involutions, i. e. the square of the antiautomorphism is the identity map; these are also called involutive antiautomorphisms. For example, in any group the map that sends x to its inverse x−1 is an involutive antiautomorphism. A ring with an involutive antiautomorphism is called a * ring, and these form an important class of examples.

If the source X or the target Y is commutative, then an antihomomorphism is the same thing as a homomorphism. 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.