antihomomorphism

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.

==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^{\text{op}}, where Y^{\text{op}} equals Y as a set, but has its multiplication reversed to that defined on Y. Denoting the (generally non-commutative) multiplication on Y by \cdot, the multiplication on Y^{\text{op}}, denoted by *, is defined by x*y := y \cdot x. The object Y^{\text{op}} is called the opposite object to Y (respectively, opposite group, opposite algebra, opposite category etc.).