antihomomorphism

In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective 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 $\phi\colon X \to Y^$, where $Y^$ 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 $Y^$, denoted by $*$, is defined by $x*y := y \cdot x$. The object $Y^$ is called the opposite object to $Y$ (respectively, opposite group, opposite algebra, opposite category