Anti-automorphism

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 being bijective it follows that it has an inverse, and that the inverse is also an antiautomorphism.

Informally, an antihomomorphism is map that switches the order of multiplication.

Formally, an antihomomorphism between X and Y is a homomorphism ${\displaystyle \phi \colon X\to Y^{\text{op}}}$, where ${\displaystyle Y^{\text{op}}}$ equals Y as a set, but has multiplication reversed: denoting the multiplication on Y as ${\displaystyle \cdot }$ and the multiplication on ${\displaystyle Y^{\text{op}}}$ as ${\displaystyle *}$, we have ${\displaystyle x*y:=y\cdot x}$. The object ${\displaystyle Y^{\text{op}}}$ is called the opposite object to Y. (Respectively, opposite group, opposite algebra, opposite category etc.)

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