Antihomomorphisms

Idea

While a homomorphism of magmas (including groups, rings, etc) must preserve multiplication, an antihomomorphism must instead reverse multiplication.

Definitions

Let $A$ and $B$ be magmas, or more generally magma objects in any symmetric monoidal category $C$. (Examples include groups, which are magmas with extra properties; rings, which are magma objects in Ab with extra proprties; etc.)

An antihomomorphism from $A$ to $B$ is a function (or $C$-morphism) $f:A\to B$ such that:

• for every two (generalised) elements $x,y$ of $A$, $f(xy)=f(y)f(x)$.

Note that for magma objects in $C$, the left-hand side of this equation is a generalised element of $B$ whose source is $\mid x\mid \otimes \mid y\mid$ (where $\mid x\mid$ and $\mid y\mid$ are the sources of the generalised elements $x$ and $y$ and $\otimes$ is the tensor product in $C$), while the right-hand side is a generalised element of $B$ whose source is $\mid y\mid \otimes \mid x\mid$. Therefore, this definition only makes unambiguous sense because $C$ is symmetric monoidal, using the unique natural isomorphism $\mid x\mid \otimes \mid y\mid \cong \mid y\mid \otimes \mid x\mid$.

An antiautomorphism is an antihomomorphism whose underlying $C$-morphism is an automorphism.

Examples

In a $*$-algebra the $*$ operator is an antiautomorphism (in fact an anti-involution).