nLab antihomomorphism

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 homomorphism $f\colon A \to B^\op$ where $B^\op$ is the opposite magma of $B$, or equivalently, it is a function (or $C$-morphism) $f\colon A \to B$ such that:

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

Note that for magma objects in $C$, the left-hand side of this equation is a generalised element of $B$ whose source is ${|x|} \otimes {|y|}$ (where ${|x|}$ and ${|y|}$ 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 ${|y|} \otimes {|x|}$. Therefore, this definition only makes unambiguous sense because $C$ is symmetric monoidal, using the unique natural isomorphism ${|x|} \otimes {|y|} \cong {|y|} \otimes {|x|}$.

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

Examples

• The inverse function of a group is a monoid anti-homomorphism, and in fact an anti-automorphism, hence an anti-involution, since $(x y) (x y)^{-1} = 1 = x x^{-1} = x y y^{-1} x^{-1}$, which means that $(x y)^{-1} = y^{-1} x^{-1}$.

• The antipode in a Hopf algebra is an anti-homomorphism (by this Prop.). The same is separately required for antipodes on associative bialgebroids.

• In a star-algebra the star-operation is an anti-homomorphism, in fact an anti-automorphism, hence an anti-involution.

• Combining these two examples, in an involutive Hopf algebra the antipode is an anti-automorphism.

Related concepts

• opposite magma