An antihomomorphism from $A$ to $B$ 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

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

Last revised on December 11, 2017 at 08:13:07.
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