Given a semigroup $S$, define that an antihomomorphism on $S$ is a function $$* :S \rightarrow S$$
satisfying $(xy)^* = y^*x^*.$ Examples abound. Consider:
- Transposition, where $S$ equals the set of $2 \times 2$ real matrices.
Conjugate-transposition, where $S$ equals the set of $2 \times 2$ complex matrices.
The map that takes a binary relation to its converse, where $S$ equals the monoid of binary relations on a set $X$.
- Inversion, in any group.
The weird thing is that in all of the above examples, the star operation is actually involutive. In fact, off the top of my head I can't think of any non-trivial antihomomorphisms that aren't also involutions.
Why do the antihomomorphisms of interest tend to be involutions?
I mean, is there some sort of "killer theorem" or something, that just makes involutive antihomomorphisms totally awesome?
Conversely, I am also interested in examples of antihomomorphisms that fail to be involutions, but which are still deemed important.
Contravariant functors are antihomomorphisms of small categories considered as semigroups.