Why do the interesting antihomomorphisms tend to be involutions?
Matthew Martinez 
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.
$\endgroup$ 12 Answers
$\begingroup$Contravariant functors are antihomomorphisms of small categories considered as semigroups.
$\endgroup$ $\begingroup$Given a semigroup $S$, define that an antihomomorphism on $S$ is a function $$* :S \rightarrow S$$
This should be called an anti-endomorphism. An antihomomorphism would rather be a function $$a :S \rightarrow T$$ Now if $S$ (or $T$) has an involutive anti-automorphism $i$, then $h=a\circ i$ (or $h=i\circ a$) is a normal homomorphism, we we get $a=h\circ i$ (or $a=i\circ h$). So in this case, you get the impression that only $i$ is an important anti-homomorphism, because every other anti-homomorphism just seems to correspond to a normal homomorphism.
If $S$ has no anti-automorphisms, then the identity $I:S \rightarrow S^{op}$ is an antihomomorphisms of interest, which cannot be written as the composition of a normal homomorphism and an involutive anti-automorphism.
The category of semigroups with homomorphisms and automorphisms as morphisms is different from the category of *-semigroups, and has different applications. The (finitely generated) free objects of this category still correspond directly $\Sigma^+$, i.e. the non-empty strings over the alphabet $\Sigma$. This category can be used to classify the classes of languages closed under reversal (of strings). (Of course, even in this case there is a trivial involutive antihomomorphism, i.e. said reversal. But it is imporant not to explicitly name it, because otherwise the free objects would be different.)
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