<s>
An	O
antiautomorphism	B-Algorithm
is	O
a	O
bijective	B-Algorithm
antihomomorphism	O
,	O
i.e.	O
</s>
<s>
From	O
bijectivity	B-Algorithm
it	O
follows	O
that	O
antiautomorphisms	B-Algorithm
have	O
inverses	O
,	O
and	O
that	O
the	O
inverse	O
of	O
an	O
antiautomorphism	B-Algorithm
is	O
also	O
an	O
antiautomorphism	B-Algorithm
.	O
</s>
<s>
Formally	O
,	O
sending	O
to	O
and	O
acting	O
as	O
the	O
identity	O
on	O
maps	O
is	O
a	O
functor	B-Language
(	O
indeed	O
,	O
an	O
involution	B-Algorithm
)	O
.	O
</s>
<s>
The	O
map	O
that	O
sends	O
x	O
to	O
x−1	O
is	O
an	O
example	O
of	O
a	O
group	O
antiautomorphism	B-Algorithm
.	O
</s>
<s>
Another	O
important	O
example	O
is	O
the	O
transpose	O
operation	O
in	O
linear	B-Language
algebra	I-Language
,	O
which	O
takes	O
row	O
vectors	O
to	O
column	O
vectors	O
.	O
</s>
<s>
With	O
matrices	O
,	O
an	O
example	O
of	O
an	O
antiautomorphism	B-Algorithm
is	O
given	O
by	O
the	O
transpose	O
map	O
.	O
</s>
<s>
Since	O
inversion	O
and	O
transposing	O
both	O
give	O
antiautomorphisms	B-Algorithm
,	O
their	O
composition	B-Application
is	O
an	O
automorphism	O
.	O
</s>
<s>
This	O
involution	B-Algorithm
is	O
often	O
called	O
the	O
contragredient	O
map	O
,	O
and	O
it	O
provides	O
an	O
example	O
of	O
an	O
outer	O
automorphism	O
of	O
the	O
general	O
linear	O
group	O
,	O
where	O
F	O
is	O
a	O
field	O
,	O
except	O
when	O
and	O
,	O
or	O
and	O
(	O
i.e.	O
,	O
for	O
the	O
groups	O
,	O
,	O
and	O
)	O
.	O
</s>
<s>
If	O
the	O
underlying	O
field	O
has	O
an	O
involution	B-Algorithm
,	O
one	O
can	O
instead	O
ask	O
φ	O
to	O
be	O
conjugate-linear	O
,	O
as	O
in	O
conjugate	O
transpose	O
,	O
below	O
.	O
</s>
<s>
It	O
is	O
frequently	O
the	O
case	O
that	O
antiautomorphisms	B-Algorithm
are	O
involutions	B-Algorithm
,	O
i.e.	O
</s>
<s>
the	O
square	O
of	O
the	O
antiautomorphism	B-Algorithm
is	O
the	O
identity	O
map	O
;	O
these	O
are	O
also	O
called	O
s	O
.	O
For	O
example	O
,	O
in	O
any	O
group	O
the	O
map	O
that	O
sends	O
x	O
to	O
its	O
inverse	O
x−1	O
is	O
an	O
involutive	O
antiautomorphism	B-Algorithm
.	O
</s>
<s>
A	O
ring	O
with	O
an	O
involutive	O
antiautomorphism	B-Algorithm
is	O
called	O
a	O
*	O
-ring	O
,	O
and	O
these	O
form	O
an	O
important	O
class	O
of	O
examples	O
.	O
</s>
<s>
The	O
composition	B-Application
of	O
two	O
antihomomorphisms	O
is	O
always	O
a	O
homomorphism	O
,	O
since	O
reversing	O
the	O
order	O
twice	O
preserves	O
order	O
.	O
</s>
<s>
The	O
composition	B-Application
of	O
an	O
antihomomorphism	O
with	O
a	O
homomorphism	O
gives	O
another	O
antihomomorphism	O
.	O
</s>
