<s>
In	O
mathematics	O
,	O
an	O
involution	B-Algorithm
,	O
involutory	O
function	O
,	O
or	O
self-inverse	B-Algorithm
function	I-Algorithm
is	O
a	O
function	O
that	O
is	O
its	O
own	O
inverse	O
,	O
</s>
<s>
for	O
all	O
in	O
the	O
domain	B-Algorithm
of	O
.	O
</s>
<s>
Any	O
involution	B-Algorithm
is	O
a	O
bijection	B-Algorithm
.	O
</s>
<s>
The	O
identity	O
map	O
is	O
a	O
trivial	O
example	O
of	O
an	O
involution	B-Algorithm
.	O
</s>
<s>
Examples	O
of	O
nontrivial	O
involutions	B-Algorithm
include	O
negation	O
(	O
)	O
,	O
reciprocation	O
(	O
)	O
,	O
and	O
complex	O
conjugation	O
(	O
)	O
in	O
arithmetic	O
;	O
reflection	B-Algorithm
,	O
half-turn	O
rotation	B-General_Concept
,	O
and	O
circle	O
inversion	O
in	O
geometry	O
;	O
complementation	O
in	O
set	O
theory	O
;	O
and	O
reciprocal	O
ciphers	O
such	O
as	O
the	O
ROT13	B-Algorithm
transformation	O
and	O
the	O
Beaufort	B-Algorithm
polyalphabetic	B-Algorithm
cipher	I-Algorithm
.	O
</s>
<s>
The	O
composition	B-Application
of	O
two	O
involutions	B-Algorithm
f	O
and	O
g	O
is	O
an	O
involution	B-Algorithm
if	O
and	O
only	O
if	O
they	O
commute	O
:	O
.	O
</s>
<s>
The	O
number	O
of	O
involutions	B-Algorithm
,	O
including	O
the	O
identity	O
involution	B-Algorithm
,	O
on	O
a	O
set	O
with	O
elements	O
is	O
given	O
by	O
a	O
recurrence	O
relation	O
found	O
by	O
Heinrich	O
August	O
Rothe	O
in	O
1800	O
:	O
</s>
<s>
The	O
number	O
of	O
fixed	O
points	O
of	O
an	O
involution	B-Algorithm
on	O
a	O
finite	O
set	O
and	O
its	O
number	B-Application
of	I-Application
elements	I-Application
have	O
the	O
same	O
parity	O
.	O
</s>
<s>
Thus	O
the	O
number	O
of	O
fixed	O
points	O
of	O
all	O
the	O
involutions	B-Algorithm
on	O
a	O
given	O
finite	O
set	O
have	O
the	O
same	O
parity	O
.	O
</s>
<s>
In	O
particular	O
,	O
every	O
involution	B-Algorithm
on	O
an	O
odd	O
number	B-Application
of	I-Application
elements	I-Application
has	O
at	O
least	O
one	O
fixed	O
point	O
.	O
</s>
<s>
These	O
are	O
not	O
the	O
only	O
pre-calculus	O
involutions	B-Algorithm
.	O
</s>
<s>
The	O
graph	O
of	O
an	O
involution	B-Algorithm
(	O
on	O
the	O
real	O
numbers	O
)	O
is	O
symmetric	O
across	O
the	O
line	O
.	O
</s>
<s>
This	O
is	O
due	O
to	O
the	O
fact	O
that	O
the	O
inverse	O
of	O
any	O
general	O
function	O
will	O
be	O
its	O
reflection	B-Algorithm
over	O
the	O
line	O
.	O
</s>
<s>
If	O
,	O
in	O
particular	O
,	O
the	O
function	O
is	O
an	O
involution	B-Algorithm
,	O
then	O
its	O
graph	O
is	O
its	O
own	O
reflection	B-Algorithm
.	O
</s>
<s>
Other	O
elementary	O
involutions	B-Algorithm
are	O
useful	O
in	O
solving	O
functional	O
equations	O
.	O
</s>
<s>
A	O
simple	O
example	O
of	O
an	O
involution	B-Algorithm
of	O
the	O
three-dimensional	O
Euclidean	O
space	O
is	O
reflection	B-Algorithm
through	O
a	O
plane	O
.	O
</s>
<s>
Performing	O
a	O
reflection	B-Algorithm
twice	O
brings	O
a	O
point	O
back	O
to	O
its	O
original	O
coordinates	O
.	O
</s>
<s>
Another	O
involution	B-Algorithm
is	O
reflection	B-Algorithm
through	O
the	O
origin	O
;	O
not	O
a	O
reflection	B-Algorithm
in	O
the	O
above	O
sense	O
,	O
and	O
so	O
,	O
a	O
distinct	O
example	O
.	O
</s>
<s>
These	O
transformations	O
are	O
examples	O
of	O
affine	O
involutions	B-Algorithm
.	O
</s>
<s>
An	O
involution	B-Algorithm
is	O
a	O
projectivity	B-Algorithm
of	O
period	O
2	O
,	O
that	O
is	O
,	O
a	O
projectivity	B-Algorithm
that	O
interchanges	O
pairs	O
of	O
points	O
.	O
</s>
<s>
Any	O
projectivity	B-Algorithm
that	O
interchanges	O
two	O
points	O
is	O
an	O
involution	B-Algorithm
.	O
</s>
<s>
The	O
three	O
pairs	O
of	O
opposite	O
sides	O
of	O
a	O
complete	O
quadrangle	O
meet	O
any	O
line	O
(	O
not	O
through	O
a	O
vertex	O
)	O
in	O
three	O
pairs	O
of	O
an	O
involution	B-Algorithm
.	O
</s>
<s>
This	O
theorem	O
has	O
been	O
called	O
Desargues	O
's	O
Involution	B-Algorithm
Theorem	O
.	O
</s>
<s>
If	O
an	O
involution	B-Algorithm
has	O
one	O
fixed	O
point	O
,	O
it	O
has	O
another	O
,	O
and	O
consists	O
of	O
the	O
correspondence	O
between	O
harmonic	O
conjugates	O
with	O
respect	O
to	O
these	O
two	O
points	O
.	O
</s>
<s>
In	O
this	O
instance	O
the	O
involution	B-Algorithm
is	O
termed	O
"	O
hyperbolic	O
"	O
,	O
while	O
if	O
there	O
are	O
no	O
fixed	O
points	O
it	O
is	O
"	O
elliptic	O
"	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
projectivities	B-Algorithm
,	O
fixed	O
points	O
are	O
called	O
double	O
points	O
.	O
</s>
<s>
Another	O
type	O
of	O
involution	B-Algorithm
occurring	O
in	O
projective	O
geometry	O
is	O
a	O
polarity	O
which	O
is	O
a	O
correlation	B-Algorithm
of	O
period	O
2	O
.	O
</s>
<s>
In	O
linear	O
algebra	O
,	O
an	O
involution	B-Algorithm
is	O
a	O
linear	O
operator	O
T	O
on	O
a	O
vector	O
space	O
,	O
such	O
that	O
.	O
</s>
<s>
Except	O
for	O
in	O
characteristic	O
2	O
,	O
such	O
operators	O
are	O
diagonalizable	O
for	O
a	O
given	O
basis	O
with	O
just	O
1s	O
and	O
1s	O
on	O
the	O
diagonal	O
of	O
the	O
corresponding	O
matrix	B-Architecture
.	O
</s>
<s>
If	O
the	O
operator	O
is	O
orthogonal	O
(	O
an	O
orthogonal	O
involution	B-Algorithm
)	O
,	O
it	O
is	O
orthonormally	O
diagonalizable	O
.	O
</s>
<s>
It	O
can	O
be	O
checked	O
that	O
for	O
all	O
x	O
in	O
V	O
.	O
That	O
is	O
,	O
f	O
is	O
an	O
involution	B-Algorithm
of	O
V	O
.	O
</s>
<s>
For	O
a	O
specific	O
basis	O
,	O
any	O
linear	O
operator	O
can	O
be	O
represented	O
by	O
a	O
matrix	B-Architecture
T	O
.	O
Every	O
matrix	B-Architecture
has	O
a	O
transpose	O
,	O
obtained	O
by	O
swapping	O
rows	O
for	O
columns	O
.	O
</s>
<s>
This	O
transposition	B-Algorithm
is	O
an	O
involution	B-Algorithm
on	O
the	O
set	O
of	O
matrices	O
.	O
</s>
<s>
Since	O
elementwise	O
complex	O
conjugation	O
is	O
an	O
independent	O
involution	B-Algorithm
,	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
or	O
Hermitian	O
adjoint	O
is	O
also	O
an	O
involution	B-Algorithm
.	O
</s>
<s>
The	O
definition	O
of	O
involution	B-Algorithm
extends	O
readily	O
to	O
modules	O
.	O
</s>
<s>
Given	O
a	O
module	O
M	O
over	O
a	O
ring	O
R	O
,	O
an	O
R	O
endomorphism	O
f	O
of	O
M	O
is	O
called	O
an	O
involution	B-Algorithm
if	O
f2	O
is	O
the	O
identity	O
homomorphism	O
on	O
M	O
.	O
</s>
<s>
Involutions	B-Algorithm
are	O
related	O
to	O
idempotents	O
;	O
if	O
2	O
is	O
invertible	O
then	O
they	O
correspond	B-Algorithm
in	O
a	O
one-to-one	O
manner	O
.	O
</s>
<s>
In	O
functional	B-Application
analysis	I-Application
,	O
Banach	O
*	O
-algebras	O
and	O
C*	B-Algorithm
-algebras	I-Algorithm
are	O
special	O
types	O
of	O
Banach	O
algebras	O
with	O
involutions	B-Algorithm
.	O
</s>
<s>
Taken	O
as	O
an	O
axiom	O
,	O
it	O
leads	O
to	O
the	O
notion	O
of	O
semigroup	O
with	O
involution	B-Algorithm
,	O
of	O
which	O
there	O
are	O
natural	O
examples	O
that	O
are	O
not	O
groups	O
,	O
for	O
example	O
square	O
matrix	B-Architecture
multiplication	O
(	O
i.e.	O
</s>
<s>
the	O
full	O
linear	O
monoid	O
)	O
with	O
transpose	O
as	O
the	O
involution	B-Algorithm
.	O
</s>
<s>
In	O
ring	O
theory	O
,	O
the	O
word	O
involution	B-Algorithm
is	O
customarily	O
taken	O
to	O
mean	O
an	O
antihomomorphism	O
that	O
is	O
its	O
own	O
inverse	O
function	O
.	O
</s>
<s>
Examples	O
of	O
involutions	B-Algorithm
in	O
common	O
rings	O
:	O
</s>
<s>
taking	O
the	O
transpose	O
in	O
a	O
matrix	B-Architecture
ring	O
.	O
</s>
<s>
In	O
group	O
theory	O
,	O
an	O
element	O
of	O
a	O
group	O
is	O
an	O
involution	B-Algorithm
if	O
it	O
has	O
order	O
2	O
;	O
i.e.	O
</s>
<s>
an	O
involution	B-Algorithm
is	O
an	O
element	O
such	O
that	O
and	O
a2	O
=	O
e	O
,	O
where	O
e	O
is	O
the	O
identity	O
element	O
.	O
</s>
<s>
Originally	O
,	O
this	O
definition	O
agreed	O
with	O
the	O
first	O
definition	O
above	O
,	O
since	O
members	O
of	O
groups	O
were	O
always	O
bijections	B-Algorithm
from	O
a	O
set	O
into	O
itself	O
;	O
i.e.	O
,	O
group	O
was	O
taken	O
to	O
mean	O
permutation	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
By	O
the	O
end	O
of	O
the	O
19th	O
century	O
,	O
group	O
was	O
defined	O
more	O
broadly	O
,	O
and	O
accordingly	O
so	O
was	O
involution	B-Algorithm
.	O
</s>
<s>
A	O
permutation	B-Algorithm
is	O
an	O
involution	B-Algorithm
precisely	O
if	O
and	O
only	O
if	O
it	O
can	O
be	O
written	O
as	O
a	O
finite	O
product	O
of	O
disjoint	O
transpositions	B-Algorithm
.	O
</s>
<s>
The	O
involutions	B-Algorithm
of	O
a	O
group	O
have	O
a	O
large	O
impact	O
on	O
the	O
group	O
's	O
structure	O
.	O
</s>
<s>
The	O
study	O
of	O
involutions	B-Algorithm
was	O
instrumental	O
in	O
the	O
classification	O
of	O
finite	O
simple	O
groups	O
.	O
</s>
<s>
An	O
element	O
of	O
a	O
group	O
is	O
called	O
strongly	O
real	O
if	O
there	O
is	O
an	O
involution	B-Algorithm
with	O
(	O
where	O
)	O
.	O
</s>
<s>
Coxeter	B-Algorithm
groups	I-Algorithm
are	O
groups	O
generated	O
by	O
involutions	B-Algorithm
with	O
the	O
relations	O
determined	O
only	O
by	O
relations	O
given	O
for	O
pairs	O
of	O
the	O
generating	O
involutions	B-Algorithm
.	O
</s>
<s>
Coxeter	B-Algorithm
groups	I-Algorithm
can	O
be	O
used	O
,	O
among	O
other	O
things	O
,	O
to	O
describe	O
the	O
possible	O
regular	O
polyhedra	O
and	O
their	O
generalizations	O
to	O
higher	O
dimensions	O
.	O
</s>
<s>
The	O
operation	O
of	O
complement	O
in	O
Boolean	O
algebras	O
is	O
an	O
involution	B-Algorithm
.	O
</s>
<s>
In	O
algebraic	O
semantics	O
,	O
such	O
a	O
negation	O
is	O
realized	O
as	O
an	O
involution	B-Algorithm
on	O
the	O
algebra	O
of	O
truth	O
values	O
.	O
</s>
<s>
Examples	O
of	O
logics	O
which	O
have	O
involutive	O
negation	O
are	O
Kleene	O
and	O
Bochvar	O
three-valued	B-Language
logics	I-Language
,	O
Łukasiewicz	B-General_Concept
many-valued	I-General_Concept
logic	I-General_Concept
,	O
fuzzy	O
logic	O
IMTL	O
,	O
etc	O
.	O
</s>
<s>
Involutive	O
negation	O
is	O
sometimes	O
added	O
as	O
an	O
additional	O
connective	O
to	O
logics	O
with	O
non-involutive	O
negation	O
;	O
this	O
is	O
usual	O
,	O
for	O
example	O
,	O
in	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
.	O
</s>
<s>
The	O
same	O
relationship	O
holds	O
also	O
between	O
MV-algebras	O
and	O
BL-algebras	O
(	O
and	O
so	O
correspondingly	O
between	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
and	O
fuzzy	O
logic	O
BL	B-General_Concept
)	O
,	O
IMTL	O
and	O
MTL	B-General_Concept
,	O
and	O
other	O
pairs	O
of	O
important	O
varieties	O
of	O
algebras	O
(	O
resp	O
.	O
</s>
<s>
Since	O
the	O
converse	O
of	O
the	O
converse	O
is	O
the	O
original	O
relation	O
,	O
the	O
conversion	O
operation	O
is	O
an	O
involution	B-Algorithm
on	O
the	O
category	O
of	O
relations	O
.	O
</s>
<s>
While	O
this	O
ordering	O
is	O
reversed	O
with	O
the	O
complementation	O
involution	B-Algorithm
,	O
it	O
is	O
preserved	O
under	O
conversion	O
.	O
</s>
<s>
The	O
XOR	B-Application
bitwise	O
operation	O
with	O
a	O
given	O
value	O
for	O
one	O
parameter	O
is	O
an	O
involution	B-Algorithm
.	O
</s>
<s>
XOR	B-Application
masks	O
were	O
once	O
used	O
to	O
draw	O
graphics	O
on	O
images	O
in	O
such	O
a	O
way	O
that	O
drawing	O
them	O
twice	O
on	O
the	O
background	O
reverts	O
the	O
background	O
to	O
its	O
original	O
state	O
.	O
</s>
<s>
The	O
NOT	O
bitwise	O
operation	O
is	O
also	O
an	O
involution	B-Algorithm
,	O
and	O
is	O
a	O
special	O
case	O
of	O
the	O
XOR	B-Application
operation	O
where	O
one	O
parameter	O
has	O
all	O
bits	O
set	O
to	O
1	O
.	O
</s>
<s>
The	O
RC4	B-Algorithm
cryptographic	O
cipher	O
is	O
an	O
involution	B-Algorithm
,	O
as	O
encryption	O
and	O
decryption	O
operations	O
use	O
the	O
same	O
function	O
.	O
</s>
<s>
Practically	O
all	O
mechanical	O
cipher	O
machines	O
implement	O
a	O
reciprocal	O
cipher	O
,	O
an	O
involution	B-Algorithm
on	O
each	O
typed-in	O
letter	O
.	O
</s>
