<s>
the	O
"	O
is	O
orthogonal	O
to	O
"	O
relation	O
in	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
Binary	O
relations	O
are	O
also	O
heavily	O
used	O
in	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
The	O
terms	O
,	O
dyadic	B-Algorithm
relation	I-Algorithm
and	O
two-place	O
relation	O
are	O
synonyms	O
for	O
binary	O
relation	O
,	O
though	O
some	O
authors	O
use	O
the	O
term	O
"	O
binary	O
relation	O
"	O
for	O
any	O
subset	O
of	O
a	O
Cartesian	O
product	O
without	O
reference	O
to	O
and	O
,	O
and	O
reserve	O
the	O
term	O
"	O
correspondence	O
"	O
for	O
a	O
binary	O
relation	O
with	O
reference	O
to	O
and	O
.	O
</s>
<s>
The	O
of	O
R	O
is	O
the	O
union	O
of	O
its	O
domain	B-Algorithm
of	I-Algorithm
definition	I-Algorithm
and	O
its	O
codomain	O
of	O
definition	O
.	O
</s>
<s>
Binary	O
relations	O
over	O
sets	O
X	O
and	O
Y	O
can	O
be	O
represented	O
algebraically	O
by	O
logical	B-Algorithm
matrices	I-Algorithm
indexed	O
by	O
X	O
and	O
Y	O
with	O
entries	O
in	O
the	O
Boolean	O
semiring	O
(	O
addition	O
corresponds	O
to	O
OR	O
and	O
multiplication	O
to	O
AND	O
)	O
where	O
matrix	O
addition	O
corresponds	O
to	O
union	O
of	O
relations	O
,	O
matrix	O
multiplication	O
corresponds	O
to	O
composition	O
of	O
relations	O
(	O
of	O
a	O
relation	O
over	O
X	O
and	O
Y	O
and	O
a	O
relation	O
over	O
Y	O
and	O
Z	O
)	O
,	O
the	O
Hadamard	O
product	O
corresponds	O
to	O
intersection	O
of	O
relations	O
,	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
corresponds	O
to	O
the	O
empty	O
relation	O
,	O
and	O
the	O
matrix	O
of	O
ones	O
corresponds	O
to	O
the	O
universal	O
relation	O
.	O
</s>
<s>
Homogeneous	O
relations	O
(	O
when	O
)	O
form	O
a	O
matrix	O
semiring	O
(	O
indeed	O
,	O
a	O
matrix	O
semialgebra	O
over	O
the	O
Boolean	O
semiring	O
)	O
where	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
corresponds	O
to	O
the	O
identity	O
relation	O
.	O
</s>
<s>
For	O
such	O
a	O
relation	O
,	O
 { Y } 	O
is	O
called	O
a	O
primary	B-Application
key	I-Application
of	O
R	O
.	O
For	O
example	O
,	O
the	O
green	O
and	O
blue	O
binary	O
relations	O
in	O
the	O
diagram	O
are	O
injective	O
,	O
but	O
the	O
red	O
one	O
is	O
not	O
(	O
as	O
it	O
relates	O
both	O
−1	O
and	O
1	O
to	O
1	O
)	O
,	O
nor	O
the	O
black	O
one	O
(	O
as	O
it	O
relates	O
both	O
−1	O
and	O
1	O
to	O
0	O
)	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
domain	B-Algorithm
of	I-Algorithm
definition	I-Algorithm
of	O
R	O
is	O
equal	O
to	O
X	O
.	O
</s>
<s>
Surjective	B-Algorithm
(	O
also	O
called	O
right-total	O
or	O
onto	O
)	O
:	O
for	O
all	O
y	O
in	O
Y	O
,	O
there	O
exists	O
an	O
x	O
in	O
X	O
such	O
that	O
xRy	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
green	O
and	O
blue	O
binary	O
relations	O
in	O
the	O
diagram	O
are	O
surjective	B-Algorithm
,	O
but	O
the	O
red	O
one	O
is	O
not	O
(	O
as	O
it	O
does	O
not	O
relate	O
any	O
real	O
number	O
to	O
−1	O
)	O
,	O
nor	O
the	O
black	O
one	O
(	O
as	O
it	O
does	O
not	O
relate	O
any	O
real	O
number	O
to	O
2	O
)	O
.	O
</s>
<s>
A	O
:	O
a	O
function	O
that	O
is	O
surjective	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
green	O
binary	O
relation	O
in	O
the	O
diagram	O
is	O
a	O
surjection	B-Algorithm
,	O
but	O
the	O
red	O
,	O
blue	O
and	O
black	O
ones	O
are	O
not	O
.	O
</s>
<s>
A	O
:	O
a	O
function	O
that	O
is	O
injective	O
and	O
surjective	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
green	O
binary	O
relation	O
in	O
the	O
diagram	O
is	O
a	O
bijection	B-Algorithm
,	O
but	O
the	O
red	O
,	O
blue	O
and	O
black	O
ones	O
are	O
not	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
homogeneous	O
relations	O
over	O
a	O
set	O
X	O
is	O
the	O
power	O
set	O
which	O
is	O
a	O
Boolean	O
algebra	O
augmented	O
with	O
the	O
involution	B-Algorithm
of	O
mapping	O
of	O
a	O
relation	O
to	O
its	O
converse	O
relation	O
.	O
</s>
<s>
Considering	O
composition	O
of	O
relations	O
as	O
a	O
binary	O
operation	O
on	O
,	O
it	O
forms	O
a	O
semigroup	O
with	O
involution	B-Algorithm
.	O
</s>
<s>
In	O
contrast	O
to	O
homogeneous	O
relations	O
,	O
the	O
composition	O
of	O
relations	O
operation	O
is	O
only	O
a	O
partial	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
Binary	O
relations	O
have	O
been	O
described	O
through	O
their	O
induced	O
concept	B-Algorithm
lattices	I-Algorithm
:	O
</s>
<s>
The	O
MacNeille	O
completion	O
theorem	O
(	O
1937	O
)	O
(	O
that	O
any	O
partial	O
order	O
may	O
be	O
embedded	O
in	O
a	O
complete	O
lattice	O
)	O
is	O
cited	O
in	O
a	O
2013	O
survey	O
article	O
"	O
Decomposition	O
of	O
relations	O
on	O
concept	B-Algorithm
lattices	I-Algorithm
"	O
.	O
</s>
<s>
The	O
"	O
induced	O
concept	B-Algorithm
lattice	I-Algorithm
is	O
isomorphic	O
to	O
the	O
cut	O
completion	O
of	O
the	O
partial	O
order	O
E	O
that	O
belongs	O
to	O
the	O
minimal	O
decomposition	O
(	O
f	O
,	O
g	O
,	O
E	O
)	O
of	O
the	O
relation	O
R.	O
"	O
</s>
<s>
Particular	O
cases	O
are	O
considered	O
below	O
:	O
E	O
total	O
order	O
corresponds	O
to	O
Ferrers	O
type	O
,	O
and	O
E	O
identity	O
corresponds	O
to	O
difunctional	B-Algorithm
,	O
a	O
generalization	O
of	O
equivalence	O
relation	O
on	O
a	O
set	O
.	O
</s>
<s>
Structural	O
analysis	O
of	O
relations	O
with	O
concepts	O
provides	O
an	O
approach	O
for	O
data	B-Application
mining	I-Application
.	O
</s>
<s>
Proposition	O
:	O
If	O
R	O
is	O
a	O
surjective	B-Algorithm
relation	O
,	O
then	O
where	O
is	O
the	O
identity	O
relation	O
.	O
</s>
<s>
The	O
idea	O
of	O
a	O
difunctional	B-Algorithm
relation	O
is	O
to	O
partition	O
objects	O
by	O
distinguishing	O
attributes	O
,	O
as	O
a	O
generalization	O
of	O
the	O
concept	O
of	O
an	O
equivalence	O
relation	O
.	O
</s>
<s>
The	O
partitioning	O
relation	O
is	O
a	O
composition	O
of	O
relations	O
using	O
relations	O
Jacques	O
Riguet	O
named	O
these	O
relations	O
difunctional	B-Algorithm
since	O
the	O
composition	O
F	O
GT	O
involves	O
univalent	O
relations	O
,	O
commonly	O
called	O
partial	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
In	O
automata	B-Application
theory	I-Application
,	O
the	O
term	O
rectangular	O
relation	O
has	O
also	O
been	O
used	O
to	O
denote	O
a	O
difunctional	B-Algorithm
relation	O
.	O
</s>
<s>
This	O
terminology	O
recalls	O
the	O
fact	O
that	O
,	O
when	O
represented	O
as	O
a	O
logical	B-Algorithm
matrix	I-Algorithm
,	O
the	O
columns	O
and	O
rows	O
of	O
a	O
difunctional	B-Algorithm
relation	O
can	O
be	O
arranged	O
as	O
a	O
block	B-Algorithm
matrix	I-Algorithm
with	O
rectangular	O
blocks	O
of	O
ones	O
on	O
the	O
(	O
asymmetric	O
)	O
main	O
diagonal	O
.	O
</s>
<s>
More	O
formally	O
,	O
a	O
relation	O
on	O
is	O
difunctional	B-Algorithm
if	O
and	O
only	O
if	O
it	O
can	O
be	O
written	O
as	O
the	O
union	O
of	O
Cartesian	O
products	O
,	O
where	O
the	O
are	O
a	O
partition	O
of	O
a	O
subset	O
of	O
and	O
the	O
likewise	O
a	O
partition	O
of	O
a	O
subset	O
of	O
.	O
</s>
<s>
In	O
1997	O
researchers	O
found	O
"	O
utility	O
of	O
binary	O
decomposition	O
based	O
on	O
difunctional	B-Algorithm
dependencies	O
in	O
database	O
management.	O
"	O
</s>
<s>
Furthermore	O
,	O
difunctional	B-Algorithm
relations	O
are	O
fundamental	O
in	O
the	O
study	O
of	O
bisimulations	B-Application
.	O
</s>
<s>
In	O
the	O
context	O
of	O
homogeneous	O
relations	O
,	O
a	O
partial	O
equivalence	O
relation	O
is	O
difunctional	B-Algorithm
.	O
</s>
<s>
The	O
corresponding	O
logical	B-Algorithm
matrix	I-Algorithm
of	O
a	O
general	O
binary	O
relation	O
has	O
rows	O
which	O
finish	O
with	O
a	O
sequence	O
of	O
ones	O
.	O
</s>
<s>
When	O
R	O
is	O
a	O
partial	O
identity	O
relation	O
,	O
difunctional	B-Algorithm
,	O
or	O
a	O
block	B-Algorithm
diagonal	I-Algorithm
relation	O
,	O
then	O
fringe(R )	O
=	O
R	O
.	O
Otherwise	O
the	O
fringe	O
operator	O
selects	O
a	O
boundary	O
sub-relation	O
described	O
in	O
terms	O
of	O
its	O
logical	B-Algorithm
matrix	I-Algorithm
:	O
fringe(R )	O
is	O
the	O
side	O
diagonal	O
if	O
R	O
is	O
an	O
upper	O
right	O
triangular	O
linear	O
order	O
or	O
strict	O
order	O
.	O
</s>
