<s>
In	O
mathematics	O
,	O
a	O
ternary	B-Algorithm
relation	I-Algorithm
or	O
triadic	B-Algorithm
relation	I-Algorithm
is	O
a	O
finitary	B-Algorithm
relation	I-Algorithm
in	O
which	O
the	O
number	O
of	O
places	O
in	O
the	O
relation	O
is	O
three	O
.	O
</s>
<s>
Ternary	B-Algorithm
relations	I-Algorithm
may	O
also	O
be	O
referred	O
to	O
as	O
3-adic	O
,	O
3-ary	O
,	O
3-dimensional	O
,	O
or	O
3-place	O
.	O
</s>
<s>
a	O
subset	O
of	O
the	O
Cartesian	O
product	O
of	O
some	O
sets	O
A	O
and	O
B	O
,	O
so	O
a	O
ternary	B-Algorithm
relation	I-Algorithm
is	O
a	O
set	O
of	O
triples	O
,	O
forming	O
a	O
subset	O
of	O
the	O
Cartesian	O
product	O
of	O
three	O
sets	O
A	O
,	O
B	O
and	O
C	O
.	O
</s>
<s>
An	O
example	O
of	O
a	O
ternary	B-Algorithm
relation	I-Algorithm
in	O
elementary	O
geometry	O
can	O
be	O
given	O
on	O
triples	O
of	O
points	O
,	O
where	O
a	O
triple	O
is	O
in	O
the	O
relation	O
if	O
the	O
three	O
points	O
are	O
collinear	O
.	O
</s>
<s>
Another	O
geometric	O
example	O
can	O
be	O
obtained	O
by	O
considering	O
triples	O
consisting	O
of	O
two	O
points	O
and	O
a	O
line	O
,	O
where	O
a	O
triple	O
is	O
in	O
the	O
ternary	B-Algorithm
relation	I-Algorithm
if	O
the	O
two	O
points	O
determine	O
(	O
are	O
incident	O
with	O
)	O
the	O
line	O
.	O
</s>
<s>
Given	O
any	O
set	O
A	O
whose	O
elements	O
are	O
arranged	O
on	O
a	O
circle	O
,	O
one	O
can	O
define	O
a	O
ternary	B-Algorithm
relation	I-Algorithm
R	O
on	O
A	O
,	O
i.e.	O
</s>
<s>
which	O
holds	O
for	O
three	O
integers	O
a	O
,	O
b	O
,	O
and	O
m	O
if	O
and	O
only	O
if	O
m	O
divides	O
a−b	O
,	O
formally	O
may	O
be	O
considered	O
as	O
a	O
ternary	B-Algorithm
relation	I-Algorithm
.	O
</s>
<s>
However	O
,	O
usually	O
,	O
this	O
instead	O
is	O
considered	O
as	O
a	O
family	O
of	O
binary	O
relations	O
between	O
the	O
a	O
and	O
the	O
b	O
,	O
indexed	O
by	O
the	O
modulus	O
m	O
.	O
For	O
each	O
fixed	O
m	O
,	O
indeed	O
this	O
binary	O
relation	O
has	O
some	O
natural	O
properties	O
,	O
like	O
being	O
an	O
equivalence	O
relation	O
;	O
while	O
the	O
combined	O
ternary	B-Algorithm
relation	I-Algorithm
in	O
general	O
is	O
not	O
studied	O
as	O
one	O
relation	O
.	O
</s>
<s>
A	O
typing	O
relation	O
indicates	O
that	O
is	O
a	O
term	O
of	O
type	O
in	O
context	O
,	O
and	O
is	O
thus	O
a	O
ternary	B-Algorithm
relation	I-Algorithm
between	O
contexts	O
,	O
terms	O
and	O
types	O
.	O
</s>
<s>
Given	O
homogeneous	O
relations	O
A	O
,	O
B	O
,	O
and	O
C	O
on	O
a	O
set	O
,	O
a	O
ternary	B-Algorithm
relation	I-Algorithm
can	O
be	O
defined	O
using	O
composition	O
of	O
relations	O
AB	O
and	O
inclusion	O
AB	O
⊆	O
C	O
.	O
Within	O
the	O
calculus	O
of	O
relations	O
each	O
relation	O
A	O
has	O
a	O
converse	O
relation	O
AT	O
and	O
a	O
complement	O
relation	O
Using	O
these	O
involutions	B-Algorithm
,	O
Augustus	O
De	O
Morgan	O
and	O
Ernst	O
Schröder	O
showed	O
that	O
is	O
equivalent	O
to	O
and	O
also	O
equivalent	O
to	O
The	O
mutual	O
equivalences	O
of	O
these	O
forms	O
,	O
constructed	O
from	O
the	O
ternary	O
are	O
called	O
the	O
Schröder	O
rules	O
.	O
</s>
