<s>
In	O
mathematics	O
,	O
a	O
t-norm	B-General_Concept
(	O
also	O
T-norm	B-General_Concept
or	O
,	O
unabbreviated	O
,	O
triangular	B-General_Concept
norm	I-General_Concept
)	O
is	O
a	O
kind	O
of	O
binary	O
operation	O
used	O
in	O
the	O
framework	O
of	O
probabilistic	O
metric	O
spaces	O
and	O
in	O
multi-valued	O
logic	O
,	O
specifically	O
in	O
fuzzy	O
logic	O
.	O
</s>
<s>
A	O
t-norm	B-General_Concept
generalizes	O
intersection	O
in	O
a	O
lattice	O
and	O
conjunction	O
in	O
logic	O
.	O
</s>
<s>
The	O
name	O
triangular	B-General_Concept
norm	I-General_Concept
refers	O
to	O
the	O
fact	O
that	O
in	O
the	O
framework	O
of	O
probabilistic	O
metric	O
spaces	O
t-norms	B-General_Concept
are	O
used	O
to	O
generalize	O
the	O
triangle	O
inequality	O
of	O
ordinary	O
metric	O
spaces	O
.	O
</s>
<s>
A	O
t-norm	B-General_Concept
is	O
a	O
function	O
T	O
:	O
 [ 0 , 1 ] 	O
 [ 0 , 1 ] 	O
→	O
 [ 0 , 1 ] 	O
that	O
satisfies	O
the	O
following	O
properties	O
:	O
</s>
<s>
Since	O
a	O
t-norm	B-General_Concept
is	O
a	O
binary	O
algebraic	O
operation	O
on	O
the	O
interval	O
[0,1],	O
infix	O
algebraic	O
notation	O
is	O
also	O
common	O
,	O
with	O
the	O
t-norm	B-General_Concept
usually	O
denoted	O
by	O
.	O
</s>
<s>
The	O
defining	O
conditions	O
of	O
the	O
t-norm	B-General_Concept
are	O
exactly	O
those	O
of	O
a	O
partially	O
ordered	O
abelian	O
monoid	O
on	O
the	O
real	O
unit	O
interval	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
The	O
monoidal	O
operation	O
of	O
any	O
partially	O
ordered	O
abelian	O
monoid	O
L	O
is	O
therefore	O
by	O
some	O
authors	O
called	O
a	O
triangular	B-General_Concept
norm	I-General_Concept
on	O
L	O
.	O
</s>
<s>
A	O
t-norm	B-General_Concept
is	O
called	O
continuous	O
if	O
it	O
is	O
continuous	O
as	O
a	O
function	O
,	O
in	O
the	O
usual	O
interval	O
topology	O
on	O
[0,1]2	O
.	O
</s>
<s>
A	O
t-norm	B-General_Concept
is	O
called	O
strict	O
if	O
it	O
is	O
continuous	O
and	O
strictly	O
monotone	O
.	O
</s>
<s>
A	O
t-norm	B-General_Concept
is	O
called	O
nilpotent	O
if	O
it	O
is	O
continuous	O
and	O
each	O
x	O
in	O
the	O
open	O
interval	O
(	O
0	O
,	O
1	O
)	O
is	O
nilpotent	O
,	O
that	O
is	O
,	O
there	O
is	O
a	O
natural	O
number	O
n	O
such	O
that	O
x	O
...	O
x	O
(	O
ntimes	O
)	O
equals0	O
.	O
</s>
<s>
A	O
t-norm	B-General_Concept
is	O
called	O
Archimedean	O
if	O
it	O
has	O
the	O
Archimedean	O
property	O
,	O
that	O
is	O
,	O
if	O
for	O
each	O
x	O
,	O
y	O
in	O
the	O
open	O
interval	O
(	O
0	O
,	O
1	O
)	O
there	O
is	O
a	O
natural	O
number	O
n	O
such	O
that	O
x	O
...	O
x	O
(	O
ntimes	O
)	O
is	O
less	O
than	O
or	O
equal	O
to	O
y	O
.	O
</s>
<s>
The	O
usual	O
partial	O
ordering	O
of	O
t-norms	B-General_Concept
is	O
pointwise	O
,	O
that	O
is	O
,	O
</s>
<s>
As	O
functions	O
,	O
pointwise	O
larger	O
t-norms	B-General_Concept
are	O
sometimes	O
called	O
stronger	O
than	O
those	O
pointwise	O
smaller	O
.	O
</s>
<s>
In	O
the	O
semantics	O
of	O
fuzzy	O
logic	O
,	O
however	O
,	O
the	O
larger	O
a	O
t-norm	B-General_Concept
,	O
the	O
weaker	O
(	O
in	O
terms	O
of	O
logical	O
strength	O
)	O
conjunction	O
it	O
represents	O
.	O
</s>
<s>
Minimum	O
t-norm	B-General_Concept
also	O
called	O
the	O
Gödel	O
t-norm	B-General_Concept
,	O
as	O
it	O
is	O
the	O
standard	O
semantics	O
for	O
conjunction	O
in	O
Gödel	O
fuzzy	O
logic	O
.	O
</s>
<s>
Besides	O
that	O
,	O
it	O
occurs	O
in	O
most	O
t-norm	B-General_Concept
based	I-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
as	O
the	O
standard	O
semantics	O
for	O
weak	O
conjunction	O
.	O
</s>
<s>
It	O
is	O
the	O
pointwise	O
largest	O
t-norm	B-General_Concept
(	O
see	O
the	O
properties	O
of	O
t-norms	B-General_Concept
below	O
)	O
.	O
</s>
<s>
Product	O
t-norm	B-General_Concept
(	O
the	O
ordinary	O
product	O
of	O
real	O
numbers	O
)	O
.	O
</s>
<s>
Besides	O
other	O
uses	O
,	O
the	O
product	O
t-norm	B-General_Concept
is	O
the	O
standard	O
semantics	O
for	O
strong	O
conjunction	O
in	O
product	O
fuzzy	O
logic	O
.	O
</s>
<s>
It	O
is	O
a	O
strict	O
Archimedean	O
t-norm	B-General_Concept
.	O
</s>
<s>
Łukasiewicz	O
t-norm	B-General_Concept
The	O
name	O
comes	O
from	O
the	O
fact	O
that	O
the	O
t-norm	B-General_Concept
is	O
the	O
standard	O
semantics	O
for	O
strong	O
conjunction	O
in	O
Łukasiewicz	B-General_Concept
fuzzy	I-General_Concept
logic	I-General_Concept
.	O
</s>
<s>
It	O
is	O
a	O
nilpotent	O
Archimedean	O
t-norm	B-General_Concept
,	O
pointwise	O
smaller	O
than	O
the	O
product	O
t-norm	B-General_Concept
.	O
</s>
<s>
The	O
name	O
reflects	O
the	O
fact	O
that	O
the	O
drastic	O
t-norm	B-General_Concept
is	O
the	O
pointwise	O
smallest	O
t-norm	B-General_Concept
(	O
see	O
the	O
properties	O
of	O
t-norms	B-General_Concept
below	O
)	O
.	O
</s>
<s>
It	O
is	O
a	O
right-continuous	O
Archimedean	O
t-norm	B-General_Concept
.	O
</s>
<s>
is	O
a	O
standard	O
example	O
of	O
a	O
t-norm	B-General_Concept
that	O
is	O
left-continuous	O
,	O
but	O
not	O
continuous	O
.	O
</s>
<s>
Despite	O
its	O
name	O
,	O
the	O
nilpotent	O
minimum	O
is	O
not	O
a	O
nilpotent	O
t-norm	B-General_Concept
.	O
</s>
<s>
is	O
a	O
strict	O
Archimedean	O
t-norm	B-General_Concept
,	O
and	O
an	O
important	O
representative	O
of	O
the	O
parametric	O
classes	O
of	O
Hamacher	O
t-norms	B-General_Concept
and	O
Schweizer	O
–	O
Sklar	O
t-norms	B-General_Concept
.	O
</s>
<s>
The	O
drastic	O
t-norm	B-General_Concept
is	O
the	O
pointwise	O
smallest	O
t-norm	B-General_Concept
and	O
the	O
minimum	O
is	O
the	O
pointwise	O
largest	O
t-norm	B-General_Concept
:	O
</s>
<s>
for	O
any	O
t-norm	B-General_Concept
and	O
all	O
a	O
,	O
b	O
in	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
For	O
every	O
t-norm	B-General_Concept
T	O
,	O
the	O
number	O
0	O
acts	O
as	O
null	O
element	O
:	O
T(a, 0 )	O
=	O
0	O
for	O
all	O
a	O
in	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
A	O
t-norm	B-General_Concept
T	O
has	O
zero	O
divisors	O
if	O
and	O
only	O
if	O
it	O
has	O
nilpotent	O
elements	O
;	O
each	O
nilpotent	O
element	O
of	O
T	O
is	O
also	O
a	O
zero	O
divisor	O
of	O
T	O
.	O
The	O
set	O
of	O
all	O
nilpotent	O
elements	O
is	O
an	O
interval	O
 [ 0 , a ] 	O
or	O
[	O
0	O
,	O
a	O
)	O
,	O
for	O
some	O
a	O
in	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
Although	O
real	O
functions	O
of	O
two	O
variables	O
can	O
be	O
continuous	O
in	O
each	O
variable	O
without	O
being	O
continuous	O
on	O
[0,1]2,	O
this	O
is	O
not	O
the	O
case	O
with	O
t-norms	B-General_Concept
:	O
a	O
t-norm	B-General_Concept
T	O
is	O
continuous	O
if	O
and	O
only	O
if	O
it	O
is	O
continuous	O
in	O
one	O
variable	O
,	O
i.e.	O
,	O
if	O
and	O
only	O
if	O
the	O
functions	O
fy(x )	O
=	O
T(x, y )	O
are	O
continuous	O
for	O
each	O
y	O
in	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
Analogous	O
theorems	O
hold	O
for	O
left	O
-	O
and	O
right-continuity	O
of	O
a	O
t-norm	B-General_Concept
.	O
</s>
<s>
A	O
continuous	O
t-norm	B-General_Concept
is	O
Archimedean	O
if	O
and	O
only	O
if	O
0	O
and	O
1	O
are	O
its	O
only	O
idempotents	O
.	O
</s>
<s>
A	O
continuous	O
Archimedean	O
t-norm	B-General_Concept
is	O
strict	O
if	O
0	O
is	O
its	O
only	O
nilpotent	O
element	O
;	O
otherwise	O
it	O
is	O
nilpotent	O
.	O
</s>
<s>
By	O
definition	O
,	O
moreover	O
,	O
a	O
continuous	O
Archimedean	O
t-norm	B-General_Concept
T	O
is	O
nilpotent	O
if	O
and	O
only	O
if	O
each	O
x1	O
is	O
a	O
nilpotent	O
element	O
of	O
T	O
.	O
Thus	O
with	O
a	O
continuous	O
Archimedean	O
t-norm	B-General_Concept
T	O
,	O
either	O
all	O
or	O
none	O
of	O
the	O
elements	O
of	O
(	O
0	O
,	O
1	O
)	O
are	O
nilpotent	O
.	O
</s>
<s>
If	O
on	O
the	O
other	O
hand	O
it	O
is	O
the	O
case	O
that	O
there	O
are	O
no	O
nilpotent	O
elements	O
of	O
T	O
,	O
the	O
t-norm	B-General_Concept
is	O
isomorphic	O
to	O
the	O
product	O
t-norm	B-General_Concept
.	O
</s>
<s>
In	O
other	O
words	O
,	O
all	O
nilpotent	O
t-norms	B-General_Concept
are	O
isomorphic	O
,	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
being	O
their	O
prototypical	O
representative	O
;	O
and	O
all	O
strict	O
t-norms	B-General_Concept
are	O
isomorphic	O
,	O
with	O
the	O
product	O
t-norm	B-General_Concept
as	O
their	O
prototypical	O
example	O
.	O
</s>
<s>
The	O
Łukasiewicz	O
t-norm	B-General_Concept
is	O
itself	O
isomorphic	O
to	O
the	O
product	O
t-norm	B-General_Concept
undercut	O
at	O
0.25	O
,	O
i.e.	O
,	O
to	O
the	O
function	O
p(x,y )	O
=	O
max( 	O
0.25	O
,	O
xy	O
)	O
on	O
[0.25,1]2	O
.	O
</s>
<s>
For	O
each	O
continuous	O
t-norm	B-General_Concept
,	O
the	O
set	O
of	O
its	O
idempotents	O
is	O
a	O
closed	O
subset	O
of	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
The	O
restriction	O
of	O
the	O
t-norm	B-General_Concept
to	O
any	O
of	O
these	O
intervals	O
(	O
including	O
its	O
endpoints	O
)	O
is	O
Archimedean	O
,	O
and	O
thus	O
isomorphic	O
either	O
to	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
or	O
the	O
product	O
t-norm	B-General_Concept
.	O
</s>
<s>
For	O
such	O
x	O
,	O
y	O
that	O
do	O
not	O
fall	O
into	O
the	O
same	O
open	O
interval	O
of	O
non-idempotents	O
,	O
the	O
t-norm	B-General_Concept
evaluates	O
to	O
the	O
minimum	O
of	O
x	O
and	O
y	O
.	O
</s>
<s>
These	O
conditions	O
actually	O
give	O
a	O
characterization	O
of	O
continuous	O
t-norms	B-General_Concept
,	O
called	O
the	O
Mostert	O
–	O
Shields	O
theorem	O
,	O
since	O
every	O
continuous	O
t-norm	B-General_Concept
can	O
in	O
this	O
way	O
be	O
decomposed	O
,	O
and	O
the	O
described	O
construction	O
always	O
yields	O
a	O
continuous	O
t-norm	B-General_Concept
.	O
</s>
<s>
A	O
t-norm	B-General_Concept
is	O
continuous	O
if	O
and	O
only	O
if	O
it	O
is	O
isomorphic	O
to	O
an	O
ordinal	O
sum	O
of	O
the	O
minimum	O
,	O
Łukasiewicz	O
,	O
and	O
product	O
t-norm	B-General_Concept
.	O
</s>
<s>
A	O
similar	O
characterization	O
theorem	O
for	O
non-continuous	O
t-norms	B-General_Concept
is	O
not	O
known	O
(	O
not	O
even	O
for	O
left-continuous	O
ones	O
)	O
,	O
only	O
some	O
non-exhaustive	O
methods	O
for	O
the	O
construction	B-General_Concept
of	I-General_Concept
t-norms	I-General_Concept
have	O
been	O
found	O
.	O
</s>
<s>
This	O
operation	O
is	O
called	O
the	O
residuum	O
of	O
the	O
t-norm	B-General_Concept
.	O
</s>
<s>
In	O
prefix	O
notation	O
,	O
the	O
residuum	O
of	O
a	O
t-norm	B-General_Concept
is	O
often	O
denoted	O
by	O
or	O
by	O
the	O
letter	O
R	O
.	O
</s>
<s>
The	O
interval	O
 [ 0 , 1 ] 	O
equipped	O
with	O
a	O
t-norm	B-General_Concept
and	O
its	O
residuum	O
forms	O
a	O
residuated	O
lattice	O
.	O
</s>
<s>
The	O
relation	O
between	O
a	O
t-norm	B-General_Concept
T	O
and	O
its	O
residuum	O
R	O
is	O
an	O
instance	O
of	O
adjunction	O
(	O
specifically	O
,	O
a	O
Galois	O
connection	O
)	O
:	O
the	O
residuum	O
forms	O
a	O
right	O
adjoint	O
R( x	O
,	O
–	O
)	O
to	O
the	O
functor	O
T( 	O
–	O
,	O
x	O
)	O
for	O
each	O
x	O
in	O
the	O
lattice	O
 [ 0 , 1 ] 	O
taken	O
as	O
a	O
poset	O
category	O
.	O
</s>
<s>
In	O
the	O
standard	O
semantics	O
of	O
t-norm	B-General_Concept
based	I-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
,	O
where	O
conjunction	O
is	O
interpreted	O
by	O
a	O
t-norm	B-General_Concept
,	O
the	O
residuum	O
plays	O
the	O
role	O
of	O
implication	O
(	O
often	O
called	O
R-implication	O
)	O
.	O
</s>
<s>
Residuum	O
of	O
the	O
Name	O
Value	O
for	O
x	O
>	O
y	O
Graph	O
Minimum	O
t-norm	B-General_Concept
Standard	O
Gödel	O
implication	O
y	O
thumb|270px|left|Standard	O
Gödel	O
implication	O
.	O
</s>
<s>
Product	O
t-norm	B-General_Concept
Goguen	O
implication	O
y	O
/	O
x	O
thumb|270px|left|Goguen	O
implication	O
.	O
</s>
<s>
Łukasiewicz	O
t-norm	B-General_Concept
Standard	O
Łukasiewicz	O
implication	O
1	O
–	O
x	O
+	O
y	O
thumb|270px|left|Standard	O
Łukasiewicz	O
implication	O
.	O
</s>
<s>
T-conorms	B-General_Concept
(	O
also	O
called	O
S-norms	O
)	O
are	O
dual	O
to	O
t-norms	B-General_Concept
under	O
the	O
order-reversing	O
operation	O
that	O
assigns	O
1	O
–	O
x	O
to	O
x	O
on	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
It	O
follows	O
that	O
a	O
t-conorm	B-General_Concept
satisfies	O
the	O
following	O
conditions	O
,	O
which	O
can	O
be	O
used	O
for	O
an	O
equivalent	O
axiomatic	O
definition	O
of	O
t-conorms	B-General_Concept
independently	O
of	O
t-norms	B-General_Concept
:	O
</s>
<s>
T-conorms	B-General_Concept
are	O
used	O
to	O
represent	O
logical	O
disjunction	O
in	O
fuzzy	O
logic	O
and	O
union	O
in	O
fuzzy	O
set	O
theory	O
.	O
</s>
<s>
Important	O
t-conorms	B-General_Concept
are	O
those	O
dual	O
to	O
prominent	O
t-norms	B-General_Concept
:	O
</s>
<s>
Maximum	O
t-conorm	B-General_Concept
,	O
dual	O
to	O
the	O
minimum	O
t-norm	B-General_Concept
,	O
is	O
the	O
smallest	O
t-conorm	B-General_Concept
(	O
see	O
the	O
properties	O
of	O
t-conorms	B-General_Concept
below	O
)	O
.	O
</s>
<s>
It	O
is	O
the	O
standard	O
semantics	O
for	O
disjunction	O
in	O
Gödel	O
fuzzy	O
logic	O
and	O
for	O
weak	O
disjunction	O
in	O
all	O
t-norm	B-General_Concept
based	I-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
.	O
</s>
<s>
Probabilistic	O
sum	O
is	O
dual	O
to	O
the	O
product	O
t-norm	B-General_Concept
.	O
</s>
<s>
It	O
is	O
also	O
the	O
standard	O
semantics	O
for	O
strong	O
disjunction	O
in	O
such	O
extensions	O
of	O
product	O
fuzzy	O
logic	O
in	O
which	O
it	O
is	O
definable	O
(	O
e.g.	O
,	O
those	O
containing	O
involutive	B-Algorithm
negation	O
)	O
.	O
</s>
<s>
Bounded	O
sum	O
is	O
dual	O
to	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
.	O
</s>
<s>
It	O
is	O
the	O
standard	O
semantics	O
for	O
strong	O
disjunction	O
in	O
Łukasiewicz	B-General_Concept
fuzzy	I-General_Concept
logic	I-General_Concept
.	O
</s>
<s>
dual	O
to	O
the	O
drastic	O
t-norm	B-General_Concept
,	O
is	O
the	O
largest	O
t-conorm	B-General_Concept
(	O
see	O
the	O
properties	O
of	O
t-conorms	B-General_Concept
below	O
)	O
.	O
</s>
<s>
is	O
a	O
dual	O
to	O
one	O
of	O
the	O
Hamacher	O
t-norms	B-General_Concept
.	O
</s>
<s>
Many	O
properties	O
of	O
t-conorms	B-General_Concept
can	O
be	O
obtained	O
by	O
dualizing	O
the	O
properties	O
of	O
t-norms	B-General_Concept
,	O
for	O
example	O
:	O
</s>
<s>
For	O
any	O
t-conorm	B-General_Concept
⊥	O
,	O
the	O
number	O
1	O
is	O
an	O
annihilating	O
element	O
:	O
⊥( a	O
,	O
1	O
)	O
=	O
1	O
,	O
for	O
any	O
a	O
in	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
Dually	O
to	O
t-norms	B-General_Concept
,	O
all	O
t-conorms	B-General_Concept
are	O
bounded	O
by	O
the	O
maximum	O
and	O
the	O
drastic	O
t-conorm	B-General_Concept
:	O
</s>
<s>
,	O
for	O
any	O
t-conorm	B-General_Concept
and	O
all	O
a	O
,	O
b	O
in	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
Further	O
properties	O
result	O
from	O
the	O
relationships	O
between	O
t-norms	B-General_Concept
and	O
t-conorms	B-General_Concept
or	O
their	O
interplay	O
with	O
other	O
operators	O
,	O
e.g.	O
</s>
<s>
A	O
t-norm	B-General_Concept
T	O
distributes	O
over	O
a	O
t-conorm	B-General_Concept
⊥	O
,	O
i.e.	O
,	O
</s>
<s>
if	O
and	O
only	O
if	O
⊥	O
is	O
the	O
maximum	O
t-conorm	B-General_Concept
.	O
</s>
<s>
Dually	O
,	O
any	O
t-conorm	B-General_Concept
distributes	O
over	O
the	O
minimum	O
,	O
but	O
not	O
over	O
any	O
other	O
t-norm	B-General_Concept
.	O
</s>
<s>
strong	O
if	O
it	O
is	O
strict	O
and	O
involutive	B-Algorithm
,	O
that	O
is	O
,	O
for	O
all	O
in	O
[	O
0	O
,	O
1 ]	O
.	O
</s>
<s>
As	O
the	O
standard	O
negator	O
is	O
used	O
in	O
the	O
above	O
definition	O
of	O
a	O
t-norm/t	O
-conorm	O
pair	O
,	O
this	O
can	O
be	O
generalized	O
as	O
follows	O
:	O
</s>
