<s>
In	O
mathematics	O
,	O
t-norms	B-General_Concept
are	O
a	O
special	O
kind	O
of	O
binary	O
operations	O
on	O
the	O
real	O
unit	O
interval	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
Various	O
constructions	O
of	O
t-norms	B-General_Concept
,	O
either	O
by	O
explicit	O
definition	O
or	O
by	O
transformation	O
from	O
previously	O
known	O
functions	O
,	O
provide	O
a	O
plenitude	O
of	O
examples	O
and	O
classes	O
of	O
t-norms	B-General_Concept
.	O
</s>
<s>
This	O
is	O
important	O
,	O
e.g.	O
,	O
for	O
finding	O
counter-examples	O
or	O
supplying	O
t-norms	B-General_Concept
with	O
particular	O
properties	O
for	O
use	O
in	O
engineering	O
applications	O
of	O
fuzzy	O
logic	O
.	O
</s>
<s>
The	O
main	O
ways	O
of	O
construction	B-General_Concept
of	I-General_Concept
t-norms	I-General_Concept
include	O
using	O
generators	O
,	O
defining	O
parametric	O
classes	O
of	O
t-norms	B-General_Concept
,	O
rotations	O
,	O
or	O
ordinal	O
sums	O
of	O
t-norms	B-General_Concept
.	O
</s>
<s>
Relevant	O
background	O
can	O
be	O
found	O
in	O
the	O
article	O
on	O
t-norms	B-General_Concept
.	O
</s>
<s>
The	O
method	O
of	O
constructing	O
t-norms	B-General_Concept
by	O
generators	O
consists	O
in	O
using	O
a	O
unary	O
function	O
(	O
generator	O
)	O
to	O
transform	O
some	O
known	O
binary	O
function	O
(	O
most	O
often	O
,	O
addition	O
or	O
multiplication	O
)	O
into	O
a	O
t-norm	B-General_Concept
.	O
</s>
<s>
The	O
construction	B-General_Concept
of	I-General_Concept
t-norms	I-General_Concept
by	O
additive	O
generators	O
is	O
based	O
on	O
the	O
following	O
theorem	O
:	O
</s>
<s>
is	O
a	O
t-norm	B-General_Concept
.	O
</s>
<s>
If	O
a	O
t-norm	B-General_Concept
T	O
results	O
from	O
the	O
latter	O
construction	O
by	O
a	O
function	O
f	O
which	O
is	O
right-continuous	O
in	O
0	O
,	O
then	O
f	O
is	O
called	O
an	O
additive	O
generator	O
of	O
T	O
.	O
</s>
<s>
The	O
function	O
f(x )	O
=	O
1	O
–	O
x	O
for	O
x	O
in	O
 [ 0 , 1 ] 	O
is	O
an	O
additive	O
generator	O
of	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
.	O
</s>
<s>
The	O
function	O
f	O
defined	O
as	O
f(x )	O
=	O
–	O
log(x )	O
if	O
0	O
x	O
≤	O
1	O
and	O
f(0 )	O
=	O
+∞	O
is	O
an	O
additive	O
generator	O
of	O
the	O
product	O
t-norm	B-General_Concept
.	O
</s>
<s>
The	O
function	O
f	O
defined	O
as	O
f(x )	O
=	O
2	O
–	O
x	O
if	O
0	O
≤	O
x	O
1	O
and	O
f(1 )	O
=	O
0	O
is	O
an	O
additive	O
generator	O
of	O
the	O
drastic	O
t-norm	B-General_Concept
.	O
</s>
<s>
Let	O
f	O
:	O
 [ 0 , 1 ] 	O
→	O
 [ 0 , + ] 	O
be	O
an	O
additive	O
generator	O
of	O
a	O
t-norm	B-General_Concept
T	O
.	O
Then	O
:	O
</s>
<s>
T	O
is	O
an	O
Archimedean	O
t-norm	B-General_Concept
.	O
</s>
<s>
(	O
Consequently	O
,	O
e.g.	O
,	O
the	O
minimum	O
t-norm	B-General_Concept
has	O
no	O
additive	O
generator	O
.	O
)	O
</s>
<s>
The	O
isomorphism	O
between	O
addition	O
on	O
 [ 0 , +∞ ] 	O
and	O
multiplication	O
on	O
 [ 0 , 1 ] 	O
by	O
the	O
logarithm	O
and	O
the	O
exponential	O
function	O
allow	O
two-way	O
transformations	O
between	O
additive	O
and	O
multiplicative	O
generators	O
of	O
a	O
t-norm	B-General_Concept
.	O
</s>
<s>
Many	O
families	O
of	O
related	O
t-norms	B-General_Concept
can	O
be	O
defined	O
by	O
an	O
explicit	O
formula	O
depending	O
on	O
a	O
parameter	O
p	O
.	O
This	O
section	O
lists	O
the	O
best	O
known	O
parameterized	O
families	O
of	O
t-norms	B-General_Concept
.	O
</s>
<s>
A	O
family	O
of	O
t-norms	B-General_Concept
Tp	O
parameterized	O
by	O
p	O
is	O
increasing	O
if	O
Tp(x, y )	O
≤	O
Tq(x, y )	O
for	O
all	O
x	O
,	O
y	O
in	O
 [ 0 , 1 ] 	O
whenever	O
p	O
≤	O
q	O
(	O
similarly	O
for	O
decreasing	O
and	O
strictly	O
increasing	O
or	O
decreasing	O
)	O
.	O
</s>
<s>
Nilpotent	O
if	O
and	O
only	O
if	O
0	O
p	O
+∞	O
(	O
for	O
p	O
=	O
1	O
it	O
is	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
)	O
.	O
</s>
<s>
The	O
family	O
of	O
Hamacher	O
t-norms	B-General_Concept
,	O
introduced	O
by	O
Horst	O
Hamacher	O
in	O
the	O
late	O
1970s	O
,	O
is	O
given	O
by	O
the	O
following	O
parametric	O
definition	O
for	O
0	O
≤	O
p	O
≤	O
+∞	O
:	O
</s>
<s>
The	O
t-norm	B-General_Concept
is	O
called	O
the	O
Hamacher	O
product	O
.	O
</s>
<s>
Hamacher	O
t-norms	B-General_Concept
are	O
the	O
only	O
t-norms	B-General_Concept
which	O
are	O
rational	O
functions	O
.	O
</s>
<s>
The	O
Hamacher	O
t-norm	B-General_Concept
is	O
strict	O
if	O
and	O
only	O
if	O
p	O
+∞	O
(	O
for	O
p	O
=	O
1	O
it	O
is	O
the	O
product	O
t-norm	B-General_Concept
)	O
.	O
</s>
<s>
The	O
family	O
of	O
Frank	O
t-norms	B-General_Concept
,	O
introduced	O
by	O
M.J.	O
Frank	O
in	O
the	O
late	O
1970s	O
,	O
is	O
given	O
by	O
the	O
parametric	O
definition	O
for	O
0	O
≤	O
p	O
≤	O
+∞	O
as	O
follows	O
:	O
</s>
<s>
The	O
Frank	O
t-norm	B-General_Concept
is	O
strict	O
if	O
p	O
+∞	O
.	O
</s>
<s>
The	O
Yager	O
t-norm	B-General_Concept
is	O
nilpotent	O
if	O
and	O
only	O
if	O
0	O
p	O
+∞	O
(	O
for	O
p	O
=	O
1	O
it	O
is	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
)	O
.	O
</s>
<s>
The	O
Aczél	O
–	O
Alsina	O
t-norm	B-General_Concept
is	O
strict	O
if	O
and	O
only	O
if	O
0	O
p	O
+∞	O
(	O
for	O
p	O
=	O
1	O
it	O
is	O
the	O
product	O
t-norm	B-General_Concept
)	O
.	O
</s>
<s>
The	O
Dombi	O
t-norm	B-General_Concept
is	O
strict	O
if	O
and	O
only	O
if	O
0	O
p	O
+∞	O
(	O
for	O
p	O
=	O
1	O
it	O
is	O
the	O
Hamacher	O
product	O
)	O
.	O
</s>
<s>
The	O
family	O
of	O
Sugeno	O
–	O
Weber	O
t-norms	B-General_Concept
was	O
introduced	O
in	O
the	O
early	O
1980s	O
by	O
Siegfried	O
Weber	O
;	O
the	O
dual	O
t-conorms	B-General_Concept
were	O
defined	O
already	O
in	O
the	O
early	O
1970s	O
by	O
Michio	O
Sugeno	O
.	O
</s>
<s>
The	O
Sugeno	O
–	O
Weber	O
t-norm	B-General_Concept
is	O
nilpotent	O
if	O
and	O
only	O
if	O
−1	O
p	O
+∞	O
(	O
for	O
p	O
=	O
0	O
it	O
is	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
)	O
.	O
</s>
<s>
The	O
ordinal	O
sum	O
constructs	O
a	O
t-norm	B-General_Concept
from	O
a	O
family	O
of	O
t-norms	B-General_Concept
,	O
by	O
shrinking	O
them	O
into	O
disjoint	O
subintervals	O
of	O
the	O
interval	O
 [ 0 , 1 ] 	O
and	O
completing	O
the	O
t-norm	B-General_Concept
by	O
using	O
the	O
minimum	O
on	O
the	O
rest	O
of	O
the	O
unit	O
square	O
.	O
</s>
<s>
Let	O
Ti	O
for	O
i	O
in	O
an	O
index	O
set	O
I	O
be	O
a	O
family	O
of	O
t-norms	B-General_Concept
and	O
(	O
ai	O
,	O
bi	O
)	O
a	O
family	O
of	O
pairwise	O
disjoint	O
(	O
non-empty	O
)	O
open	O
subintervals	O
of	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
is	O
a	O
t-norm	B-General_Concept
.	O
</s>
<s>
Ordinal	O
sums	O
of	O
t-norms	B-General_Concept
enjoy	O
the	O
following	O
properties	O
:	O
</s>
<s>
Each	O
t-norm	B-General_Concept
is	O
a	O
trivial	O
ordinal	O
sum	O
of	O
itself	O
on	O
the	O
whole	O
interval	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
The	O
empty	O
ordinal	O
sum	O
(	O
for	O
the	O
empty	O
index	O
set	O
)	O
yields	O
the	O
minimum	O
t-norm	B-General_Concept
Tmin	O
.	O
</s>
<s>
Summands	O
with	O
the	O
minimum	O
t-norm	B-General_Concept
can	O
arbitrarily	O
be	O
added	O
or	O
omitted	O
without	O
changing	O
the	O
resulting	O
t-norm	B-General_Concept
.	O
</s>
<s>
An	O
ordinal	O
sum	O
of	O
t-norm	B-General_Concept
is	O
continuous	O
if	O
and	O
only	O
if	O
each	O
summand	O
is	O
a	O
continuous	O
t-norm	B-General_Concept
.	O
</s>
<s>
An	O
ordinal	O
sum	O
is	O
Archimedean	O
if	O
and	O
only	O
if	O
it	O
is	O
a	O
trivial	O
sum	O
of	O
one	O
Archimedean	O
t-norm	B-General_Concept
on	O
the	O
whole	O
unit	O
interval	O
.	O
</s>
<s>
If	O
is	O
a	O
left-continuous	O
t-norm	B-General_Concept
,	O
then	O
its	O
residuum	O
R	O
is	O
given	O
as	O
follows	O
:	O
</s>
<s>
The	O
ordinal	O
sum	O
of	O
a	O
family	O
of	O
continuous	O
t-norms	B-General_Concept
is	O
a	O
continuous	O
t-norm	B-General_Concept
.	O
</s>
<s>
By	O
the	O
Mostert	O
–	O
Shields	O
theorem	O
,	O
every	O
continuous	O
t-norm	B-General_Concept
is	O
expressible	O
as	O
the	O
ordinal	O
sum	O
of	O
Archimedean	O
continuous	O
t-norms	B-General_Concept
.	O
</s>
<s>
Since	O
the	O
latter	O
are	O
either	O
nilpotent	O
(	O
and	O
then	O
isomorphic	O
to	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
)	O
or	O
strict	O
(	O
then	O
isomorphic	O
to	O
the	O
product	O
t-norm	B-General_Concept
)	O
,	O
each	O
continuous	O
t-norm	B-General_Concept
is	O
isomorphic	O
to	O
the	O
ordinal	O
sum	O
of	O
Łukasiewicz	O
and	O
product	O
t-norms	B-General_Concept
.	O
</s>
<s>
Important	O
examples	O
of	O
ordinal	O
sums	O
of	O
continuous	O
t-norms	B-General_Concept
are	O
the	O
following	O
ones	O
:	O
</s>
<s>
Dubois	O
–	O
Prade	O
t-norms	B-General_Concept
,	O
introduced	O
by	O
Didier	O
Dubois	O
and	O
Henri	O
Prade	O
in	O
the	O
early	O
1980s	O
,	O
are	O
the	O
ordinal	O
sums	O
of	O
the	O
product	O
t-norm	B-General_Concept
on	O
 [ 0 , p ] 	O
for	O
a	O
parameter	O
p	O
in	O
 [ 0 , 1 ] 	O
and	O
the	O
(	O
default	O
)	O
minimum	O
t-norm	B-General_Concept
on	O
the	O
rest	O
of	O
the	O
unit	O
interval	O
.	O
</s>
<s>
The	O
family	O
of	O
Dubois	O
–	O
Prade	O
t-norms	B-General_Concept
is	O
decreasing	O
and	O
continuous	O
with	O
respect	O
to	O
p	O
..	O
</s>
<s>
Mayor	O
–	O
Torrens	O
t-norms	B-General_Concept
,	O
introduced	O
by	O
Gaspar	O
Mayor	O
and	O
Joan	O
Torrens	O
in	O
the	O
early	O
1990s	O
,	O
are	O
the	O
ordinal	O
sums	O
of	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
on	O
 [ 0 , p ] 	O
for	O
a	O
parameter	O
p	O
in	O
 [ 0 , 1 ] 	O
and	O
the	O
(	O
default	O
)	O
minimum	O
t-norm	B-General_Concept
on	O
the	O
rest	O
of	O
the	O
unit	O
interval	O
.	O
</s>
<s>
The	O
family	O
of	O
Mayor	O
–	O
Torrens	O
t-norms	B-General_Concept
is	O
decreasing	O
and	O
continuous	O
with	O
respect	O
to	O
p	O
..	O
</s>
<s>
The	O
construction	B-General_Concept
of	I-General_Concept
t-norms	I-General_Concept
by	O
rotation	O
was	O
introduced	O
by	O
Sándor	O
Jenei	O
(	O
2000	O
)	O
.	O
</s>
<s>
Let	O
T	O
be	O
a	O
left-continuous	O
t-norm	B-General_Concept
without	O
zero	O
divisors	O
,	O
N	O
:	O
 [ 0 , 1 ] 	O
→	O
 [ 0 , 1 ] 	O
the	O
function	O
that	O
assigns	O
1	O
−	O
x	O
to	O
x	O
and	O
t	O
=	O
0.5	O
.	O
</s>
<s>
is	O
a	O
left-continuous	O
t-norm	B-General_Concept
,	O
called	O
the	O
rotation	O
of	O
the	O
t-norm	B-General_Concept
T	O
.	O
</s>
<s>
Geometrically	O
,	O
the	O
construction	O
can	O
be	O
described	O
as	O
first	O
shrinking	O
the	O
t-norm	B-General_Concept
T	O
to	O
the	O
interval	O
 [ 0.5 , 1 ] 	O
and	O
then	O
rotating	O
it	O
by	O
the	O
angle	O
2π/3	O
in	O
both	O
directions	O
around	O
the	O
line	O
connecting	O
the	O
points	O
(	O
0	O
,	O
0	O
,	O
1	O
)	O
and	O
(	O
1	O
,	O
1	O
,	O
0	O
)	O
.	O
</s>
<s>
The	O
theorem	O
can	O
be	O
generalized	O
by	O
taking	O
for	O
N	O
any	O
strong	O
negation	O
,	O
that	O
is	O
,	O
an	O
involutive	B-Algorithm
strictly	O
decreasing	O
continuous	O
function	O
on	O
[0,1],	O
and	O
for	O
t	O
taking	O
the	O
unique	O
fixed	O
point	O
ofN	O
.	O
</s>
<s>
The	O
resulting	O
t-norm	B-General_Concept
enjoys	O
the	O
following	O
rotation	O
invariance	O
property	O
with	O
respect	O
toN	O
:	O
</s>
