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the only idempotent elements are
b
1
,
b
2
,
b
3
,
b
4
, etc., as well as 0 and 1, that is, the
[
,
]
points giving the partition of
0
1
.
Remark 2.2.37
}
are taken into account in both theoretic fuzzy logic and its applications, it should
be pointed out that provided there are, at least, two statements 'x is
P
' and 'x is
Q
' such that
Although currently only continuous t-norms in
{
min
}∪{
pr od
}∪{
W
μ
P
(
x
), μ
Q
(
x
)
∈{
0
,
1
}
and
μ
Pand P
=
μ
P
, μ
Qand Q
=
μ
Q
,
the only
possibility for representing
μ
·
˃
=
T
ⓦ
(μ
×
˃)
, is by taking as continuous t-norm
T
an ordinal-sum with the single interval
(
min
(μ
P
(
x
), μ
Q
(
x
)),
max
(μ
P
(
x
), μ
Q
(
x
)))
.
Remark 2.2.38
Which t-norms are strictly non-decreasing in the sense that if 0
<
a
<
b
<
1, then
T
(
a
,
c
)<
T
(
b
,
c
)
for all
c
∈[
0
,
1
]
?
•
If
T
=
min, the answer is negative. For example, 0
.
3
<
0
.
5, but min
(
0
.
2
,
0
.
3
)
=
min
(
0
.
2
,
0
.
5
)
=
0
.
2
•
If
T
=
W
˕
, the answer is also negative. For example,0
.
3
<
0
.
5, but
W
(
0
.
2
,
0
.
3
)
=
W
(
0
.
2
,
0
.
5
)
=
0
•
If
T
=
prod
, the answer is positive, since:
a
<
b
⃒
˕(
a
)<˕(
b
)
⃒
˕(
a
)
·
˕
)
⃒
˕
−
1
))) < ˕
−
1
˕(
c
)<˕(
b
)
·
˕(
c
(˕(
a
)
·
˕(
c
(˕(
b
)
·
˕(
c
)))
,or
pr od
˕
(
a
,
c
)<
pr od
˕
(
b
,
c
)
, because
˕(
c
)
∈
(
0
,
1
]
.
•
If
T
is an ordinal-sum, it can't be strictly non-decreasing because of the values it
takes with min.
Analogously, the only t-conorms that are strictly non-decreasing are those in
{
pr od
˕
}
.
2.2.8 Laws of Fuzzy Sets
X
As it was said, in all standard algebras
(
[
0
,
1
]
,
T
,
S
,
N
)
of fuzzy sets the triplet
(
T
,
S
,
N
)
share the following common properties:
1.
T
and
S
are commutative and associative
2. 1 is neutral for
T
, and 0 is neutral for
S
3. 0 is absorbent for
T
, and 1 is absorbent for
S
4. For all
T
and
S
,itis
T
min
<
max
S
(
)
5. Each
T
is non decreasing in the two variables
6.
N
is involutive, strictly decreasing and such that
N
S
(
)
=
1,
a list of properties that gives some basic laws for fuzzy sets in the standard algebras,
like
0
•
μ
·
˃
=
˃
·
μ, μ
+
˃
=
˃
+
μ
,
•
μ
+
(˃
+
ʻ)
=
(μ
+
˃)
+
ʻ
=
(˃
+
μ)
+
ʻ
=
ʻ
+
(˃
+
μ)
•
μ
·
μ
1
=
μ, μ
+
μ
0
=
μ, μ
+
μ
1
=
μ
1
, μ
·
μ
0
=
μ
0
•
If
μ
˃
, then
μ
·
ʻ
˃
·
ʻ
, and
ʻ
+
μ
˃
+
ʻ
•
etc.
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