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2.2.4 Decomposable Algebras
X
X
X
Definition 2.2.17
An operation with fuzzy sets
∗:[
0
,
1
]
×[
0
,
1
]
ₒ[
0
,
1
]
is
decomposable, or functionally expressible
, if it exists a numerical operation
∗:
[
0
,
1
]×[
0
,
1
]ₒ[
0
,
1
]
, such that
(μ
∗
˃)(
x
)
= ∗
(μ(
x
), ˃(
x
)),
X
for all
μ, ˃
in
[
0
,
1
]
and all
x
in
X
. Of course, by this formula, a numerical operation
∗
allows to define an operation
∗
for fuzzy sets.
X
For example, the operation min in
[
0
,
1
]
is decomposable since, by definition, it is
(
min
(μ, ˃))(
x
)
=
min
(μ(
x
), ˃(
x
))
X
for all
μ, ˃
in
[
0
,
1
]
and all
x
in
X
.
X
X
Definition 2.2.18
A function
f
:[
0
,
1
]
ₒ[
0
,
1
]
is
decomposable, or function-
f
ally expressible
, if it exists a numerical function
:[
0
,
1
]ₒ[
0
,
1
]
, such that
)
=
f
(
f
(μ))(
x
(μ(
x
)),
X
μ
[
,
]
for all
in
0
1
and all
x
in
X
.
For example, the function
defined by
μ
(
x
)
=
1
−
μ(
x
),
μ
(
is decomposable because of
N
0
=
1
−
id gives
x
)
=
N
0
(μ(
x
))
. With
X
=[
0
,
1
]
,
μ
∗
(
the function defined by
x
)
=
1
−
μ(
1
−
x
)
is not decomposable, since if it were
such, that is, if there is
N
:[
0
,
1
]ₒ[
0
,
1
]
such that 1
−
μ(
1
−
x
)
=
N
(μ(
x
))
,it
X
suffices to take
μ, ˃
∈[
0
,
1
]
such that
•
μ(
0
)
=
0
, μ(
1
)
=
1
⃒
N
(
0
)
=
1
−
μ(
1
−
0
)
=
1
−
μ(
1
)
=
0
,
N
(
1
)
=
1
−
μ(
1
−
1
)
=
1
−
μ(
0
)
=
1
•
˃(
0
)
=
1
, ˃(
1
)
=
0
⃒
N
(
0
)
=
1
−
˃(
1
−
0
)
=
1
−
˃(
1
)
=
1
,
N
(
1
)
=
1
−
˃(
1
−
1
)
=
1
−
˃(
0
)
=
0
that is absurd.
The algebras of fuzzy sets
X
,
·
,
+
,
)
(
[
0
,
1
]
, can be
·
,
+
,
are decomposable
Decomposable
if the three operation
·
,
+
,
is decom-
Partially decomposable
if at least one of the three operations
posable
Non decomposable
if no one of the three operations is decomposable
In what follows we will only deal with decomposable algebras
, that is, such that:
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