Information Technology Reference
In-Depth Information
Example 1.4.
in the set
R
of all real numbers. It will stay an
MV-algebra, if we shall define two binary operations
Consider the unit interval
[
0, 1
]
⊕
[
]
¬
,
on
0, 1
, one unary operation
≤
and the usual ordering
by the following way:
⊕
=
(
+
)
a
b
min
a
b
,1
,
a
b
=
max
(
a
+
b
−
1, 0
)
,
¬
=
−
a
1
a
.
⊕
It is easy to imagine that
a
b
corresponds to the disjunction of the assertions
a
,
b
,
a
b
to the
conjunction of
a
,
b
and
¬
a
to the negation of
a
.
By the Mundici theorem ([48])any MV-algebra can be defined similarly as in Example 1.4, only
the group
R
must be substitute by an arbitrary
l
-group.
Definition 1.2.
(
G
,
+
,
≤
)
such that
By an
l
-group we consider an algebraic system
(
+)
(i)
G
,
is an Abelian group,
(
≤
)
(ii)
G
,
is a lattice,
≤
=
⇒
+
≤
+
(iii)
a
b
a
c
b
c
.
Definition 1.3.
By an
MV
-algebra we consider an algebraic system
(
M
,0,
u
,
⊕
,
)
such that
M
=[
0,
u
]
⊂
G
, where
(
G
,
+
,
≤
)
is an
l
-group, 0 its neutral element,
u
a positive element, and
a
⊕
b
=(
a
+
b
)
∧
u
,
=(
+
−
)
∨
a
b
a
b
u
0,
¬
=
−
a
u
a
.
(Ω
S
)
S
Example 1.5.
Let
,
be a measurable space,
a
σ
-algebra,
G
=
{
A
=(
μ
A
,
ν
A
)
;
μ
A
,
ν
A
:
Ω
→
R
}
,
A
+
B
=(
μ
A
+
μ
B
,
ν
A
+
ν
B
−
1
)=(
μ
A
+
μ
B
,1
−
(
1
−
ν
A
+
1
−
ν
B
))
,
A
≤
B
⇐⇒
μ
A
≤
μ
B
,
ν
A
≥
ν
B
.
Then
(
G
,
+
,
≤
)
is an
l
-group with the neutral element
0
=(
0, 1
)
,
A
−
B
=(
μ
A
−
μ
B
,
ν
A
−
+
)
ν
B
1
, and the lattice operations
∨
=(
μ
A
∨ μ
B
,
v
A
∧ ν
B
)
A
B
,
∧
=(
μ
A
∧ μ
B
,
ν
A
∨ ν
B
)
A
B
.
Put
u
=(
1, 0
)
and define the MV-algebra
M
=
{
A
∈
G
;
(
0, 1
)=
0
≤
A
≤
u
=(
1, 0
)
}
,
A
⊕
B
=(
A
+
B
)
∧
u
=
=(
μ
A
+
μ
B
,
ν
A
+
ν
B
−
)
∧
(
)=
1
1, 0
=((
μ
A
+
μ
B
)
∧
(
ν
A
+
ν
B
−
)
∨
1,
1
0,
=(
+
−
)
∨
(
)
A
B
A
B
u
0, 1
=
