Information Technology Reference
In-Depth Information
Example 6.1.
Let
M⊃F
be the MV-algebra defined in Theorem 1.1 (Example1.5). Then
M
=(
μ
A
μ
B
,
ν
A
+
ν
B
−
ν
A
ν
B
)
with the product
A
.
B
is an MV-algebra with product. Indeed,
(
1, 0
)
.
(
μ
A
,
ν
A
)=(
1.
μ
A
,0
+
ν
A
−
0.
ν
A
)=(
μ
A
,
ν
A
)
.
Moreover
(
μ
A
,
ν
A
)
.
((
μ
B
,
ν
B
)
(
1
−
μ
C
,1
−
ν
C
)) =
=(
μ
A
((
μ
B
− μ
C
)
∨
)
+(
ν
B
− ν
C
+
)
∧
− ν
A
((
ν
B
+
− ν
C
)
∧
))
0
,
ν
A
1
1
1
1
.
On the other hand
((
μ
A
,
ν
A
)
.
(
μ
B
,
ν
B
))
(
¬
(
μ
A
,
ν
A
)
.
(
μ
C
,
ν
C
)) =
((
μ
A
(
μ
B
−
μ
C
))
∨
0,
(
ν
A
+(
ν
B
−
ν
C
+
1
)
−
ν
A
(
ν
B
+
1
−
ν
C
))
∧
1
)
.
Denote
− ν
C
+
=
ν
B
1
k
.
≤
If 1
k
, then
ν
A
+
k
∧
1
−
ν
A
(
k
∧
1
)=
ν
A
+
1
−
ν
A
=
1,
(
ν
A
+
k
−
ν
A
k
)
∧
1
=(
ν
A
+
k
(
1
−
ν
A
))
∧
1
=
1.
<
If
k
1, then
ν
A
+
∧
− ν
A
k
∧
=
ν
A
+
− ν
A
k
,
k
1
1
k
(
ν
A
+
k
−
ν
A
k
)
∧
1
=
ν
A
+
k
−
ν
A
k
,
hence actually
A
.
(
B
¬
C
)=(
A
.
B
)
(
¬
(
A
.
C
))
.
Similarly as in Section 1 we can define a product in D-posets, we shall name such D-posets
Kôpka D-posets.
Definition 6.2.
A Kôpka D-poset is a pair
(
D
,
∗
)
, where D is a D-poset, and
∗
is a commutative
and associative operation on D satisfying the following conditions:
1.
∀
a
∈
D
:
a
∗
1
=
a
;
2.
∀
a
,
b
∈
D
,
a
≤
b
,
∀
c
∈
D
:
a
∗
c
≤
b
∗
c
;
∀
∈
−
(
∗
)
≤
−
3.
a
,
b
D
:
a
a
b
1
b
;
∀
(
)
⊂
∀
∈
∗
∗
4.
a
n
D
,
a
n
a
,
b
D
:
a
n
b
a
b
.
n
Evidently every IF-family
can be embedded to an MV-algebra with product and it is a
special case of a Kôpka D-poset, hence any result from the Kôpka D-poset theory can be
applied to our IF-events theory ([26], [64]).
F
Now let us consider a theory dual to the IF-events theory, theory of IV-events. A prerequisity
of IV-theory is in the fact that it considers natural ordering and operations of vectors. On the
other hand the IV-theory is isomorphic to the IF-theory ([65],[43]).
Definition 6.3.
Let
(
Ω
,
S
)
be a measurable space,
S
be a
σ
-algebra.
By an IV-event a pair
2
A
=(
μ
A
,
ν
A
)
:
Ω
→
[
0, 1
]
is considered such that
A
≤
B
⇐⇒
μ
A
≤
μ
B
,
ν
A
≤
ν
B
;
