Information Technology Reference
In-Depth Information
⊆
v
(
=
)
(
=
)
(
∀
∈
)(
∀
∈
)(
(
a
k
,
l
)
≤
(
b
k
,
l
)).
A
B
iff
K
P
&
L
Q
&
k
K
l
L
V
V
The strict relation “inclusion”
is
A
⊂
∗
B
iff
(((
K
⊂
P
)
&
(
L
⊂
Q
))
∨
((
K
⊆
P
)
&
(
L
⊂
Q
))
∨
((
K
⊂
P
)
&
(
L
⊆
Q
)))
&
(
∀
k
∈
K
)(
∀
l
∈
L
)(
V
(
a
k
,
l
)<
V
(
b
k
,
l
)).
The non-strict relation “inclusion”
is
A
⊆
∗
B
iff
(
K
⊆
P
)
&
(
L
⊆
Q
)
&
(
∀
k
∈
K
)(
∀
l
∈
L
)(
V
(
a
k
,
l
)
≤
V
(
b
k
,
l
)).
1.6 Operations “Reduction” Over an
R
-IM,
(
0
,
1
)
-IM and L-IM
Here and below we use symbol “
” for lack of some component in the separate
definitions. In some cases, it is suitable to change this symbol with “0”.
Now, we introduce operations
⊥
(
k
,
⊥
)
- and
(
⊥
,
l
)
-reduction of a given IM
A
=
[
K
,
L
,
{
a
k
i
,
l
j
}]
:
A
(
k
,
⊥
)
=[
−{
}
,
,
{
c
t
u
,v
w
}]
K
k
L
where
c
t
u
,v
w
=
a
k
i
,
l
j
for
t
u
=
k
i
∈
K
−{
k
}
and
v
w
=
l
j
∈
L
and
A
)
=[
K
,
L
−{
l
}
,
{
c
t
u
,v
w
}]
,
(
⊥
,
l
where
c
t
u
,v
w
=
a
k
i
,
l
j
for
t
u
=
k
i
∈
K
and
v
w
=
l
j
∈
L
−{
l
}
.
Second, we define
A
)
=
(
A
,
⊥
)
)
(
⊥
,
l
)
=
(
A
)
)
(
k
,
⊥
)
,
(
k
,
l
(
k
(
⊥
,
l
i.e.,
A
(
k
,
l
)
=[
K
−{
k
}
,
L
−{
l
}
,
{
c
t
u
,v
w
}]
,
where
c
t
u
,v
w
=
a
k
i
,
l
j
for
t
u
=
k
i
∈
K
−{
k
}
and
v
w
=
l
j
∈
L
−{
l
}
.
Theorem 2
For every IM A and for every k
1
,
k
2
∈
K, l
1
,
l
2
∈
L,
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