Information Technology Reference
In-Depth Information
Then we obtain sequentially 1
α,β,...,μ ∈{
,
} .
where
0
1
de f
de f
a
αβγ
a
αβγ
B 2
=
B
( min , max )
B
=
b
δεζ
( min , max )
b
δεζ
e
ηθ ι
e
ηθ ι
d
κλμ
d
κλμ
d
e
...
a min
(
max
(α, κ)),
max
(β, η))
min
(
max
(α, λ),
max
(β, θ)) . . .
=
b min
(
max
(δ, κ)),
max
(ε, η))
min
(
max
(δ, λ),
max
(ε, θ))
. . .
e min
(
max
(η, κ)),
max
(θ, η))
min
(
max
(η, λ), θ)
. . .
(κ,
(λ, η))
(
(κ, λ),
(ε, θ))
. . .
d
min
max
min
max
max
...
f
a
...
min
(
max
(α, μ),
max
(β, ι))
b
...
min
(
max
(δ, μ),
max
(ε, ι))
.
e
...
min
(
max
(η, μ),
max
(θ, ι))
d
...
min
(
max
(κ, μ),
max
(λ, ι))
Now, we can directly see that when K
=
P
={
1
,
2
,...,
m
}
and L
=
Q
=
{
we obtain the definitions for standard matrix operations.
In the IMcase, we can use different symbols as indices of the rows and columns and
they, as we saw above, give us additional information and possibilities for description.
Let
1
,
2
,...,
n
}
IM R
be the set of all
R
-IMs and let
I =[∅ , , ⊥] ,
where symbol “
” denotes the lack of IM-elements. Let, as above,
, ∗∈{+ , × ,
max
,
min
.
The following assertions for the IM are discussed in [3].
}
and
( , ) ∈{ ( + , × ), (
max
,
min
), (
min
,
max
) }
Theorem 1 (a)
IM R ,
is a commutative semigroup,
(b)
IM R ,
is a commutative semigroup,
(c)
IM R , , +
is a semigroup,
(d)
IM R , ,
I
is a commutative monoid.
L be two indices. Now, following
the paper of E. Sotirova, V. Bureva and the author [24], we introduce the following
eight aggregation operations over it:
Let the IM A be given and let k 0
K and l 0
1 Here, in the IM we use symbol “
” in the end of the rows of the first IM and in the beginning of
the second IM to denote that the IM is divided into two parts, because of a lack of place.
...
 
Search WWH ::




Custom Search