Introduction Fuzzy number min-plus algebra and matrix.
International Seminar on Innovation in Mathematics and Mathematics Education 1
st
ISIM-MED 2014 Department of Mathematics Education,Yogyakarta State University,Yogyakarta,
November 26-30, 2014
AP-2
that R
, , is a commutative idempotent semiring [1] with neutral element =
and unity element e = 0. Moreover, R
, , is a semifield, that is R
, , is a
commutative semiring [1], where for every a R there exist a such that a a = 0. Thus, R
, , is a min-plus algebra, and is written as R
min
[2]. One can define
x
:= 0,
k
x
:= x
1
k
x
,
: = 0 and
k
: = , for k = 1, 2, ... .
The operations and in R
min
can be extend to the matrices operations in
n m
min
R
, with
n m
min
R : = {A = A
ij
A
ij
R
min
, for i = 1, 2, ..., m and j = 1, 2, ..., n}, the set of all matrices over max-plus algebra. Specifically, for A, B
n n
min
R
we define A B
ij
= A
ij
B
ij
and A B
ij
=
kj ik
n k
B A
1
. We also define matrix E
n n
min
R , E
ij
: =
j
i ,
j i
, if
if
and
n m
min
R
,
ij
:= for every i and j . For any matrices A
n n
min
R
, one can define
A
= E
n
and
k
A
= A
1
k
A
for k = 1, 2, ... . We can show that
n m
min
R ,
is an idempotent commutative semigroup,
n n
min
R
, , is an idempotent semiring with matrix
as neutral element and matrix E as unity element, and
n m
min
R
is a
semimodule over R
min
[1] and [6]. Definitions and concepts in the interval min-plus algebra are analogous to the
concepts in the interval max-plus algebra which can be seen in [6]. The closed interval
x in R
min
is a subset of R
min
of the form x = [
x
,
x
] = {x R
min
x
m
x
m
x
}.
The interval x in R
min
is called min-plus interval, which is in short is called interval. Define
I R
:= { x = [
x
,
x
]
x
,
x
R ,
m
x
m
x
} { }, where := [, ].
In the IR
, define operation
and
as x
y = [
x
y
,
x
y
] and x
y = [
x
y
,
x
y
] , x, y IR
. Since R
, , is an idempotent semiring and it has no zero divisors, with neutral
element , we can show that IR
is closed with respect to the operation
and
.
Moreover, IR
,
,
is a comutative idempotent semiring with neutral element = [, ] and unity element 0 = [0, 0]. This comutative idempotent semiring IR
,
,
is called interval min-plus algebra which is written as IR
min
.
Define IR
n m
min
:= {A = A
ij
A
ij
IR
min
, for i = 1, 2, ..., m and j = 1, 2, ..., n
}. The element of IR
n m
min
are called matrix over interval min-plus algebra. Furthermore, this matrix is called interval matrix. The operations
and
in IR
min
can be extended to the matrices operations of in IR
n m
max
. Specifically, for A, B IR
n n
min
and IR
min
we define
A
ij
=
A
ij
, A
B
ij
= A
ij
B
ij
and A
B
ij
=
kj ik
n k
B A
1
.