Strongly regular matrices and simple image set in interval max-plus algebra.

Published Online: January 2016

http://dx.doi.org/10.17654/NT038010063

Volume 38, Number 1, 2016, Pages 63-78 ISSN: 0972-5555

Received: September 7, 2015; Revised: September 25, 2015; Accepted: October 9, 2015 2010 Mathematics Subject Classification: 15A15, 05C50, 15A30.

Keywords and phrases: image set, strongly regular matrix, simple image set. Communicated by K. K. Azad

STRONGLY REGULAR MATRICES AND SIMPLE

IMAGE SET IN INTERVAL MAX-PLUS ALGEBRA

Siswanto1,2, Ari Suparwanto2 and M. Andy Rudhito3

1

Mathematics Department

Faculty of Mathematics and Natural Sciences Sebelas Maret University

Indonesia

e-mail: sis.mipauns@yahoo.co.id

2

Mathematics Department

Faculty of Mathematics and Natural Sciences Gadjah Mada University

Indonesia

e-mail: ari_suparwanto@yahoo.com

3

The Study Program in Mathematics Education Faculty of Teacher Training and Education Sanata Dharma University

Indonesia

e-mail: arudhito@yahoo.co.id

Abstract

Suppose that R is the set of real numbers and R_ε = RU{ }ε with .

−∞ =

ε Max-plus algebra is the set R_ε that is equipped with two

are matrices whose elements belong to R_ε. The set I( )R_ε = [ ]

{x = x,x x,x∈R,ε< x≤ x} { }U ε with ε =[ ]ε,ε which is equipped with maximum and addition operations is known as interval max-plus algebra. Matrices over interval max-plus algebra are

matrices whose elements belong to I( )R_ε . The research about image

set, strongly regular matrix and simple image set in max-plus algebra have been done. In this paper, we investigate image set, strongly regular matrix and simple image set in interval max-plus algebra.

1. Introduction

According to Baccelli et al. [1], max-plus algebra has been applied for

modelling and analyzing problems in the area of planning, communication,

production, queuing system with finite capacity, parallel computation and

traffic. Max-plus algebra is the set of R_ε = RU

{ }

ε equipped with two

operations, ⊗ (maximum) and ⊗ (addition), where R is the set of real

numbers with ε = −∞ (Baccelli et al. [1]). According to Tam [7], max-plus

algebra is a linear algebra over semiring R_ε that is equipped with ⊕

(maximum) and ⊗ (addition) operations, while min-plus algebra is a linear

algebra over semiring R_ε′ = RU

{ }

ε′ , ε′ = ∞ equipped with⊕′(minimum)

and ⊗′ (addition) operations. Tam [7] also defined complete max-plus

algebra and complete min-plus algebra. Complete max-plus algebra is a

linear algebra over semiring R_ε,ε′ = R_ε U

{ }

ε′ equipped with ⊕ (maximum) and ⊗ (addition) operations, while complete min-plus algebra is a linear

algebra over semiring R_ε′,ε = R_ε′ U

{ }

ε equipped with ⊕′ (minimum) and ⊗′ (plus) operations. Following the definition of max-plus algebra given by Baccelli et al. [1], we define min-plus algebra is the set of R_ε′ equipped with ⊕′ (minimum) and ⊗′ (plus) operations. Besides that, we define complete max-plus algebra is the set R_ε,ε′ = R_ε U

{ }

ε′ equipped with ⊕ (maximum) and ⊗ (addition) operations, while complete min-plus algebra is the set

{ }

ε = _ε′

ε

ε′ R U

It shows that R_ε,ε′ =R_εU

{ }

ε′ =RU

{ }

ε,ε′ =R_ε′U

{ }

ε′ =R_ε′,ε. Furthermore,

ε′ ε,

R and R_ε′,ε are written as .R It can be formed a set of matrices in the

size m×n of the sets R_ε, R_ε′, and .R The set of matrices with components

in R_ε is denoted by R_εm×n. The matrix A∈Rm_ε×n is called matrix over

max-plus algebra. The matrix over min-plus algebra, over complete

max-plus algebra and over min-max-plus algebra can also be defined (Tam [7]).

In 2010, Tam [7] discussed the linear system in max-plus algebra, image set and strongly regular matrix. Tam [7] and Butkovic [2, 3] mentioned that the solution of the linear system in max-plus algebra is closely related to image set, simple image set and strongly regular matrix. Butkovic [3-5] have shown the characteristics of the strongly regular matrix, that is matrix has strong permanent.

Rudhito [6] has generalized max-plus algebra into interval max-plus algebra and fuzzy number max-plus algebra. Moreover, Siswanto et al. [8] have discussed linear equation in interval max-plus algebra. Based on the discussion on the max-plus algebra that is connected to the system of linear equation, in this paper we discuss image set, strongly regular matrix and simple image set in interval max-plus algebra. Furthermore, it will also discuss matrix has strong permanent. The matrix has strong permanent is the characteristic of strongly regular matrix.

Before discussing the result of the research, the necessary concepts are discussed. These are interval max-plus algebra, system of linear equation in max-plus algebra and system of linear equation and the system of linear inequation in interval max-plus algebra.

Related to the system of linear equation A⊗x =b, Tam [7] defined the

solution set of the system of linear equation and image set of matrix A.

Definition 1.1. Given A∈R_εm×n and b∈Rm_ε. The solution set of the

system of linear equation A⊗x = b is S

(

A, b

) {

= x∈Rn_ε|A⊗x = b

}

and

Based on the above definition, the following theorem and corollary are

given.

Theorem 1.2. Given A∈Rm_ε×n and b∈R_εm. Vector b∈Im

( )

A if and

only if S

(

A, b

)

≠ ∅.

Corollary 1.3. Given A∈Rm_ε×n and b∈R_εm. Vector b∈Im

( )

A if and

only if A⊗

(

A∗⊗′b

)

= b.

The following is the definitions of subspace and theorem which states

that for A∈R_εm×n, Im

( )

A is a subspace.

Definition 1.4. Suppose that S ⊆ R_εn. If for each x, y∈S and for each

,

ε′

∈

α R α⊗x∈S and ,x⊕ y∈S then S is known as subspace of max-plus algebra.

Theorem 1.5. Suppose that A∈Rm_ε×n. If α,β∈R and u, v∈Im

( )

A

then α⊗u⊕β⊗v∈Im

( )

A .

In max-plus algebra there are the definitions of strongly linearly

independent and strongly regular matrix.

Definition 1.6. The vectors A₁,..., A_n∈Rm_ε are said to be linearly dependent if one of those vectors can be expressed as a linear combination of

other vectors. Otherwise A₁,..., A_n are considered linearly independent.

Definition 1.7. The vectors A₁,..., A_n ∈Rm_ε are said to be strongly linearly independent if there is b∈Rm as such so that b can be uniquely

expressed as a linear combination of A₁,..., A_n.

Below is the definition of simple image set.

Definition 1.9. Given .A∈Rm_ε×n The simple image set of matrix A is

defined as S_A =

{

b∈Rm|A⊗x =b has unique solution

}

.

Theorem 1.10. Suppose A∈R_εm×n is double R-astic and b∈Rm,

then S

(

A, b

) {

∈ 0,1, ∞

}

.

Definition 1.11. Suppose A∈Rm_ε×n, T

( )

A is defined as

{ (

S A, b

)

}

. m b∈R |

Theorem 1.12. If A∈R_εm×n is double R-astic, then T

( )

A is

{

0,∞

}

or

{

0,1, ∞

}

.

Based on Definition 1.7, the column vectors of A are strongly linearly

independent if and only if 1∈T

( )

A . In max-plus algebra, a matrix that has

strong permanent is also defined (Butkovic [4]).

Definition 1.13. Given A∈R_εm×n and π∈P_n, where P_n is the set of

all permutation of N =

{

1, 2, ...,n

}

. ω

(

A, π

)

is defined as

∏

⊗_{∈ π( )} n

j Aj, j and the permanent of matrix A is

( )

=

∑

⊕_π∈ ω

(

π

)

n P A

A , .

maper

Tam [7] defined that a matrix has strong permanent as given below.

Definition 1.14. The set of all permutation is written as ap

( )

A , and is

defined as ap

( )

A =

{

π∈P_n|maper

( )

A = ω

(

A, π

)

}

. Matrix A is said to have

strong permanent if ap

( )

A =1.

Butkovic [4] and Tam [7] explained the connection between strong

regular matrix and matrix has strong permanent as below.

Theorem 1.15. Suppose that A∈Rm_ε×n is double R-astic. Matrix A is

In the following section, interval max-plus algebra, matrix over interval max-plus algebra and the system of linear equation in interval max-plus algebra is in (Rudhito [6], Siswanto et al. [8]) are presented.

Closed interval x in R_ε is a subset of R_ε in the form of x =

[ ]

x, x =

{

x∈R_ε|x ≤ x ≤ x

}

. The interval x in R_ε is known as interval max-plus. A number x∈R_ε can be expressed as interval

[ ]

x, x .

Definition 1.16. I

( ) {

R _ε = x=

[ ]

x,x |x,x∈R,ε<x≤ x

} { }

U ε is formed,

with ε =

[ ]

ε, ε . On the set I

( )

R _ε, the operations ⊕ and ⊗ with x ⊕ y =

[

x⊕ y, x ⊕ y

]

and x ⊗ y =

[

x⊕ y, x ⊕ y

]

for every x, y∈I

( )

R _ε are defined. Furthermore they are known as interval max-plus algebra and

notated as I

( )

ℜ _max =

( ( )

I R _ε; ⊕, ⊗

)

.

Definition 1.17. The set of matrices in the size m×n with the elements

in I

( )

R _ε is notated as I

( )

R m_ε×n, namely:

( )

{

A

[ ]

A A I

( )

;for i 1, 2,..., m and j 1, 2,..., n

}

.

I R m_ε×n = = _ij | _ij ∈ R _ε = =

Matrices which belong to I

( )

R m_ε×n are known as matrices over interval

max-plus algebra or shortly interval matrices.

Definition 1.18. For A∈I

( )

R m_ε×n, matrix A =

[ ]

A_ij ∈Rm_ε×n and A =

[ ]

m n

ij

A ∈R_ε× are defined, as a lower bound matrix and an upper bound

matrix of the interval matrix A, respectively.

Definition 1.19. Given that interval matrix A∈I

( )

R m_ε×n, and A and

A are a lower bound matrix and an upper bound matrix of the interval

matrix A, respectively. The matrix interval of A is defined, namely

[

A, A

]

=

{

A= R_εm×n|A≤ A≤ A

}

and I

(

R_εm×n

)

_b =

{

[

A, A

]

|A∈

( ) }

R _εm×n .

max-plus algebra. Furthermore, A ≈

[

A, A

]

, x ≈

[ ]

x, x and b ≈

[ ]

b,b with

[

A, A

]

∈I

(

Rn_ε×n

)

_b,

[ ] [ ]

x, x , b,b ∈I

( )

Rn_{ε b}. To find the solution of the system of linear equation in interval max-plus algebra, the solution of the

system of linear equation in max-plus algebra A⊗x = bdanA⊗x = b

should be determined.

Definition 1.20. Given the system of linear equation A⊗ x = b with

( )

m n

I

A∈ R _ε× and b∈I

( )

R m_ε. IS

(

A,b

) {

= x∈I

( )

R m_ε |A⊗x = b

}

is

defined.

Along with the discussion the system of linear equation in max-plus

algebra, without loss of generality, the system of linear equation in interval

max-plus algebra A⊗x = b with A∈I

( )

R _εm×n double I

( )

R -astic and

( )

m I

b∈ R is discussed.

Theorem 1.21. Suppose that A∈I

( )

R m_ε×n double I

( )

R -astic and

( )

m. I

b∈ R Vector y∈IS

(

A,b

)

if and only if y∈S

(

A, b

)

, y ∈S

(

A, b

)

and y ≤ y. Furthermore y can be expressed as y ≈

[

y, y

]

.

Each of the following two corollaries is a criterion of the system of linear equation which has solution and a criterion the system of linear

equation which has unique solution. The notations of M =

{

1, 2,..., m

}

and

{

n

}

N = 1, 2,..., are used to simplify.

Corollary 1.22. Suppose A∈I

( )

R m_ε×n is double I

( )

R -astic and

( )

m, I

b∈ R then the following three statements are equivalent:

a. IS

(

A, b

)

≠ ∅.

b. yˆ∈IS

(

A,b

)

.

Corollary 1.23. Suppose A∈I

( )

R m_ε×n is double I

( )

R -astic and

( )

m, I

b∈ R then IS

(

A, b

) { }

= yˆ if and only if

a. U_j∈N M_j

(

A, b

)

= M and U_j∈N M_j

(

A, b

)

= M.

b. U_j∈N′ M_j

(

A, b

)

≠ M for every N′ ⊆ N, N′ ≠ N and

(

A b

)

M M_j

N

j∈ ′′ , =

U for every N′′ ⊆ N, N′′ ≠ N.

2. Main Results

In this section, we present new results of the research. The results that are

obtained, are related with the system of linear equation in interval max-plus

algebra A⊗ x = b, A∈I

( )

R _εm×n and b∈I

( )

R m.

We will start the definition of the image set of matrix A.

Definition 2.1. Given A∈I

( )

R m_ε×n. The image set of matrix A is

( ) {

A A x x I

( ) }

n .

IIm = ⊗ | ∈ R _ε

Based on the definition of the image set of a matrix, a theorem which

demonstrated the necessary and sufficient conditions of the elements of the

image set is presented.

Theorem 2.2. Given A∈I

( )

R m_ε×n and b∈I

( )

R m_ε. Vector b∈IIm

( )

A

if and only if IS

(

A, b

)

≠ ∅.

Proof. Suppose that A≈

[

A, A

]

and b ≈

[ ]

b,b with A,A∈R_εm×n and

.

,b m

b ∈R_ε Since b∈IIm

( )

A , it means that b∈Im

( )

A and b ∈Im

( )

A .

Therefore, based on Theorem 1.2, i.e., S

(

A, b

)

≠ ∅ and S

(

A,b

)

≠ ∅, then

(

A, b

)

≠ ∅.

IS On the other hand, since IS

(

A, b

)

≠ ∅, S

(

A,b

)

≠ ∅ and

(

A,b

)

≠ ∅.

S Thus, based on Theorem 1.2 that is b∈Im

( )

A and

( )

A , Im

With regard to Theorem 2.2, the following corollary is obtained:

Corollary 2.3. Given A∈I

( )

R _εm×n and b∈I

( )

R _εm. Vector b∈IIm

( )

A

if and only if b = A⊗

(

A∗⊗′b

)

and b = A⊗

(

A∗⊗′b

)

.

Proof. Suppose that A ≈

[

A, A

]

and b ≈

[ ]

b, b . Since b∈IIm

( )

A , it

means that b∈Im

( )

A and b ∈Im

( )

A . Based on Corollary 1.3, b =

(

A b

)

A⊗ ∗⊗′ and b = A ⊗

(

A∗ ⊗′b

)

. On the other hand, since

(

A b

)

A

b = ⊗ ∗⊗′ and b = A ⊗

(

A∗⊗′b

)

, based on Corollary 1.3,

( )

A

Im

b∈ and b ∈Im

( )

A are obtained. Therefore, b ≈

[ ]

b, b ∈IIm

( )

A . ~

The following theorem presents that IIm

( )

A is a subspace of I

( )

R m_ε.

Theorem 2.4. Given A∈I

( )

R m_ε×n, IIm

( )

A is a subspace of I

( )

R _εm.

Proof. Since IIm

( ) {

A = A⊗ x|x∈ I

( ) }

R _εn , IIm

( )

A ⊆ I

( )

R _εm and

( )

A ≠∅.

Im Taking p,q∈IIm

( )

A and α,β∈I

( )

R , accordingly,

[

p q p q

]

q

p ⊗β⊗ ≈ α⊗ ⊕β⊗ α⊗ ⊕β⊗

⊗

α , with α ⊗ p⊕β⊗

( )

A Im

q∈ and α⊗ p⊕β⊗q ∈Im

( )

A . It means that α⊗p⊕β⊗

( )

A .

IIm

q∈ In other words, α ⊗ p⊕β⊗q∈IIm

( )

A and hence IIm

( )

A is

a subspace of I

( )

R _εm. ~

Several definitions and theorems are presented to support the discussion on strongly regular matrix.

Definition 2.5. The vectors A_1,,..., A_n ∈I

( )

R m_ε is said to be linearly dependent if one of those vectors can be said to be as linear combination of

other vectors. Otherwise then, A ...,₁, A_n is said to be linearly independent.

Theorem 2.6. If the vectors A₁,..., A_n∈I

( )

R m_ε with A₁ ≈

[

A₁, A₁

]

,

[

A A

]

An

[

An An

]

A₂ ≈ ₂, ₂,..., ≈ , are linearly dependent, then A₁, A₂, n

A

Proof. It is known that A₁,..., A_n ∈I

( )

R m_ε is linearly dependent. It

means that one of those vectors can be stated as a linear combination of

the others. Without losing the generality of the proof, for example

(

i i

)

n

i A

A₁ = ⊕₌₂ α ⊗ with α_i ∈I

( )

R _m, since A₁ ≈

[

A₁, A₁

]

, A₂ ≈

[

A₂, A₂

]

,..., A_n ≈

[

A_n, A_n

]

, A₁=⊕n_i=2

(

α_i⊗A_i

)

and A₁=⊕_in₌₂

(

α_i⊗A_i,

)

. In other words, A₁, A₂,..., A_n and A₁, A₂,..., A_n are linearly dependent.

Definition 2.7. The vectors A₁,..., A_n ∈I

( )

R m_ε is said to be strongly linearly independent if there is b∈I

( )

R m, bcan be uniquely expressed as a

linear combination of A₁,..., A_n, that is b = ⊕_in₌₁

(

α_i ⊗ A_i

)

, α_i ∈I

( )

R . The following is the definition of strongly regular matrix.

Definition 2.8. Matrix A =

(

A_1,A₂KA_n

)

∈I

( )

R _εn×n is said to be strongly regular if the vectors A₁,..., A_n ∈I

( )

R n_ε are strongly linearly independent.

Based on the definition of strongly linearly independent in the max-plus algebra, the following theorem is formulated.

Theorem 2.9. If the vectors A_1,,..., A_n ∈I

( )

R _εm with A₁ ≈

[

A₁, A₁

]

,

[

A A

]

An

[

An An

]

A₂ ≈ ₂, ₂,..., ≈ , are strongly linearly independent, then

n A A

A₁, ₂,..., and A₁, A₂,..., A_n are strongly linearly independent.

Proof. It is known that A₁ ≈

[

A₁, A₁

]

, A₂ ≈

[

A₂, A₂

]

,..., A_n ≈

[

A ,_n A_n

]

are strongly linearly independent, then there is b ≈

[ ]

b,b ∈I

( )

R m so that b

can be uniquely expressedaslinearcombination of A₁,..., A_n. Suppose that

(

)

;

1 i i

n

i A

b = ⊕₌ α ⊗ since

(

)

[

₁

(

)

,

1 i i ni i i

n

i α ⊗ A ≈ ⊕ α ⊗ A

(

)

]

,

1 i i

n

i α ⊗A

⊕₌ b = ⊕ni=1

(

αi ⊗Ai

)

and b = ⊕in=1

(

αi ⊗ Ai

)

, It means that b can be uniquely expressed as a linear combination of A₁, A₂, ..., A_n and b can be uniquely expressed as a linear combination A₁, A₂,..., A_n. In other words, A₁, A₂,..., A_n and A₁, A₂, ..., A_n are strongly linearly

independent. ~

The following is the definition of simple image set.

Definition 2.10. Given A∈I

( )

R _εm×n. The set IS_A =

{

b∈I

( )

R m|

b x

A⊗ = has a unique solution is defined as a simple image set of

}

matrix A.

The regularity properties of the matrix are closely related to the number

of the solutions of the system of linear equation. It can be stated in the

following definitions and theorems.

Theorem 2.11. Suppose that A∈I

( )

R m_ε×n is a double I

( )

R -astic and

( )

m I

b∈ R is the number of the elements of IS

(

A, b

)

is IS

(

A, b

)

∈

{

01, ∞

}

.

Proof. Suppose that A∈I

( )

R _εm×n is double I

( )

R -astic and b∈I

( )

R m.

Suppose that A ≈

[

A, A

]

and b ≈

[ ]

b,b with each of A and A is double

R-astic, while b,b ∈Rm. Based on the Theorem 1.10, S

(

A, b

)

∈

{

0,1, ∞

}

and S

(

A,b

) {

∈ 0,1, ∞

}

. Therefore, following the Definition 1.18, it is confirmed that IS

(

A,b

) {

∈ 0,1, ∞

}

. ~

Definition 2.12. Suppose that A∈I

( )

R m_ε×n, T

( )

A is defined as

{ (

IS A, b

)

|b∈Rm

}

.

Proof. Suppose that A∈I

( )

R _εm×n, A ≈

[

A, A

]

with A∈R_εm×n and

. n m

A∈R_ε× Based on the theorem for the case of max-plus algebra,

( ) {

A = 0,1, ∞

}

T or T

( ) {

A = 0, ∞

}

and T

( ) {

A = 0, ∞

}

or T

( ) {

A = 0,1,∞

}

.

Therefore, T

( ) {

A = 0, ∞

}

or

( ) {

A = 0,1, ∞

}

. ~

Based on Theorem 2.13, A is a strongly regular matrix if and only if

( )

.

1∈T A Then, the criteria of strongly regular matrix is discussed.

Definition 2.14. Given A =

( )

a_ij ∈I

( )

R _εn×n and π∈P_n, with P_n is the

set of all permutations of

{

1, 2,...,n

}

. ω

(

A, π

)

is defined as ⊗_j∈n a_{j,π( )j} .

Theorem 2.15. If A∈I

( )

R _εn×n double I

( )

R -astic and π∈P_n with

[

A, A

]

,

A ≈ then ω

(

A, π

)

=

[

ω

(

A, x

) (

ω A, π

)

]

.

Proof. If A∈I

( )

R m_ε×n and π∈P_n with A ≈

[

A, A

]

, it means that

(

A π

)

= ⊗j∈naj π_{( )}j = ⊗j∈n

[

aj π_{( )}j aj π_{( )}j

]

ω , , , , ,

[

⊗ _{, ( )}, ⊗ _, ( )

] [

= ω

(

, π

) (

, ω , π

)

]

.

= _j∈_n a_j π _{j j}∈_n a_j π _j A A ~

Definition 2.16. The permanent of matrix A is defined as

(

,

)

.

maper⊗_π∈Pn ω A π

Next discussion is matrix over interval max-plus algebra which has weak permanent and strong permanent.

Definition 2.17. Suppose that A∈I

( )

R _εn×n double I

( )

R -astic with

[

A, A

]

.

A ≈ The set of all optimal permutations is ap

( )

A with ap

( )

A =

{

π∈P_n|maper

( )

A = ω

(

A, π

)

or maper

( )

A =ω

(

A, π

)}

. Matrix A is said to have strong permanent if ap

( )

A =1 and it is said to have weak permanent

if ap

( )

A =1and ap

( )

A =1.

Theorem 2.18. Suppose that A∈I

( )

R n_ε×n is double I

( )

R -astic. If

Proof. Suppose that A∈I

( )

R _εn×n, A≈

[

A, A

]

with A∈

( )

R n_ε×n and

. n n

A∈R_ε× Since A has strong permanent, it means that ap

( )

A =1.

Suppose that π₁∈ap

( )

A , then maper

( )

A = ω

(

A, π₁

)

and maper

( )

A =

(

, π₁

)

.

ω A Therefore, ap

( )

A =1 and ap

( )

A =1. In other words, A has

weak permanent. ~

Theorem 2.19. Suppose that A∈I

( )

R n_ε×n is double I

( )

R -astic with

[

A, A

]

.

A ≈ Matrix A has weak permanent if and only if A and A have

strong permanent.

Proof. Suppose that A∈I

( )

R _εn×n, A≈

[

A, A

]

with A∈Rn_ε×n and

. n n

A∈R_ε× It is known that matrix A has weak permanent. It means that

( )

A =1

ap and ap

( )

A =1. Therefore, matrix A and A have strong

permanent. On the other hand, suppose that Aand A have strong permanent,

it means that ap

( )

A =1 and ap

( )

A = A. Therefore, matrix A with

[

A A

]

A ≈ , has weak permanent. ~

Theorem 2.20. Suppose that A∈I

( )

R n_ε×n is double I

( )

R -astic with

[

A, A

]

.

A ≈ Matrix A has strong permanent if and only if A and A have

strong permanent as such that if maper

( )

A = ω

(

A, π₁

)

and maper

( )

A =

(

, π₂

)

,

ω A then π₁ = π₂.

Proof. Suppose that A∈I

( )

R _εn×n with A ≈

[

A, A

]

. It is known that

matrix A has strong permanent. Since matrix A has strong permanent, it

means that ap

( )

A =1. Suppose that π∈ap

( )

A it means that π∈P_n so

that maper

( )

A = ω

(

A, π

)

and maper

( )

A = ω

(

A, π

)

. Thus, ap

( )

A =1 and

( )

A =1.

ap Therefore, A and A have strong permanent. On the other hand,

(

, π1

)

ω A andmaper

( )

A = ω

(

A, π₂

)

, then π₁ = π₂. Therefore, ap

( )

A =1 and ap

( )

A =1 with π₁∈ ap

( )

A , π₂∈ ap

( )

A and .π₁=π₂ Since

( ) {

A P

( )

A

(

A

)

( )

A

ap = π∈ _n|maper =ω ,π ormaper = ω

(

A, π

)}

, ap

( )

A =1.

Therefore, matrix A has strong permanent.

The following is theorem which has the characteristics of strongly

regular matrix.

Theorem 2.21. Suppose that A∈I

( )

R n_ε×n is double I

( )

R -astic. Matrix

A is strongly regular if and only if matrix A has strong permanent.

Proof. Suppose that A∈I

( )

R n_ε×n is double I

( )

R -astic, with

[

A, A

]

.

A ≈ It is known that A is strongly regular matrix. Suppose that

n A

A ...,₁, are the columns of matrix A. Since A is strongly regular matrix, there is b∈I

( )

R n so that b can be uniquely expressed as linear combination

of ,A_1,,..., A_n that is b = ⊕_in₌₁

(

x_i ⊗ A_i

)

, x_i ∈I

( )

R . The equation

(

)

,

1 x A A x b

b = ⊕_in_{= i} ⊗ _i ⇔ ⊗ = is the system of linear equation in

interval max-plus algebra which has a unique solution. Based on

Corollary 1.23, ap

( )

A =1. Therefore, matrix A has strong permanent.

The converse, it is known that matrix A has strong permanent. Based on

Theorem 2.20, matrix A and A have such strong permanent, so that if

( )

(

, 1

)

maper A = ω A π and maper

( )

A = ω

(

A, π₂

)

, π₁ = π₂. Therefore, A

matrix and A are both strongly regular matrix, and thus, there are b

and b which can be uniquely written as the linear combinations

(

i i

)

n

i x A

b = ⊕₌₁ ⊗ and b = ⊕n_i=1

(

x_i ⊗A_i

)

, in which A₁, A₂,..., A_n are the columns of matrix ,A while A₁, A₂,..., A_n are the columns of matrix .A That b=⊕_in₌₁

(

x_i⊗A_i

)

⇔A⊗x=b and b = ⊕_in₌₁

(

x_i ⊗ A_i

)

, ⇔ A ⊗x =

.

unique solution. Since maper

( )

A = ω

(

A, π₁

)

and maper

( )

A = ω

(

A, π₂

)

,

2 1 = π

π so that

[

b, b

]

≈ b. As the result,

[

A⊗x, A ⊗x

] [ ]

= b, b and

since A⊗x ≈

[

A⊗x, A ⊗x

]

, A⊗ x =b or b = ⊕_in₌₁

(

x_i ⊗ A_i

)

with n

A

A_1,,..., as the columns of matrix A. It means that there is b which can be uniquely expressed as linear combination of A₁,..., A_n. This matrix A is

known as strongly regular matrix. ~

3. Concluding Remarks

Based on the discussion, the following conclusions are obtained:

1. The necessary and sufficient condition of matrix A∈I

( )

R n_ε×n, which

is double I

( )

R -astic is strongly regular matrix.

2. The system of linear equation in the interval max-plus algebra

, b x

A⊗ = A∈I

( )

R n_ε×n which is double I

( )

R -astic and b∈I

( )

R n has

unique solution if and only if matrix A has strong permanent.

Acknowledgment

The authors thank the anonymous referees for their valuable suggestions which let to the improvement of the manuscript.

References

[1] F. Baccelli, G. Cohen, J. G. Older and J. P. Quadat, Synchronization and Linearity, John Wiley & Sons, New York, 2001.

[2] P. Butkovic, Strongly regularity of matrices - a survey of results, Discrete Applied Mathematics 48 (1994), 45-68.

[3] P. Butkovic, Simple image set of (max, +) linear mappings, Discrete Applied Mathematics 105 (2000), 73-86.

[5] P. Butkovic, Max Linear Systems: Theory and Algorithm, Springer, London, 2010.

[6] M. A. Rudhito, Fuzzy number max-plus algebra and its application to fuzzy scheduling and queuing network problems, Doctoral Program of Mathematics, Faculty of Mathematics and Natural Sciences, Gadjah Mada University, 2011 (Indonesian).

[7] K. P. Tam, Optimizing and approximating eigenvectors in max algebra (Thesis), A Thesis Submitted to The University of Birmingham for the Degree of Doctor of Philosophy (Ph.D.), 2010.