On Matrices Over Path Algebra

Vol. 11, No. 2, 2016, 45-55

ISSN: 2279-087X (P), 2279-0888(online) Published on 25 April 2016

www.researchmathsci.org

45

Annals of

On Matrices Over Path Algebra

Kanak Ray Chowdhury1, Md. Yasin Ali2, Abeda Sultana3, Nirmal Kanti Mitra4 and A F M khodadad Khan5

1

Department of Mathematics, Mohammadpur Model School and College Mohammadpur, Dhaka. e-mail : [email protected]

2

School of Science and Engineering

University of Information Technology & Sciences, Dhaka 3

Department of Mathematics, Jahangirnagar University, Savar, Bangladesh 4

Mathematical and Physical Sciences

Bangladesh University of Business and Technology, Dhaka 5

School of Engineering and Computer Science, Independent University Bashundhara R/A, Dhaka, Bangladesh

Received 23 March 2016; accepted 4 April 2016

Abstract. In this paper, we consider a path algebra and discuss about matrices over path algebra. Some investigations on transitivity over path algebra are performed. Some properties and characterization for permanent of matrices over path algebra are established. Several inequalities over permanent are discussed. Also the adjoint matrix over path algebra are studied.

Keywords: Commutative, Idempotent, Transitive, Permanent, Incline, Adjoint. AMS Mathematics Subject Classification (2010): 16Y60

1. Introduction

Path algebras are useful in many areas such as design of switching circuits, automata theory, information system, dynamic programming and decision theory. For further examples [3, 8]. Boolean matrix, incline matrix [13] are the prototypical examples of matrices over path algebra. In 1999, Golan [4] worked on semirings and matrices over semirings. Transitive matrices are important type of matrices. Since the beginning of 1980s, many authors have studied this types for some special cases of path algebra e.g. [5]. In [3], Hasimoto considered transitivity of matrices over a general path algebra. A large number of work on permanent theory have been published [12, 13].

2. Preliminaries

In this section, we present some definitions and examples of algebraic structures of semiring. We support these definitions by some examples.

Definition 2.1. Let S be a non empty set with two binary operations + and . . Then the algebraic structure (S; +, .) is called asemiring iff a,b,cS,

46

(iii) (S; .) is monoid with identity element 1. (iv) a. (b+c) = a.b + a.c and (b+c).a = b.a + c.a. (v) 0. a = 0. a = 0 and

0



1

.

Example 2.1(a). (B = {0,1}, +, .) is a semiring, where + and . are defined by

Example 2.1(b). (I =[0,1]; +, .) is a semiring, where order in [0,1] is usual



and + and . are defined as follows:

a + b = max{a,b}, a.b = min{a,b}.

Example 2.1(c). (L = {1, 2, 3, 4, 6, 8, 12, 24};│, +, .) is a semiring, where a + b = lcm{a , b}, a.b = gcd{a , b}.

Definition 2.2. Let (S; +, .) be a semiring. Then S is called commutative iff x,yS, x.y y.x.

Example 2.2(a).



₀

 

{

x



:

x



0}

,



₀

 

{

x



:

x



0}

,



₀

 

{

x



:

x



0}

are commutative semirings .

Definition 2.3. Let (S; +, .) be a commutative semiring. It is called Boolean semiring iff



x



S

,

. .x x

x 

Example 2.3(a).

(i) (B = {0,1}; +, .) is a Boolean semiring, where + and . are defined in Example 2.1(a). (ii) (I =[0,1]; +, .) is a Boolean semiring, where order in [0,1] is usual



and + and . are defined as follows:

a + b = max{a,b}, a.b = min{a,b}.

(iii) Let X 



and P(X) is power set of X. + and . are defined by AB AB

and A.B AB;A,BP(X). Then (P(X); +, .) is a Boolean semiring, where



and X are zero and identity of P(X) respectively.

3. Transitive matrix over path algebra

In this section, we discuss path algebra, incline and some elementary properties of transitivity of matrices over path algebra.

Definition 3.1. Let (S; +, .) be a semiring. Then S is called path algebra iff xxx ;xS.

Example 3.1(a).

(1) (B={0, 1}; +, . ) is a path algebra, where + and . are defined in Example 2.1(a). (2) The fuzzy algebra (F [0.1];,T) ; where



= max and T is a t-norm is a path algebra.

+ 0 1 0 0 1 1 1 1

47

(3) Every bounded distributive lattice is a path algebra.

Definition 3.2. Let (S; +, .) be a path algebra. Then S is called commutativepath algebra iff

x.y y.x;x,yS.

Example 3.2(a).

(1) (B={0, 1}; +, . ) is a commutative path algebra, where + and . are defined in Example 2.1(a).

(2) The fuzzy algebra

(

F



[

0

.

1

]

;



,

T

)

; where



= max and T is a t-norm is a commutative path algebra.

(3) Every bounded distributive lattice is commutative path algebra.

Definition 3.3. Let (S; +, .) be a semiring. Then S is called an incline iff

x



1



1

;



x



S

.

Example 3.3(a). The fuzzy algebra (F [0.1];,T) ; where



= max and T is a t-norm is an incline.

Proposition 3.4. Any incline is a path algebra. Proof : Let (S; +, .) be an incline.

Then 1 + 1 = 1. Now xS,

x x.1



x

.

(

1



1

)



x



x

So

x



x



x

; xS,

Hence S is a path algebra. 

Remark 3.4(a). For any a,bS; ababb.

Definition 3.5. Let (S; +, .) be a path algebra and xS.Then

x



S

is called transitive element iff

x2  x.

Proposition 3.6. . Let (S; +, .) be an incline. Then every element in S is transitive. Proof : Let (S; +, .) be an incline.

Then

x

.



1



1

;



x



S

,

Now x2 x x(x1)

1 . x

  x

Therefore x2 x.

48

The least positive integer k is called the index and the least positive integer d is called the period of A. It is denoted by k(A) and d(A).

Definition 3.8. Let (S; +, .) be a commutative path algebra and AMn(S). Then A

is called transitive element iff

A

2



A

.

Definition 3.9. Let (S; +, .) be a path algebra and AMn(S). Then A is called

invertible matrix iff GMn(S)such that AGGAIn, where In stands for identity matrix of order n.

Definition 3.10. Let (S ; +,.) be a path algebra and AMn(S). Then A is called a

permutation matrix if every element of A is either 0 or 1 and each row and each column contains exactly one 1.

Example 3.10(a).



























0

1

0

0

0

0

1

0

1

0

0

0

0

0

0

1

A

is a permutation matrix.

Remark 3.10(b). A permutation matrix AM_n(S) is clearly invertible over S and T

A is inverse of A.

From Example 3.10(a),



























0

1

0

0

0

0

1

0

1

0

0

0

0

0

0

1

A



1

0

0

0

0

0

1

0

0

0

0

1

0

1

0

0

T

A



























Clearly ATA AAT  I

Therefore ATis inverse of A.

Proposition 3.11. Let (S; +, .) be a path algebra andAMn(S). Then

(1) A is transitive iff

a

_ik

a

_kj



a

_ijfor all i, j, k{1,2,3,...,n}

(2) A is transitive iff PAPTis transitive for any nn permutation matrix P. Proof:

(1) Let A be transitive. Then A2  A

A

AA



49

) (

1

ij kj ik n

k

a a

a 







; where i, j,k{1,2,3,...,n}

)

(

)

)(

(

a

_ik

a

_kj



a

_ij



Conversely suppose

(

a

_ik

)(

a

_kj

)



(

a

_ij

)

Now AA = ik kj n

k

a

a



1



(

a

_ik

)(

a

_kj

)



(

a

_ij

)

= A

Hence A2  A. (2) Let A be transitive. Then A2  A

Now (PAPT)2 (PAPT)(PAPT)

(PA)(PTP)(APT)



(

PA

)

I

_n

(

AP

T

)



P

(

AI

_n

A

)

P

T P(AA)PT

PA2PT

Hence (PAPT)2 PAPT

Conversely

(PAPT)2 PAPT

Then (PAPT)(PAPT)PAPT

(PA)(PTP)(APT) PAPT



(

PA

)

I

_n

(

AP

T

)



PAP

T



P

(

AI

_n

A

)

P

T



PAP

T PA2PT PAPT

PTPA2PTPPTPAPTP

(PTP)A2(PTP)(PTP)A(PTP)



I

_n

A

2

I

_n



I

_n

AI

_n  A2  A

Hence A is transitive. 

4. Permanent and adjoint of matrix over path algebra

In this section, we discuss permanent and adjoint of matrices over path algebra. Some properties also established.

Definition 4.1. Let Let (S; +, .) be a path algebra and AMn(S). Then the permanent

50

_{1 (1) 2 (2) 3 (3)}... _{n (n)}

S a a a a perA n   



   

Where S_n denotes the symmetric group of the set

i



{

1

,

2

,

3

,

...,

n

}

. Remark 4.2. The partial relation



over M_n(S) is defined as:

A



B



a

_ij



b

_ij

;



i

,

j

A   B A B B.

Proposition 4.3. Let (S; +, .) be a commutative path algebra and A,BMn(S). Then ).

( ). ( )

(AB per A per B

per 

Proof: per (AB) = ( ... ( ))

1 ) 2 ( 2 1 2 ) 1 ( 1 1

1 n n

n n n S n b a b a b a n        

 

  









1 2 3 1 2 3

1 2

1 2 3 (1) ( 2) (3) ( )

, ,...

(

...

...

)

n n n n n n S

a a

_{ }

a

_

a

_

b

_{ }

b

_{ }

b

_{ }

b

_{ }

   







( 1 (1) 2 (2) 3 (3)... ( ) (1) (1) (2) (2) (3) (3)... (n) (n))

S n n S b b b b a a a a n n             



 



 

a_{1 (1)}a_{2 (2)}a_{3 (3)}...a_{n (n)} per(B)

Sn    



  

 per(A).per(B)

Hence per(AB) per(A).per(B). 

Definition 4.4. Let (S; +, .) be a commutative path algebra and AMn(S). Then A is

called idempotent iff A2  A.

Proposition 4.5. Let (S; +, .) be a commutative path algebra and AMn(S). If A is

idempotent with per(A)1,then per(A) is idempotent.

Proof : Since A is idempotent, so A2  A.

By Proposition 4.3 we get

per(AB) per(A).per(B). ....(i) Putting A for B in (i)

per

(

AA

)



per

(

A

).

per

(

A

)

 per(A2)(per(A))2

 per(A)(per(A))2

(per(A))2  per(A) ....(ii) We have

per(A)1

 per(A)per(A) per(A)

51

(per(A))2  per(A). 

Definition 4.6. Let (S; +, .) be a commutative path algebra andAMn(S). Then A is

said to be nilpotent matrix if kZ such that

A

k



0

.

Definition 4.7. Let (S; +, .) be a commutative path algebra and AMn(S). Then A

is called irreflexive element iff aii 0 ; where i{1,2,3,...,n}.

Proposition 4.8. Let (S; +, .) be a commutative path algebra andAMn(S).

(i) If

(

A

k

)

_ii



0

, for all i, k{1,2,3,...,n}, then A is nilpotent. (ii) If A is irreflexive and transitive, then A is nilpotent.

Proof : Trivial.

Definition 4.9. Let (S; +, .) be a commutative path algebra andAM_n(S);n2.

The matrix B is said to be adjoint matrix of matrix A if b_ij  A_ji ; 1i,j n, where ji

A

is matrix of order

n



1

formed by delating row j and column i from A. It is denoted by

adj

(

A

).

Proposition 4.10. Let (S; +, .) be a commutative path algebra andA,BMn(S) and A

B . Then (1) adjBadjA

(2) if A is nilpotent then B is nilpotent and h(B)h(A). Proof: (1)

Let A,BMn(S) and B A.

Then

(

perB i

( | j))

T_{n n}



(

perA i

( | j))

_{n n}T_



adjBadjA

(2) Let A be nilpotent.

So



k



Z

 such that

A

k



0

. Let

l



k

. Since B A, so Bl  Ak

 Bl 0



l



0

B

.

Hence B is nilpotent. Again

Bl  Ak

ij k ij l

A

B ) ( )

( 

 ; where i, j,k,l{1,2,3,...,n}

From this we get

nilpotent index of B



nilpotent index of A

) ( )

(B k A

l 

52

) ( )

(B h A

h 

 .  Proposition 4.11. Let (S; +, .) be a commutative path algebra andA,BMn(S). Then

(1) adj(A)adj(B)adj(AB)

(2) (adj(A))T adj(AT). Proof:

(1) We have A AB and B AB. By Proposition 4.10 (1), we get

adj(A)adj(AB) ….(i) and adj(B)adj(AB) …(ii) From (i) and (ii) we get

adj

(

A

)



adj

(

B

)



adj

(

A



B

)

.

(2) It is obvious. 

Definition 4.12. Let (S; +, .) be a commutative path algebra and AMn(S). Then A

is called symmetric iff

A

T



A

.

Definition 4.13. Let (S; +, .) be a commutative path algebra and A,B,CMn(S).

Then

A



B



C

iff ( )

1

kj ik n

k

ij a b

c 







for any i,j{1,2,3,...,n}.

Proposition 4.14. Let (S; +, .) be a path algebra andA,B,C,DMn(S). Then

(i) (BC)T CT BT

(ii) If AB then DADB and

A



C



B



C

. Proof : (i) ( )

1

kj ik n

k

c b C

B 







 ,

For i = 1,2, …..,m j = 1, 2, …..,l.

ik kj T n

k T

c b C

B ) ( ( ))

(

1







 

= ( )

1

ki jk n

k

b

c 





Now CT c_jk,

B

T



b

_ki ki jk T T

b c B

C   

= ( )

1

ki jk n

k

b

c 





53 (ii) Let

AB.

Then

a

_ij



b

_ij

;

for i = 1, 2,…..,m j = 1, 2, …..,n



A

D ( ) ( )

1 1

ij ti m

i ij ti m

i

b d a

d  







 

Therefore DADB. Again

A



C



( ) ( )

1 1

jk ij n

j jk ij n

j

c b c

a  







 

Therefore

A



C



B



C

.

∆ Definition 4.15. Let (S; +, .) be a commutative path algebra and AM_n(S). Then A

is called reflexive element iff a_ii 1 ; where i{1,2,3,...,n}.

Definition 4.16. Let (S; +, .) be a commutative path algebra and AM_n(S). Then A

is called weeklyreflexive element iff

a

_ii



a

_ij ; where i,j{1,2,3,...,n}.

Definition 4.17. Let (S; +, .) be a commutative path algebra and AMn(S). Then A

is called nearlyirreflexive element iff

a

_ii



a

_ij ; where i,j{1,2,3,...,n}. Proposition 4.18. Let (S; +, .) be a path algebra andAM_n(S). If A is nearly irreflexive and symmetric, then

(1) AA A

(2) AA is symmetric and nearly irreflexive. (3) A2 is weekly reflexive.

Proof : (1) Let

T  AA

Then





  n

k

kj ik

ij a a

t

1

)

( .…(i)



(

a

_ii



a

_ij

)



a

_ij

Hence AA A. (2) Now





  n

k

ki jk

ji a a

t

1

54



   n k ik kj a a 1 )

( [  A is Symmetric]



   n k kj ik a a 1 ) (



t

_ij Hence T is symmetric. Again









n k ki ik

ii

a

a

t

1

)

(



   n k ik ik a a 1 ) (



  n k ik a 1 ( ) 1 kj n k ik a a  







t

_ij

Hence AA is nearly irreflexive. (3) Let S  A2.

Then ki

n

k ik

ii a a

s



  1 ik n k ik

a

a







1 ik n k a



  1



  n k kj ika a 1



s

_ij

Hence A2 is weekly reflexive. ∆

Proposition 4.19. Let (S; +, .) be a path algebra andAM_n(S). If A is nearly

irreflexive and symmetric, then AAT is symmetric and nearly irreflexive. Proof : Let

S



A



A

T.

Then



   n k ik ik

ii a a

s 1 ) (



  n k ik a 1





  n k jk ik a a 1 )

55 Hence AAT is nearly irreflexive.

By Proposition 4.14, (AAT)T (AT)T AT  AAT. ∆

Proposition 4.20. Let (S; +, .) be a path algebra andAMn(S). If A is irreflexive and

transitive, then (1) A AT 0 (2) ATA0

Proof : Let S  AAT

Then





  n

k

jk ik

ij a a

s

1

)

(



(

a

_ij



a

_ji

)(

a

_ij



a

_jj

)



a

_ij

a

_ji



a

_ii



0

Hence AAT 0. The proof of (2) is similar.

Ackowledgements: The authors thank to the referees for their suggestions which have made the paper more readable.

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