On Matrices Over Semirings

Transcription

1 Aals of Pure ad Applied Mathematics Vol. 6, No. 1, 14, 1-1 ISSN: 79-87X (P, (olie Pulished o 16 April 14 Aals of O Matrices Over Semirigs K. R.Chowdhury 1, Aeda Sultaa, N.K.Mitra 3 ad A.F.M.Khodadad Kha 4 1 Departmet of Mathematics, Mohammadpur Model School ad College Mohammadpur, Dhaka, Bagladesh, krchowdhury1975@yahoo.com Departmet of Mathematics, Jahagiragar Uiversity, Savar, Bagladesh 3 Mathematical ad Physical Scieces, Bagladesh Uiversity of Busiess ad Techology, Dhaka, Bagladesh 4 School of Egieerig ad Computer Sciece, Idepedet Uiversity, Bagladesh, Bashudhara R/A, Dhaka, Bagladesh Received 7 Feruary 14; accepted 14 March 14 Astract. I this paper, matrices over semirigs are ivestigated. This is doe y itroducig some examples of semirigs ad presetig some results o regular ad ivertile matrix semirigs. These iclude coditios for regularity ad ivertiility of matrices over semirigs as geeralizatio of correspodig results o matrices over rigs. Examples ad results are illustrated y computig usig MATLAB. Keywords: Idempotet, Additively commutative semirig, Regular AMS Mathematics Suject Classificatio (1: 16Y6 1. Itroductio The otio of semirig was first itroduced y H.S. Vadiver i H.S. Vadiver itroduced a algeraic system, which cosists of o empty set S with two iary operatios additio ( ad multiplicatio (.. The system (S;,. satisfies oth distriutive laws ut does ot satisfy cacellatio law of additio. The system he costructed was rig like ut ot exactly a rig. Vadiver called this system a Semirig. The study of matrices over geeral semirigs has a log history. I 1964, Rutherford [3] gave a proof of Cayley Hamilto theorem for a commutative semirig avoidig the use of determiats. Sice the, a umer of works o theory of matrices over semirigs were pulished [1, 1]. I 1999, J S Gola descried semirigs ad matrices over semirigs i his work [5] comprehesively. The techiques of matrices have importat applicatios i optimizatio theory, models of discrete evet etwork ad graph theory. Luce [1] characterized the ivertile matrices over a Boolea algera of at least two elemets. Rutherford [] has itroduced that a square matrix over a Boolea algera of elemets is ivertile. Additively iverse semirigs are studied y Karvellas [1]. Kaplasky [4], Petrich [9], Goodearl [6], Reuteauer [1], Fag [8] have studied semirig.. Prelimiaries I this sectio, we preset some defiitios ad examples of semirig. MATLAB fuctio scripts are ot preseted i this paper ut iputs ad outputs from the computer are give. 1

2 K.R.Chowdhury, Aeda Sultaa, N.K.Mitra ad A.F.M. Khodadad Kha Defiitio.1. Let S e o empty set with two iary operatios ad.. The the algeraic structure (S;,. is called a semirig iff a,, c S ; (i (S; is a semigroup (ii (S;. is a semigroup (iii a. (c = a. a.c ad (c.a =.a c.a. Example.1(a. (L = {5, 1,, 5, 4, 5, 1, };,,. is a semirig, where a = lcm{a, }, a. = gcd{a, }. The MATLAB fuctio scripts are ot show. Outputs are preseted elow. >> A=[ ]; A = >> joi(a as = >> meet(a as = Defiitio.. Let (S;,. e a semirig. The S is called (i additively commutative iff x, y S, x y = y x.

3 O Matrices Over Semirigs (ii multiplicatively commutative iff x, y S, x. y = y. x. (S;,. is called a commutative semirig iff oth (i ad (ii hold. Example. (a. Every ouded distriutive lattice is a commutative semirig uder joi ad meet. Defiitio.3. Let (S;,. e a semirig. The a elemet of S iff x S, x = x = x ad x. = =. x. S is called zero Example.3 (a. Cosider the set of positive itegers Z with the operatios a = lcm{a, } ad a. = a. The ( Z ;,. is a semirig with zero elemet 1, ut 1 is ot zero, sice 1.a = a.1 = a 1 for ay a Z ad a 1. Defiitio.4. Let (S;,. e a semirig. The a elemet idetity of S iff x S, x. 1 = x = 1. x. 1 S is called Example.4 (a. Let X φ ad P(X is power set of X. ad. are defied y A B = A B ad A.B = A B; A, B P( X. The (P(X;,. is a semirig, where φ ad X are zero ad idetity of P(X respectively. Defiitio.5. Let (S;,. e a commutative semirig with zero ( ad idetity (1. The (S;,. is called idempotet semirig iff x S, x x = x = xx. Example.5 (a. (I =[,1];,. is a a idempotet semirig, where order i [,1] is usual ad ad. are defied as follows: a = max{a, }, a. = mi{a, }. Propositio.6. Let (S;,. e a idempotet semirig with zero ( ad idetity (1. The (a x, y S, x y = x = = y ( x, y S, xy = 1 x = 1 = y. Proof: (a By the defiitio of idempotet semirig x S, x x = x = x. Let x y =... (i Puttig x for y i (i x x = x = Puttig y for x i (i Hece x = = y y y = y = 3

4 K.R.Chowdhury, Aeda Sultaa, N.K.Mitra ad A.F.M. Khodadad Kha ( Let xy = 1. (ii Puttig x for y i (ii xx=1 x =1 x = 1 Puttig y for x i (ii yy =1 y = 1 y = 1 Hece x = 1 = y 3. Matrices over semirigs Throughout this sectio (S;,. is a additively commutative semirig with zero ( ad idetity (1 (1. is positive iteger ad M is set of all matrices over S. Propositio 3.1. For ay semirig S, ( M ;,. is a semirig. Further if S is additively commutative the M is additive commutative. If S has zero the has zero. If S has zero as well as idetity the M has idetity. I M, ad. are defied y [ a ij ] [ ij ] = [ aij ij ] ad [ aij ][ ij ] = aik. kj. Proof: We kow 4 k = 1 A, B M ( S A B M. Agai is associative o the set of matrices, so for all A, B, C M ( S, A (B C = (A B C. Therefore ( M ; is semigroup. Similarly A, B M ( S AB M. Agai. is associative o the set of matrices, so for all A, B, C M ( S, A. (B.C = (A.B.C. Therefore ( M ;. is semigroup. Moreover for all A, B, C M ( S, A.(B C = A.B A.C ad (A B.C = A.C B.C. Therefore ( M ;,. is a semirig. Agai is commutative o the set of matrices i.e. AB = BA. Sice,1 S, so M ( Sad I M ( S. We have A = A= A ad AI = A = IA. M

5 O Matrices Over Semirigs Hece ( M ;,. is additively commutative semirig with zero ad idetity. Example 3.1(a. (B = {,1},,. is a commutative semirig, where ad. are defied as i [7]. The M (B, set of all matrices over B is a additively commutative semirig. Here M ( B ==. For example M ( B has = 16 elemets: >> MatList 1 O =, C =, D =, E =, F =, 1 1 G =, H =, P =, Q =, I =, S =, T =, U =, V =, W =, X = geerated y MATLAB fuctio script. The additio tale also geerated y MATLAB fuctio scripts is show elow: >> MatSum MatSum = Here zero, O = ad idetity, I =. 5

6 K.R.Chowdhury, Aeda Sultaa, N.K.Mitra ad A.F.M. Khodadad Kha Propositio 3.. M ( Z, M ( Q, M ( R are additively commutative semirig with zero uder usual additio ad multiplicatio.these are ot multiplicatively commutative. Also, oe of them are rigs. Proof : A, B, C M ( Z Clearly M ( Z is a additively commutative semirig with zero. Let us show y a example M ( Z is ot multiplicatively commutative: Let 3 1 A = ad B = Now AB = = ad BA = = Therefore AB BA Hece M ( Z is ot multiplicatively commutative. Last Part: We have 1 A = M ( Z The 1 A = M ( Z Such that A (-A =. Hece M ( Z is ot a rig. The case for M ( Q ad M ( R are similar. Defiitio 3.3. Let (S;,. e a semirig. The S is called a regular semirig iff x S, y S such that xyx = x. Propositio 3.4. Let (S;,. e a additively commutative semirig with zero ( ad a positive iteger. If M is a regular semirig, the so is S. Proof: Let M e a regular semirig. The A M ( S, B M ( S such that ABA = A. 6

7 Defie A M ( S y O Matrices Over Semirigs, i, j N with ( i, j (1,1 A( i, j = a, for ( i, j = (1,1 a The A = For this A, let B(1,1 B(1, B(1,3... B(1, B(,1 B(, B(,3... B(, B = B(3,1 B(3, B(3,3... B(3,... B(,1 B(, B(,3... B(, e such that ABA=A. a. B(1, Now AB = a. B(1,1. a ABA = Now ABA = A a. B(1,1. a = a. Remark 3.5. The coverse of the aove Propositio 3.4 is ot ecessarily true for =. Let us show it y a example: (S = {1,, 3, 6};,,. is a regular semirig, where a = lcm{a, }, a. = gcd{a, }; a, S. But M ( S is ot regular semirig. Let A = ( 3 6 M S 7

8 K.R.Chowdhury, Aeda Sultaa, N.K.Mitra ad A.F.M. Khodadad Kha 1 For this A, let B = such that ABA=A Now ABA = = (1. =.( ( ( (1..( ( ( (1. =.( ( ( (1..( (1. 1 ( , so A=ABA follows that 1.( (1. 1. = 1. (i.( ( = 6....(ii From (i we get 3.(. = 1 = 1 From (ii, clearly.( From (ii we get ( = 6 But =1 6 = = = , which is a cotradictio. So M ( is ot regular. S Example 3.7. (B = {,1};,. is a commutative semirig with zero, where ad. are defied as i [7]. The M (, set of all matrices over B is regular semirig. B Remark 3.8. (B = {,1};,. is a commutative semirig, where ad. are defied as i [7]. The M (, set of all matrices over B is ot idempotet semirig. B From Example 3.1(a, the Matlist geerated y MATLAB fuctio script. The multiplicatio tale also geerated y MATLAB fuctio scripts is show elow: 8

9 >> Matmult O Matrices Over Semirigs Matmult = From Matmult Tale we see that D = D, E = E, S = I S, T = X T, W = X W. Ackowledgemet. The authors thak to the referees for their suggestios which have made the paper more readale. REFERENCES 1. C. Reuteauer ad H. Strauig, Iversio of matrices over a commutative semirig, J. Algera, 88 ( D.E. Rutherford, Iverses of Boolea matrices, Proc Glasgow Math Asssoc., 6 ( D.E. Rutherford, The Cayley Hamilto theorem for semirig, Proc. Roy. Soc. Ediurgh. Soc., A66 ( I. Kaplasky, Fields ad Rigs, The Uiversity of Chicago Press, Chicago, J. S. Gola, Semirigs ad their Applicatios, Kluwer Academic Pulishers, K.R. Goodearl, Vo Neuma Regular Rigs, Pitma, Lodo, K. Ray Chowdhury, A. Sultaa, N.K. Mitra ad A.F.M. Khodadad Kha, Some structural properties of semirigs, Aals of Pure ad Applied Mathematics, 5( ( Li Fag, Regularity of semigroup rigs, Semigroup Forum, 53 ( M. Petrich, Itroductio to Semirig, Charles E Merrill Pulishig Compay, Ohio,

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