- Email: [email protected]
Matrices over semirings
NORrH-~
Matrices over Semirings SHAMIK GHOSH
Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Calcutta 700 019, India
Communicated by John Mordeson
ABSTRACT Introducing the concept of semi-invertibility of square matrices over semirings, some sufficient conditions for semi-invertibility of square matrices over various kinds of semirings are obtained. Also a necessary and sufficient condition for semi-invertibility of square matrices over semirings is obtained.
0.
INTRODUCTION
A semiring is an algebraic system (S, + , .) in which (S, + ) is an abelian monoid with identity element 0 and (S,-) is another semigroup, connected by ring-like distributivity. Also, Or = r0 = 0 for all r ~ S. An idealI of a semiring S is a subsemiring of S such that IS, SIc_I. A k-ideal K of S is an ideal of S in which x + y = z , x ~ S , y , z ~ K implies x~K. In this paper, all semirings are commutative (i.e., ab = ba for all a, b ~ S ), except the semirings of matrices over commutative sembings with multiplicatire identity, and every semiring contains the multiplicative identity 1, except the subsemirings and the ideals of sembings. To avoid the trivial case S = {0}, we also assume that 1 ~ O. Many aspects of the theory of matrices and determinants over semirings have been studied by Beasley and Pullman [1, 2], Cunninghame-Green [4], Zimmermann [10], and others. In the first section of this paper, we study some properties of determinants of square matrices over semirings. In the second section, we introduce the concept of semi-invertibility of square matrices over semirings and obtain some sufficient conditions for the semi-invertibility of square matriINFORMATION SCIENCES 90, 221-230 (1996) © Elsevier Science Inc. 1996 655 Avenue of the Americas, New York, NY 10010
0020-0255/96/$15.00 SSDI 0020-0255(95)00283-9
222
S. GHOSH
ces over various kinds of semirings which are generalizations of a field. In that section, we also obtain a necessary and sufficient condition for semi-invertibility of square matrices over semirings. The section ends with an application in solving a system of simultaneous linear equations over a semiring which cannot be embedded in a ring. 1.
PRELIMINARIES
Let R be a semiring and Mn(R) be the semiring of n × n square matrices over R. Let A = (aij)~M,(R). Then the positive determinant IA[ + and the negative determinant [A I- of A are defined as follows. Let S, and A , be the symmetric group and the alternating group, respectively, on {1, 2 .... , n}. Then ([5]) [A[ +=
~_, al,~o)a2,,(2)...an~,(n) o'EA n
and IA[ =
~
al,r(1)a2cr(2)...an,r(n).
ere Sn\A n
The pair (IAI+,[A[ -) is called the bideterminant of A [5]. For each i , j ~ { 1 , 2 .... ,n}, let Agj be the matrix in Mn_I(R) obtained by deleting the ith row and the jth column of A. ]Aijl + and IA~j[- are called the positive minor and negative minor, respectively [5]. It is known [5] that [AI += ~ a i i l A i j [ (+' and i~1
IA[ - = ~ a i j [ A i j l c-',
(I)
i=1
where [A/j[ (+) []Zi2[ (-)] means IAij[+ [[Aiil-] when i+j is even, and it means IAijl- [IAijl ~-] otherwise. Following the proof described in [5], one can easily show that IZ[ +=
~ a~jlA~jl (÷) j=l
and
[AI - = ~ aijlAijl (-).
(II)
j=l
The following properties of determinants are well known [5]: (i) Let A, B ~Mn(R) be such that B is formed by multiplying all the entries of one row or one column of A by an element r~R, then (IBI+,IBI -) = (rlZl +, rl AI-).
M A T R I C E S O V E R SEMIRINGS
223
(ii) If one column [resp. row] of A ~M~(R) is a linear combination of other columns [resp. rows] of A, then IAI + = IAI-. We denote the transpose of a matrix A ~Mn(R) by A r. LEMMA 1.1. LetA ~M,(R), then (IArl +, IATI-) = (IAI +, IAI-).
Proof. Let 0 = (i 1,i 2. . . . . i~) ~ S,. We consider the product ai,lai22...ai.~. We rearrange the product in the ascending order of i 1, i 2.... , i, and obtain aljlazj2.., a~jo (say). Then clearly (Jl, J2..... j,) = O- 1. Now O~A, if and only if 0 -1 ~ A , . The above arguments show that each term in the summation of fA[ + [resp. IA[-] is also present in the summation of IAr[ + [resp. IArl -] and vice-versa. [] LEMMA 1.2. Let A =(aii)~M~(R), then for re:t,
• ark]Atkl (+)= ~ k=l
ark]A,kt~-).
k=l
Proof. Let C be the matrix obtained from A by replacing the tth row by its rth row. Since the matrix C has two rows in common, we have IC[+=lCI -. Now
ICl+= ~
C,klC, kl ~+~ and
ICI = ~
k=l
c,klC,~l ~ ~.
k=l
Also, it is clear that Ark =Ctk for all k = 1,2 ..... n. This, with ark =Ctk for all k = 1, 2 ..... n, completes the proof. [] 2.
INVERSE OF A SQUARE MATRIX
Let R be a semiring, and A ~Mn(R) is said to be Owertible if AB = I = BA for some B~Mn(R). For any two matrices A , B ~ M , ( R ) , we have IABI+=IAI+[BI++[AI [ B l - + r and IAB[-=IAI+IBI +lhl-lnl++r for some r ~ R [5]. PROPOSITION 2.1. Let A ~ M, ( R ) be an invertible matrix. Then IAI + 4: IAI-.
Proof. By hypothesis, A B = I for some B~M,(R). Let IAI+=IAI -. lAB[ +=IABI-=IAI+(IB[ + + l n l - ) + r . But IABI + = [ I I + = 1 and IABI- = I I [ - = 0, which is a contradiction. Thus IAI + 4: [A]-. []
Then
224
S. G H O S H
We know that, in the case of a field, a square matrix is invertible iff its determinant is nonzero. Proposition 2.1 is only a necessary condition for invertibility of square matrices over semirings (see Example 2.8). Now we consider various special kinds of semirings which arise as generalizations of fields. We know that the commutative rings with identity which are free of ideals (i.e., the only ideals are trivial) are the fields. A semiring S is called a semifield [5] if, for any a ¢ 0 in S, there is b ~ S such that a b - - 1 . Semifields are the ideal-free semirings. But it is well known that k-ideals are precisely the kernels of semiring homomorphisms, and hence in the case of semirings, k-ideals play more important roles than ideals (See, for example, [3] and [7]). A semiring S is said to satisfy condition (C) if, for each a ~ 0 in S, there exist r, s ~ S such that 1 + ra = sa ([8, 5]) (in [5], Golan used the term austere semiring). The above class of semirings are exactly the class of semirings which are free of k-ideals [5], i.e., a semiring S satisfies condition (C) iff every homomorphism from S to another semiring S' has trivial kernel. Clearly, semifields satisfy condition (C), but the converse is not true (see Example 2.3). DEFINITION 2.2. A semiring S is said to satisfy condition (D) if, for each pair a :~ b in S, there exist r, s ~ S such that 1 + ra + sb = sa + rb. Obviously, the semiring satisfying condition (D) also satisfies condition (C), and here also the converse is not true, as the following example shows. E X A M P L E 2.3. Let S = Q + x {1} u @ + x {2}, where Q+ is the set of positive rationals. We define (a, i ) + (b, j ) = (a + b, max(i, j)) and (a, i ) . ( b , j ) = (ab, max(i, j ) ) f o r all (a, i), (b, j ) ~ S. Let R = S U {0}, where 0 is an element outside S satisfying O + ( a , i ) = ( a , i ) and 0 . ( a , i ) = 0 for all ( a , i ) ~ S . Also, 0 + 0 = 0.0 = 0. Then (R, + , .) is a semiring with additive identity 0 and multiplicative identity (1,1). R is not a semifield and does not satisfy condition (D); but R satisfies condition (C). We know that a semiring S can be embedded in a ring iff S is additicely cancellatiue (i.e., a + c = b + c implies a = b for any a, b, c ~ S) [9]. The smallest ring (unique up to isomorphisms over S) in which S can be embedded is called the difference ring [9] of S and is denoted by S. S consists of all differences a - b for a, b ~ S, where elements of the image of S under the isomorphism of embedding are identified with the elements of S. PROPOSITION 2.4. Let S be an additively cancellative semiring. Then S is a field iff S satisfies condition ( D ).
MATRICES OVER SEMIRINGS
225
Proof. Let S be a field and a 4=b in S. Then 0 # a - b ~ S, and so there is 2 ~ S such that $ ( a - b ) = l . But Y c = s - r for some r , s ~ S . Thus 1 = ( s - r)(a - b), which implies 1 + ra + sb = sa + rb, as required. Conversely, let S satisfy condition (D) and ~ 4:0 in S. Then ~ = a - b for some a , b ~ S such that a4=b. Thus l + r a + s b = s a + r b for some r , s ~ S . This implies ( s - r ) ( a - b ) = l , i.e., 2 ~ = 1 , where f f = s - r ~ S . Hence S is a field. [] Z s = { x ~ S : x + z = z for some z ~ S } S. S is called zeroic if S = Z s [5].
is called the zeroid of a semiring
PROPOSITION 2.5. Let S be a semiring satisfying condition (D). Then S is either additively cancellative or zeroic. Proof. Let S -4=Z s and a, b, c ~ S such that a + c = b + c. If a # b, then there are r, s ~ S such that 1 + ra + sb = sa + rb. Adding (r + s)c with both sides, we get l + ( r + s ) ( b + c ) = ( r + s ) ( b + c ) . This implies I ~ Z s and hence S = Z s, which is a contradiction. Thus a = b and S is additively cancellative. [] PROPOSITION 2.6. Let S be a zeroic semiring. Then S is free o f k-ideals iff S satisfies condition ( D ). Proof. In view of the above discussions, we have only to show that if S is k-ideal-free, then S satisfies condition (D). Let S be free of k-ideals and ~,/3 ~ S such that a 4:/3. Let I = { x E S : x + r a + s / 3 = s o ~ + r / 3 , for some r , s ~ S } . It is easy to show that I is a k-ideal of S. Then either I={0}, the zero-ideal, or I = S. If we can show that I contains a nonzero element, then we are done. Since S = Z s, there is z ~ S such that 1 + z =z. Then oi + z ~ + z / 3 = ( 1 + z)ol + z / 3 = z o l + z/3. Similarly, /3 + z a + z / 3 = z a + z/3. Thus a,/3 e l . Since ol 4:/3, one of them must be nonzero. Hence the proof. [] We may summarize the above discussions in the following diagrams: , condition (C) (k-ideal-free) '
semifield condition (D)
'either
condition(D) condition (D)
&
&
zeroic
zeroic~
or
condition (D) additively cancellative
,condition(C)
additively cancellative,
&
zeroic
, difference ring is a field
E X A M P L E 2.7. (i) S = ( Q ~ ' , m a x , -) is a zeroic semifield of nonnegative rationals satisfying condition (D).
226
S. G H O S H
( i i ) S = Q0~(v~-) = {a + bf2-: a, b ~ Q~-} with usual multiplication and addition is a semiring that satisfies condition (D); but S is not a semifield. Also, Z s = {0} 4: S and S is the field Q(v/2-), where Q is the field of rational numbers. (iii) Let S = Q + × Q+ U{(0,0)}. Then S is a semifield with pointwise usual addition and multiplication; but S does not satisfy condition (D). Also, note that S = Q × Q, which is not a field.
Now turning back to the question of invertibility of a square matrix
A ~M,(R) over a semiring R, we first note that the condition IAl+4: IAIis not, in general, sufficient even when R is a semifield or R satisfies condition (D) or condition (C). E X A M P L E 2.8. Let R be a semiring as described in Example 2.7(i). Let /
A = {l3
]}~M2(R) .Then
AI+=44:6=IAI
. B u t there is no such B ~
\
M2(R) satisfying A B = I 2 =
1
o
01. 1]
In [2], Beasley and Pullman have shown that invertible matrices over a
zero-sum-free (i.e., a + b = 0 implies a = 0 = b [5]) and zero-divisor-free (i.e., ab = 0 implies either a = 0 or b = 0; also called entire [5]) semiring are very rare (see [2, theorem 3.1]). Also, we see that semifields and semirings satisfying condition (D) or condition (C), which are not rings, are zerosum-free and zero-divisor-free. Thus we introduce the concept of semiinvertibility. DEFINITION 2.9. A ~Mn(R) is said to be semi-invertible if there exist A1,Az~Mn(R) such that I + A A 1 =AA 2 and I + A I A = A z A . PROPOSITION 2.10. Let R be a semiring satisfying condition (D) and A ~M,(R). If IAI + 4: IZl-, then A is semi-invertible.
Proof. Let B = A r, ~,j=lBijl (+) and ~ij=lBijl ~ ). Let A+=(aij) and A - = (/3i). Now since IAI + ~ IAI-, there exist r, s ~ R such that 1 + rlA] + + siAl =slAl++rlAI -. Let Al=rA++sA and A2=sA++rA -. If A I = (ci), then AIA = ( f / ) , where n
fij= ~ ci~akj = Y'~ akjcik k=l
k~l
= ~ ak~(rlBi,[ '+' + slBi~(-') k=l
=r ~_~bjklBi~l(+'+s ~ b]klBikl'-', k=l
k=l
227
MATRICES OVER SEMIRINGS where B = (bij). Then by (II), L e m m a 1.2, and L e m m a 1.1, we get
l =
tl bjklBikl ~+~, ( r + s ) ~_~
i4=j,
rlBl+ +slBl-=r]Al+ +slAI ,
i=j.
k=l
Similarly, A2A = (gij), where
I
( r + s ) k~=lbJk[Bik[(+),
gij =
( slAl+ + rlAI ,
i4:j, i=j.
Thus l+fii=gii for all i = 1 , 2 , . . . , n and fij=gij for all i--gj. Therefore, I + A 1 A =A2A. Similarly, one can show that I + A A 1 = A A 2. [] The following example shows that the condition [A[ + ~ [ A [ necessary for a matrix A to be semi-invertible.
is not
E X A M P L E 2.11. Let R be a semiring as described in Example 2.7(i) and /
A=[
\
\
1
2
24 ~ M 2 ( R ) . Then [ A ] + = 4 = I A [ - ; but A is semi-invertible with l
PROPOSZTION 2.12. Let R be a semiring satisfying condition (C) [or a semifieldl and A ~Mn(R). If only one of the elements IAf + or IAI- is nonzero, then A is semi-invertible. The proof, being similar to that of Proposition 2.10, is omitted. A semiring S is called the additively inversive semiring [6] if for each a ~ S , there exists a unique element a ' ~ S such that a + a ' + a = a and a ' + a + a ' = a ' . It is well known that for any a,b e S , (a +b)' =a' +b' and (ab)' =a'b=ab' [6]. It may be noted that in a semifield S for an element a ~ S, there exists b ~ S such that a + b = 0 iff S is a ring. PROPOSmON 2.13. Let R be an additively inversive semifield which is not a ring andA ~Mn(R). If ( [ A I + , ] A I ) 4 : ( 0 , 0 ) , then there existA1,C~Mn(R) such that A ~A + C = I + C and AA ~+ C = I + C.
228
S. G H O S H
Proof. By the given condition, d=IAI+ +(IAI-) ' 4=0. Let B = A r and A 1 = ( a i ) , where aii =d-l[lBijl(+)+(]Bij[(-))']. Then A1A =(fij), where
fii = ~_~ Otikak) k=l
=d-l[k~lb]klBikl(+) + (k~lbjklBik[( ))']" Thus, by (II), L e m m a 1.2, and L e m m a 1.1, fii=d-~(lBl++(lBl-)')= d ~(IAI++(IAI ) ' ) = d - l d = l , and if i ~ j , fo=rij+r;j, where rij= d I E~=lbjk[Bikl n (+) . Let C 1=(c/j), where
cij=
rij, O,
i :l=j, i=j.
Thus fo + ciJ = cij for all i C j and fii + % = 1 = 1 + % for all i. Therefore, A ~ A + C 1 = I + C l. Similarly, one can show that A A j + C 2 = I + C 2 for some C2~Mn(R). Then A ~ A + C = I + C and A A ~ + C = I + C , where C = C l +C 2. [] Now we obtain a necessary and sufficient condition for semi-invertibility of square matrices over a semiring. Let R be a semiring. A (right) semimodule [(/eft) sernimodule] over R is defined in a natural way [5]. Let R n be the set of all matrices of order 1 × n over R. Then R ~ is a (left) semimodule over R. Let A ~ M , ( R ) and R"A ={rA: r ~ R n} (the product rA is the usual multiplication of matrices of order 1 × n and n x n). Then we see that RnA is a (left) subsemimodule of the (left) R-semimodule R n. Similarly, if we consider R , as the set of all matrices of order n × 1 over R, then R , is a (right) semimodule over R and ARn = {At: r ~ R , } is a (right) subsemimodule of the (right) R-semimodule R n. Let M be a (right) R-semimodule [(left) R-semimodule]. A k-subsemimodule N of M is a subsemimodule of M in which a + b = c , a E M, b, c ~ N implies a ~ N. As in the case of k-ideals, here also k-subsemimodules are precisely the kernels of R-semimodule homomorphisms. Let T be a subsemimodule of M. The least k-subsemimodule containing T is called
229
MATRICES OVER SEMIRINGS the k-closure of T and is d e n o t e d by (T) k. Then
(T)k={a~M:a+b=c
forsomeb,c~T}.
It is well known that if R is a commutative ring with identity and A ~ M n ( R ) , then A is invertible iff A R n =Rn and R"A = R n. The following t h e o r e m is analogous to this result in the case of semirings. THEOREM 2.14. Let R be a semiring and A c M n ( R ) . Then A is semiinvertible iff ( A R n ) k = R , and (R~A)k = R ~.
Proof. Let A be a semi-invertible matrix. Clearly, ( A R ~ ) k C R ~. Let r ~ R , . Since A is semi-invertible, there are A 1 , A 2 ~ M , ( R ) such that I + A . 4 1 = A A 2. This implies r + A ( A I r ) = A ( A 2 r ) . Thus r ~ ( A R , , ) k. Similarly, one can show that (R"A)k = R ~. Conversely, let ( A R n ) k =R~. Let e i be the ith column of the identity t t! matrix I ~ M , ( R ) . N o w the above relation shows that there exist ei, e~ R , such that e i +Ae'i = A e ' . Let A 1 = (e]e~... e',) and A 2 = (e'~e~ ... e~). T h e n I + A A 1 = A A 2. Similarly, ( R " A ) k = R " gives I + A ' I A =A'2A. Simple calculations show that / + A A 3 =AM 4 and I + A 3 A = A 4 A , where A t = A j + A ~ +A'~A.41 and A 4 = A 2 + A " +A!lMA1, as required. [] PROPOSITION 2.15. Let R be a semiring and A E M , ( R ) be a semi-invertible matrix. I f rl, r 2 ~ R~ be such that A r l = A r 2, then r 1 + t = r 2 + t for some t ~ R~. Moreover, i f R is additively cancellatic, e, then A r 1 = A t 2 implies rl = r~ and A R , [ R~A] is isomorphic to R , [R "] as (right) R-semimodules [(left) R-semimodules].
Proof. W e have I + A I A = A 2 A , A1,A2~M,,(R). Then ri+Al(Ari) = Az(Ar/), i = 1,2. Thus r 1 + ( A 1 + A 2 ) ( A r j ) = r 2 + ( A I + A z ) ( A r l ) . Moreover, if R is additively cancellative, so is Rn and then r 1 = r 2. Also, R~ is then isomorphic to A R , as (right) R-semimodules through an isomorphism qt : Rn ~ AR~ defined by ",It(r) = A r for all r ~ R,,. [] A PPLICA TION PROBLEM 2.16. Let R be a semiring as described in Example 2.7(i). Solve: x + 2 y + z = 9 , 7x + 4 y + 3 z = 3 5 , 3x + 5 y + 2 z = 2 0 , where x , y , z ~ R .
Solution. Let A =
7 \3
4
3 , X=
y , and B =
5
2
z
~5 . T h e n by the \2;]
given system of equations, we have A X = B . Since IAI + = 35 ~ 28 = [ A I , by Proposition 2.10, A is semi-invertible, so that I + A ~ A = A 2 A , where
230
S. G H O S H
A 1 = r A + W s A -, A 2 = s A + W r A -, and r , s ~ R 1+35r+28s W e have X + A 1 A X = A 2 A X
such that
=28r+35s.
(1)
, which implies X+A1B=A2B.
Here A +=
9 2 35
, A-=
6
14 3 12
5
3 , A+B = 14
(2)
14(t/, a n d A - B 315]
=
126 / . 280]
(4,0)(i)
Now (1) is satisfied by r = 1 / 3 1 5 , s = 1 / 3 5 . P u t t i n g these values of r and s in (2), we get
+
3.6 =
, which implies x = 5, y = 4, z = 9.
8"0
The author is grateful to Professor T. K~ Mukherjee and Dr. M. K. Sen for their kind help and guidance in this work. The author also expresses his deep gratitude to the learned referee for his valuable suggestions. REFERENCES 1. Le Roy B. Beasley and N. J. Pullman, Operators that preserve semiring matrix functions, Lin. Alg. Appl. 99:199-216 (1988). 2. Le Roy B. Beasley and N. J. Pullman, Linear operators strongly preserving idempotent matrices over semirings. Lin. Alg. Appl. 160:217-229 (1992). 3. S. Bourne, The Jacobson radical of a semiring, Proc. Nat. Acad. Sci. U.S.A. 37:163-170 (1951). 4. R. A. Cunninghame-Green, The characteristic maxpolynomial of a matrix, J. Math. Anal. Appl. 95:110-116 (1983). 5. J. S. Golan, The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, Longman Scientific and Technical, U.K., 1992. 6. P. H. Karvellas, Inversive semirings, J. Austral. Math. Soc. 18(3):277-288 (1974). 7. D. R. La Torte, On h-ideals and k-ideals in hemirings. Publ. Math Debrecen 12:219-226 (1965). 8. M. K. Sen and M. R. Adhikari, On maximal k-ideals of semirings, Proc. Amer. Math. Soc. 118(3):699-703 (1993). 9. H. J. Weinert, /tiber halbringe und halbk6rper II, Acta. Math. Acad. Sci. Hungar. 14:209-227 (1963). 10. U. Zirnmermann, Linear and Combinatorial Optimization in Ordered Algebraic Structures, North Holland, Amsterdam, 1981. Received 1 January 1995; revised 24 August 1995
Matrices over Semirings SHAMIK GHOSH
Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Calcutta 700 019, India
Communicated by John Mordeson
ABSTRACT Introducing the concept of semi-invertibility of square matrices over semirings, some sufficient conditions for semi-invertibility of square matrices over various kinds of semirings are obtained. Also a necessary and sufficient condition for semi-invertibility of square matrices over semirings is obtained.
0.
INTRODUCTION
A semiring is an algebraic system (S, + , .) in which (S, + ) is an abelian monoid with identity element 0 and (S,-) is another semigroup, connected by ring-like distributivity. Also, Or = r0 = 0 for all r ~ S. An idealI of a semiring S is a subsemiring of S such that IS, SIc_I. A k-ideal K of S is an ideal of S in which x + y = z , x ~ S , y , z ~ K implies x~K. In this paper, all semirings are commutative (i.e., ab = ba for all a, b ~ S ), except the semirings of matrices over commutative sembings with multiplicatire identity, and every semiring contains the multiplicative identity 1, except the subsemirings and the ideals of sembings. To avoid the trivial case S = {0}, we also assume that 1 ~ O. Many aspects of the theory of matrices and determinants over semirings have been studied by Beasley and Pullman [1, 2], Cunninghame-Green [4], Zimmermann [10], and others. In the first section of this paper, we study some properties of determinants of square matrices over semirings. In the second section, we introduce the concept of semi-invertibility of square matrices over semirings and obtain some sufficient conditions for the semi-invertibility of square matriINFORMATION SCIENCES 90, 221-230 (1996) © Elsevier Science Inc. 1996 655 Avenue of the Americas, New York, NY 10010
0020-0255/96/$15.00 SSDI 0020-0255(95)00283-9
222
S. GHOSH
ces over various kinds of semirings which are generalizations of a field. In that section, we also obtain a necessary and sufficient condition for semi-invertibility of square matrices over semirings. The section ends with an application in solving a system of simultaneous linear equations over a semiring which cannot be embedded in a ring. 1.
PRELIMINARIES
Let R be a semiring and Mn(R) be the semiring of n × n square matrices over R. Let A = (aij)~M,(R). Then the positive determinant IA[ + and the negative determinant [A I- of A are defined as follows. Let S, and A , be the symmetric group and the alternating group, respectively, on {1, 2 .... , n}. Then ([5]) [A[ +=
~_, al,~o)a2,,(2)...an~,(n) o'EA n
and IA[ =
~
al,r(1)a2cr(2)...an,r(n).
ere Sn\A n
The pair (IAI+,[A[ -) is called the bideterminant of A [5]. For each i , j ~ { 1 , 2 .... ,n}, let Agj be the matrix in Mn_I(R) obtained by deleting the ith row and the jth column of A. ]Aijl + and IA~j[- are called the positive minor and negative minor, respectively [5]. It is known [5] that [AI += ~ a i i l A i j [ (+' and i~1
IA[ - = ~ a i j [ A i j l c-',
(I)
i=1
where [A/j[ (+) []Zi2[ (-)] means IAij[+ [[Aiil-] when i+j is even, and it means IAijl- [IAijl ~-] otherwise. Following the proof described in [5], one can easily show that IZ[ +=
~ a~jlA~jl (÷) j=l
and
[AI - = ~ aijlAijl (-).
(II)
j=l
The following properties of determinants are well known [5]: (i) Let A, B ~Mn(R) be such that B is formed by multiplying all the entries of one row or one column of A by an element r~R, then (IBI+,IBI -) = (rlZl +, rl AI-).
M A T R I C E S O V E R SEMIRINGS
223
(ii) If one column [resp. row] of A ~M~(R) is a linear combination of other columns [resp. rows] of A, then IAI + = IAI-. We denote the transpose of a matrix A ~Mn(R) by A r. LEMMA 1.1. LetA ~M,(R), then (IArl +, IATI-) = (IAI +, IAI-).
Proof. Let 0 = (i 1,i 2. . . . . i~) ~ S,. We consider the product ai,lai22...ai.~. We rearrange the product in the ascending order of i 1, i 2.... , i, and obtain aljlazj2.., a~jo (say). Then clearly (Jl, J2..... j,) = O- 1. Now O~A, if and only if 0 -1 ~ A , . The above arguments show that each term in the summation of fA[ + [resp. IA[-] is also present in the summation of IAr[ + [resp. IArl -] and vice-versa. [] LEMMA 1.2. Let A =(aii)~M~(R), then for re:t,
• ark]Atkl (+)= ~ k=l
ark]A,kt~-).
k=l
Proof. Let C be the matrix obtained from A by replacing the tth row by its rth row. Since the matrix C has two rows in common, we have IC[+=lCI -. Now
ICl+= ~
C,klC, kl ~+~ and
ICI = ~
k=l
c,klC,~l ~ ~.
k=l
Also, it is clear that Ark =Ctk for all k = 1,2 ..... n. This, with ark =Ctk for all k = 1, 2 ..... n, completes the proof. [] 2.
INVERSE OF A SQUARE MATRIX
Let R be a semiring, and A ~Mn(R) is said to be Owertible if AB = I = BA for some B~Mn(R). For any two matrices A , B ~ M , ( R ) , we have IABI+=IAI+[BI++[AI [ B l - + r and IAB[-=IAI+IBI +lhl-lnl++r for some r ~ R [5]. PROPOSITION 2.1. Let A ~ M, ( R ) be an invertible matrix. Then IAI + 4: IAI-.
Proof. By hypothesis, A B = I for some B~M,(R). Let IAI+=IAI -. lAB[ +=IABI-=IAI+(IB[ + + l n l - ) + r . But IABI + = [ I I + = 1 and IABI- = I I [ - = 0, which is a contradiction. Thus IAI + 4: [A]-. []
Then
224
S. G H O S H
We know that, in the case of a field, a square matrix is invertible iff its determinant is nonzero. Proposition 2.1 is only a necessary condition for invertibility of square matrices over semirings (see Example 2.8). Now we consider various special kinds of semirings which arise as generalizations of fields. We know that the commutative rings with identity which are free of ideals (i.e., the only ideals are trivial) are the fields. A semiring S is called a semifield [5] if, for any a ¢ 0 in S, there is b ~ S such that a b - - 1 . Semifields are the ideal-free semirings. But it is well known that k-ideals are precisely the kernels of semiring homomorphisms, and hence in the case of semirings, k-ideals play more important roles than ideals (See, for example, [3] and [7]). A semiring S is said to satisfy condition (C) if, for each a ~ 0 in S, there exist r, s ~ S such that 1 + ra = sa ([8, 5]) (in [5], Golan used the term austere semiring). The above class of semirings are exactly the class of semirings which are free of k-ideals [5], i.e., a semiring S satisfies condition (C) iff every homomorphism from S to another semiring S' has trivial kernel. Clearly, semifields satisfy condition (C), but the converse is not true (see Example 2.3). DEFINITION 2.2. A semiring S is said to satisfy condition (D) if, for each pair a :~ b in S, there exist r, s ~ S such that 1 + ra + sb = sa + rb. Obviously, the semiring satisfying condition (D) also satisfies condition (C), and here also the converse is not true, as the following example shows. E X A M P L E 2.3. Let S = Q + x {1} u @ + x {2}, where Q+ is the set of positive rationals. We define (a, i ) + (b, j ) = (a + b, max(i, j)) and (a, i ) . ( b , j ) = (ab, max(i, j ) ) f o r all (a, i), (b, j ) ~ S. Let R = S U {0}, where 0 is an element outside S satisfying O + ( a , i ) = ( a , i ) and 0 . ( a , i ) = 0 for all ( a , i ) ~ S . Also, 0 + 0 = 0.0 = 0. Then (R, + , .) is a semiring with additive identity 0 and multiplicative identity (1,1). R is not a semifield and does not satisfy condition (D); but R satisfies condition (C). We know that a semiring S can be embedded in a ring iff S is additicely cancellatiue (i.e., a + c = b + c implies a = b for any a, b, c ~ S) [9]. The smallest ring (unique up to isomorphisms over S) in which S can be embedded is called the difference ring [9] of S and is denoted by S. S consists of all differences a - b for a, b ~ S, where elements of the image of S under the isomorphism of embedding are identified with the elements of S. PROPOSITION 2.4. Let S be an additively cancellative semiring. Then S is a field iff S satisfies condition ( D ).
MATRICES OVER SEMIRINGS
225
Proof. Let S be a field and a 4=b in S. Then 0 # a - b ~ S, and so there is 2 ~ S such that $ ( a - b ) = l . But Y c = s - r for some r , s ~ S . Thus 1 = ( s - r)(a - b), which implies 1 + ra + sb = sa + rb, as required. Conversely, let S satisfy condition (D) and ~ 4:0 in S. Then ~ = a - b for some a , b ~ S such that a4=b. Thus l + r a + s b = s a + r b for some r , s ~ S . This implies ( s - r ) ( a - b ) = l , i.e., 2 ~ = 1 , where f f = s - r ~ S . Hence S is a field. [] Z s = { x ~ S : x + z = z for some z ~ S } S. S is called zeroic if S = Z s [5].
is called the zeroid of a semiring
PROPOSITION 2.5. Let S be a semiring satisfying condition (D). Then S is either additively cancellative or zeroic. Proof. Let S -4=Z s and a, b, c ~ S such that a + c = b + c. If a # b, then there are r, s ~ S such that 1 + ra + sb = sa + rb. Adding (r + s)c with both sides, we get l + ( r + s ) ( b + c ) = ( r + s ) ( b + c ) . This implies I ~ Z s and hence S = Z s, which is a contradiction. Thus a = b and S is additively cancellative. [] PROPOSITION 2.6. Let S be a zeroic semiring. Then S is free o f k-ideals iff S satisfies condition ( D ). Proof. In view of the above discussions, we have only to show that if S is k-ideal-free, then S satisfies condition (D). Let S be free of k-ideals and ~,/3 ~ S such that a 4:/3. Let I = { x E S : x + r a + s / 3 = s o ~ + r / 3 , for some r , s ~ S } . It is easy to show that I is a k-ideal of S. Then either I={0}, the zero-ideal, or I = S. If we can show that I contains a nonzero element, then we are done. Since S = Z s, there is z ~ S such that 1 + z =z. Then oi + z ~ + z / 3 = ( 1 + z)ol + z / 3 = z o l + z/3. Similarly, /3 + z a + z / 3 = z a + z/3. Thus a,/3 e l . Since ol 4:/3, one of them must be nonzero. Hence the proof. [] We may summarize the above discussions in the following diagrams: , condition (C) (k-ideal-free) '
semifield condition (D)
'either
condition(D) condition (D)
&
&
zeroic
zeroic~
or
condition (D) additively cancellative
,condition(C)
additively cancellative,
&
zeroic
, difference ring is a field
E X A M P L E 2.7. (i) S = ( Q ~ ' , m a x , -) is a zeroic semifield of nonnegative rationals satisfying condition (D).
226
S. G H O S H
( i i ) S = Q0~(v~-) = {a + bf2-: a, b ~ Q~-} with usual multiplication and addition is a semiring that satisfies condition (D); but S is not a semifield. Also, Z s = {0} 4: S and S is the field Q(v/2-), where Q is the field of rational numbers. (iii) Let S = Q + × Q+ U{(0,0)}. Then S is a semifield with pointwise usual addition and multiplication; but S does not satisfy condition (D). Also, note that S = Q × Q, which is not a field.
Now turning back to the question of invertibility of a square matrix
A ~M,(R) over a semiring R, we first note that the condition IAl+4: IAIis not, in general, sufficient even when R is a semifield or R satisfies condition (D) or condition (C). E X A M P L E 2.8. Let R be a semiring as described in Example 2.7(i). Let /
A = {l3
]}~M2(R) .Then
AI+=44:6=IAI
. B u t there is no such B ~
\
M2(R) satisfying A B = I 2 =
1
o
01. 1]
In [2], Beasley and Pullman have shown that invertible matrices over a
zero-sum-free (i.e., a + b = 0 implies a = 0 = b [5]) and zero-divisor-free (i.e., ab = 0 implies either a = 0 or b = 0; also called entire [5]) semiring are very rare (see [2, theorem 3.1]). Also, we see that semifields and semirings satisfying condition (D) or condition (C), which are not rings, are zerosum-free and zero-divisor-free. Thus we introduce the concept of semiinvertibility. DEFINITION 2.9. A ~Mn(R) is said to be semi-invertible if there exist A1,Az~Mn(R) such that I + A A 1 =AA 2 and I + A I A = A z A . PROPOSITION 2.10. Let R be a semiring satisfying condition (D) and A ~M,(R). If IAI + 4: IZl-, then A is semi-invertible.
Proof. Let B = A r, ~,j=lBijl (+) and ~ij=lBijl ~ ). Let A+=(aij) and A - = (/3i). Now since IAI + ~ IAI-, there exist r, s ~ R such that 1 + rlA] + + siAl =slAl++rlAI -. Let Al=rA++sA and A2=sA++rA -. If A I = (ci), then AIA = ( f / ) , where n
fij= ~ ci~akj = Y'~ akjcik k=l
k~l
= ~ ak~(rlBi,[ '+' + slBi~(-') k=l
=r ~_~bjklBi~l(+'+s ~ b]klBikl'-', k=l
k=l
227
MATRICES OVER SEMIRINGS where B = (bij). Then by (II), L e m m a 1.2, and L e m m a 1.1, we get
l =
tl bjklBikl ~+~, ( r + s ) ~_~
i4=j,
rlBl+ +slBl-=r]Al+ +slAI ,
i=j.
k=l
Similarly, A2A = (gij), where
I
( r + s ) k~=lbJk[Bik[(+),
gij =
( slAl+ + rlAI ,
i4:j, i=j.
Thus l+fii=gii for all i = 1 , 2 , . . . , n and fij=gij for all i--gj. Therefore, I + A 1 A =A2A. Similarly, one can show that I + A A 1 = A A 2. [] The following example shows that the condition [A[ + ~ [ A [ necessary for a matrix A to be semi-invertible.
is not
E X A M P L E 2.11. Let R be a semiring as described in Example 2.7(i) and /
A=[
\
\
1
2
24 ~ M 2 ( R ) . Then [ A ] + = 4 = I A [ - ; but A is semi-invertible with l
PROPOSZTION 2.12. Let R be a semiring satisfying condition (C) [or a semifieldl and A ~Mn(R). If only one of the elements IAf + or IAI- is nonzero, then A is semi-invertible. The proof, being similar to that of Proposition 2.10, is omitted. A semiring S is called the additively inversive semiring [6] if for each a ~ S , there exists a unique element a ' ~ S such that a + a ' + a = a and a ' + a + a ' = a ' . It is well known that for any a,b e S , (a +b)' =a' +b' and (ab)' =a'b=ab' [6]. It may be noted that in a semifield S for an element a ~ S, there exists b ~ S such that a + b = 0 iff S is a ring. PROPOSmON 2.13. Let R be an additively inversive semifield which is not a ring andA ~Mn(R). If ( [ A I + , ] A I ) 4 : ( 0 , 0 ) , then there existA1,C~Mn(R) such that A ~A + C = I + C and AA ~+ C = I + C.
228
S. G H O S H
Proof. By the given condition, d=IAI+ +(IAI-) ' 4=0. Let B = A r and A 1 = ( a i ) , where aii =d-l[lBijl(+)+(]Bij[(-))']. Then A1A =(fij), where
fii = ~_~ Otikak) k=l
=d-l[k~lb]klBikl(+) + (k~lbjklBik[( ))']" Thus, by (II), L e m m a 1.2, and L e m m a 1.1, fii=d-~(lBl++(lBl-)')= d ~(IAI++(IAI ) ' ) = d - l d = l , and if i ~ j , fo=rij+r;j, where rij= d I E~=lbjk[Bikl n (+) . Let C 1=(c/j), where
cij=
rij, O,
i :l=j, i=j.
Thus fo + ciJ = cij for all i C j and fii + % = 1 = 1 + % for all i. Therefore, A ~ A + C 1 = I + C l. Similarly, one can show that A A j + C 2 = I + C 2 for some C2~Mn(R). Then A ~ A + C = I + C and A A ~ + C = I + C , where C = C l +C 2. [] Now we obtain a necessary and sufficient condition for semi-invertibility of square matrices over a semiring. Let R be a semiring. A (right) semimodule [(/eft) sernimodule] over R is defined in a natural way [5]. Let R n be the set of all matrices of order 1 × n over R. Then R ~ is a (left) semimodule over R. Let A ~ M , ( R ) and R"A ={rA: r ~ R n} (the product rA is the usual multiplication of matrices of order 1 × n and n x n). Then we see that RnA is a (left) subsemimodule of the (left) R-semimodule R n. Similarly, if we consider R , as the set of all matrices of order n × 1 over R, then R , is a (right) semimodule over R and ARn = {At: r ~ R , } is a (right) subsemimodule of the (right) R-semimodule R n. Let M be a (right) R-semimodule [(left) R-semimodule]. A k-subsemimodule N of M is a subsemimodule of M in which a + b = c , a E M, b, c ~ N implies a ~ N. As in the case of k-ideals, here also k-subsemimodules are precisely the kernels of R-semimodule homomorphisms. Let T be a subsemimodule of M. The least k-subsemimodule containing T is called
229
MATRICES OVER SEMIRINGS the k-closure of T and is d e n o t e d by (T) k. Then
(T)k={a~M:a+b=c
forsomeb,c~T}.
It is well known that if R is a commutative ring with identity and A ~ M n ( R ) , then A is invertible iff A R n =Rn and R"A = R n. The following t h e o r e m is analogous to this result in the case of semirings. THEOREM 2.14. Let R be a semiring and A c M n ( R ) . Then A is semiinvertible iff ( A R n ) k = R , and (R~A)k = R ~.
Proof. Let A be a semi-invertible matrix. Clearly, ( A R ~ ) k C R ~. Let r ~ R , . Since A is semi-invertible, there are A 1 , A 2 ~ M , ( R ) such that I + A . 4 1 = A A 2. This implies r + A ( A I r ) = A ( A 2 r ) . Thus r ~ ( A R , , ) k. Similarly, one can show that (R"A)k = R ~. Conversely, let ( A R n ) k =R~. Let e i be the ith column of the identity t t! matrix I ~ M , ( R ) . N o w the above relation shows that there exist ei, e~ R , such that e i +Ae'i = A e ' . Let A 1 = (e]e~... e',) and A 2 = (e'~e~ ... e~). T h e n I + A A 1 = A A 2. Similarly, ( R " A ) k = R " gives I + A ' I A =A'2A. Simple calculations show that / + A A 3 =AM 4 and I + A 3 A = A 4 A , where A t = A j + A ~ +A'~A.41 and A 4 = A 2 + A " +A!lMA1, as required. [] PROPOSITION 2.15. Let R be a semiring and A E M , ( R ) be a semi-invertible matrix. I f rl, r 2 ~ R~ be such that A r l = A r 2, then r 1 + t = r 2 + t for some t ~ R~. Moreover, i f R is additively cancellatic, e, then A r 1 = A t 2 implies rl = r~ and A R , [ R~A] is isomorphic to R , [R "] as (right) R-semimodules [(left) R-semimodules].
Proof. W e have I + A I A = A 2 A , A1,A2~M,,(R). Then ri+Al(Ari) = Az(Ar/), i = 1,2. Thus r 1 + ( A 1 + A 2 ) ( A r j ) = r 2 + ( A I + A z ) ( A r l ) . Moreover, if R is additively cancellative, so is Rn and then r 1 = r 2. Also, R~ is then isomorphic to A R , as (right) R-semimodules through an isomorphism qt : Rn ~ AR~ defined by ",It(r) = A r for all r ~ R,,. [] A PPLICA TION PROBLEM 2.16. Let R be a semiring as described in Example 2.7(i). Solve: x + 2 y + z = 9 , 7x + 4 y + 3 z = 3 5 , 3x + 5 y + 2 z = 2 0 , where x , y , z ~ R .
Solution. Let A =
7 \3
4
3 , X=
y , and B =
5
2
z
~5 . T h e n by the \2;]
given system of equations, we have A X = B . Since IAI + = 35 ~ 28 = [ A I , by Proposition 2.10, A is semi-invertible, so that I + A ~ A = A 2 A , where
230
S. G H O S H
A 1 = r A + W s A -, A 2 = s A + W r A -, and r , s ~ R 1+35r+28s W e have X + A 1 A X = A 2 A X
such that
=28r+35s.
(1)
, which implies X+A1B=A2B.
Here A +=
9 2 35
, A-=
6
14 3 12
5
3 , A+B = 14
(2)
14(t/, a n d A - B 315]
=
126 / . 280]
(4,0)(i)
Now (1) is satisfied by r = 1 / 3 1 5 , s = 1 / 3 5 . P u t t i n g these values of r and s in (2), we get
+
3.6 =
, which implies x = 5, y = 4, z = 9.
8"0
The author is grateful to Professor T. K~ Mukherjee and Dr. M. K. Sen for their kind help and guidance in this work. The author also expresses his deep gratitude to the learned referee for his valuable suggestions. REFERENCES 1. Le Roy B. Beasley and N. J. Pullman, Operators that preserve semiring matrix functions, Lin. Alg. Appl. 99:199-216 (1988). 2. Le Roy B. Beasley and N. J. Pullman, Linear operators strongly preserving idempotent matrices over semirings. Lin. Alg. Appl. 160:217-229 (1992). 3. S. Bourne, The Jacobson radical of a semiring, Proc. Nat. Acad. Sci. U.S.A. 37:163-170 (1951). 4. R. A. Cunninghame-Green, The characteristic maxpolynomial of a matrix, J. Math. Anal. Appl. 95:110-116 (1983). 5. J. S. Golan, The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, Longman Scientific and Technical, U.K., 1992. 6. P. H. Karvellas, Inversive semirings, J. Austral. Math. Soc. 18(3):277-288 (1974). 7. D. R. La Torte, On h-ideals and k-ideals in hemirings. Publ. Math Debrecen 12:219-226 (1965). 8. M. K. Sen and M. R. Adhikari, On maximal k-ideals of semirings, Proc. Amer. Math. Soc. 118(3):699-703 (1993). 9. H. J. Weinert, /tiber halbringe und halbk6rper II, Acta. Math. Acad. Sci. Hungar. 14:209-227 (1963). 10. U. Zirnmermann, Linear and Combinatorial Optimization in Ordered Algebraic Structures, North Holland, Amsterdam, 1981. Received 1 January 1995; revised 24 August 1995