Linear systems in (max,+) algebra

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1 Linear systems in (ma,+) algebra Ma Plus INRIA Le Chesnay, France Proceedings of the 29th Conference on Decision and Control Honolulu, Dec. 99 Abstract In this paper, we study the general system of linear equations in the (ma; +) algebra. We introduce a symmetrization of this algebra and a new notion called balance which generalizes classical equations. This construction results in the linear closure of the (ma; +) algebra in the sense that every non-degenerate system of linear balances has a unique solution given by Cramer s rule. I. Introduction The (ma; +) algebra plays a crucial role in at least two fields: path algebra (research of the path of maimal weight in a graph). performance evaluation of Discrete Event Dynamic Systems (DEDS). In this paper, we eamine a fundamental problem in this algebra: solving systems of linear equations. Let us start by introducing the notation used throughout this paper. We shall denote ma by (i.e. ma(a; b) is noted ab), and use instead of the usual addition + (e.g. 2 3 = 5). (also denoted by ") is the null element for ( = ) and is absorbing for the product ( = ). is the unit element: =. (R[f g; ; ) is called the (ma; +) algebra, or simply R ma. Usual computational rules hold in R ma (for instance a (b c) = (b c) a = (b a) (c a)). This in particular allows us to define and manipulate vectors and matrices as usual. For simplicity, we sometimes omit (we write ab instead of a b). A general account of this kind of algebraic structures can be found in Gondran and Minou [7] for the graph theoretic point of view and Cohen, Moller, Quadrat and Viot [4] for the Discrete Event Systems point of view. For more than thirty years, it has been known that the implicit vector equation = A b (A being a n n matri) can be solved by iteration, leading to the study of A = Id A A 2 : : :. Other vector equations of the type H = b (H not being necessarily a square matri) also can be dealt with by using residuation theory (see Blyth []). But for the most general This is a collective name for a working group on Discrete Event Systems Theory, at INRIA (META2 Project) composed of Marianne Akian, Guy Cohen, Stéphane Gaubert, Ramine Nikoukhah, and Jean Pierre Quadrat. Guy Cohen is also with Ecole des Mines. Address for correspondence: S. Gaubert, INRIA, Domaine de Voluceau-Rocquencourt, B.P. 5, 753 Le Chesnay Cede, France. gaubert@seti.inria.fr (moved to Stephane.Gaubert@inria.fr). system of n linear equations with n unknowns A b = C d () where A and C are n n matrices with entries in R ma and b,d are vectors of (R ma ) n, no result eisted until now to the best of the authors knowledge. In section 2, we first eplain why the general equation () is essential for the study of Discrete Event Systems. Then, we embed the (ma; +) algebra into a symmetrized algebra (cf. Section 3-B), where the balance relation plays the role of equality (Definition 3.). The original elements are identified with positive elements in this new algebra. We associate the system of balances (A C) (d b) with system (). Among the many solutions balances may have, a restricted class can be associated with solutions in the (ma; +) algebra: these are signed solutions (i.e. positive, negative, or null). The main result of this paper states that non-degenerate systems of linear balances always have a unique signed solution, given by Cramer s rule (Theorem 6.). When this solution is positive, it determines the unique solution of system (). Eample. Find the solutions of: ma(; y 4; ) = ma( ; y + ; 2) ma( + 3; y + 2; 5) = ma(y + 2; 7) or in matri form y 5 = 2 y 2 : 7 This problem is solved in Section 6-A. Before going into further details, let us make a simple remark: if a > b, the equation a = b is equivalent to a =. This suggests the (2) naive rule: a b = a if a > b. (3) We can now try to solve system (2) using this naive rule: (i) ( 4)y = ( ) y 2 (2), (ii) 3 2y ( 5) = 2y 7 (i) ) [ ( )] = [ ( 4)]y (2 ) ) (i ) : = y 2 (i ) and (ii) ) 3(y 2) 2y ( 5) = 2y 7 ) (4 2 2)y = 7 ( 5) 5 ) 4y = 7 ) y = 3 : Together with (i), we get = 32 = 4, and it is immediate to check that (; y) = (4; 3) is a solution of system (2). Our goal is to make it clear when and why these calculations are valid.

2 2 II. Linear System Theory for Discrete Event Systems To see why equations of type () are fundamental in the (ma; +) algebra and its applications, we need to review some of the work done in developing a system theory in this algebra. In the contet of DEDS, Cohen, Dubois, Moller, Quadrat and Viot (see [2], [4]) have developed a system theory, analogous to the conventional system theory for linear differential and recurrent equations. They have shown that a restricted class of deterministic Petri nets, timed event graphs, can be described by linear recurrent equations in the R ma algebra. This class can be used to model fleible workshops, some distributed processing systems, and in particular systems involving synchronization constraints. A complete study, leading to the concepts of state-space representation and transfer function can be found in [4]. The study of general linear equations of type () appears to be the theoretical background needed to deal with the following interesting topics:..a Notions of rank. Many notions of rank and of linear dependence in vector structures over the (ma; +) algebra can be found in the literature. The following is a conventional one. Definition 2.: fu i g i2i is free iff the canonical map f i g i2i 7! L i2i iu i is one to one. This condition is practically never fulfilled. The following notion of weak independence has been studied by Moller [9] and Wagneur [3]. Definition 2.2: fu i g i2i is weakly independent iff no vector u i is spanned by the others fu j g j6=i. Weak independence has some pathological features: for instance, there eists an infinite family weakly independent in R 3 ma (see Cuninghame Green [5]). From the early work of Gondran and Minou [6], there is a more appealing definition of linear dependence, bearing a strong resemblance with classical linear algebra: Definition 2.3: fu i g i2i is dependent iff there eists a partition I = J [ K and a non-trivial family of scalars f i g such that L j2j ju j = L k2k ku k. We have: fu i g free ) fu i g non-dependent ) fu i g weakly independent. In the case of Definition 2.3, the column vector defined by = ( i ) is solution of an equation of the type U = U which is a homogeneous form of equation ()...b Controllability, Observability. It is well known that the only invertible matrices with entries in R ma can be written as M = DS where D is a diagonal matri with invertible entries and S is a permutation matri (see e.g. [5]). This means that endomorphisms of (R ma ) n are never invertible. As a consequence, defining an effective notion of controllability or observability is far from being obvious and the only notions studied so far are structural (see [3]). A sharper theory should be based on the notion 2.3 of dependence...c Minimality. Some attempts (cf. Olsder []) have already been made suggesting that minimal realizations may be related to two-sided ARMA models: Y (n) A + Y (n ) : : : A+ k Y (n k) B U(n) : : : : : : B U (n k) = A k Y (n ) : : : A Y (n k) k B + U(n) : : : B+ U(n k) k which clearly have the form of equation (). III. Symmetrizing the (ma; +) algebra A natural approach to our problem may have been to embed the (ma; +) algebra into a structure in which every non-trivial scalar linear equation has at least one solution. Indeed, solving a = " means symmetrizing a. Because ma is idempotent (i.e. a a = ma(a; a) = a), the following remark shows how hopeless it is to adapt the classical symmetrization of monoids (e.g. the way we buildzfrom N) to the R ma contet: Proposition 3.: Every idempotent group is reduced to the null element. Proof Assume the group (G; ) is idempotent with null element ". Let b be the symmetric element of a 2 G. Then a = a " = a (a b) = (a a) b = a b = ". A. The algebra of pairs Let us now consider the set of pairs R 2 ma endowed with the natural dioid structure: ( ; ) (y ; y ) = ( y; y ) ( ; ) (y ; y ) = ( y y ; y y ) with ("; ") as the null element and (; ") as the unit element. Let = ( ; ) and define the minus sign as = ( ; ). The absolute value of is denoted by jj =. The balance operator () is defined by = = (jj; jj). It is immediate to check that j j and are dioid morphisms (respectively onto R ma and the diagonal of R 2 ma ). In addition, we have the following properties: (i) a = ( a) (ii) idempotence a = a (iii) absorpsion ab = (ab) (iv) involution ( a) = a (v) additive morphism (a b) = ( a) ( b) (vi) sign rule (a b) = ( a) b : In particular (iv) (vi) allow us to write as usual a ( b) = a b. B. Quotient structure Definition 3.: [Balance relation] Let = ( ; ) and y = (y ; y ). We say that balances y (denoted by y) if and only if y = y. It is fundamental to notice that is not transitive. For instance, consider (; ) (; ), (; ) (; ) but (; ) 6 (; )! Since cannot be an equivalence relation, there is no point to define the quotient structure ofr 2 ma by (opposed to the conventional algebra in whichn 2 = ' Z). However, we can introduce a new relation R on R 2 ma ( ; ) R (y ; y ), < : 6= ; y 6= y ; and y = y ( ; ) = (y ; y ) otherwise Recall that a dioid is a set D together with two laws and such that (i): (D; ) is associative, commutative, idempotent (i.e. a a a = a), with null element ", (ii): (D; ) is associative with unit element e, (iii): product is distributive over addition and (iv): the null element is absorbing ( a a " = " a = "). (4)

3 3 which is an equivalence relation, stronger than. It is easy to check that R is compatible with the structure laws ofr 2 ma, with the balance relation and also with the, j j and operators. Definition 3.2: We set S ma = R 2 ma= R and we call it the symmetrized algebra of R ma. We distinguish three kinds of equivalence classes: (t; ) = f(t; ); < tg ( ; t) = f( ; t); < tg (t; t) = f(t; t)g called positive called negative called balanced. By associating (t; ) with t 2 R ma, we can identify R ma with the subdioid of positive or null classes, R ma. The set of negative or null classes (of the form for 2 R ma ) will be denoted byr ma, the set of balanced classes (of the form ) by R ma. This yields the decomposition S ma = R ma [R ma [R ma (5) " being the only element common to any two of these three sets. This should be compared withz= N + [N. These conventions allow us to write 3 2 instead of (3; ) ( ; 2). We thus have 3 2 = (3; 2) = (3; ) = 3. More generally, calculations in S ma can be summarized as follows a b = a if a > b b a = a if a > b a a = a : This includes and generalizes the initial naive rule (3). Because of its importance, we introduce the notationr _ ma for are called signed the set R ma [ R ma. The elements of R_ ma elements. They are either positive, negative or null. We have: Proposition 3.2: R _ man f"g = S ma n R ma is the set of all invertible elements ofs ma. Proof t ( t) = ( t) ( t) = for t 2 R ma n f"g obviously shows that every non-null element ofr _ ma is invertible. Moreover, formula (4,(iii)) shows that R ma is absorbing for the product. Thus, y 6= for all y 2 S ma since 62 R ma. Remark 3. It can be proved that R is the weakest equivalence relation stronger than. In this sense R is natural. Remark 3.2 There is a nicer algebraic way to introduce the relation R. Let Sol (a) = f 2 R 2 ma ; ag, then it can easily be verified that (6) R y () Sol () = Sol (y) (7) which makes it clear that R is an equivalence relation. This leads to a very simple proof of the compatibility of R with addition. Because the following propositions are equivalent: 2 Sol (a c) a c c a c 2 Sol (a) Sol (a) = Sol (b) implies that Sol (a c) = Sol (b c). Remark 3.3 The equivalent formulation (7) allows etending symmetrization to more general dioids than the (ma; +) algebra. In fact, it is always possible to define the quotient of the additive monoid of pairs by the map a 7! Sol (a), but this quotient may not be compatible with multiplication! What is specific to the total order structure ofr ma is the decomposition (5). Since our goal here is to give an eistence and uniqueness theorem, we only consider the case of a totally ordered multiplicative group, the generic case of which being the (ma; +) algebra. But a more general theory can be developed along the same lines. A. General properties IV. Linear balances Before solving general linear balances, we need to eplain why balances in S ma generalize equations in R ma. The main algebraic features of balances are: Properties 4.: (i) a a (refleivity) (ii) a b, b a (symmetry) (iii) a b, a b " (iv) a b; c d ) a c b d (v) a b ) ac bc Let us prove (v): a b, a b 2 R ma and as R ma is absorbing, (a b)c = ac bc 2 R ma, i.e. ac bc. Although is not transitive, when some variables are signed, we can manipulate balances in the same way as we manipulate equations: Property 4.2: [Weak substitution] a c b and 2 R_ ma ) ca b Proof We have 2 R ma or 2 R ma. Assume for instance that 2 R ma, = ( ; "). With obvious notations: a = a and c b = c b. Adding c a c a to the last equality, we get c c a c a b = c c a c a b, which yields c a c a b = c a c a b, i.e. ca b. By taking c =, the weak substitution property 4.2 becomes: Property 4.3: [Weak transitivity] a ; b and 2 R _ ma ) a b We conclude by a simple remark which allows translating balances into equalities: Property 4.4: [Reduction of balances] ) = y y and (; y) 2 (R _ ma )2 It is immediate to etend balances to the vector case. Properties (4.,i v), 4.2, 4.3 and 4.4 still hold when a; b; ; y and c are matrices with appropriate dimensions, provided we replace 2 R _ ma by every entry 2 R _ ma. Therefore, we say a vector is signed iff every entry is signed. B. From equations to balances We now consider a solution of the equation () inr ma. We have A b C d (refleivity), and by (4.,iii): (A C) (b d) " : () Conversely, assuming that is a positive solution of (), we get A b C d with A b and C d 2 R ma R_ ma. Using 4.4, we get A b = C d. So, we have:

4 4 Proposition 4.5: The set of solutions of the general linear system of equations () inr ma and the set of positive solutions of the associated linear balance () in S ma coincide. Thus, the original problem reduces to studying linear balances in S ma. Remark 4. The case where a solution of () has some negative and some positive entries is also of interest. We write = + with + ; 2 (R ma )n. Partitioning the columns of A and C according to the sign of the entries of : A = A + A,C = C + C (in such a way that A = A + + A and C = C + + C ), we can affirm the eistence of a solution to the new problem A + + C b = A C + + d : C. The scalar linear balance Theorem 4.6: Let a 2 R _ ma n f"g and b 2 R _ ma, then the balance a b " (9) has the unique signed solution: [ = a b. Proof From properties 4.,v and 4.,iii, ab " is equivalent to a b. Using the reduction property 4.4 and a b 2 R _ ma, we get = a b. Remark 4.2 Non-trivial linear balances always have solutions in S ma, that is whys ma may be considered as a linear closure of R ma. Remark 4.3 We can describe all the solutions of (9). For all t 2 R ma, we have obviously at ". Adding this balance to a [ b ", we get a( [ t ) b ". Thus, t = [ t () is solution of (9). If t j [ j, then t = t is balanced. Conversely, it can be checked that every solution of (9) may be written as in (). The unique signed solution [ is also the least solution. Remark 4.4 If b 62 R _ ma, we lose uniqueness of signed solutions. Every such that jaj jbj (i.e. jj ja bj) is solution of balance (9). Remark 4.5 If a 62 R _ ma, we again lose uniqueness. Assume b 2 R _ ma (otherwise, the balance holds for all value of ), then every such that jaj jbj is a solution. V. A fundamental identity Before dealing with general systems, we need to etend the determinant machinery to the S ma contet. We define the sign of a permutation by sgn() = if is even and sgn() = if is odd. Then the determinant of an n n matri A = (a i;j ) is given (as usual) by det A = M sgn() no i= a i;(i) : det remains an n-linear antisymmetric function of the rows (or columns). det A is balanced (but non-null in general) if two rows (or columns) of the matri A are identical. We denote by A adj the transpose of the matri of cofactors ([A adj ] i;j = cof j;i (A)), and by Id the identity matri (with on the diagonal and " elsewhere). The following is just a combinatorial identity, that can be shown by adapting a result by Reutenauer & Straubing [2] or the usual demonstration: Theorem 5.: AA adj det A :Id : Remark 5. The formulation of Reutenauer and Straubing consists in defining a positive determinant det + A (where the sum is taken over all even permutations) and a negative determinant det A (odd permutations). The matri of positive cofactors is defined by [A adj+ ] i;j = det + A(jji) if i + j even det A(jji) if i + j odd where A(ijj) denotes the matri A from which row i and column j are removed, and the matri of negative cofactors A adj is defined similarly. With these notations, Theorem 5. can be rewritten as follows: AA adj+ det A:Id = AA adj det + A:Id : This formula does not use the sign and is valid in any semiring. The symmetrized algebra appears as a natural way of handling (and proving in an algebraic way) such identities. A. Cramer s rule VI. Solving systems of linear balances Because of the remarks of section 4, we only consider the solutions of balances in (R _ ma )n, that is signed solutions. We can now state the fundamental result for the eistence and uniqueness of signed solutions of linear systems: Theorem 6. (Cramer system) Let A be an n n matri with entries in S ma and b 2 (S ma ) n. Then every signed solution of satisfies: A b () det A : A adj b : (2) Conversely, assume that A adj b is signed and det A is invertible, then the Cramer solution [ = (det A) A adj b is the unique signed solution of (). Proof Assume det A is invertible and A adj b is signed. By rightmultiplying the identity AA adj det A Id by (det A) b we easily see that the Cramer signed solution [ satisfies (). This proves the converse implication. We shall consider the direct implication only when det A is invertible. The proof is by induction on the size of the matri. Let us prove (2) for the last row, i.e. det A n (A L adj b) n. Developing with respect to the last n column, det A = k= a k;ncof k;n (A) we get that at least one term is invertible, say a ;n cof ;n (A). We now partition in an obvious way A, b and : A; a A ;n b = A ; b = A n;n b ; = n

5 5 where A ; is an (n ) matri, A is an (n ) (n ) matri, etc... A; A b () a ;n n b () A A n;n n b () Since det A = ( ) n+ cof ;n (A) is invertible, we apply the induction hypothesis to (), ( ) : A b A n;n n which implies that (det A ) A adj (b A n;n n ). Using the weak substitution property 4.2, we replace 2 (R _ ma )n in (): i.e. A ; (det A ) A adj (b A n;n n ) a ;n n b [det A :a ;n A ; A adj An;n ] n det A :b A ; A adj b : Here, we recognize the developments of ( ) n+ det A and ( ) n+ (A adj b) n. Thus det A : n (A adj b) n : Since the same result holds when developing with respect to any column k other than n, this concludes the proof. Remark 6. Let D i be the determinant of the matri obtained by replacing the i-th column of A with b, then (A adj b) i = D i. Assume det A invertible, then the equation (2) is equivalent to: (i) i (det A) D i : If A adj b 2 (R _ ma )n, then i = (det A) D i, which is eactly the classical i-th Cramer formula. Eample 6. Let us go back to our original problem (Eample ). The balance corresponding to equation (2) is 3 2 y with determinant D = 4 (invertible). So, = D D = D = ; D y = A adj b = D D y = = 4 = 4; y = D y 7 D (R _ ma )2 : = 7 (3) = 7 4 = 3 gives the unique positive solution in S ma of balance (3). Thus, it is the unique solution of equation (2) in R ma. Eample 6.2 In the two dimensional case, the condition A adj b 2 (R _ ma )n has a very clear geometric interpretation. The following picture represents the solutions to balances in the plane of signed coordinates (R _ ma )2. From Theorem 6., we easily see that the two lines L and L 2 meet at a single point: (; ). However, L 2, which is parallel to L 2 has a degenerate intersection with L because the second Cramer determinant of system L ; L 2 is balanced. R ma L : y 2 R ma 2 2 " R ma L 2 : y 2 L R 2 : ma y 2 Remark 6.2 det A being invertible is not a necessary condition for system A b to have a signed solution for all values of b! Consider A = 2 4 " " " det A = ". Let t 2 R _ ma such that jb ij jtj for all coordinate i, and let = t t " t. Then A b. Remark 6.3 As already noticed by Gondran and Minou (see [6]), determinants have a natural interpretation in terms of assignment problems. So the Cramer computations have the same compleity as n + assignment problems, which can be solved using flow algorithms. B. General case We can even solve A b in some degenerate cases: Theorem 6.2: Assume that det A 6= " (but possibly det A ") then for all values of b there eists a signed solution of A b such that jj =j det Aj ja adj bj. It is remarkable that the classical Gauss-Seidel and Jacobi algorithms can be adapted to the S ma case, for which we have convergence after n iterations (!). This in particular provides an algorithmic proof of Theorem 6.2. We write A = DU L, with U upper-triangular, L lower-triangular and D diagonal. Let us introduce the notation jj y for y and jj = jyj. We now state: Theorem 6.3: [Jacobi N algorithm] Assume the domination n property j det Aj = j i= a i;ij 6= " then: / There eists a (perhaps non-unique) sequence of signed vectors f p g such that (i) " = : : : p : : : (ii) D p+ jj (U L) p b : 2/ Such a sequence is stationary after n iterations ( n = n+ = : : :) and n is a solution of A b 3/ j n j = j det Aj ja adj bj. Sketch of proof / can be shown by an induction argument which is omited due to the lack of space. To understand why p is stationary, we introduce ^ p = j p j. (ii) yields ^ p+ = M ^ p+ jdj jbj with M = jdj ju Lj. We have ^ p+ = (Id M : : : M p )jdj jbj. The domination hypothesis implies that M has no circuits with weight >, which implies that the series M = IdM M 2 : : :M p : : : is stationary after step n, i.e. ^ n = ^ n+ = : : : (this is a classical result, cf. [7], p.72, Theorem ). Because n and n+ are signed, n n+ 3 5

6 6 and j n j = j n+ j imply n = n+. Replacing in (ii), we get D n (U L) n b which is equivalent to A n b. This concludes the proof of 2=. Statement 3= follows: Lemma 6.4: (jdj ju Lj) jdj = j det Aj ja adj j. This can be deduced from a theorem due to Yoeli ([4], Theorem 4). Eample 6.3 We apply the Jacobi algorithm to " with j det Aj ja adj bj = 5 jj 3 2 jj 4 3 jj #" = 4 ) 2 = 3 = or ) and ) 5 2 jj = jj 3 4 = jj = jj jj jj 3 3 # t. " 4 # ; say 3 = 2 = = 2 3 = 2 or 2; say 2 3 = 2 3 = 3 2 = 3 3 = 2 Different choices for 3 and 2 3 yield another solution: 3 = ; 3 2 = ; 3 3 = 2. For the homogeneous system, the analogy with the classical situation is complete. The following result generalizes a theorem of Gondran and Minou [6]: Theorem 6.5: [Homogeneous case] Let A be an nn matri with entries in S ma. Then the equation A " has a signed non-null solution if and only if det A ". References [] T.S. Blyth. Matrices over ordered algebraic structures, J. of London Mathematical Society, Vol. 39, pp , 964. [2] G. Cohen, D. Dubois, J.P. Quadrat and M. Viot. Analyse du comportement périodique des systèmes de production par la théorie des dioides, INRIA Report No. 9, Le Chesnay, France, 93. [3] G. Cohen, P. Moller, J.P. Quadrat and M. Viot. Linear system theory for discrete-event systems, 23rd IEEE Conf. on Decision and Control, Las Vegas, Nevada, 94. [4] G. Cohen, P. Moller, J.P. Quadrat and M. Viot. Algebraic Tools for the Performance Evaluation of Discrete Event Systems, IEEE Proceedings: Special issue on Discrete Event Systems, 77, No., 99. [5] R.A. Cuninghame-Green, Minima Algebra, Lectures notes in Economics and Mathematical Systems No. 66, Springer, Berlin, 979. [6] M. Gondran and M. Minou, L indépendance linéaire dans les dioïdes, E.D.F., Bulletin de la Direction des Etudes et Recherches, Série C, Mathématiques, Informatique, No., pp. 67 9, 97. [7] M. Gondran and M. Minou, Graphes et algorithmes, Eyrolles, Paris, France, 979. [] M. Plus, L algèbre (ma; +) et sa symétrisation ou l algèbre des équilibres, Comptes Rendus à l Académie des Sciences, Section Automatique, 99 (to appear). [9] P. Moller, Notions de rang dans les dioides vectoriels, in Algèbres Eotiques et Systèmes à Evénements Discrets, Séminaire CNRS/CNET/INRIA, 3-4 Juin 97, CNET Issy les Moulineau. [] G.J. Olsder, R.E. de Vries. On an analogy of minimal realizations in conventional and discrete-event dynamic systems, in Algèbres Eotiques et Systèmes à Evénements Discrets, Séminaire CNRS/CNET/INRIA, 3-4 Juin 97, CNET Issy les Moulineau. [] G.J. Olsder, C. Roos, Cramer and Cayley-Hamilton in the maalgebra, Linear Algebra and its Applications,, pp. 7, 9. [2] C. Reutenauer, H. Straubing. Inversion of matrices over a commutative semiring, J. Algebra,, No. 2, 94, pp [3] E. Wagneur. Finitely generated moduloids: the eistence and unicity problem for bases, in Analysis and Optimization of Systems, J.L. Lions, A. Bensoussan Eds, Lecture notes in Control and Information Sciences No., pp , Springer, 9. [4] M. Yoeli. A note on a generalization of Boolean matri theory, Amer. Math. Monthly, No. 6, pp , 96. Erratum and Further References Remark 3.. Read congruence instead of equivalence relation. Remark 6.3. The author is grateful to Peter Butkovič for having pointed out a serious error here. Given a square matri A with entries in S ma, the computation of j det Aj is equivalent to an assignment problem. But, it is not clear whether or not the sign (positive,negative,balanced) of det A can be computed in polynomial time. When A has its entries in R ma, Peter Butkovič proved that the computation of det A reduces in polynomial time to the detection of an even cycle in a digraph, a problem which is not known to be polynomial (P.B. Regularity of Matrices in Min-Algebra and its Time compleity, preprint, Sep. 993). The theory initiated in this paper has been developed in:. Synchronization and Linearity, F. Baccelli, G. Cohen, G.J. Olsder and J.P. Quadrat, Wiley, Théorie des systèmes linéaires dans les dioïdes, S. Gaubert, Thèse de l École des Mines de Paris, July, 992.