International Journal of Algebra, Vol. 12, 2018, no. 6, 247-255 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2018.8727 A Note on Linearly Independence over the Symmetrized Max-Plus Algebra Gregoria Ariyanti Department of Mathematics Education Catholic University of Widya Mandala Madiun 63131 Madiun, Indonesia Ari Suparwanto Department of Mathematics Gadjah Mada University 55281 Yogyakarta, Indonesia Budi Surodjo Department of Mathematics Gadjah Mada University 55281 Yogyakarta, Indonesia Copyright c 2018 Gregoria Ariyanti, Ari Suparwanto and Budi Surodjo. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The symmetrized max-plus algebra is an algebraic structure which is a commutative semiring, has a zero element ɛ =, the identity element e = 0, and an additively idempotent. Motivated by the the previous study as in conventional linear algebra, in this paper will be described the necessary and sufficient condition of linear independent over the symmetrized max-plus algebra. We show that a columns of a matrix over the symmetrized max-plus algebra are linear dependent if and only if the determinat of that matrix is ɛ. Mathematics Subject Classification: 15A80
248 Gregoria Ariyanti, Ari Suparwanto and Budi Surodjo Keywords: the symmetrized max-plus algebra, linearly independence, determinant 1 Introduction The system max-plus algebra lacks an additive inverse. Therefore, some equations do not have a solution. For example, the equation 3 x= 2 has no solution since there is no x such that max(3,x = 2. In De Schutter [3] and Singh [6] state that one way of trying to solve this problem is to extend the max-plus algebra to a larger system which will include and additive inverse in the same way that the natural numbers were extended to the larger system of integers. Therefore, we have that the system (S,, is called the symmetrized max-plus algebra and S = R 2 ɛ/b with B is an equivalence relation. As in conventional linear algebra we can define the linearly dependence and independence of vectors in the max-plus sense [4]. However the definitions are a little more complex. Akian et al. [1] and Farlow [4] state that a family m 1, m 2,..., m k of elements of a semimodule M over a semiring S is linearly dependent in the Gondran-Minoux sense if there exist two subset I, J K := 1, 2,..., k, I J =, I J = K, and scalars α 1, α 2,..., α k S, not all equal to 0, such that i I α im i = j J α jm j. Motivated by linearly dependence and independence over the max-plus algebra, in this paper will be described linearly independence over the symmetrized max-plus algebra. In Gaubert [5] and De Schutter [3], a matrix over the symmetrized max-plus algebra can be characterized with determinant. In this paper, we show that the necessary and sufficient condition of linearly independece over the symmetrized max-plus algebra. In Section 2, we will review some basic facts for the symmetrized max-plus algebra and a matrix over the symmetrized max-plus algebra. In Section 3, we show linearly independence of a vector set over the symmetrized max-plus algebra. 2 Some Preliminaries on The Symmetrized Max- Plus Algebra In this section, we review some basic facts for the symmetrized max-plus algebra and a matrix over the symmetrized max-plus algebra, following [3] and [6].
Linearly independence over the symmetrized max-plus algebra 249 2.1 The Symmetrized Max-Plus Algebra Let the set of all real numbers R ɛ = R {ɛ} with ɛ := and e := 0. For all a, b R ɛ, the operations and are defined as a b = max (a, b and a b = a + b. Then,(R ɛ,, is called the max-plus algebra. Definition 2.1. [3] Let u = (x, y, v = (w, z R 2 ɛ. 1. Two unary operators and (. are defined as u = (y, x and u = u ( u. 2. An element u is called balances with v, denoted by u v, if x z = y w. 3. A relation B is defined as follows : { (x, y (w, z, if x y and w z (x, yb(w, z if (x, y = (w, z, otherwise Because B is an equivalence relation, we have the set of factor S = R 2 ɛ/b and the system (S,, is called the symmetrized max-plus algebra, with the operations of addition and multiplication on S is defined as follows (a, b (c, d = (a c, b d (a, b (c, d = (a c b d, a d b c for (a, b, (c, d S. The system (S,, is a semiring, because (S, is associative, (S, is associative, and (S,, satisfies both the left and right distributive. Lemma 2.2. [2] Let (S,, be the symmetrized max-plus algebra. Then the following statements holds. 1. (S,, is commutative. 2. An element (ɛ, ɛ is a zero element and an absorbent element. 3. An element (e, ɛ is an identity element. 4. (S,, is an additively idempotent. The system S is divided into three classes, there are S consists of all positive elements or S = {(t, ɛ t R ɛ } with (t, ɛ = {(t, x R 2 ɛ x < t}, S consists of all negative elements or S = {(ɛ, t t R ɛ } with (ɛ, t = {(x, t R 2 ɛ x < t}, and S consists of all balanced elements or S = {(t, t t R ɛ } with (t, t = {(t, t R 2 ɛ}. Because S isomorfic with R ɛ, so it will be shown that for a R ɛ, can be expressed by (a, ɛ S. Furthermore, it is easy to verify that for a R ɛ we have that a = (a, ɛ with (a, ɛ S, a = (a, ɛ = (a, ɛ = (ɛ, a with (ɛ, a S, and a = a a = (a, ɛ (a, ɛ = (a, ɛ (ɛ, a = (a, a S.
250 Gregoria Ariyanti, Ari Suparwanto and Budi Surodjo Lemma 2.3. For a,b R ɛ,a b = (a, b. Proof. a b = (a, ɛ (b, ɛ = (a, ɛ (ɛ, b = (a, b. Lemma 2.4. For (a, b S with a,b R ɛ, the following statements hold : 1. If a>b then (a, b = (a, ɛ. 2. If a<b then (a, b = (ɛ, b. 3. If a=b then (a, b = (a, a or (a, b = (b, b. Proof. 1. For a > b we have that a b = a. In other words, a ɛ = a b. The result that (a, b (a, ɛ. So it follows that (a, bb(a, ɛ. Therefore (a, b = (a, ɛ. 2. For a < b we have that a b = b. In other words, a b = b ɛ. The result that (a, b (ɛ, b. So it follows that (a, bb(ɛ, b. Therefore (a, b = (ɛ, b. Corollary 2.5. For a, b R ɛ, a b = a, if a > b b, if a < b a, if a = b 2.2 Matrices over The Symmetrized Max-Plus Algebra Let S the symmetrized max-plus algebra, n a positive integer greater than 1 and M n (S is the set of all nxn matrices over S. Operation and for matrices over the symmetrized max-plus algebra are defined as C = A B where c ij = a ij b ij and C = A B where c ij = l a il b lj. Zero matrix nxn over S { is ɛ n with (ɛ n ij = ɛ and identity matrix nxn over S is E n with e, if i = j [E n ] ij = ɛ, if i j. Definition 2.6. We say that the matrix A M n (S is invertible over S if A B E n dan B A E n for any B M n (S. Definition 2.7. [5] Let a matrix A M n (S. The determinant of A is defined by det A = σ S n sgn(σ ( n i=1 A iσ(i with S n is the set of all { 0, if σ is even permutation permutations of {1, 2,..., n}, and sgn(σ = 0, if σ is odd permutation. Note that the operator and the systems of max-linear balances hold :
Linearly independence over the symmetrized max-plus algebra 251 Lemma 2.8. [2] 1. a, b, c S, a c b a b c 2. a, b S S, a b a = b 3. Let A M n (S. The homogeneous linear balance A x ɛ nx1 has a non trivial solution in S or S if and only if det(a ɛ. 3 Linear Independence over The Symmetrized Max-Plus Algebra The symmetrized max-plus algebra is an idempotent semiring, so in order to define rank, linear combination, linear dependence, and independence we need definition of a semimodule. A semimodule is essentially a linear space over a semiring. Let S be a semiring with ɛ as a zero. A (left semimodule M n 1 (S over S is a commutative monoid (S, with zero element ɛ M n 1 (S, together with an S multiplication S M n 1 (S M n 1 (S, (r, x r.x = r x such that, for all r, s S and x, y M n 1 (S, we have 1. r (s x = (r s x 2. (r s x = r x s x 3. e x = x 4. r ɛ = ɛ 5. r (x y = r x r y In a similar way, a right semimodule can be defined. The rank, linear combination, and linear independent are given in the next definitions. Definition 3.1. [3] Let A M mxn (S. The max-algebraic minor rank of A, rank (A, is the dimension of the largest square submatrix of A the maxalgebraic determinant of which is not balanced. Definition 3.2. Let a 1, a 2,..., a n M nx1 (S and α 1, α 2,..., α m S. The expression m i=1 α i a i is called a linear combination of {a 1, a 2,..., a n }. Definition 3.3. A set of vectors {a i M nx1 (S i = 1, 2,..., m} is said to be a linear independent set whenever the only solution for the scalars α i in m i=1 α i a i ɛ nx1 is the trivial solution α i = ɛ.
252 Gregoria Ariyanti, Ari Suparwanto and Budi Surodjo The relation between linear dependent and linear combination are given in the following theorem. Lemma 3.4. If the set of vectors {a i M nx1 (S i = 1, 2,..., n} is linear dependent then one of the vectors can be presented as a linear combination of the other vectors in the set. Proof. Let α 1 a 1 α 2 a 2... α n a n ɛ. Because {a i M nx1 (S i = 1, 2,..., n} is a linear dependent set, without loss of generality, we can take α 1 ɛ. So, there is a scalar α 1 1 such that α 1 1 (α 1 a 1 α 2 a 2... α n a n α 1 1 ɛ a 1 α1 1 α 2 a 2... α1 1 α n a n ɛ n a 1 β 2 a 2 β 3 a 3... β n a n or a 1 β i a i i=2 This leads to a characterization of linear dependent in term of determinants. Theorem 3.5. Let {a i M nx1 (S i = 1, 2,..., n} be a vector set. Construct a matrix A such that A = ( a 1 a 2... a n. The set of vectors {a i M nx1 (S i = 1, 2,..., n} are linear dependent if and only if det(a ɛ. Proof. ( Because {a i M nx1 (S i = 1, 2,..., n} is a linear dependent set, this implies that one of the vectors (without loss of generality, let s say its the vector a 1 can be presented as a linear combination of the other vectors in the set. Or, a 1 β 2 a 2 β 3 a 3... β n a n or a 1 n β i a i Next, take matrix A and subtract a 1 with n i=2 β i a i. This results in another matrix (say A whose last column is a ɛ vector. A = ( a a 2... a n ( ɛ a2... a n Because det ( ɛ a 2... a n ɛ so, we now have that det(a ɛ. It follows from the fact that one of the columns of the matrix being ɛ, that det(a ɛ. ( Let {a i M nx1 (S i = 1, 2,..., n} is a linear independent. We can show that det(a not balanced with ɛ. Construct α 1 a 1 α 2 a 2... α n a n ɛ. i=2
Linearly independence over the symmetrized max-plus algebra 253 We have that a 1 α 1 a 2 α 2... a n α n ɛ. α 1 Consequently, ( α 2 a 1 a 2... a n. ɛ. Because {a i M nx1 (S i = 1, 2,..., n} is a linear independent, so we have α n α 1 = α 2 =... = α n = ɛ We can see, that homogenous linear balance A x ɛ has a trivial solution x = ɛ. Since it follows from lemma 2.8 that the homogeneous linear balance A x ɛ has a non trivial signed solution if and only if deta ɛ, so we have det(a not balanced with ɛ. Example 3.6. Let {v 1, v 2, v 3, v 4 } M 4 1 (S with 0 2 ɛ 7 ɛ, v 2 = 9 5, v 3 = 3 6 2 4 0 v 1 =, and v 4 = 0 7 9 3 We have, v 2 and v 4 are linear combinations from {v 1, v 3 } such that and v 2 = 2 v 1 ( 1 v 3 v 4 = v 1 3 v 3. We have det ( v 1 v 2 v 3 v 4 = det 0 2 ɛ 0 7 9 3 7 ɛ 5 6 9 2 4 0 3 = (18 16 ɛ 12 ( 10 17 18 ɛ = 18 18 = 18 ɛ. From Theorem 3.5, we have that {v 1, v 2, v 3, v 4 } is linearly dependent. The following theorem shows that relation between non trivial solution and the rank of matrix in the homogeneous linear balance. Theorem 3.7. Let A M mxn (S. The homogeneous linear balance A x ɛ has a non trivial signed solution (i.e. x / M n 1 (S if and only if rank (A < n. Proof. ( Let rank(a = n = r. We have A r x ɛ can be represented as ( Er A r x = x ɛ. ɛ
254 Gregoria Ariyanti, Ari Suparwanto and Budi Surodjo So, the only solution of A r x ɛ is x ɛ. ( Let A M mxn (S and rank(a = r < n. Suppose A r, the row echelon form of A can be represented as a form e ɛ... ɛ... ɛ e... ɛ........ ( Er C A r = ɛ ɛ... e... = ɛ ɛ ɛ ɛ... ɛ ɛ... ɛ..... ɛ ɛ... ɛ ɛ... ɛ Without loss of generality, the rank A is r and there is at least ( r + 1 columns. Er C Therefore, C has at least one columns and A r x = x = ( ( ɛ ɛ Er C z ɛ. Therefore, E ɛ ɛ y r z C y ɛ or z = C y. We ( C y have shown that x = is non trivial solution for y not balanced y with ɛ. 2 1 0 ɛ Example 3.8. Let A x ɛ with A = ɛ 1 0 ɛ. 1 0 ɛ 1 ( 2 With row echelon form, we have rank(a = 3 < 4 and C = ( 1 ( 2 1 We can show that for y = 1, we have x = 0 ( 1 is nontrivial solution. 1 Theorem 3.9. Let the set of vectors S = {a i M nx1 (S i = 1, 2,..., n}. Construct a matrix A such that A = ( a 1 a 2... a n with m n. If n > m then S is linear dependent. Proof. Let {a i M nx1 (S i = 1, 2,..., n} and construct α 1 a 1 α 2 a 2... α n a n ɛ. We have that a 1 α 1 a 2 α 2... a n α n ɛ. α 1 So, ( α 2 a 1 a 2... a n. ɛ. Because A = ( a 1 a 2... a n with α n m n, m < n and rank(a n, this implies that rank(a m < n.
Linearly independence over the symmetrized max-plus algebra 255 Since it follows from Theorem 3.7, so ( a 1 a 2... a n α 1 α 2. α n ɛ has a non trivial solution. Therefore, there is α i ɛ. This means that S = {a i M nx1 (S i = 1, 2,..., n} is linearly dependent. References [1] M. Akian, S. Gaubert and A. Guterman, Linear Independence over Tropical Semirings and Beyond, American Mathematical Society, 2008. [2] B. De Schutter, Max-Algebraic System Theory for Discrete Event Systems, Diss., Faculty of Applied Sciences, K.U. Leuven, Leuven, Belgium, 1996. [3] B. De Schutter and B. De Moor, A Note on The Characteristic Equation in The Max-Plus Algebra, Linear Algebra and Its Applications, 261 (1997, no. 1 3, 237 250. https://doi.org/10.1016/s0024-3795(9600407-7 [4] Kasie G. Farlow, Max Plus Algebra, Diss., Faculty of the Virginia Polytechnic Institute and State University in Partial Fulfillment of the Requirements for the Degree of Masters, 2009. [5] S. Gaubert, Théorie des Systèmes Linéaires dans les Dioïdes, Diss., Ecole Nationale Supérieure des Mines de Paris, France, 1992. [6] D. Singh, M. Ibrahim and J.N. Singh, A Note on Symmetrized Max-Plus Algebra, Journal of Mathematical Sciences and Mathematics Education, 5 (2008, no. 1, 1-10. Received: July 25, 2018; Published: September 22, 2018