- Email: [email protected]
On nilpotency of generalized fuzzy matrices
Fuzzy Sets and Systems 161 (2010) 2213 – 2226 www.elsevier.com/locate/fss
On nilpotency of generalized fuzzy matrices夡 Yi-Jia Tan∗ Department of Mathematics, Fuzhou University, Fuzhou 350108, China Received 1 April 2009; received in revised form 12 March 2010; accepted 31 March 2010 Available online 8 April 2010
Abstract In this paper, generalized fuzzy matrices are considered as matrices over a special type of semiring which is called path algebra. Some properties and characterizations for nilpotent generalized fuzzy matrices are established, reduction of generalized fuzzy matrices is defined and some properties of reduction of nilpotent generalized fuzzy matrices are obtained. Partial results obtained in this paper generalize the corresponding ones on fuzzy matrices, on lattice matrices and on incline matrices. © 2010 Elsevier B.V. All rights reserved. Keywords: Algebra; Generalized fuzzy matrix; Nilpotent matrix; Reduction of matrix; Path algebra
1. Introduction Nilpotency of generalized fuzzy matrices [17] over a special type of semiring is considered. The semiring is called path algebra (see [16]) (or additively idempotent semiring (see [10]). Boolean or fuzzy algebra, bounded distributive lattices and inclines are special cases of path algebra. Boolean matrices, fuzzy matrices, lattice matrices (see [9]) and incline matrices (see [2]) are the prototypical examples of matrices over path algebras. Path algebras are useful tools in diverse areas such as design of switching circuits, automata theory, information systems, dynamic programming and decision theory. For further examples, see [3,11]. The study of matrices over general semirings has long history. In 1964, Rutherford [25] gave a proof of the Cayley– Hamilton theorem for a commutative semiring avoiding the use of determinants. Since then, a number of works on theory of matrices over semirings were published (see e.g. [8,12,21,22,24]). In 1999, Golan described semirings and matrices over semirings in his work [10] comprehensively. The techniques of matrices over semirings have important applications in optimization theory, models of discrete event networks and graph theory. For further examples, see [1,7]. Nilpotent matrices are an important type of matrices. Since the beginning of the 1960s, many authors have studied this type of matrices for some special cases of path algebras (see e.g. [9,13,15,19,23,26,28,30]). In 1964, Give’on [9] proved that an n × n lattice matrix A is nilpotent if and only if An = O. In [15], Hashimoto considered the reduction of nilpotent fuzzy matrices and obtained some properties of the reduction. Nilpotent fuzzy matrices represent acyclic fuzzy graphs, and in general, acyclic graphs are used to represent consistent systems and are important in the representation of precedence relations [6,14,20]. In [19], Li gave some characterizations for nilpotent fuzzy matrices. The results by 夡 Supported by Natural Science Foundation of Fujian province (2008J0194), China. ∗ Tel.: +86 591 88174165.
E-mail address: [email protected]. 0165-0114/$ - see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2010.03.019
2214
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
Hashimoto [15] and Li [19] were generalized to nilpotent lattice matrices by Tan [28]. Ren et al. [23] showed that a fuzzy matrix A is nilpotent if and only if every principal minor of A is 0. This result was generalized to lattice matrices by Tan [26] and Zhang [30], independently. Recently, Han et al. [13] characterized the nilpotent matrices over an incline without nonzero nilpotent elements in terms of principal minors, main diagonals, nilpotent indices and adjoint matrices, and established some properties of the reduction of nilpotent matrices over an additively residuated incline without nonzero nilpotent elements. In the present work, we will consider the nilpotent matrices over more general path algebras, namely over a class of commutative path algebras. In Section 3, we will give some properties and characteristics for the nilpotent matrices. In Section 4, we consider the reduction of the nilpotent matrices over an additively residuated commutative path algebra and obtain some properties of the reduction. Partial results in this work generalize and develop the corresponding results on fuzzy matrices in [15,19,23], on lattice matrices in [9,26,28,30] and on incline matrices in [13]. 2. Definitions and preliminary lemmas In this section, we shall give some definitions and lemmas. For convenience, we use n to denote the set {1,2, . . . ,n}. Definition 2.1 (Golan [10]). An algebraic system (L , +, ·) is called a semiring if (L,+) is an Abelian monoid with identity element 0 and (L , ·) is another monoid with identity element 1 and 0 1, connected by ring-like distributivity, and 0r = r 0 = 0 for all r ∈ L. A semiring L is called commutative if ab = ba for all a, b ∈ L; L is called an antiring if a + b = 0 implies that a = b = 0 for all a, b ∈ L (see [29]). Antirings were studied in [10] under the name of zerosumfree semirings. Definition 2.2 (Carrè [3], Hashimoto [16]). A semiring L is called a path algebra if a + a = a for all a ∈ L. Note that every path algebra is an antiring. Indeed, if L is a path algebra and a + b = 0 for a, b ∈ L then a = a + 0 = a + (a + b) = (a + a) + b = a + b = 0 and similarly b = 0. Path algebras were studied in [10] under the name of additively idempotent semiring. Path algebras are abundant: for example, every Boolean algebra, the fuzzy algebra F = ([0, 1], ∨, T ), where ∨ = max and T is a t-norm (for t-norm, refer to [18]), and every bounded distributive lattice are commutative path algebras. In addition, the max-plus algebra (R ∪ {−∞}, max, +) and the min-plus algebra (R ∪ {+∞}, min, +) (see [4,31]) are commutative path algebras. A semiring L is called an incline (see [2,10]) if a + 1 = 1 for all a ∈ L. Note that any incline is a path algebra. In fact, if L is an incline then 1 + 1 = 1, and so for any a ∈ L, a = a(1 + 1) = a + a. Throughout this paper, L is always supposed to be a commutative path algebra. Definition 2.3. An element a in a path algebra L is said to be nilpotent if a k = 0 for some positive integer k. The least positive integer k satisfying a k = 0 is called the nilpotent index of a and denoted by h(a). The set of all nilpotent elements in L is denoted by Z(L). It is clear that if L is a Boolean algebra or a bounded distributive lattice, then Z (L) = {0}, that is, L has no nonzero nilpotent elements. Example 2.1. Let L = ([0, 1], ∨, ·), where [0, 1] is the unit interval, a ∨ b = max{a, b} and a · b = (a + b − 1) ∨ 0 for a, b ∈ [0, 1]. It is easy to verify that L is a commutative path algebra. For any a ∈ [0, 1], we can see that a k = (ka − k + 1) ∨ 0. Obviously, a k = 0 if and only if a ≤ 1 − (1/k). Hence Z (L) = [0, 1). For any path algebra L, we define a relation “≤ ” on L as follows, a ≤ b if and only if a + b = b for all a,b in L. The following properties are derived immediately from above definition. (1) The relation ≤ is a partial order on L. (2) The element 0 is the least element in L, i.e., 0 ≤ a for all a ∈ L. (3) For any a, b, c, d ∈ L, a ≤ b and c ≤ d implies that a + c ≤ b + d and ac ≤ bd. In the following, we consider matrices over commutative path algebras as generalized fuzzy matrices.
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
2215
Let L be a commutative path algebra. Denote by Mm×n (L) the set of all m × n matrices over L. Especially, we put Mn (L) = Mn×n (L). For A ∈ Mm×n (L), we denote by ai j or Ai j the element of L which stands in the (i,j)-entry of A, and denote by A T the transpose of A. For any A, B ∈ Mm×n (L) and C ∈ Mn×l (L), we define: A+B = (ai j + bi j )m×n ; AC = ( nk=1 aik ck j )m×l ; A ≤ B if ai j ≤ bi j for i ∈ m and j ∈ n; It is easy to verify that (Mn (L), +, ·) is a path algebra, and A+C ≤ B + D and AC ≤ B D for all A, B, C, D ∈ Mn (L) with A ≤ B and C ≤ D. For A ∈ Mn (L), the powers of A are defined as follows: A0 = In , Al = Al−1 A, l = 1,2, . . . . where In is the identity matrix of order n. Definition 2.4. Let A ∈ Mn (L). A is said to be nilpotent if Ak = O for some positive integer k, where O denotes the zero matrix. The least positive integer k satisfying Ak = O is called the nilpotent index of A, and denoted by h(A). Let A ∈ Mn (L). If A2 ≤ A then A is called transitive (see [16]); if aii is nilpotent for any i ∈ n then A is called weakly irreflexive, especially, if aii = 0 for any i ∈ n then A is called irreflexive. If there exists a B ∈ Mn (L) such that A ≤ B and B is transitive and B ≤ C for any transitive matrix C in Mn (L) satisfying A ≤ C, then the matrix B is called a transitive closure of A. It is clear that if A has a transitive closure then it is unique. The transitive closure of A is denoted by t(A). Hashomoto [16] discussed some compositions of transitive matrices over a path algebra and obtained some important properties. Duan [5] considered the transitive closure of matrices over an incline and proved that any n × n matrix A over an incline has transitive closure and t(A) = nk=1 Ak . However, for general path algebra L, there may be some matrices over L which have no transitive closure. Example 2.2. Let L = ([0, +∞), ∨, ·), where ∨ = max and “·” is the usual multiplication of real numbers. Then L is a commutative but not an incline. path algebra 2 k 0 Let A = 0.4 , B = k 0 k ∈ M2 (L), where k ∈ L. Then it is easy to verify that A has transitive closure 0.4 0.4 0.8 2 and t( A) = 0.4 0.8 , but Bk have no transitive closure for all k > 1. Definition 2.5. Let A ∈ Mn (L). The permanent per A of A is defined as follows: per A = a1(1) a2(2) · · · an (n) ∈Sn
where Sn denotes the symmetric group of the set n. Let U = {i 1 , i 2 , . . . , ir }, V = { j1 , j2 , . . . , jr } ⊆ n. For A ∈ Mn (L), we denote by A[U |V ] the r × r submatrix of A whose (u, v)-entry is equal to aiu jv (u, v ∈ r ) and by A(U |V ) the (n−r )×(n−r ) submatrix of A obtained by deleting rows i 1 , i 2 , . . . , ir and columns j1 , j2 , . . . , jr from A, where 1 ≤ i 1 < i 2 < · · · < ir ≤ n and 1 ≤ j1 < j2 < · · · < jr ≤ n. The matrix A[U |U ] is called a principal submatrix of order r of A, and per A[U |U ] is called a principal minor of order T . The matrix adjA is called the adjoint matrix of A. r of A. Let ad j A = ( per A(i| j))n×n Lemma 2.1. Let L be a commutative path algebra, a, b, a1 , a2 , . . . , am ∈ L. Then (1) if a is nilpotent and b ≤ a then b is nilpotent; (2) if a is nilpotent then ar and ra are nilpotent for any r in L; m (3) i=1 ai is nilpotent if and only if a1 , a2 , . . . , am are nilpotent.
2216
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
Proof. (1) and (2) are clear. (3) By binomial theorem and idempotency of the operation “+” in L, we have (x + y) = k
k
x k−l y l
(2.1)
l=0 0 for any x,y in L and any m mpositive integer k (Note that x = 1 for any element x in L). ai is nilpotent. Then, by (1), we have that each ai is nilpotent since ai ≤ i=1 ai for any Suppose now that i=1 m i ∈ m. In the following we will prove that if a1 , a2 , . . . , am are nilpotent then i=1 ai is nilpotent. We shall prove the statement by induction on m. It is clear if m = 1, and we may assume it holds for m − 1(m − m−1 1 ≥ 1). Suppose that a1 , a2 , . . . , am are nilpotent. Then i=1 ai is nilpotent (by the induction hypothesis). Let m−1 m m−1 k m−1 k−l l ai ) + am )k = l=0 ( a ) am k1 = h( i=1 ai ) and k2 = h(am ) and k = k1 + k2 . Then ( i=1 ai )k = (( i=1 i=1 i l = 0, and if l < k then k−l ≥ k and so ( m−1 a )k−l = 0. This implies ( m−1 a )k−l a l = 0 (by (2.1)). If l ≥ k2 then am 2 m i=1 i i=1 i m m 1 for all l ∈ {0, 1, . . . , k}. Then ( i=1 ai )k = 0, i.e., i=1 ai is nilpotent. This proves (3).
Lemma 2.2. Let A, B ∈ Mn (L). Then (1) A is nilpotent if and only if P A P T is nilpotent for any n × n permutation matrix P; (2) if A is nilpotent and B ≤ A then B is nilpotent and h(B) ≤ h( A). The proof is omitted. Lemma 2.3. Let A ∈ Mn (L). Then T (1) per A = per n(A ); (2) per A = j=1 ai j per A(i| j) for any i ∈ n; (3) ad j( A T ) = (ad j A)T .
The proof is trivial. Lemma 2.4. Let A ∈ Mn (L) be nilpotent. Then t( A) = Proof. Let B = B2 =
h(A)−1 k=1
2h(A)−2
h(A)−1 k=1
Ak .
Ak . Obviously, A ≤ B. Since (Mn (L), +, ·) is a path algebra, we have
Ak
k=2
=
h(A)−1
Ak (because Al = O for all l ≥ h( A))
k=2
≤ B. i.e., B is transitive. If there exists a transitive matrix C in Mn (L) such that A ≤ C then C 2 ≤ C and so C k ≤ C for all positive integers k. Hence Ak ≤ C k ≤ C, and so B = A + A2 + · · · + Ah(A)−1 ≤ C. By the definition of transitive closure, we obtain t( A) = B. This proves the lemma. Remark 2.1. Lemma 2.4 shows that any nilpotent matrix over a path algebra has transitive closure. A path algebra L is said to be additively residuated if for any a, b ∈ L there exists an element r ∈ L such that a + x ≥ b if and only if x ≥ r . Obviously, r is the least element x ∈ L satisfying a + x ≥ b. The element r will be denoted by b−a, i.e., a + x ≥ b if and only if x ≥ b − a. Remark 2.2. The fuzzy algebra ([0, 1], ∨, ∧), where ∨ = max and ∧ = min, is an additively residuated path algebra and in this case the operation “–” coincides with the operation “d” defined in [15], i.e., a if b < a a − b = adb = (1) 0 if a ≤ b where b < a means that b ≤ a and b a.
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
2217
Remark 2.3. Any Boolean algebra L is an additively residuated path algebra and in this case a − b = abc , where bc denotes the complement element of b in L. Remark 2.4. Every dually Brouwerian lattice (see [27]) and any additively residuated incline (see [13]) are additively residuated path algebras. Lemma 2.5. Let L be an additively residuated path algebra. Then for any a, b, c, a1 , a2 , . . . , al , b1 , b2 , . . . , bl ∈ L(l ≥ 2), we have (1) (2) (3) (4) (5) (6)
(a − b) + b ≥ a and a−a = 0, and a−0 = a; a(b − c) ≥ ab − ac and (a − b)c ≥ ac − bc; (a − b) − c ≥ a − (b + c); if al ≤ b then a − c≤l b − c and l c − a ≥ c − b; (a − b ) ≥ a − i i i=1 i i=1 bi ; i=1 l−1 ( i=1 (ai − ai+1 )) + al = li=1 ai .
Proof. (1) and (4) are clear, we will prove (2), (3), (5) and (6). (2) Since c + (b − c) ≥ b (by (1)), we have ac + a(b − c) ≥ ab, and so a(b − c) ≥ ab − ac. Similarly, we can prove that (a − b)c ≥ ac − bc. This proves (2). (3) Since (b + c) + ((a − b) − c) = b + (c + ((a − b) − c)) ≥ b + (a − b) ≥ a, we have (a − b) − c ≥ a − (b + c). This proves (3). (5) Since li=1 bi + li=1 (ai − bi ) =
l
(bi + (ai − bi ))
i=1
≥
l
ai (by (1)),
i=1
we have l
(ai − bi ) ≥
i=1
l i=1
ai −
l
bi .
i=1
This proves (5). (6). We will prove (6) by induction on l. By (1), we have (a1 − a2 ) + a2 ≥ a1 , and so (a1 − a2 ) + a2 = (a1 − a2 ) + (a2 + a2 ) = ((a1 − a2 ) + a2 ) + a2 ≥ a1 + a2 . On the other hand, since a1 ≥ a1 − a2 , we have a1 + a2 ≥ (a1 − a2 ) + a2 . Hence (a1 − a2 ) + a2 = a1 + a2 and (6) holds for l = 2. Assume that (6) holds for l − 1(l − 1 ≥ 2). Then l−1 (ai − ai+1 ) + al = ((a1 − a2 ) + (a2 − a3 ) + · · · + (al−1 − al )) + al i=1
= (a1 − a2 ) + ((a2 − a3 ) + · · · + (al−1 − al ) + al ) = (a1 − a2 ) + (a2 + a3 + · · · + al ) (by the induction hypothesis) = ((a1 − a2 ) + a2 ) + (a3 + · · · + al ) = a1 + a2 + · · · + al =
l
ai .
i=1
This proves (6).
Let L be an additively residuated path algebra. Given A, B ∈ Mm×n (L), we define: A − B = (ai j − bi j )m×n .
2218
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
Lemma 2.6. Let L be a commutative additively residuated path algebra. Then for any A, B, C, A1 , A2 , . . . , Al , B1 , B2 , . . . , Bl ∈ Mn (L), we have (1) (2) (3) (4) (5) (6)
(A − B) + B ≥ A and A−A = O, and A−O = A; A(B − C) ≥ AB − AC and ( A − B)C ≥ AC − BC; (A − B) − C ≥ A − (B + C); if A ≤ B then A − C ≤ B − C and C − A ≥ C − B; l (Ai − Bi ) ≥ li=1 Ai − li=1 Bi ; i=1 l ( l−1 i=1 Ai . i=1 (Ai − Ai+1 )) + Al =
Proof. (1) and (3)–(6) can be obtained from (1) and (3)–(6) in Lemma 2.5, we only prove (2). (2) For any i, j ∈ n, we have n aik (bk j − ck j ) (A(B − C))i j = k=1
≥
n
(aik bk j − aik ck j ) (by Lemma 2.5(2))
k=1
≥
n
aik bk j
−
k=1
n
aik ck j
(by Lemma 2.5(5))
k=1
= (AB)i j − ( AC)i j = ( AB − AC)i j . Therefore A(B − C) ≥ AB − AC. Similarly, we can prove (A − B)C ≥ AC − BC. 3. Some properties and characteristics of nilpotent matrices In [13], Han et al. gave some important features of nilpotent matrices over an incline without nonzero nilpotent elements (see Theorems 3.4 and 3.7 in [13]). In this section, we will extend these results to matrices over a general path algebra and obtain some properties and characteristics of the nilpotent matrices. Proposition 3.1. Let A = (ai j ) ∈ Mn (L). If every element of A is nilpotent then A is nilpotent. Proof. Suppose that every element of A is nilpotent. Let h(ai j ) = ki j for any i, j ∈ n and k = max{ki j |i, j ∈ n}, and put l = n 2 k. Then aikj = 0 for any i, j ∈ n. In the following we will prove Al = O. It is clear that (Al )i j = i ,i , ...,i ∈n aii1 ai1 i2 · · · ail−1 j for any i, j ∈ n. Let T = aii1 ai1 i2 · · · ail−1 j be any term of 1 2
l−1
( Al )i j . Since aii1 , ai1 i2 , . . . , ail−1 j ∈ {ast |s, t ∈ n} and the set {ast |s, t ∈ n} contains at most n 2 different elements, there exist t0 , t1 , . . . , tk−1 ∈ {0, 1, . . . , l − 1} such that t0 < t1 < · · · < tk−1 and ait0 it0 +1 = ait1 it1 +1 = · · · = aitk−1 itk−1 +1 (taking i 0 = i and il = j) (note that l = n 2 k), and so T has the factor (ait0 it0 +1 )k . But (ait0 it0 +1 )k = 0, we have T = 0. Since T is any term of ( Al )i j , we have ( Al )i j = 0 for any i, j ∈ n. This means that Al = O. Proposition 3.2. Let A ∈ Mn (L). If all main diagonal entries of Ak are nilpotent for any k ∈ n, then: (1) all elements of An are nilpotent; (2) A is nilpotent. Proof. (1) Let A = (ai j ). Then (An )i j =
a a i 1 ,i 2 , ...,i n−1 ∈n ii 1 i 1 i 2
· · · ain−1 j for any i, j ∈ n. Let T = aii1 ai1 i2 · · · ain−1 j
be any term of Since i, i 1 , . . . , i n−1 , j ∈ n, there exist s, t ∈ {0, 1, . . . , n} such that i s = i t with s < t (taking i 0 = i and i n = j), and so ais is+1 · · · ait−1 it = ais is+1 · · · ait−1 is . But ais is+1 · · · ait−1 is is a term of (At−s )is is and (At−s )is is is nilpotent, we have ais is+1 · · · ait−1 it is nilpotent (by Lemma 2.1(3)). Since ais is+1 · · · ait−1 it is a factor of T, T is nilpotent (by Lemma 2.1(2)), and so (An )i j is nilpotent for any i, j ∈ n (by Lemma 2.1(3)). This proves (1). (2) By (1) and Proposition 3.1, An is nilpotent, and so A is nilpotent. This proves (2). ( An )i j .
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
2219
Corollary 3.1. If A ∈ Mn (L) is weakly irreflexive and transitive, then A is nilpotent. Proof. Put A = (ai j ). Since A is transitive, A2 ≤ A and so Ak ≤ A for all k ∈ n. Then ( Ak )i j ≤ ai j for any i, j ∈ n. Since A is weakly irreflexive, we have aii is nilpotent for any i ∈ n, and so ( Ak )ii is nilpotent for any i, k ∈ n (by Lemma 2.1(1)), i.e., all main diagonal entries of Ak are nilpotent for any k ∈ n. By Proposition 3.2, we have that all elements of An are nilpotent. Then An is nilpotent (by Proposition 3.1) and so A is nilpotent. Proposition 3.3. Let A ∈ Mn (L). If all principal minors of A are nilpotent then (1) all elements of An are nilpotent; (2) A is nilpotent. Proof. (1) Put A = (ai j ) and consider any term T = aii1 ai1 i2 · · · ain−1 j of (An )i j for any i, j ∈ n. Since i, i 1 , . . . , i n−1 , j ∈ n, there exist s, t ∈ {0, 1, . . . , n} such that s < t and i s = i t (taking i 0 = i and i n = j). Let K = {( p, q)| p < q, i p = i q }. Then K ⭋ and there exists a (k, l) ∈ K satisfying l − k = min{q − p|( p, q) ∈ K }. But the product aik ik+1 · · · ail−1 il = aik ik+1 · · · ail−1 ik is a term of the principal minor per A[i k , i k+1 , . . . , il−1 |i k , i k+1 , . . . , il−1 ] and per A[i k , i k+1 , . . . , il−1 |i k , i k+1 , . . . , il−1 ] is nilpotent, the product aik ik+1 · · · ail−1 il is nilpotent (by Lemma 2.1(3)). Since aik ik+1 · · · ail−1 il is a factor of aii1 ai1 i2 · · · ain−1 j , we have that T = aii1 ai1 i2 · · · ain−1 j is nilpotent (by Lemma 2.1(2)). But T is any term of ( An )i j , we have ( An )i j is nilpotent for any i, j ∈ n (by Lemma 2.1(3)). This proves (1). (2) By (1) and Proposition 3.1, An is nilpotent, and so A is nilpotent. This proves (2). Proposition 3.4. If A ∈ Mn (L) is nilpotent, then (1) all main diagonal entries of Am are nilpotent for any positive integer m; (2) ai1 i2 ai2 i3 · · · aim i1 is nilpotent for any positive integer m and any i 1 , i 2 , . . . , i m ∈ n; (3) per A is nilpotent. Proof. (1) Since A is nilpotent, Al = O for some positive integer l, and so (Am )l = Aml = O for any positive integer m. Then ( Aml )ii = 0 for any i ∈ n. But (( Am )ii )l is some term of (Aml )ii , we have that ((Am )ii )l = 0 (by Lemma 2.1(3)), i.e., (Am )ii is nilpotent for any i ∈ n and any positive integer m. This proves (1). (2) Since ai1 i2 ai2 i3 · · · aim i1 is a term of (Am )i1 i1 , by (1) and Lemma 2.1(3), we have that ai1 i2 ai2 i3 · · · aim i1 is nilpotent. This proves (2). (3) Consider any term T = a1(1) a2(2) · · · an (n) of per A, where ∈ Sn . Since m (1) ∈ n for all positive integer m, there exist s, t ∈ {0, 1, . . . , n} such that s < t and s (1) = t (1) and s (1), s+1 (1), . . . , t−1 (1) are mutually different (note that 0 (1) = 1). Then as (1)s+1 (1) · · · at−1 (1)s (1) = as (1)s+1 (1) · · · at−1 (1)t (1) is a factor of T . But as (1)s+1 (1) · · · at−1 (1)s (1) is nilpotent (by (2)), T is nilpotent for any ∈ Sn (by Lemma 2.1(2)). Then per A is nilpotent (by Lemma 2.1(3)). This proves (3). Theorem 3.1. For any A ∈ Mn (L), the following statements are equivalent. (1) (2) (3) (4) (5)
A is nilpotent. All principal submatrices of A are nilpotent. All principal minors of A are nilpotent. All elements of An are nilpotent. For any k ∈ n, all main diagonal entries of Ak are nilpotent.
Proof. (1) ⇒ (2). Suppose that A is nilpotent. Let B be any principal submatrix of order r of A. Then there exists an n × n permutation matrix P such that B C B O O C = + P APT = E D O O E D T T where C ∈ Mr ×(n−r ) (L), E ∈ M(n−r )×r (L) and D ∈ Mn−r (L). This implies that OB O O ≤ P A P . But P A P is nilpotent (by Lemma 2.2(1)), the matrix OB O O is nilpotent (by Lemma 2.2(2)), and so B is nilpotent.
2220
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
(2) ⇒ (3). It follows from Proposition 3.4(3). (3) ⇒ (4). It follows from Proposition 3.3(1). (4) ⇒ (5). Suppose that all elements of An are nilpotent. Then, by Proposition 3.1, An is nilpotent and so A is nilpotent, and by Proposition 3.4(1), all main diagonal entries of Ak are nilpotent for any k ∈ n. (5) ⇒ (1). It follows from Propositions 3.2(2). If L has no nonzero nilpotent elements, then Z (L) = {0}. By Theorem 3.1, we have Corollary 3.2. If L has no nonzero nilpotent elements, then for any A ∈ Mn (L), the following statements are equivalent. (1) (2) (3) (4) (5)
A is nilpotent. All principal submatrices of A are nilpotent. All principal minors of A are 0. An = O. All main diagonal entries of Ak are 0 for any k ∈ n.
Remark 3.1. Corollary 3.2 shows that if L has no nonzero nilpotent elements and A ∈ Mn (L) is nilpotent then An = O (or h(A) n), but this result is not true for nilpotent matrices over a general path algebra. 0.7 Example 3.1. Consider the path algebra L in Example 2.1. Let A = 0.8 0.9 0.6 ∈ M2 (L). Then, we have 2 0.8 0.7 A2 = 0.9 0.6 ((0.8 + 0.8 − 1) ∨ 0) ∨ ((0.7 + 0.9 − 1) ∨ 0) ((0.8 + 0.7 − 1) ∨ 0) ∨ ((0.7 + 0.6 − 1) ∨ 0) = ((0.9 + 0.8 − 1) ∨ 0) ∨ ((0.6 + 0.9 − 1) ∨ 0) ((0.9 + 0.7 − 1) ∨ 0) ∨ ((0.6 + 0.6 − 1) ∨ 0) 0.6 0.5 = , 0.7 0.6 0.2 0.1 0 0 0 0 , A5 = and A6 = . A4 = 0.3 0.2 0.1 0 0 0 Clearly, A2 O and h( A) = 6 > 2. Remark 3.2. Corollary 3.2 generalizes and develops Corollary 5.2 of Give’on [9], Theorem 1 of Li [19], Theorem 1 of Ren et al. [23], Theorem 6.6 of Tan [26], Theorem 1 of Zhang [30] and Theorem 3.4 of Han et al. [13]. Given a matrix A ∈ Mn (L), we denote by A(i ⇒ j) the matrix obtained from A by replacing the row j with the row i. Theorem 3.2. If A ∈ Mn (L) is nilpotent, then (1) per A(i ⇒ j) is nilpotent for any i, j ∈ n; (2) All elements of both AadjA and (adjA)A are nilpotent; (3) All elements of (ad j A)2 are nilpotent. Proof. Let A ∈ Mn (L) be nilpotent. (1) For any i ∈ n, we have per A(i ⇒ i) = per A is nilpotent (by Proposition 3.4(3)). For any i, j ∈ n with i j, we have a1(1) · · · ai (i) · · · ai ( j) · · · an (n) . per A(i ⇒ j) = ∈Sn
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
2221
For any ∈ Sn , if l (i) j for all l ≥ 1, then there exists a d such that d (i) = i with 1 ≤ d ≤ n and i, (i), . . . , d−1 (i) are mutually different. Hence ai (i) a(i)2 (i) · · · ad−1 (i)i is nilpotent (by Proposition 3.4(2)). Since the product ai (i) a(i)2 (i) · · · ad−1 (i)i is a factor of the product a1(1) · · · ai (i) · · · ai ( j) · · · an (n) , we have that a1(1) · · · ai (i) · · · ai ( j) · · · an (n) is nilpotent (by Lemma 2.1(2)). If there exists a positive integer l such that l (i) = j, then there must be a positive integer d such that i = d ( j) with 1 ≤ d ≤ n and i, ( j), . . . , d−1 ( j) are mutually different. Thus ai ( j) a( j)2 ( j) · · · ad−1 ( j)i is nilpotent (by Proposition 3.4(2)). But the product ai ( j) a( j)2 ( j) · · · ad−1 ( j)i is a factor of the product a1(1) · · · ai (i) · · · ai ( j) · · · an (n) , again, we have a1(1) · · · ai (i) · · · ai ( j) · · · a n (n) is nilpotent (by Lemma 2.1(2)). Consequently, per A(i ⇒ j) = ∈Sn a1(i) · · · ai (i) · · · ai ( j) · · · an (n) is nilpotent (by Lemma 2.1(3)). This proves (1). (2) Let B = A · ad j A. Then, by (1) and Lemma 2.3(2), we have that bi j = k∈n aik per A( j|k) = per A(i ⇒ j) is nilpotent for all i, j ∈ n. By Lemma 2.3(3), we have ((ad j A)A)T = A T · (ad j A)T = A T · ad j( A T ). Then all elements of the matrix ((ad j A)A)T are nilpotent and so all elements of (adjA)A are nilpotent. This proves (2). (3) Let C = (ad j A)2 . For any i, j ∈ n, we have ci j = per A( j|k) per A(k|i). k∈n
Let c¯i j =
per A( j|k) per A(k|i)
k∈n i
where n i = n − {i}. Then ci j = per A( j|i) per A(i|i) + c¯i j . Since per A(i|i) is nilpotent (by Theorem 3.1), we have per A( j|i) per A(i|i) is nilpotent (by Lemma 2.1(2)). When k i, we have ⎛ ⎞ ⎝ per A(k|i) ≤ ast ≤ ast ⎠ . t∈n i
Therefore c¯i j ≤
k∈n i
=
k∈n i
=
k∈n i
s∈n k
⎛ ⎝( per A( j|k)) ·
t∈n i
⎛
⎝
⎛
⎝⎝ per A( j|k) · ⎝
⎞⎞ ast ⎠⎠
s∈n
t∈n i
⎛⎛
s∈n
⎞⎞ ask ⎠⎠ ·
s∈n
⎛⎛
⎝⎝
s∈n
⎞ ask per A( j|k)⎠ ·
t∈n ik
t∈n ik
⎛
⎝
⎛
⎝
⎞⎞ ast ⎠⎠ , (where n ik = n − {i,k})
s∈n
⎞⎞ ast ⎠⎠ .
s∈n
On the other hand, since per A(s ⇒ j) = for any s, j ∈ n (by (1)), we have k∈n ask per A( j|k) is nilpotent that ask per A( j|k) is nilpotent for any s, j, k ∈ n (by Lemma 2.1(3)), and so s∈n ask per A( j|k) is nilpotent for any j, k ∈ n (by Lemma 2.1(3)). Then ( s∈n ask per A( j|k)) · t∈n ik ( s∈n ast ) is nilpotent (by Lemma 2.1(2)), and so k∈n i (( s∈n ask per A( j|k)) · t∈n ik ( s∈n ast )) is nilpotent for any i, j ∈ n (by Lemma 2.1(3)). This implies that c¯i j is nilpotent for any i, j∈ n (by Lemma 2.1(1)). Since per A( j|i) per A(i|i) is nilpotent and ci j = per A( j|i) per A(i|i) + c¯i j , we have ci j is nilpotent for any i, j ∈ n (by Lemma 2.1(3)), i.e., all elements of (ad j A)2 are nilpotent. This proves (3).
2222
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
Corollary 3.3. If L has no nonzero nilpotent elements and A ∈ Mn (L) is nilpotent, then (1) per A(i ⇒ j) = 0 for any i, j ∈ n; (2) Aad j A = (ad j A)A = O; (3) (ad j A)2 = O. Remark 3.3. Corollary 3.3 generalizes and develops Proposition 3.4 of Tan [28] and Theorem 3.7 of Han et al. [13]. 4. Reduction of nilpotent matrices Let L be an additively residuated path algebra and A, B ∈ Mn (L). We define A/B = A−AB. The matrix A/A = A−A2 is called the reduction of A. It is clear that A/A ≤ A for all A ∈ Mn (L). Hashimoto [15] discussed the reduction of irreflexive and transitive fuzzy matrices and obtained some important properties, and applied these properties to nilpotent fuzzy matrices in order to obtain some properties of reduction of nilpotent fuzzy matrices. Tan [28] considered the reduction of nilpotent lattice matrices and extended the results in [15] to nilpotent matrices over dually Brouwerian lattices, and furthermore, Han et al. [13] extended the results in [28] to nilpotent matrices over additively residuated inclines without nonzero nilpotent elements. In this section, we will consider the reduction of nilpotent matrices over an additively residuated path algebra in general. Partial results obtained in this section generalize the previous results on nilpotent fuzzy matrices by Hashimoto [15] and on nilpotent lattice matrices by Tan [28] and on nilpotent incline matrices by Han et al. [13]. Theorem 4.1. If A ∈ Mn (L) is nilpotent, then t(A/A) = t(A). Proof. If A = O, then t( A/A) = t( A) = O. In the following, we assume A O (or h( A)2). Let B = A/A. Then B ≤ A and so B is nilpotent. Thus t(B) = t(A/A) ≤ t( A). To prove t(A) ≤ t( A/A) = t(B), we first prove that B l ≥ Al − Al+1
(4.1)
for all l ≥ 1. We will prove (4.1) by induction on l. For l = 1, B = A−A2 . Assume that (4.1) holds for l − 1(l ≥ 2). Then B l = B · B l−1 ≥ B( Al−1 − Al ) ≥ B Al−1 − B Al (by Lemma 2.6(2)). Since B = A − A2 ≤ A, we have B l ≥ ( A − A2 )Al−1 − Al+1 (by Lemma 2.6(4)) ≥ (Al − Al+1 ) − Al+1 (by Lemma 2.6(2)) ≥ Al − ( Al+1 + Al+1 ) (by Lemma 2.6(3)) = Al − Al+1 . Now t(A/A) = t(B) =
h(B)−1
B k (by Lemma 2.4)
k=1
=
h(A)−1
B k (by Lemma 2.2(2))
k=1
≥
h(A)−1
( Ak − Ak+1 ) (by (4.1))
k=1
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
=
h(A)−2
2223
( Ak − Ak+1 ) + Ah(A)−1 (because Ah(A) = O)
k=1
=
h(A)−1
Ak (by Lemma 2.6(6))
k=1
= t(A). Therefore t(A/A) = t( A). This completes the proof. Remark 4.1. Theorem 4.1 generalizes Theorem 4.1 of Tan [28] and Theorem 4.1 of Han et al. [13]. If A ∈ Mn (L) is weakly irreflexive and transitive then A is nilpotent (by Corollary 3.1) and t( A) = A. By Theorem 4.1, we have Corollary 4.1. If A ∈ Mn (L) is weakly irreflexive and transitive then t(A/A) = A. Theorem 4.2. If A ∈ Mn (L) is nilpotent, then (1) t(A)/t( A) ≤ A ≤ t(A); (2) t(A)/t( A) = A/t( A) = t(A)/A; (3) t(t(A)/t(A))=t(A). Proof. It is clear that (1)–(3) hold for A = O. In the following, we assume A O (or h( A) 2). Since A is nilpotent, we have t(A) = A + A2 + · · · + Ah(A)−1 = A + A2 + · · · + Ah(A)−1 + Ah(A) (because Ah(A) = O). = A + At( A). Then t( A) = A + At( A) (1) We have t(A)/t( A) = t( A) − (t(A))2 = t( A) − (A2 + A3 + · · · + A2h(A)−2 ) = t( A) − ( A2 + A3 + · · · + Ah(A) ) (because Al = O for all l ≥ h( A) + 1) = ( A + At( A)) − (O + At( A)) (by (4.2)) ≤ (A − O) + ( At(A) − At(A)) (by Lemma 2.6(5)) = A (by Lemma 2.6(1)). i.e., t( A)/t( A) ≤ A. Since A ≤ t( A), we have that (1) holds. (2) First, we have A/t(A) = A − At(A) = (A − At(A)) + ( At(A) − At(A)) (by Lemma 2.6(1)) ≥ (A + At( A)) − ( At(A) + At( A)) (by Lemma 2.6(5)) = t(A) − At( A) (by (4.2))
(4.2)
2224
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
= t(A) − t( A)A (because At(A) = t( A)A) = t( A)/A, and t( A)/A = t(A) − t( A)A ≥ A − At( A) = A/t( A). Then A/t( A) = t( A)/A. Second, we have t(A)/t( A) = t( A) − (t(A))2 = t(A) − ( A2 + A3 + · · · + A2h(A)−2 ) = t( A) − ( A2 + A3 + · · · + Ah(A) ) (because Al = O for all l ≥ h( A) + 1) = t( A) − t( A)A = t(A)/A. Therefore, we have t(A)/t(A) = A/t( A) = t(A)/A and (2) holds. (3) Since A is nilpotent, we have t(A) is transitive and nilpotent. By Theorem 4.1, we have t(t( A)/t( A)) = t(t( A)) = t(A) and (3) holds. Theorem 4.3. If A ∈ Mn (L) is nilpotent then for S ∈ Mn (L) the following statements are equivalent. (1) A/t( A) ≤ S ≤ t( A); (2) t(A) = t(S); (3) A/t( A) = S/t(A). Proof. (1) ⇒ (2). Suppose that A/t(A) ≤ S ≤ t( A). Since A is nilpotent, we have t(A) is nilpotent, and so A/t(A) and S are nilpotent. Then t( A/t(A)) ≤ t(S) ≤ t(t(A)) = t( A). Since t(A/t( A)) = t(t(A)/t(A)) (by Theorem 4.2(2)) = t( A) (by Theorem 4.2(3)), we have t(A) ≤ t(S) ≤ t( A), and so t( A) = t(S) and (2) holds. (2) ⇒ (3). Suppose that t(A) = t(S). Then S ≤ t(S) = t(A), and so S is nilpotent since t(A) is nilpotent. Thus A/t(A) = t(A)/t(A) (by Theorem 4.2(2)) = t(S)/t(S) = S/t(S) (by Theorem 4.2(2)) = S/t( A), and so (3) holds. (3) ⇒ (1). Suppose that A/t( A) = S/t( A). Then A/t(A) ≤ S and S − St(A) = S/t( A) ≤ A. By S − St( A) ≤ A, we have (S − St(A)) + St(A) ≤ A + St( A). But (S − St( A)) + St( A) ≥ S (by Lemma 2.6(1)), we have S ≤ A + St(A). Furthermore, we have S ≤ A + ( A + St( A))t( A) = A + At( A) + S(t( A))2 . Repeating this manner, we can obtain that S ≤ A + At(A) + · · · + A(t(A))k−1 + S(t( A))k for any positive integer k.
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
2225
Since (t(A))h(A) = Ah(A) (I + A + · · · + Ah(A)−2 )h(A) = O, we have S ≤ A + At( A) + · · · + A(t( A))h(A)−1 = A + t( A)A (because t(A) is transitive) = t(A) (by (4.2)). Therefore, we have A/t( A) ≤ S ≤ t(A) and (1) holds. By Corollary 3.1 and Theorem 4.3, we have Corollary 4.2. If A ∈ Mn (L) is weakly irreflexive and transitive then for S ∈ Mn (L) the following statements are equivalent. (1) A/A ≤ S ≤ A; (2) t(S) = A; (3) A/A = S/A.
Remark 4.2. Corollary 4.2 shows that under the assumption of the corollary A/A is just the least matrix whose transitive close coincides with A, and ( A/A)/A = A/A. Remark 4.3. Corollary 4.2 generalizes Theorems 2 and 3 of Hashimoto [15] and Theorems 4.3 and 4.4 of Tan [28] and Theorems 4.3 and 4.4 of Han et al. [13]. 5. Conclusions A path algebra is a special type of semiring but it contains Boolean algebra, fuzzy algebra, distributive lattice and incline. This paper studied nilpotent matrices and their reduction over a path algebra. Some results about nilpotent matrices over max–min fuzzy algebra or distributive lattice or incline are proved to be tenable over a path algebra. This paper illustrates some problems about fuzzy matrices or lattice matrices or incline matrices can be discussed on a general path algebra. Acknowledgment The author would like to thank the referees for a number of constructive comments and valuable suggestions. References [1] F. Baccelli, J. Mairesse, Ergodic theorems for stochastic operators and discrete event networks, in: J. Gunawardena (Ed.), Idempotency (Bristol, 1994), Publ. Newton Inst., Vol. 11, Cambridge University Press, Cambridge, 1998, pp. 171–208. [2] Z.Q. Cao, K.H. Kim, F.W. Roush, Incline Algebra and Applications, John Wiley, New York, 1984. [3] B. Carrè, Groups and Networks, Clarendon Press, Oxford, 1979. [4] R.A. Cuninghame-Green, Minimax algebra, in: Lecture Notes in Economics and Mathematical Systems, Vol. 166, Springer-Verlag, Berlin, 1979. [5] J.S. Duan, The transitive closure, convergence of powers and adjoint of generalized fuzzy matrices, Fuzzy Sets and Syst. 145 (2004) 301–311. [6] A.C. Fisher, J.S. Liebmain, G.L. Nemhauser, Computer construction of project networks, Commun. ACM 11 (1968) 493–497. [7] B. Gaujal, J.M. Alain, Computational issues in recursive stochastic systems, in: J. Gunawardena (Ed.), Idempotency (Bristol, 1994), Publ. Newton Inst., Vol. 11, Cambridge University Press, Cambridge, 1998, pp. 209–230. [8] S. Ghosh, Matrices over semirings, Inf. Sci. 90 (1996) 221–230. [9] Y. Give’on, Lattice matrices, Inf. Control 7 (1964) 477–484. [10] J.S. Golan, Semirings and their Applications, Kluwer Academic Publishers, Dordrecht, 1999. [11] M. Gondran, Path Algebra and Algorithms, in: B. Roy (Ed.), Combinatorial Programming: Methods and Applications, Reidel, Dordrecht, 1975, pp. 137–148. [12] D.A. Gregory, N.J. Pullman, Semiring rank: Boolean rank and nonnegative rank factorization, J. Combinations Inf. Syst. Sci. 8 (3) (1983) 223–233. [13] S.C. Han, H.X. Li, J.Y. Wang, On nilpotent incline matrices, Linear Algebra Appl. 406 (2005) 201–217.
2226 [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226 F. Harary, On the consistency of precedence matrices, J. Assoc. Comput. Math. 7 (1960) 255–259. H. Hashimoto, Reduction of a nilpotent fuzzy matrices, Inf. Sci. 27 (1982) 223–243. H. Hashimoto, Transitivity of generalized fuzzy matrices, Fuzzy Sets and Syst. 17 (1985) 83–90. K.H. Kim, F.W. Roush, Generalized fuzzy matrices, Fuzzy Sets and Syst. 4 (1980) 293–315. E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer, Dordrecht, 2000. J.X. Li, Some results on the nilpotent fuzzy matrices, Fuzzy Syst. Math. 3 (1) (1989) 52–55 (in Chinese). R.B. Marimont, A new method of checking the consistency of precedence matrices, J. Assoc. Comput. Math. 6 (1959) 162–171. M. Minoux, Bideterminants, arborescences and extension of the matrix-tree theorem to semirings, Discrete Math. 171 (1997) 191–200. M. Minoux, A generalization of the all minors matrix tree theorem to semirings, Discrete Math. 199 (1999) 139–150. H.T. Ren, Z. Zhang, Z.S. Liu, A necessary and sufficient condition for nilpotent fuzzy matrices, Acta Sci. Nat. Univ. Neimonggol 27 (1996) 594–596 (in Chinese). C. Reutenauer, H. Straubing, Inversion of matrices over a commutative semiring, J. Algebra 88 (1984) 350–360. D.E. Rutherford, The Cayley–Hamilton theorem for semiring, Proc. R. Soc. Edinburgh Sect. A 66 (1964) 211–215. Y.J. Tan, On the powers of matrices over a distributive lattice, Linear Algebra Appl. 336 (2001) 1–14. Y.J. Tan, On compositions of lattice matrices, Fuzzy sets and Syst. 129 (2002) 19–28. Y.J. Tan, On nilpotent matrices over distributive lattices, Fuzzy Sets and Syst. 151 (2005) 421–433. E.M. Vechtomov, Two general structure theorems on submodules, Abelian Groups and Modules (in Russian), Tomsk State University, Tomsk, Vol. 15, 2000, pp. 17–23. K.L. Zhang, On the nilpotent matrices over D01 -lattices, Fuzzy Sets and Syst. 117 (2001) 403–406. U. Zimmermann, Linear and combinatorial optimization in ordered algebraic structures, Annals of Discrete Mathematics, Vol. 10, North Holland, 1981.
On nilpotency of generalized fuzzy matrices夡 Yi-Jia Tan∗ Department of Mathematics, Fuzhou University, Fuzhou 350108, China Received 1 April 2009; received in revised form 12 March 2010; accepted 31 March 2010 Available online 8 April 2010
Abstract In this paper, generalized fuzzy matrices are considered as matrices over a special type of semiring which is called path algebra. Some properties and characterizations for nilpotent generalized fuzzy matrices are established, reduction of generalized fuzzy matrices is defined and some properties of reduction of nilpotent generalized fuzzy matrices are obtained. Partial results obtained in this paper generalize the corresponding ones on fuzzy matrices, on lattice matrices and on incline matrices. © 2010 Elsevier B.V. All rights reserved. Keywords: Algebra; Generalized fuzzy matrix; Nilpotent matrix; Reduction of matrix; Path algebra
1. Introduction Nilpotency of generalized fuzzy matrices [17] over a special type of semiring is considered. The semiring is called path algebra (see [16]) (or additively idempotent semiring (see [10]). Boolean or fuzzy algebra, bounded distributive lattices and inclines are special cases of path algebra. Boolean matrices, fuzzy matrices, lattice matrices (see [9]) and incline matrices (see [2]) are the prototypical examples of matrices over path algebras. Path algebras are useful tools in diverse areas such as design of switching circuits, automata theory, information systems, dynamic programming and decision theory. For further examples, see [3,11]. The study of matrices over general semirings has long history. In 1964, Rutherford [25] gave a proof of the Cayley– Hamilton theorem for a commutative semiring avoiding the use of determinants. Since then, a number of works on theory of matrices over semirings were published (see e.g. [8,12,21,22,24]). In 1999, Golan described semirings and matrices over semirings in his work [10] comprehensively. The techniques of matrices over semirings have important applications in optimization theory, models of discrete event networks and graph theory. For further examples, see [1,7]. Nilpotent matrices are an important type of matrices. Since the beginning of the 1960s, many authors have studied this type of matrices for some special cases of path algebras (see e.g. [9,13,15,19,23,26,28,30]). In 1964, Give’on [9] proved that an n × n lattice matrix A is nilpotent if and only if An = O. In [15], Hashimoto considered the reduction of nilpotent fuzzy matrices and obtained some properties of the reduction. Nilpotent fuzzy matrices represent acyclic fuzzy graphs, and in general, acyclic graphs are used to represent consistent systems and are important in the representation of precedence relations [6,14,20]. In [19], Li gave some characterizations for nilpotent fuzzy matrices. The results by 夡 Supported by Natural Science Foundation of Fujian province (2008J0194), China. ∗ Tel.: +86 591 88174165.
E-mail address: [email protected]. 0165-0114/$ - see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2010.03.019
2214
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
Hashimoto [15] and Li [19] were generalized to nilpotent lattice matrices by Tan [28]. Ren et al. [23] showed that a fuzzy matrix A is nilpotent if and only if every principal minor of A is 0. This result was generalized to lattice matrices by Tan [26] and Zhang [30], independently. Recently, Han et al. [13] characterized the nilpotent matrices over an incline without nonzero nilpotent elements in terms of principal minors, main diagonals, nilpotent indices and adjoint matrices, and established some properties of the reduction of nilpotent matrices over an additively residuated incline without nonzero nilpotent elements. In the present work, we will consider the nilpotent matrices over more general path algebras, namely over a class of commutative path algebras. In Section 3, we will give some properties and characteristics for the nilpotent matrices. In Section 4, we consider the reduction of the nilpotent matrices over an additively residuated commutative path algebra and obtain some properties of the reduction. Partial results in this work generalize and develop the corresponding results on fuzzy matrices in [15,19,23], on lattice matrices in [9,26,28,30] and on incline matrices in [13]. 2. Definitions and preliminary lemmas In this section, we shall give some definitions and lemmas. For convenience, we use n to denote the set {1,2, . . . ,n}. Definition 2.1 (Golan [10]). An algebraic system (L , +, ·) is called a semiring if (L,+) is an Abelian monoid with identity element 0 and (L , ·) is another monoid with identity element 1 and 0 1, connected by ring-like distributivity, and 0r = r 0 = 0 for all r ∈ L. A semiring L is called commutative if ab = ba for all a, b ∈ L; L is called an antiring if a + b = 0 implies that a = b = 0 for all a, b ∈ L (see [29]). Antirings were studied in [10] under the name of zerosumfree semirings. Definition 2.2 (Carrè [3], Hashimoto [16]). A semiring L is called a path algebra if a + a = a for all a ∈ L. Note that every path algebra is an antiring. Indeed, if L is a path algebra and a + b = 0 for a, b ∈ L then a = a + 0 = a + (a + b) = (a + a) + b = a + b = 0 and similarly b = 0. Path algebras were studied in [10] under the name of additively idempotent semiring. Path algebras are abundant: for example, every Boolean algebra, the fuzzy algebra F = ([0, 1], ∨, T ), where ∨ = max and T is a t-norm (for t-norm, refer to [18]), and every bounded distributive lattice are commutative path algebras. In addition, the max-plus algebra (R ∪ {−∞}, max, +) and the min-plus algebra (R ∪ {+∞}, min, +) (see [4,31]) are commutative path algebras. A semiring L is called an incline (see [2,10]) if a + 1 = 1 for all a ∈ L. Note that any incline is a path algebra. In fact, if L is an incline then 1 + 1 = 1, and so for any a ∈ L, a = a(1 + 1) = a + a. Throughout this paper, L is always supposed to be a commutative path algebra. Definition 2.3. An element a in a path algebra L is said to be nilpotent if a k = 0 for some positive integer k. The least positive integer k satisfying a k = 0 is called the nilpotent index of a and denoted by h(a). The set of all nilpotent elements in L is denoted by Z(L). It is clear that if L is a Boolean algebra or a bounded distributive lattice, then Z (L) = {0}, that is, L has no nonzero nilpotent elements. Example 2.1. Let L = ([0, 1], ∨, ·), where [0, 1] is the unit interval, a ∨ b = max{a, b} and a · b = (a + b − 1) ∨ 0 for a, b ∈ [0, 1]. It is easy to verify that L is a commutative path algebra. For any a ∈ [0, 1], we can see that a k = (ka − k + 1) ∨ 0. Obviously, a k = 0 if and only if a ≤ 1 − (1/k). Hence Z (L) = [0, 1). For any path algebra L, we define a relation “≤ ” on L as follows, a ≤ b if and only if a + b = b for all a,b in L. The following properties are derived immediately from above definition. (1) The relation ≤ is a partial order on L. (2) The element 0 is the least element in L, i.e., 0 ≤ a for all a ∈ L. (3) For any a, b, c, d ∈ L, a ≤ b and c ≤ d implies that a + c ≤ b + d and ac ≤ bd. In the following, we consider matrices over commutative path algebras as generalized fuzzy matrices.
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
2215
Let L be a commutative path algebra. Denote by Mm×n (L) the set of all m × n matrices over L. Especially, we put Mn (L) = Mn×n (L). For A ∈ Mm×n (L), we denote by ai j or Ai j the element of L which stands in the (i,j)-entry of A, and denote by A T the transpose of A. For any A, B ∈ Mm×n (L) and C ∈ Mn×l (L), we define: A+B = (ai j + bi j )m×n ; AC = ( nk=1 aik ck j )m×l ; A ≤ B if ai j ≤ bi j for i ∈ m and j ∈ n; It is easy to verify that (Mn (L), +, ·) is a path algebra, and A+C ≤ B + D and AC ≤ B D for all A, B, C, D ∈ Mn (L) with A ≤ B and C ≤ D. For A ∈ Mn (L), the powers of A are defined as follows: A0 = In , Al = Al−1 A, l = 1,2, . . . . where In is the identity matrix of order n. Definition 2.4. Let A ∈ Mn (L). A is said to be nilpotent if Ak = O for some positive integer k, where O denotes the zero matrix. The least positive integer k satisfying Ak = O is called the nilpotent index of A, and denoted by h(A). Let A ∈ Mn (L). If A2 ≤ A then A is called transitive (see [16]); if aii is nilpotent for any i ∈ n then A is called weakly irreflexive, especially, if aii = 0 for any i ∈ n then A is called irreflexive. If there exists a B ∈ Mn (L) such that A ≤ B and B is transitive and B ≤ C for any transitive matrix C in Mn (L) satisfying A ≤ C, then the matrix B is called a transitive closure of A. It is clear that if A has a transitive closure then it is unique. The transitive closure of A is denoted by t(A). Hashomoto [16] discussed some compositions of transitive matrices over a path algebra and obtained some important properties. Duan [5] considered the transitive closure of matrices over an incline and proved that any n × n matrix A over an incline has transitive closure and t(A) = nk=1 Ak . However, for general path algebra L, there may be some matrices over L which have no transitive closure. Example 2.2. Let L = ([0, +∞), ∨, ·), where ∨ = max and “·” is the usual multiplication of real numbers. Then L is a commutative but not an incline. path algebra 2 k 0 Let A = 0.4 , B = k 0 k ∈ M2 (L), where k ∈ L. Then it is easy to verify that A has transitive closure 0.4 0.4 0.8 2 and t( A) = 0.4 0.8 , but Bk have no transitive closure for all k > 1. Definition 2.5. Let A ∈ Mn (L). The permanent per A of A is defined as follows: per A = a1(1) a2(2) · · · an (n) ∈Sn
where Sn denotes the symmetric group of the set n. Let U = {i 1 , i 2 , . . . , ir }, V = { j1 , j2 , . . . , jr } ⊆ n. For A ∈ Mn (L), we denote by A[U |V ] the r × r submatrix of A whose (u, v)-entry is equal to aiu jv (u, v ∈ r ) and by A(U |V ) the (n−r )×(n−r ) submatrix of A obtained by deleting rows i 1 , i 2 , . . . , ir and columns j1 , j2 , . . . , jr from A, where 1 ≤ i 1 < i 2 < · · · < ir ≤ n and 1 ≤ j1 < j2 < · · · < jr ≤ n. The matrix A[U |U ] is called a principal submatrix of order r of A, and per A[U |U ] is called a principal minor of order T . The matrix adjA is called the adjoint matrix of A. r of A. Let ad j A = ( per A(i| j))n×n Lemma 2.1. Let L be a commutative path algebra, a, b, a1 , a2 , . . . , am ∈ L. Then (1) if a is nilpotent and b ≤ a then b is nilpotent; (2) if a is nilpotent then ar and ra are nilpotent for any r in L; m (3) i=1 ai is nilpotent if and only if a1 , a2 , . . . , am are nilpotent.
2216
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
Proof. (1) and (2) are clear. (3) By binomial theorem and idempotency of the operation “+” in L, we have (x + y) = k
k
x k−l y l
(2.1)
l=0 0 for any x,y in L and any m mpositive integer k (Note that x = 1 for any element x in L). ai is nilpotent. Then, by (1), we have that each ai is nilpotent since ai ≤ i=1 ai for any Suppose now that i=1 m i ∈ m. In the following we will prove that if a1 , a2 , . . . , am are nilpotent then i=1 ai is nilpotent. We shall prove the statement by induction on m. It is clear if m = 1, and we may assume it holds for m − 1(m − m−1 1 ≥ 1). Suppose that a1 , a2 , . . . , am are nilpotent. Then i=1 ai is nilpotent (by the induction hypothesis). Let m−1 m m−1 k m−1 k−l l ai ) + am )k = l=0 ( a ) am k1 = h( i=1 ai ) and k2 = h(am ) and k = k1 + k2 . Then ( i=1 ai )k = (( i=1 i=1 i l = 0, and if l < k then k−l ≥ k and so ( m−1 a )k−l = 0. This implies ( m−1 a )k−l a l = 0 (by (2.1)). If l ≥ k2 then am 2 m i=1 i i=1 i m m 1 for all l ∈ {0, 1, . . . , k}. Then ( i=1 ai )k = 0, i.e., i=1 ai is nilpotent. This proves (3).
Lemma 2.2. Let A, B ∈ Mn (L). Then (1) A is nilpotent if and only if P A P T is nilpotent for any n × n permutation matrix P; (2) if A is nilpotent and B ≤ A then B is nilpotent and h(B) ≤ h( A). The proof is omitted. Lemma 2.3. Let A ∈ Mn (L). Then T (1) per A = per n(A ); (2) per A = j=1 ai j per A(i| j) for any i ∈ n; (3) ad j( A T ) = (ad j A)T .
The proof is trivial. Lemma 2.4. Let A ∈ Mn (L) be nilpotent. Then t( A) = Proof. Let B = B2 =
h(A)−1 k=1
2h(A)−2
h(A)−1 k=1
Ak .
Ak . Obviously, A ≤ B. Since (Mn (L), +, ·) is a path algebra, we have
Ak
k=2
=
h(A)−1
Ak (because Al = O for all l ≥ h( A))
k=2
≤ B. i.e., B is transitive. If there exists a transitive matrix C in Mn (L) such that A ≤ C then C 2 ≤ C and so C k ≤ C for all positive integers k. Hence Ak ≤ C k ≤ C, and so B = A + A2 + · · · + Ah(A)−1 ≤ C. By the definition of transitive closure, we obtain t( A) = B. This proves the lemma. Remark 2.1. Lemma 2.4 shows that any nilpotent matrix over a path algebra has transitive closure. A path algebra L is said to be additively residuated if for any a, b ∈ L there exists an element r ∈ L such that a + x ≥ b if and only if x ≥ r . Obviously, r is the least element x ∈ L satisfying a + x ≥ b. The element r will be denoted by b−a, i.e., a + x ≥ b if and only if x ≥ b − a. Remark 2.2. The fuzzy algebra ([0, 1], ∨, ∧), where ∨ = max and ∧ = min, is an additively residuated path algebra and in this case the operation “–” coincides with the operation “d” defined in [15], i.e., a if b < a a − b = adb = (1) 0 if a ≤ b where b < a means that b ≤ a and b a.
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
2217
Remark 2.3. Any Boolean algebra L is an additively residuated path algebra and in this case a − b = abc , where bc denotes the complement element of b in L. Remark 2.4. Every dually Brouwerian lattice (see [27]) and any additively residuated incline (see [13]) are additively residuated path algebras. Lemma 2.5. Let L be an additively residuated path algebra. Then for any a, b, c, a1 , a2 , . . . , al , b1 , b2 , . . . , bl ∈ L(l ≥ 2), we have (1) (2) (3) (4) (5) (6)
(a − b) + b ≥ a and a−a = 0, and a−0 = a; a(b − c) ≥ ab − ac and (a − b)c ≥ ac − bc; (a − b) − c ≥ a − (b + c); if al ≤ b then a − c≤l b − c and l c − a ≥ c − b; (a − b ) ≥ a − i i i=1 i i=1 bi ; i=1 l−1 ( i=1 (ai − ai+1 )) + al = li=1 ai .
Proof. (1) and (4) are clear, we will prove (2), (3), (5) and (6). (2) Since c + (b − c) ≥ b (by (1)), we have ac + a(b − c) ≥ ab, and so a(b − c) ≥ ab − ac. Similarly, we can prove that (a − b)c ≥ ac − bc. This proves (2). (3) Since (b + c) + ((a − b) − c) = b + (c + ((a − b) − c)) ≥ b + (a − b) ≥ a, we have (a − b) − c ≥ a − (b + c). This proves (3). (5) Since li=1 bi + li=1 (ai − bi ) =
l
(bi + (ai − bi ))
i=1
≥
l
ai (by (1)),
i=1
we have l
(ai − bi ) ≥
i=1
l i=1
ai −
l
bi .
i=1
This proves (5). (6). We will prove (6) by induction on l. By (1), we have (a1 − a2 ) + a2 ≥ a1 , and so (a1 − a2 ) + a2 = (a1 − a2 ) + (a2 + a2 ) = ((a1 − a2 ) + a2 ) + a2 ≥ a1 + a2 . On the other hand, since a1 ≥ a1 − a2 , we have a1 + a2 ≥ (a1 − a2 ) + a2 . Hence (a1 − a2 ) + a2 = a1 + a2 and (6) holds for l = 2. Assume that (6) holds for l − 1(l − 1 ≥ 2). Then l−1 (ai − ai+1 ) + al = ((a1 − a2 ) + (a2 − a3 ) + · · · + (al−1 − al )) + al i=1
= (a1 − a2 ) + ((a2 − a3 ) + · · · + (al−1 − al ) + al ) = (a1 − a2 ) + (a2 + a3 + · · · + al ) (by the induction hypothesis) = ((a1 − a2 ) + a2 ) + (a3 + · · · + al ) = a1 + a2 + · · · + al =
l
ai .
i=1
This proves (6).
Let L be an additively residuated path algebra. Given A, B ∈ Mm×n (L), we define: A − B = (ai j − bi j )m×n .
2218
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
Lemma 2.6. Let L be a commutative additively residuated path algebra. Then for any A, B, C, A1 , A2 , . . . , Al , B1 , B2 , . . . , Bl ∈ Mn (L), we have (1) (2) (3) (4) (5) (6)
(A − B) + B ≥ A and A−A = O, and A−O = A; A(B − C) ≥ AB − AC and ( A − B)C ≥ AC − BC; (A − B) − C ≥ A − (B + C); if A ≤ B then A − C ≤ B − C and C − A ≥ C − B; l (Ai − Bi ) ≥ li=1 Ai − li=1 Bi ; i=1 l ( l−1 i=1 Ai . i=1 (Ai − Ai+1 )) + Al =
Proof. (1) and (3)–(6) can be obtained from (1) and (3)–(6) in Lemma 2.5, we only prove (2). (2) For any i, j ∈ n, we have n aik (bk j − ck j ) (A(B − C))i j = k=1
≥
n
(aik bk j − aik ck j ) (by Lemma 2.5(2))
k=1
≥
n
aik bk j
−
k=1
n
aik ck j
(by Lemma 2.5(5))
k=1
= (AB)i j − ( AC)i j = ( AB − AC)i j . Therefore A(B − C) ≥ AB − AC. Similarly, we can prove (A − B)C ≥ AC − BC. 3. Some properties and characteristics of nilpotent matrices In [13], Han et al. gave some important features of nilpotent matrices over an incline without nonzero nilpotent elements (see Theorems 3.4 and 3.7 in [13]). In this section, we will extend these results to matrices over a general path algebra and obtain some properties and characteristics of the nilpotent matrices. Proposition 3.1. Let A = (ai j ) ∈ Mn (L). If every element of A is nilpotent then A is nilpotent. Proof. Suppose that every element of A is nilpotent. Let h(ai j ) = ki j for any i, j ∈ n and k = max{ki j |i, j ∈ n}, and put l = n 2 k. Then aikj = 0 for any i, j ∈ n. In the following we will prove Al = O. It is clear that (Al )i j = i ,i , ...,i ∈n aii1 ai1 i2 · · · ail−1 j for any i, j ∈ n. Let T = aii1 ai1 i2 · · · ail−1 j be any term of 1 2
l−1
( Al )i j . Since aii1 , ai1 i2 , . . . , ail−1 j ∈ {ast |s, t ∈ n} and the set {ast |s, t ∈ n} contains at most n 2 different elements, there exist t0 , t1 , . . . , tk−1 ∈ {0, 1, . . . , l − 1} such that t0 < t1 < · · · < tk−1 and ait0 it0 +1 = ait1 it1 +1 = · · · = aitk−1 itk−1 +1 (taking i 0 = i and il = j) (note that l = n 2 k), and so T has the factor (ait0 it0 +1 )k . But (ait0 it0 +1 )k = 0, we have T = 0. Since T is any term of ( Al )i j , we have ( Al )i j = 0 for any i, j ∈ n. This means that Al = O. Proposition 3.2. Let A ∈ Mn (L). If all main diagonal entries of Ak are nilpotent for any k ∈ n, then: (1) all elements of An are nilpotent; (2) A is nilpotent. Proof. (1) Let A = (ai j ). Then (An )i j =
a a i 1 ,i 2 , ...,i n−1 ∈n ii 1 i 1 i 2
· · · ain−1 j for any i, j ∈ n. Let T = aii1 ai1 i2 · · · ain−1 j
be any term of Since i, i 1 , . . . , i n−1 , j ∈ n, there exist s, t ∈ {0, 1, . . . , n} such that i s = i t with s < t (taking i 0 = i and i n = j), and so ais is+1 · · · ait−1 it = ais is+1 · · · ait−1 is . But ais is+1 · · · ait−1 is is a term of (At−s )is is and (At−s )is is is nilpotent, we have ais is+1 · · · ait−1 it is nilpotent (by Lemma 2.1(3)). Since ais is+1 · · · ait−1 it is a factor of T, T is nilpotent (by Lemma 2.1(2)), and so (An )i j is nilpotent for any i, j ∈ n (by Lemma 2.1(3)). This proves (1). (2) By (1) and Proposition 3.1, An is nilpotent, and so A is nilpotent. This proves (2). ( An )i j .
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
2219
Corollary 3.1. If A ∈ Mn (L) is weakly irreflexive and transitive, then A is nilpotent. Proof. Put A = (ai j ). Since A is transitive, A2 ≤ A and so Ak ≤ A for all k ∈ n. Then ( Ak )i j ≤ ai j for any i, j ∈ n. Since A is weakly irreflexive, we have aii is nilpotent for any i ∈ n, and so ( Ak )ii is nilpotent for any i, k ∈ n (by Lemma 2.1(1)), i.e., all main diagonal entries of Ak are nilpotent for any k ∈ n. By Proposition 3.2, we have that all elements of An are nilpotent. Then An is nilpotent (by Proposition 3.1) and so A is nilpotent. Proposition 3.3. Let A ∈ Mn (L). If all principal minors of A are nilpotent then (1) all elements of An are nilpotent; (2) A is nilpotent. Proof. (1) Put A = (ai j ) and consider any term T = aii1 ai1 i2 · · · ain−1 j of (An )i j for any i, j ∈ n. Since i, i 1 , . . . , i n−1 , j ∈ n, there exist s, t ∈ {0, 1, . . . , n} such that s < t and i s = i t (taking i 0 = i and i n = j). Let K = {( p, q)| p < q, i p = i q }. Then K ⭋ and there exists a (k, l) ∈ K satisfying l − k = min{q − p|( p, q) ∈ K }. But the product aik ik+1 · · · ail−1 il = aik ik+1 · · · ail−1 ik is a term of the principal minor per A[i k , i k+1 , . . . , il−1 |i k , i k+1 , . . . , il−1 ] and per A[i k , i k+1 , . . . , il−1 |i k , i k+1 , . . . , il−1 ] is nilpotent, the product aik ik+1 · · · ail−1 il is nilpotent (by Lemma 2.1(3)). Since aik ik+1 · · · ail−1 il is a factor of aii1 ai1 i2 · · · ain−1 j , we have that T = aii1 ai1 i2 · · · ain−1 j is nilpotent (by Lemma 2.1(2)). But T is any term of ( An )i j , we have ( An )i j is nilpotent for any i, j ∈ n (by Lemma 2.1(3)). This proves (1). (2) By (1) and Proposition 3.1, An is nilpotent, and so A is nilpotent. This proves (2). Proposition 3.4. If A ∈ Mn (L) is nilpotent, then (1) all main diagonal entries of Am are nilpotent for any positive integer m; (2) ai1 i2 ai2 i3 · · · aim i1 is nilpotent for any positive integer m and any i 1 , i 2 , . . . , i m ∈ n; (3) per A is nilpotent. Proof. (1) Since A is nilpotent, Al = O for some positive integer l, and so (Am )l = Aml = O for any positive integer m. Then ( Aml )ii = 0 for any i ∈ n. But (( Am )ii )l is some term of (Aml )ii , we have that ((Am )ii )l = 0 (by Lemma 2.1(3)), i.e., (Am )ii is nilpotent for any i ∈ n and any positive integer m. This proves (1). (2) Since ai1 i2 ai2 i3 · · · aim i1 is a term of (Am )i1 i1 , by (1) and Lemma 2.1(3), we have that ai1 i2 ai2 i3 · · · aim i1 is nilpotent. This proves (2). (3) Consider any term T = a1(1) a2(2) · · · an (n) of per A, where ∈ Sn . Since m (1) ∈ n for all positive integer m, there exist s, t ∈ {0, 1, . . . , n} such that s < t and s (1) = t (1) and s (1), s+1 (1), . . . , t−1 (1) are mutually different (note that 0 (1) = 1). Then as (1)s+1 (1) · · · at−1 (1)s (1) = as (1)s+1 (1) · · · at−1 (1)t (1) is a factor of T . But as (1)s+1 (1) · · · at−1 (1)s (1) is nilpotent (by (2)), T is nilpotent for any ∈ Sn (by Lemma 2.1(2)). Then per A is nilpotent (by Lemma 2.1(3)). This proves (3). Theorem 3.1. For any A ∈ Mn (L), the following statements are equivalent. (1) (2) (3) (4) (5)
A is nilpotent. All principal submatrices of A are nilpotent. All principal minors of A are nilpotent. All elements of An are nilpotent. For any k ∈ n, all main diagonal entries of Ak are nilpotent.
Proof. (1) ⇒ (2). Suppose that A is nilpotent. Let B be any principal submatrix of order r of A. Then there exists an n × n permutation matrix P such that B C B O O C = + P APT = E D O O E D T T where C ∈ Mr ×(n−r ) (L), E ∈ M(n−r )×r (L) and D ∈ Mn−r (L). This implies that OB O O ≤ P A P . But P A P is nilpotent (by Lemma 2.2(1)), the matrix OB O O is nilpotent (by Lemma 2.2(2)), and so B is nilpotent.
2220
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
(2) ⇒ (3). It follows from Proposition 3.4(3). (3) ⇒ (4). It follows from Proposition 3.3(1). (4) ⇒ (5). Suppose that all elements of An are nilpotent. Then, by Proposition 3.1, An is nilpotent and so A is nilpotent, and by Proposition 3.4(1), all main diagonal entries of Ak are nilpotent for any k ∈ n. (5) ⇒ (1). It follows from Propositions 3.2(2). If L has no nonzero nilpotent elements, then Z (L) = {0}. By Theorem 3.1, we have Corollary 3.2. If L has no nonzero nilpotent elements, then for any A ∈ Mn (L), the following statements are equivalent. (1) (2) (3) (4) (5)
A is nilpotent. All principal submatrices of A are nilpotent. All principal minors of A are 0. An = O. All main diagonal entries of Ak are 0 for any k ∈ n.
Remark 3.1. Corollary 3.2 shows that if L has no nonzero nilpotent elements and A ∈ Mn (L) is nilpotent then An = O (or h(A) n), but this result is not true for nilpotent matrices over a general path algebra. 0.7 Example 3.1. Consider the path algebra L in Example 2.1. Let A = 0.8 0.9 0.6 ∈ M2 (L). Then, we have 2 0.8 0.7 A2 = 0.9 0.6 ((0.8 + 0.8 − 1) ∨ 0) ∨ ((0.7 + 0.9 − 1) ∨ 0) ((0.8 + 0.7 − 1) ∨ 0) ∨ ((0.7 + 0.6 − 1) ∨ 0) = ((0.9 + 0.8 − 1) ∨ 0) ∨ ((0.6 + 0.9 − 1) ∨ 0) ((0.9 + 0.7 − 1) ∨ 0) ∨ ((0.6 + 0.6 − 1) ∨ 0) 0.6 0.5 = , 0.7 0.6 0.2 0.1 0 0 0 0 , A5 = and A6 = . A4 = 0.3 0.2 0.1 0 0 0 Clearly, A2 O and h( A) = 6 > 2. Remark 3.2. Corollary 3.2 generalizes and develops Corollary 5.2 of Give’on [9], Theorem 1 of Li [19], Theorem 1 of Ren et al. [23], Theorem 6.6 of Tan [26], Theorem 1 of Zhang [30] and Theorem 3.4 of Han et al. [13]. Given a matrix A ∈ Mn (L), we denote by A(i ⇒ j) the matrix obtained from A by replacing the row j with the row i. Theorem 3.2. If A ∈ Mn (L) is nilpotent, then (1) per A(i ⇒ j) is nilpotent for any i, j ∈ n; (2) All elements of both AadjA and (adjA)A are nilpotent; (3) All elements of (ad j A)2 are nilpotent. Proof. Let A ∈ Mn (L) be nilpotent. (1) For any i ∈ n, we have per A(i ⇒ i) = per A is nilpotent (by Proposition 3.4(3)). For any i, j ∈ n with i j, we have a1(1) · · · ai (i) · · · ai ( j) · · · an (n) . per A(i ⇒ j) = ∈Sn
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
2221
For any ∈ Sn , if l (i) j for all l ≥ 1, then there exists a d such that d (i) = i with 1 ≤ d ≤ n and i, (i), . . . , d−1 (i) are mutually different. Hence ai (i) a(i)2 (i) · · · ad−1 (i)i is nilpotent (by Proposition 3.4(2)). Since the product ai (i) a(i)2 (i) · · · ad−1 (i)i is a factor of the product a1(1) · · · ai (i) · · · ai ( j) · · · an (n) , we have that a1(1) · · · ai (i) · · · ai ( j) · · · an (n) is nilpotent (by Lemma 2.1(2)). If there exists a positive integer l such that l (i) = j, then there must be a positive integer d such that i = d ( j) with 1 ≤ d ≤ n and i, ( j), . . . , d−1 ( j) are mutually different. Thus ai ( j) a( j)2 ( j) · · · ad−1 ( j)i is nilpotent (by Proposition 3.4(2)). But the product ai ( j) a( j)2 ( j) · · · ad−1 ( j)i is a factor of the product a1(1) · · · ai (i) · · · ai ( j) · · · an (n) , again, we have a1(1) · · · ai (i) · · · ai ( j) · · · a n (n) is nilpotent (by Lemma 2.1(2)). Consequently, per A(i ⇒ j) = ∈Sn a1(i) · · · ai (i) · · · ai ( j) · · · an (n) is nilpotent (by Lemma 2.1(3)). This proves (1). (2) Let B = A · ad j A. Then, by (1) and Lemma 2.3(2), we have that bi j = k∈n aik per A( j|k) = per A(i ⇒ j) is nilpotent for all i, j ∈ n. By Lemma 2.3(3), we have ((ad j A)A)T = A T · (ad j A)T = A T · ad j( A T ). Then all elements of the matrix ((ad j A)A)T are nilpotent and so all elements of (adjA)A are nilpotent. This proves (2). (3) Let C = (ad j A)2 . For any i, j ∈ n, we have ci j = per A( j|k) per A(k|i). k∈n
Let c¯i j =
per A( j|k) per A(k|i)
k∈n i
where n i = n − {i}. Then ci j = per A( j|i) per A(i|i) + c¯i j . Since per A(i|i) is nilpotent (by Theorem 3.1), we have per A( j|i) per A(i|i) is nilpotent (by Lemma 2.1(2)). When k i, we have ⎛ ⎞ ⎝ per A(k|i) ≤ ast ≤ ast ⎠ . t∈n i
Therefore c¯i j ≤
k∈n i
=
k∈n i
=
k∈n i
s∈n k
⎛ ⎝( per A( j|k)) ·
t∈n i
⎛
⎝
⎛
⎝⎝ per A( j|k) · ⎝
⎞⎞ ast ⎠⎠
s∈n
t∈n i
⎛⎛
s∈n
⎞⎞ ask ⎠⎠ ·
s∈n
⎛⎛
⎝⎝
s∈n
⎞ ask per A( j|k)⎠ ·
t∈n ik
t∈n ik
⎛
⎝
⎛
⎝
⎞⎞ ast ⎠⎠ , (where n ik = n − {i,k})
s∈n
⎞⎞ ast ⎠⎠ .
s∈n
On the other hand, since per A(s ⇒ j) = for any s, j ∈ n (by (1)), we have k∈n ask per A( j|k) is nilpotent that ask per A( j|k) is nilpotent for any s, j, k ∈ n (by Lemma 2.1(3)), and so s∈n ask per A( j|k) is nilpotent for any j, k ∈ n (by Lemma 2.1(3)). Then ( s∈n ask per A( j|k)) · t∈n ik ( s∈n ast ) is nilpotent (by Lemma 2.1(2)), and so k∈n i (( s∈n ask per A( j|k)) · t∈n ik ( s∈n ast )) is nilpotent for any i, j ∈ n (by Lemma 2.1(3)). This implies that c¯i j is nilpotent for any i, j∈ n (by Lemma 2.1(1)). Since per A( j|i) per A(i|i) is nilpotent and ci j = per A( j|i) per A(i|i) + c¯i j , we have ci j is nilpotent for any i, j ∈ n (by Lemma 2.1(3)), i.e., all elements of (ad j A)2 are nilpotent. This proves (3).
2222
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
Corollary 3.3. If L has no nonzero nilpotent elements and A ∈ Mn (L) is nilpotent, then (1) per A(i ⇒ j) = 0 for any i, j ∈ n; (2) Aad j A = (ad j A)A = O; (3) (ad j A)2 = O. Remark 3.3. Corollary 3.3 generalizes and develops Proposition 3.4 of Tan [28] and Theorem 3.7 of Han et al. [13]. 4. Reduction of nilpotent matrices Let L be an additively residuated path algebra and A, B ∈ Mn (L). We define A/B = A−AB. The matrix A/A = A−A2 is called the reduction of A. It is clear that A/A ≤ A for all A ∈ Mn (L). Hashimoto [15] discussed the reduction of irreflexive and transitive fuzzy matrices and obtained some important properties, and applied these properties to nilpotent fuzzy matrices in order to obtain some properties of reduction of nilpotent fuzzy matrices. Tan [28] considered the reduction of nilpotent lattice matrices and extended the results in [15] to nilpotent matrices over dually Brouwerian lattices, and furthermore, Han et al. [13] extended the results in [28] to nilpotent matrices over additively residuated inclines without nonzero nilpotent elements. In this section, we will consider the reduction of nilpotent matrices over an additively residuated path algebra in general. Partial results obtained in this section generalize the previous results on nilpotent fuzzy matrices by Hashimoto [15] and on nilpotent lattice matrices by Tan [28] and on nilpotent incline matrices by Han et al. [13]. Theorem 4.1. If A ∈ Mn (L) is nilpotent, then t(A/A) = t(A). Proof. If A = O, then t( A/A) = t( A) = O. In the following, we assume A O (or h( A)2). Let B = A/A. Then B ≤ A and so B is nilpotent. Thus t(B) = t(A/A) ≤ t( A). To prove t(A) ≤ t( A/A) = t(B), we first prove that B l ≥ Al − Al+1
(4.1)
for all l ≥ 1. We will prove (4.1) by induction on l. For l = 1, B = A−A2 . Assume that (4.1) holds for l − 1(l ≥ 2). Then B l = B · B l−1 ≥ B( Al−1 − Al ) ≥ B Al−1 − B Al (by Lemma 2.6(2)). Since B = A − A2 ≤ A, we have B l ≥ ( A − A2 )Al−1 − Al+1 (by Lemma 2.6(4)) ≥ (Al − Al+1 ) − Al+1 (by Lemma 2.6(2)) ≥ Al − ( Al+1 + Al+1 ) (by Lemma 2.6(3)) = Al − Al+1 . Now t(A/A) = t(B) =
h(B)−1
B k (by Lemma 2.4)
k=1
=
h(A)−1
B k (by Lemma 2.2(2))
k=1
≥
h(A)−1
( Ak − Ak+1 ) (by (4.1))
k=1
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
=
h(A)−2
2223
( Ak − Ak+1 ) + Ah(A)−1 (because Ah(A) = O)
k=1
=
h(A)−1
Ak (by Lemma 2.6(6))
k=1
= t(A). Therefore t(A/A) = t( A). This completes the proof. Remark 4.1. Theorem 4.1 generalizes Theorem 4.1 of Tan [28] and Theorem 4.1 of Han et al. [13]. If A ∈ Mn (L) is weakly irreflexive and transitive then A is nilpotent (by Corollary 3.1) and t( A) = A. By Theorem 4.1, we have Corollary 4.1. If A ∈ Mn (L) is weakly irreflexive and transitive then t(A/A) = A. Theorem 4.2. If A ∈ Mn (L) is nilpotent, then (1) t(A)/t( A) ≤ A ≤ t(A); (2) t(A)/t( A) = A/t( A) = t(A)/A; (3) t(t(A)/t(A))=t(A). Proof. It is clear that (1)–(3) hold for A = O. In the following, we assume A O (or h( A) 2). Since A is nilpotent, we have t(A) = A + A2 + · · · + Ah(A)−1 = A + A2 + · · · + Ah(A)−1 + Ah(A) (because Ah(A) = O). = A + At( A). Then t( A) = A + At( A) (1) We have t(A)/t( A) = t( A) − (t(A))2 = t( A) − (A2 + A3 + · · · + A2h(A)−2 ) = t( A) − ( A2 + A3 + · · · + Ah(A) ) (because Al = O for all l ≥ h( A) + 1) = ( A + At( A)) − (O + At( A)) (by (4.2)) ≤ (A − O) + ( At(A) − At(A)) (by Lemma 2.6(5)) = A (by Lemma 2.6(1)). i.e., t( A)/t( A) ≤ A. Since A ≤ t( A), we have that (1) holds. (2) First, we have A/t(A) = A − At(A) = (A − At(A)) + ( At(A) − At(A)) (by Lemma 2.6(1)) ≥ (A + At( A)) − ( At(A) + At( A)) (by Lemma 2.6(5)) = t(A) − At( A) (by (4.2))
(4.2)
2224
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
= t(A) − t( A)A (because At(A) = t( A)A) = t( A)/A, and t( A)/A = t(A) − t( A)A ≥ A − At( A) = A/t( A). Then A/t( A) = t( A)/A. Second, we have t(A)/t( A) = t( A) − (t(A))2 = t(A) − ( A2 + A3 + · · · + A2h(A)−2 ) = t( A) − ( A2 + A3 + · · · + Ah(A) ) (because Al = O for all l ≥ h( A) + 1) = t( A) − t( A)A = t(A)/A. Therefore, we have t(A)/t(A) = A/t( A) = t(A)/A and (2) holds. (3) Since A is nilpotent, we have t(A) is transitive and nilpotent. By Theorem 4.1, we have t(t( A)/t( A)) = t(t( A)) = t(A) and (3) holds. Theorem 4.3. If A ∈ Mn (L) is nilpotent then for S ∈ Mn (L) the following statements are equivalent. (1) A/t( A) ≤ S ≤ t( A); (2) t(A) = t(S); (3) A/t( A) = S/t(A). Proof. (1) ⇒ (2). Suppose that A/t(A) ≤ S ≤ t( A). Since A is nilpotent, we have t(A) is nilpotent, and so A/t(A) and S are nilpotent. Then t( A/t(A)) ≤ t(S) ≤ t(t(A)) = t( A). Since t(A/t( A)) = t(t(A)/t(A)) (by Theorem 4.2(2)) = t( A) (by Theorem 4.2(3)), we have t(A) ≤ t(S) ≤ t( A), and so t( A) = t(S) and (2) holds. (2) ⇒ (3). Suppose that t(A) = t(S). Then S ≤ t(S) = t(A), and so S is nilpotent since t(A) is nilpotent. Thus A/t(A) = t(A)/t(A) (by Theorem 4.2(2)) = t(S)/t(S) = S/t(S) (by Theorem 4.2(2)) = S/t( A), and so (3) holds. (3) ⇒ (1). Suppose that A/t( A) = S/t( A). Then A/t(A) ≤ S and S − St(A) = S/t( A) ≤ A. By S − St( A) ≤ A, we have (S − St(A)) + St(A) ≤ A + St( A). But (S − St( A)) + St( A) ≥ S (by Lemma 2.6(1)), we have S ≤ A + St(A). Furthermore, we have S ≤ A + ( A + St( A))t( A) = A + At( A) + S(t( A))2 . Repeating this manner, we can obtain that S ≤ A + At(A) + · · · + A(t(A))k−1 + S(t( A))k for any positive integer k.
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226
2225
Since (t(A))h(A) = Ah(A) (I + A + · · · + Ah(A)−2 )h(A) = O, we have S ≤ A + At( A) + · · · + A(t( A))h(A)−1 = A + t( A)A (because t(A) is transitive) = t(A) (by (4.2)). Therefore, we have A/t( A) ≤ S ≤ t(A) and (1) holds. By Corollary 3.1 and Theorem 4.3, we have Corollary 4.2. If A ∈ Mn (L) is weakly irreflexive and transitive then for S ∈ Mn (L) the following statements are equivalent. (1) A/A ≤ S ≤ A; (2) t(S) = A; (3) A/A = S/A.
Remark 4.2. Corollary 4.2 shows that under the assumption of the corollary A/A is just the least matrix whose transitive close coincides with A, and ( A/A)/A = A/A. Remark 4.3. Corollary 4.2 generalizes Theorems 2 and 3 of Hashimoto [15] and Theorems 4.3 and 4.4 of Tan [28] and Theorems 4.3 and 4.4 of Han et al. [13]. 5. Conclusions A path algebra is a special type of semiring but it contains Boolean algebra, fuzzy algebra, distributive lattice and incline. This paper studied nilpotent matrices and their reduction over a path algebra. Some results about nilpotent matrices over max–min fuzzy algebra or distributive lattice or incline are proved to be tenable over a path algebra. This paper illustrates some problems about fuzzy matrices or lattice matrices or incline matrices can be discussed on a general path algebra. Acknowledgment The author would like to thank the referees for a number of constructive comments and valuable suggestions. References [1] F. Baccelli, J. Mairesse, Ergodic theorems for stochastic operators and discrete event networks, in: J. Gunawardena (Ed.), Idempotency (Bristol, 1994), Publ. Newton Inst., Vol. 11, Cambridge University Press, Cambridge, 1998, pp. 171–208. [2] Z.Q. Cao, K.H. Kim, F.W. Roush, Incline Algebra and Applications, John Wiley, New York, 1984. [3] B. Carrè, Groups and Networks, Clarendon Press, Oxford, 1979. [4] R.A. Cuninghame-Green, Minimax algebra, in: Lecture Notes in Economics and Mathematical Systems, Vol. 166, Springer-Verlag, Berlin, 1979. [5] J.S. Duan, The transitive closure, convergence of powers and adjoint of generalized fuzzy matrices, Fuzzy Sets and Syst. 145 (2004) 301–311. [6] A.C. Fisher, J.S. Liebmain, G.L. Nemhauser, Computer construction of project networks, Commun. ACM 11 (1968) 493–497. [7] B. Gaujal, J.M. Alain, Computational issues in recursive stochastic systems, in: J. Gunawardena (Ed.), Idempotency (Bristol, 1994), Publ. Newton Inst., Vol. 11, Cambridge University Press, Cambridge, 1998, pp. 209–230. [8] S. Ghosh, Matrices over semirings, Inf. Sci. 90 (1996) 221–230. [9] Y. Give’on, Lattice matrices, Inf. Control 7 (1964) 477–484. [10] J.S. Golan, Semirings and their Applications, Kluwer Academic Publishers, Dordrecht, 1999. [11] M. Gondran, Path Algebra and Algorithms, in: B. Roy (Ed.), Combinatorial Programming: Methods and Applications, Reidel, Dordrecht, 1975, pp. 137–148. [12] D.A. Gregory, N.J. Pullman, Semiring rank: Boolean rank and nonnegative rank factorization, J. Combinations Inf. Syst. Sci. 8 (3) (1983) 223–233. [13] S.C. Han, H.X. Li, J.Y. Wang, On nilpotent incline matrices, Linear Algebra Appl. 406 (2005) 201–217.
2226 [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
Y.-J. Tan / Fuzzy Sets and Systems 161 (2010) 2213 – 2226 F. Harary, On the consistency of precedence matrices, J. Assoc. Comput. Math. 7 (1960) 255–259. H. Hashimoto, Reduction of a nilpotent fuzzy matrices, Inf. Sci. 27 (1982) 223–243. H. Hashimoto, Transitivity of generalized fuzzy matrices, Fuzzy Sets and Syst. 17 (1985) 83–90. K.H. Kim, F.W. Roush, Generalized fuzzy matrices, Fuzzy Sets and Syst. 4 (1980) 293–315. E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer, Dordrecht, 2000. J.X. Li, Some results on the nilpotent fuzzy matrices, Fuzzy Syst. Math. 3 (1) (1989) 52–55 (in Chinese). R.B. Marimont, A new method of checking the consistency of precedence matrices, J. Assoc. Comput. Math. 6 (1959) 162–171. M. Minoux, Bideterminants, arborescences and extension of the matrix-tree theorem to semirings, Discrete Math. 171 (1997) 191–200. M. Minoux, A generalization of the all minors matrix tree theorem to semirings, Discrete Math. 199 (1999) 139–150. H.T. Ren, Z. Zhang, Z.S. Liu, A necessary and sufficient condition for nilpotent fuzzy matrices, Acta Sci. Nat. Univ. Neimonggol 27 (1996) 594–596 (in Chinese). C. Reutenauer, H. Straubing, Inversion of matrices over a commutative semiring, J. Algebra 88 (1984) 350–360. D.E. Rutherford, The Cayley–Hamilton theorem for semiring, Proc. R. Soc. Edinburgh Sect. A 66 (1964) 211–215. Y.J. Tan, On the powers of matrices over a distributive lattice, Linear Algebra Appl. 336 (2001) 1–14. Y.J. Tan, On compositions of lattice matrices, Fuzzy sets and Syst. 129 (2002) 19–28. Y.J. Tan, On nilpotent matrices over distributive lattices, Fuzzy Sets and Syst. 151 (2005) 421–433. E.M. Vechtomov, Two general structure theorems on submodules, Abelian Groups and Modules (in Russian), Tomsk State University, Tomsk, Vol. 15, 2000, pp. 17–23. K.L. Zhang, On the nilpotent matrices over D01 -lattices, Fuzzy Sets and Syst. 117 (2001) 403–406. U. Zimmermann, Linear and combinatorial optimization in ordered algebraic structures, Annals of Discrete Mathematics, Vol. 10, North Holland, 1981.