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Abstract
For an arbitrary n X n fuzzy matrix R we define its period of oscillation d* and index k*. We present some properties of d* and k*, and show that d* / [n] (where [n] is the least common multiple of 1, 2, . . . , n). Then we obtain an algorithm to find d* and k*. The results are useful for examining periodicity of powers of finite fuzzy relations, and they also hold for a Boolean matrix.
... Fan and Liu [9] studied the properties of oscillating power sequence of fuzzy matrix and showed that the period set of n × n fuzzy matrices is not bounded from above by a power of n for all integers n. Li [14] presented some properties of index and period of a fuzzy matrix and showed that the period of n × n fuzzy matrix is a divisor of [n]. Li [15] proved that the index of n × n fuzzy matrix is less than or equal to (n − 1)[n]. ...
... "⇐" It is trivial. Lemma 2.1 generalizes Theorem 3.5 in [14] and Theorem 3.7 in [16]. Proof. ...
... Theorem 4.1 generalizes Theorem 3.7(1) in [14] and Theorem 4.3 in [20], and Corollary 4.1 is just Theorem 4.3 in [20]. However, it is impossible to prove Theorem 4.1 by the methods in [14,20]. ...
Inclines are the additively idempotent semirings in which products are less than or equal to factors. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. And the Boolean matrices, the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices (i.e., the matrices over inclines).This paper studies the power sequence of incline matrices in detail. A necessary and sufficient condition for the incline matrix to have index is given and the indices of some incline matrices with indices are estimated. It is proved that the period of n×n incline matrix with index is a divisor of [n] and that the set of periods of n×n incline matrices with indices is not bounded from above in the sense of a power of n for all n. An equivalent condition and some sufficient conditions for the incline matrix to converge in finite steps are established. The stability of the orbits of an incline matrix is considered and a theorem in [Fuzzy Sets and Systems 81 (1996) 227] is pointed out being false. The results in the present paper include some previous results in the literatures which were obtained for the Boolean matrices, the fuzzy matrices and the lattice matrices among their special cases.
... In particular, for the description of the behaviour of fuzzy systems, whose states are expressed using fuzzy values taken from the real interval 0; 1 or an arbitrary distributive lattice, the powers of lattice matrices are of special importance. Among the structures, over which the powers of matrices are computed, are max–addition (linear systems with synchronization [3,11,24,39,45]) max–multiplication [12,13], max–min (fuzzy systems, [9,16,17,18,20,35,36,53]), and their generalizations to max–t-norm [14,15] and to general sup–inf in a distributive lattice [52]. Results include proving convergence for special types of matrices [16,29,53], proving upper bounds [36] or computing the length of the oscillation period of the matrix power sequence [20], estimating its exponent, investigating connections between power sequence and eigenvectors [8,51], etc. ...
... The most cited reference in the papers on powers of fuzzy matrices is [53] (often considered to be the origin of this topic), which proves convergence of fuzzy matrices fulÿlling some relatively strong conditions. In [35] it was shown, using purely algebraic methods, that the period length of a matrix A ∈ M n (F) divides [n], the least common multiple of all integers not greater than n. The author also gives an algorithm for computing the period length, which, in the worst case, involves computing the power of A as high as the [n]th. ...
In this paper we review the results on powers of matrices over a distributive lattice. We stress the significance of the graph-theoretical approach and show how several previous results for matrices of special types can be obtained in a unified way.
... By replacing L m with the closed interval ½0; 1 (a infinite chain with the usual ordering) in the above definitions, we may get so-called n  n fuzzy matrices and fuzzy matrix semirings. The fuzzy matrices have been extensively studied [1][2][3][4][5][8][9][10][11][12][13][14]. We notice that the some problems in theory of fuzzy matrices may be translated into the corresponding problems in the theory of semilattice-ordered semirings M n ðL m Þ. ...
... In some papers in literature (for example, [8,11,12] ...
Let LmLm denote the chain {0,1,2,…,m-1}{0,1,2,…,m-1} with the usual ordering and Mn(Lm)Mn(Lm) the matrix semiring of all n×nn×n matrices with elements in LmLm. We firstly introduce some order-preserving semiring homomorphisms from Mn(Lm)Mn(Lm) to M(Lk)M(Lk). By using these homomorphisms, we show that a matrix over the finite chain LmLm can be decomposed into the sum of some matrices over the finite chain LkLk, where k<mk<m. As a result, cut matrices decomposition theorem of a fuzzy matrix (Theorem 4 in [Z.T. Fan, Q.S. Cheng, A survey on the powers of fuzzy matrices and FBAMs, International Journal of Computational Cognition 2 (2004) 1–25 (invited paper)]) is generalized and extended. Further, we study the index and periodicity of a matrix over a finite chain and get some new results. On the other hand, we introduce a semiring embedding mapping from the semiring Mn(Lm)Mn(Lm) to the direct product of the h copies of the semiring Mn(Lk)Mn(Lk) and discuss Green’s relations on the multiplicative semigroup of the semiring Mn(Lm)Mn(Lm). We think that some results obtained in this paper is useful for the study of fuzzy matrices.
... The periodicity of these special matrices is 1 or 2 [44]. Li [43] has studied periodicity and index of fuzzy matrices in the general case. In [41] Fan has proved that the periodicity of a fuzzy matrix is the least common multiple (l.c.m.) of periodicity of its cut matrices, and the index of a fuzzy matrix is not greater than the maximum index of its cut matrices. ...
The iterates of Fuzzy Neutrosophic Soft Circulant Matrices (FNSCMs) under the max-min product is examined right now. It is indicated that on the off chance that the first row of a FNSCM is in decreasing order, at that point the iterates of the circulant converge and on the off chance that the first row is in increasing order, at that point the iterates oscillate.
In this paper, we study the convergence of infinite products of a finite number of fuzzy matrices, where the operations involved are max-min algebra. Two types of convergences in this context will be discussed: the weak convergence and strong convergence. Since any given fuzzy matrix can be "decomposed" of the sum of its associated Boolean matrices, we shall show that the weak convergence of infinite products of a finite number of fuzzy matrices is equivalent to the weak convergence of infinite products of a finite number of the associated Boolean matrices. Further characterizations regarding the strong convergence will be established. On the other hand, sufficient conditions for the weak convergence of infinite products of fuzzy matrices are proposed. A necessary condition for the weak convergence of infinite products of fuzzy matrices is presented as well.
In the literature, the limiting behavior of powers of a fuzzy matrix has been studied with max-product composition. Pang and Guu proposed a simple and effective characterization for the limiting behavior with the notion of asymptotic period. In this paper, we shall extend Pang and Guu's work to the realm with the max-Archimedean-t-norm composition, to which the max-product belongs. Moreover, some sufficient conditions for the powers to converge in finitely many steps will be established.
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 generalized fuzzy matrices with periods are established and an expression for the transitive closure of a generalized fuzzy matrix with a period as a sum of its powers is shown. Reduction of generalized fuzzy matrices is defined and some properties for reduction of generalized fuzzy matrices with periods are obtained. Also, the structure of all standard eigenvectors for a generalized fuzzy matrix with a period is determined.
In this paper, the notion of minimal strong component of a fuzzy matrix is proposed. By using this concept, the periodicity of the power sequence of a fuzzy matrix is described on the basis of the relation of periodicity between a fuzzy matrix and its cut matrices. Further the greatest value max∑li=n[li] of periodicity of all n×n fuzzy matrices is obtained.
Natural powers, via max-min multiplication, of s-transitive fuzzy matrices are considered. It is shown that they oscillate with period equal to 2. This is an extension of the convergence property concerning max-min transitive fuzzy matrices.
We give a systematic development of fuzzy matrix theory. Many of our results generalize to matrices over the two element Boolean algebra, over the nonnegative real numbers, over the nonnegative integers, and over the semirings, and we present these generalizations. Our first main result is that while spaces of fuzzy vectors do not have a unique basis in general they have a unique standard basis, and the cardinality of any two bases are equal. Thus concepts of row and column basis, row and column rank can be defined for fuzzy matrices. Then we study Green's equivalence classes of fuzzy matrices. New we give criteria for a fuzzy matrix to be regular and prove that the row and column rank of any regular fuzzy matrix are equal. Various inverses are also studied. In the next section, we obtain bounds for the index and period of a fuzzy matrix.
Convergence of powers of a fuzzy transitive matrix is considered and some conditions for convergence are given under the max-min operation on fuzzy matrices. A fuzzy transitive matrix is a matrix which represents a fuzzy transitive relation, and has many interesting properties. The conditions for convergence of fuzzy matrices are examined under a special operation which is essential to reduction of fuzzy matrices or fuzzy systems. Given results show graph-theoretic properties of fuzzy matrices and are useful when we consider convergence of systems represented by fuzzy matrices.
A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
A Boolean matrix is a matrix with elements having values of either 1 or 0; a fuzzy matrix is a matrix with elements having values in the closed interval [0, 1]. Fuzzy matrices occur in the modeling of various fuzzy systems, with products usually determined by the “max(min)” rule arising from fuzzy set theory. In this paper, some sufficient conditions for convergence under “max(min)” products of the powers of a square fuzzy matrix and of a fuzzy state process are established.