Computing matrix period in max-min algebra

Powers of matrices over distributive lattices?a review

“…Here, the representation of matrices by the join-irreducible elements of F and hence its constituents is useless (in fact, each element of a linear lattice is join-irreducible), so instead binary cut matrices A for ∈ F deÿned by (A ) ij = 1 if a ij ¿ ; 0 otherwise (5) are used. (Cut matrices are called by some authors its sections or zero patterns [33] or threshold matrices [9,20] etc.) Notice, that although for A ∈ M n (F) theoretically inÿnitely many cut matrices exist, in practice only at most n 2 + 1 of them may be di erent, and the decisive changing points correspond to the entries of A. Matrix A can again be uniquely expressed as a linear combination of its cut matrices:…”

“…However, the graph-theoretical approach gives the following theorem, proved independently in [33,20,17] (the references are given chronologically).…”

“…Some of them are in existing SCCs or they merge previous SCCs; in both cases, the periods of new SCCs divide the periods of SCCs existing in digraphs with greater cut values. So what only really matters, are the periods of minimal (with respect to vertex inclusion) SCCs and the number of those is at most n (see [20]). Hence using the Balcar-Veinott's algorithm for computing the period of each minimal SCC yields and algorithm with complexity O(n 3 ).…”

“…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.…”

Robustness of Interval Toeplitz Matrices in Fuzzy Algebra

“…Periodic behaviour of fuzzy matrices with corresponding polynomial algorithms were studied in [5] and [10]. However, in practice we deal often with inexact input data.…”