Powers of matrices over distributive lattices?a review

Abstract

In this paper we review the results on powers of matrices over a distributive lattice. We stress the signiÿ-cance of the graph-theoretical approach and show how several previous results for matrices of special types can be obtained in a uniÿed way.

On the λ-robustness of matrices over fuzzy algebra

“…It was shown in [6] that each λ ∈ B is an eigenvalue of a given matrix A and the sequence e λ , A ⊗ e λ , A 2 ⊗ e λ , . .…”

“…These relations allow us to formulate the next assertion for λ ≥ max i,j∈N a ij using the known results on the ultimately periodic matrices (see [4,6,10,9]). Proof.…”

“…Convergence and periodicity of matrix powers and the relations between the matrix and orbit periods in fuzzy algebra have been studied in [4][5][6]8,9,16,19,20].…”

On the convergence exponent of decomposable relations

“…Powers of real matrices computed with at least one of the addition/multiplication operations replaced by maximum or minimum have been extensively studied, since they have applications in various types of discrete systems [2]. In this section, we assume that all the fuzzy relations R ∈ F (X × X) and X is a finite set.…”

“…Thomason [11]). Let R ∈ F (X × X), define R 1 = R, R k+1 = R k R for all positive integer k. [2], Thomason [11]). A fuzzy relation R ∈ F (X × X) is called convergent if there exists a positive integer k such that R k+1 = R k .…”