Max-plus algebra is the algebraic structure in which classical addition and multiplication are replaced by a⊕b = max{a, b} and a⊗b = a+b, respectively. Each system of linear equation in max-plus algebra we can write in the matrix form A⊗ x = b, where A and b are matrix and vector of suitable size. If we replace the matrix elements with matrix interval A = [A,A] and vector elements by vector int...

We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a notion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrizati...

We describe an application of the Legendre transform to communication networks. The Legendre transform applied to max-plus algebra linear systems corresponds to the Fourier transform applied to conventional linear systems. Hence, it is a powerful tool that can be applied to max-plus linear systems and their identification. Linear max-plus algebra has been already used to describe simple data co...

The max-Łukasiewicz semiring is defined as the unit interval [0, 1] equipped with the arithmetics “a + b” = max(a, b) and “ab” = max(0, a + b− 1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Łukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, w...

This paper discusses the usefulness of (max, +) algebra as a mathematical modelling framework for discrete event systems (DESs). A Max-Plus Algebra Toolbox developed at Lehrstuhl für Systemtheorie technischer Prozesse is presented. This software package is a set of functions to take advantage of the (max, +) algebra in the Matlab environment for rapid prototyping, design, and analysis of DESs. ...

The network calculus is a framework for the analysis of communication net-works, which exploits that many computer network models become tractablefor analysis if they are expressed in a min-plus or max-plus algebra. In amin-plus algebra, the network calculus characterizes amounts of traffic andavailable service as functions of time. In a max-plus algebra, the network cal-cul...

A fast learning method for morphological neural networks which are formulated based on max plus algebra, is proposed. The proposed fast learning method is deduced from the idempotent properties of max plus algebra and cost matrices. Through experiments using artificial training data sets, it is conformed that the computation time of the proposed method is decreased into 60.9-68.6% and 40.9 – 48...

The eigenspace structure for a given n × n concave Monge matrix in a max-plus algebra is described. Based on the description, an O(n) algorithm for computing the eigenspace dimension is formulated, which is faster than the previously known algorithms. Analogous results for convex Monge matrices have been published in [M. Gavalec, J. Plavka, Structure of the eigenspace of a Monge matrix in max-p...