This paper summarizes results on some topics in the max-plus convex geometry, mainly concerning the role of multiorder, Kleene stars and cyclic projectors, and relates them to some topics in max algebra. The multiorder principle leads to max-plus analogues of some statements in the finite-dimensional convex geometry and is related to the set covering conditions in max algebra. Kleene stars are ...

Exotic semirings such as the “(max;+) semiring” (R[f 1g;max;+), or the “tropical semiring” (N[f+1g;min;+), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis (lo...

A multidimensional optimization problem is considered, which is formulated in terms of tropical mathematics as to minimize a nonlinear objective function subject to linear inequality constraints. The optimization problem is motivated by a problem in project scheduling when an optimal schedule is given by minimizing the flow time of activities in a project under various activity precedence const...

Consider the set B = [0, 1] endowed with the operations⊕ = max,∧ = min. This is a well-known distributive lattice, and like any distributive lattice it can be considered as a semiring equipped with addition ⊕ and multiplication ⊗ := ∧. Importantly, both operations are idempotent, a⊕a = a and a⊗a = a∧a = a, and closely related to the order: a⊕ b = b ⇔ a ≤ b ⇔ a∧ b = a. For standard literature on...

We examine a multidimensional optimisation problem in the tropical mathematics setting. The problem involves the minimisation of a nonlinear function defined on a finite-dimensional semimodule over an idempotent semifield subject to linear inequality constraints. We start with an overview of known tropical optimisation problems with linear and nonlinear objective functions. A short introduction...

In this paper, we introduce a family of graphs which is a generalization of zero-divisor graphs and compute an upper-bound for the diameter of such graphs. We also investigate their cycles and cores

Erhard Aichinger (Johannes Kepler University Linz) A remark on the composition of polynomial functions over algebraically closed fields In 1969, M. D. Fried and R. E. MacRae proved that for univariate polynomials p, q, f, g ∈ K[t] (K a field) with p, q nonconstant, p(x)− q(y) divides f(x)− g(y) in K[x, y] if and only if there is h ∈ K[t] such that f = h(p(t)) and g = h(q(t)). In 1995, F. Binder...