Multiorder, Kleene stars and cyclic projectors in the geometry of max cones

TL;DR

This paper explores the geometric structure of max cones using multiorder, Kleene stars, and cyclic projectors, linking max-plus convex geometry with max algebra and providing tools for solving max-linear systems.

Contribution

It introduces new geometric insights into max cones, relating multiorder principles, Kleene stars, and cyclic projectors, and connects these to max algebra and convex decomposition.

Findings

Max-plus analogues of convex geometry statements are established.

Kleene stars describe eigenspaces and convex regions in max algebra.

Cyclic projectors help solve homogeneous max-linear systems.

Abstract

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 fundamental for max algebra, as they accumulate the weights of optimal paths and describe the eigenspace of a matrix. On the other hand, the approach of tropical convexity decomposes a finitely generated semimodule into a number of convex regions, and these regions are column spans of uniquely defined Kleene stars. Another recent geometric result, that several semimodules with zero intersection can be separated from each other by max-plus halfspaces, leads to investigation of specific nonlinear operators called cyclic projectors. These nonlinear operators can be used to find a solution to homogeneous multi-sided systems of max-linear equations. The results are presented in the setting of max cones, i.e., semimodules over the max-times semiring.