Tropical linear algebra with the Lukasiewicz T-norm

TL;DR

This paper explores the max-Lukasiewicz semiring, demonstrating its equivalence to tropical linear algebra, and develops a theory of matrix powers and eigenproblems within this framework.

Contribution

It introduces a novel connection between max-Lukasiewicz and tropical linear algebra, and develops foundational theory for matrix powers and eigenproblems in this semiring.

Findings

Max-Lukasiewicz linear algebra can be reformulated as tropical linear algebra.

A theory of matrix powers over the max-Lukasiewicz semiring is established.

Eigenproblem analysis is extended to this semiring context.

Abstract

The max-Lukasiewicz 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-Lukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Lukasiewicz semiring.