Tropical linear algebra with the Lukasiewicz T-norm
Martin Gavalec, Zuzana Nemcova, Sergei Sergeev

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.
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