Tropical linear algebra with the Łukasiewicz T-norm

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, we develop a theory of the matrix powers and the eigenproblem over the max-Łukasiewicz semiring. © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).