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.