A walk on max-plus algebra

Max-plus algebra is a kind of idempotent semiring over R_max := R ∪ {-∞} with two operations ⊕:= max and ⊗:= +. In this paper, we introduce a new model of a walk on one dimensional lattice on Z, as an analogue of the quantum walk, over the max-plus algebra and we call it max-plus walk. In the conventional quantum walk, the summation of the ℓ²-norm of the states over all the positions is a conserved quantity. In contrast, the summation of eigenvalues of state decision matrices is a conserved quantity in the max-plus walk. Moreover, spectral analysis on the total time evolution operator is also given.