A congruence on the semiring of normal tropical matrices

Orthogonality for (0, −1) tropical normal matrices

“…Normal matrices (and a slightly weaker notion called definite matrices) over different sets of numbers have been studied for more than fifty years, under different names, beginning with Yoeli in [27]. The notion appears in connection with tropical algebra and geometry [6,7,8,18,22,24,25,28]. In computer science they have been called DBM (difference bound matrices).…”

“…Semirings are widely used in discrete event systems, dynamic programming and linguistics [10,11,21]. Recent attention has been paid to the semiring of normal matrices over R ∪ {−∞} with tropical operations, in [28].…”

Isocanted alcoved polytopes

“…(1) 0 ∈ p(A) if and only if A is normal (N) (meaning a i,i = 0, a i,j ≤ 0, ∀i, j), (see [10,38]) (2) if 0 ∈ p(A), then A describes p(A) optimally (or tightly) if and only if A is normal idempotent (NI) (meaning that, in addition to normality, we have A ⊙ A = A, which requires that a i,j + a jk ≤ a ik , ∀i, j, k) 1 (see [24,30,36]). (3) for each alcoved polytope P containing 0 there exist a unique NI matrix A such that P = p(A) (see Lemma 2.6 in [24] and [30,36]).…”

“…General references for polytopes are [3,4,13,19,29,33,39,40]. Normal idempotent matrices have been used in [26,38]. Idempotent matrices, also called Kleene stars, have been used in [24,30,36] in connection to polytopes.…”

Quasi-Euclidean classification of alcoved convex polyhedra

“…Normal matrices (and slightly weaker notions) have been studied for more than fifty years (see the pioneer works [1,8,55]), under various different names. In connection with Tropical Mathematics, they appear in [12,13,14,15,39,48] and very recently in [56]. In connection with Automata Theory, Scheduling Theory, Computer Science, Discrete Event Systems and Dynamical Programming, normal matrices appear in [6,7,22,36] among other, where they are named difference bounded matrices (DBM).…”