The max-plus algebra of exponent matrices of tiled orders

Abstract

Abstract. An exponent matrix is an n×n matrix A = (a ij ) over N 0 satisfying (1) a ii = 0 for all i = 1, . . . , n and (2) a ij + a jk ≥ a ik for all pairwise distinct i, j, k ∈ {1, . . . , n}. In the present paper we study the set E n of all non-negative n × n exponent matrices as an algebra with the operations ⊕ of component-wise maximum and ⊙ of component-wise addition. We provide a basis of the algebra (E n , ⊕, ⊙, 0) and give a row and a column decompositions of a matrix A ∈ E n with respect to this basis. This structure result determines all n × n tiled orders over a fixed discrete valuation ring. We also study automorphisms of E n with respect to each of the operations ⊕ and ⊙ and prove that Aut(E n , ⊙) = Aut(E n , ⊕) = Aut(E n , ⊙, ⊕, 0) ≃ S n × C 2 , n > 2.

Orders and polytropes: matrix algebras from valuations

“…The orders M in Proposition 6 are called graduated orders in (Plesken, 1983, Remark II.4). They are also known as tiled orders (Dokuchaev et al 2017;Jategaonkar 1974), split orders (Shemanske 2010) or monomial orders (Yang and Chia-Fu 2015). A graduated order M is in standard form if M ≥ 0 and m i j + m ji > 0 for i = j.…”

“…A variant of P d that assumes nonnegativity constraints was studied in (Deza et al 2002), which offers additional data. We also refer to (Dokuchaev et al 2017) for a study of the cone of polytropes from the perspective of semiring theory.…”

The cone of quasi-semimetrics and exponent matrices of tiled orders

“…We will need a description of E n which was developed by Dokuchaev et al [11]. For each proper subset I of {1, .…”

“…Order of "tiled form" appeared as essential ingredients in the study of large classes of orders in algebras, gaining special significance in the theory of orders of finite representation type and in the investigation of global dimension. Since then tiled orders (also called Schurian orders and monomial orders) were studied from various points of view, and, apart from their structural, representation theoretic and homological applications, they turned out to be useful to additive categories [1] and, more recently, to the connection between Cohen-Macaulay representation theory and cluster categories [4] (see also the bibliography in [11]).…”

“…A natural question is whether new automorphisms can crop up if we consider each of those monoids. That is not the case: it was proved in [11] that the automorphism group of each is exactly S n .…”