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The max-plus algebra of exponent matrices of tiled orders

Mikhailo Dokuchaev, Vladimir V. Kirichenko, Ganna Kudryavtseva, Makar Plakhotnyk

Abstract

An exponent matrix is an $n\times n$ matrix $A=(a_{ij})$ over ${\mathbb N}^0$ satisfying (1) $a_{ii}=0$ for all $i=1,\ldots, n$ and (2) $a_{ij}+a_{jk}\geq a_{ik}$ for all pairwise distinct $i,j,k\in\{1,\dots, n\}$. In the present paper we study the set ${\mathcal E}_n$ of all non-negative $n\times n$ exponent matrices as an algebra with the operations $\oplus$ of component-wise maximum and $\odot$ of component-wise addition. We provide a basis of the algebra $({\mathcal E}_n, \oplus, \odot,0)$ and give a row and a column decompositions of a matrix $A\in {\mathcal E}_n$ with respect to this basis. This structure result determines all $n\times n$ tiled orders over a fixed discrete valuation ring. We also study automorphisms of ${\mathcal E}_n$ with respect to each of the operations $\oplus$ and $\odot$ and prove that ${\rm Aut}(\mathcal{E}_n,\, \odot ) = {\rm Aut}(\mathcal{E}_n,\, \oplus ) = {\rm Aut}(\mathcal{E}_n,\, \odot ,\oplus ,0) \simeq {\mathcal{S}}_n \times C_2,$$n>2.$

The max-plus algebra of exponent matrices of tiled orders

Abstract

An exponent matrix is an matrix over satisfying (1) for all and (2) for all pairwise distinct . In the present paper we study the set of all non-negative exponent matrices as an algebra with the operations of component-wise maximum and of component-wise addition. We provide a basis of the algebra and give a row and a column decompositions of a matrix with respect to this basis. This structure result determines all tiled orders over a fixed discrete valuation ring. We also study automorphisms of with respect to each of the operations and and prove that

Paper Structure

This paper contains 6 sections, 15 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Proof of the Structure Theorem
  3. Automorphisms of $({\mathcal{E}}_n, \odot)$
  4. Automorphisms of $({\mathcal{E}}_n, \oplus)$
  5. Strict downsets of elements of ${\mathcal{E}}_n$
  6. Automorphisms of $({\mathcal{E}}_n, \oplus)$

Key Result

Theorem 1.1

The matrices $T_I$, where $I$ runs through the proper subsets of the set $\{1,2,\dots, n\}$, form a basis of the algebra $({\mathcal{E}}_n, \odot, \oplus, 0)$. That is, any matrix $A\in {\mathcal{E}}_n$ can be written in the form where all the matrices $B_1,\dots, C_m$ are blocks (as usual $\odot$ is performed prior to $\oplus$). Moreover, this basis is the only minimal basis of the algebra $({\m

Theorems & Definitions (28)

  • Theorem 1.1: Structure Theorem
  • Remark 2.1
  • Remark 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Example 2.5
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • ...and 18 more