Ordered Structures of Polynomials over Max-Plus Algebra

Wang, Cailu; Xia, Yuanqing; Tao, Yuegang

doi:10.3390/sym13071137

Cited by 5 publications

(4 citation statements)

References 33 publications

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“…Similarly to [7,[9][10][11][12][13][14][15][16][17][20][21][22], by an interval of R(m, n), we mean a subset of R(m, n) of the form Y = (y ij ) for i ∈ M, j ∈ N, where each y ij is an arbitrary interval belonging to R. For each i, j we denote y ij := inf y ij and y ij = sup y ij . Then, we also have…”

Section: Interval Solutionsmentioning

confidence: 99%

“…Moreover, some of the equivalent conditions of strong versions of solvability can be checked in a polynomial number of arithmetic operations. Systems of two-sided (max, plus)-linear equations A ⊗ x = B ⊗ x have been studied by many authors [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The motivation for this research are papers [8,10].…”

Section: Introductionmentioning

confidence: 99%

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Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

Myšková

Plávka

2021

Mathematics

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Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there are some pseudopolynomial algorithms, but a polynomial algorithm is still waiting for an appearance. The paper deals with the analysis of solvability of two-sided (max, plus)-linear equations with inexact (interval) data. The purpose of the paper is to get efficient necessary and sufficient conditions for solvability of the interval systems using the property of the solution set of the non-interval system Ax = Bx. The main contribution of the paper is a transformation of weak versions of solvability to either subeigenvector problems or to non-interval two-sided (max, plus)-linear systems and obtaining the equivalent polynomially checked conditions for the strong versions of solvability.

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Section: Interval Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

Myšková

Plávka

2021

Mathematics

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“…This set is known as the set of matrices over max-plus algebra, denoted ℝ 𝜀 𝑚×𝑛 (Jones, 2021). In max-plus algebra, a form of polynomial is also known, which is called maxpolynomial (Butcovic, 2010;Wang et al, 2021).…”

Section: A Introductionmentioning

confidence: 99%

Characteristic Min-Polynomial and Eigen Problem of a Matrix over Min-Plus Algebra

Maghribi,

Siswanto,

Sutrima

2023

JTAM

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Let R_ε=R∪{-∞}, with R being a set of all real numbers. The algebraic structure (R_ε,⊕,⊗) is called max-plus algebra. The task of finding the eigenvalue and eigenvector is called the eigenproblem. There are several methods developed to solve the eigenproblem of A∈R_ε^(n×n), one of them is by using the characteristic max-polynomial. There is another algebraic structure that is isomorphic with max-plus algebra, namely min-plus algebra. Min-plus algebra is a set of R_(ε^' )=R∪{+∞} that uses minimum (⊕^' ) and addition (⊗) operations. The eigenproblem in min-plus algebra is to determine λ∈R_(ε^' ) and v∈R_(ε^')^n such that A⊗v=λ⊗v. In this paper, we provide a method for determining the characteristic min-polynomial and solving the eigenproblem by using the characteristic min-polynomial. We show that the characteristic min-polynomial of A∈R_(ε^')^(n×n) is the permanent of I⊗x⊕^' A, the smallest corner of χ_A (x) is the principal eigenvalue (λ(A)), and the columns of A_λ^+ with zero diagonal elements are eigenvectors corresponding to the principal eigenvalue.

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“…Some researchers also expand the concept in max-plus algebra into interval max-plus algebra, see [4] [5]. ℝ 𝑚𝑎𝑥 is idempotent commutative semiring [6] and can be referred to as idempotent semifield [7]. If ℝ 𝑚𝑎𝑥 𝑛 is defined as ℝ 𝑚𝑎𝑥 𝑛 = {(𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) 𝑇 |𝑥 𝑖 ∈ ℝ 𝑚𝑎𝑥 , 𝑖 = 1,2, … , 𝑛}, ℝ 𝑚𝑎𝑥 𝑛 is a semimodules over the semiring ℝ 𝑚𝑎𝑥 [6].…”

Section: Introductionmentioning

confidence: 99%

Bases in Min-Plus Algebra

Rosyada¹,

Siswanto²,

Kurniawan³

2021

Advances in Social Science, Education and Humanities Research

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In classical linear algebra, a basis is a vector set that generates all elements in the vector space and that vector set is a linear independence set. However, the definitions of the linear dependence and independence in min-plus algebra are little more complex given that the min-plus algebra is the linear algebra over the commutative idempotent semiring. The definition of the linear dependence (independence) is used in this paper is Gondran-Minoux linear dependence (independence). A finite set is Gondran-Minoux linearly dependent if the set can be divided into two sets that form a linear space with an intersection which is not a zero vector. We will define the concept of the bases in min-plus algebra. In this paper also defined the concept of a weak bases and will be shown that the linear dependence (independence) is not needed to form a weak basis. In the last part of the research result's are proven that every basis in a semimodules in min-plus algebra is a weak basis.

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Ordered Structures of Polynomials over Max-Plus Algebra

Cited by 5 publications

References 33 publications

Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

Characteristic Min-Polynomial and Eigen Problem of a Matrix over Min-Plus Algebra

Bases in Min-Plus Algebra

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