Bases in Min-Plus Algebra

Abstract

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.

Terminologi Dasar Pada Himpunan Simetrisasi Aljabar Min-Plus

“…Penerapan aljabar min-plus pada berbagai bidang sudah banyak dilakukan, antara lain masalah rute tercepat distribusi susu (Suwanti et al, 2017), masalah simulasi produksi susu (Pramesthi, 2021), masalah distribusi kentang (Putri, 2016), masalah jaringan angkutan (Susilowati, 2018). Penelitian pengembangan aljabar min-plus juga sudah beberapa dilakukan antara lain masalah dualitas max-plus dan min-plus jaringan (Liebeherr, 2017), masalah arsitektur jaringan, (Darbon et al, 2021), nilai eigen pada matriks atas aljabar min-plus (Hook, 2017), (Rahayu et al, 2021), masalah aljabar min-plus interval (Awallia et al, 2020), dan basis atas aljabar min-plus (Rosyada et al, 2021).…”

Determining the Inverse of a Matrix over Min-Plus Algebra

“…If 𝑎, 𝑏 ∈ ℝ 𝜀′ , 𝑎 ⊕ ′ 𝑏 = min (𝑎, 𝑏), 𝑎 ⊗ ′ 𝑏 = 𝑎 + 𝑏 (Suroto, 2022). Furthermore, min-plus algebra has an idempotent commutative semiring structure and represented as ℝ 𝑚𝑖𝑛 = (ℝ 𝜀 ′ ; ⊕ ′ ,⊗ ′ ) (Akian et al, 2007;Rosyada et al, 2021). Min-plus algebra can also be applied in everyday life, this has been discussed by several researchers including the fastest route modeling (Suprayitno, 2017;Susilowati & Fitriani, 2019), fiber networks (Li & Zhao, 2012), and petri nets (Farhi et al, 2009).…”