Edelweiss Applied Science and Technology 
ISSN: 2576-8484 
Vol. 8, No. 6, 9544-9554 
2024 
Publisher: Learning Gate 
DOI: 10.55214/25768484.v8i6.4034 
© 2024 by the authors; licensee Learning Gate 

© 2024 by the authors; licensee Learning Gate 

* Correspondence:  gregoria.ariyanti@ukwms.ac.id 

 
 
 
 
 

Notes on matrix inverse over min-plus algebra 

 
Gregoria Ariyanti1*, Ana Easti Rahayu Maya Sari2, Christina Manurung3 
1,2,3Department of Mathematics Education, Widya Mandala Surabaya Catholic University; gregoria.ariyanti@ukwms.ac.id 
(G.A.) 

 

 
Abstract: One of the semiring structures is the max-plus algebra, a set with entries ℝ𝜀 = ℝ ∪ {−∞}  equipped with the 

operation ⊕, which represents the maximum value, and the operation ⊗, which means addition. Another semiring 

structure is the min-plus algebra, a set with entries ℝ𝜀 = ℝ ∪ {+∞}  equipped with the operation ⊕, representing the 

minimum value, and the operation ⊗, which means addition. Matrices over min-plus algebras can have inverses determined 
by certain conditions. The general inverse type can define the inverse of matrices over min-plus algebras. In this paper, we 
will develop the characteristics of general inverse matrices over min-plus algebras. The research method used is the 
literature study method sourced from books and journal articles. The main result of this study is that the generalized 

inverse of the matrix 𝐴 ∈  𝑅𝑚𝑖𝑛
𝑛×𝑛 can be obtained by determining the matrix 𝑋 with entry 𝑋𝑘𝑙 =

𝑛
𝑚𝑖𝑛
𝑖 = 1

𝑛
𝑚𝑖𝑛
𝑗 = 1

(−𝑎𝑖𝑘 + 𝑎𝑖𝑗 −

𝑎𝑙𝑗) which satisfies 𝐴 ⊗ 𝑋 ⊗ 𝐴 = 𝐴. 

Keywords: Generalized, Inverse, Min-plus algebra. 

 
1. Introduction  

Semiring is an algebraic structure obtained from rings with the condition that a ring is weakened by 
eliminating several ring conditions. Other algebraic structures known as Semigroups and Semirings will 
emerge if some properties of Groups and Rings are weakened. This shows that the algebraic structures 
created are Semigroups and then Semirings if some Group or Ring conditions are removed ([1], [2]). 
The main difference between semiring and ring structures can be seen from the existence of an inverse 
element for the addition operation ([3], [4]). 

A structure (S,+,×) with S, a non-empty set, + addition operation, and × multiplication operation, is 
semiring if it fulfils the commutative and associative properties of addition, multiplication associativity, 
distributive, has a zero element, and a unit element ([4], [5]). It is well known that a semigroup is formed 
by a non-empty set S and the associative binary operation ×. Thus, the structure (S,+,×) is said to be 
semiring if (S,+) is a commutative semigroup, (S,×) is a semigroup, distributive, has element 0 and has a 
unit element ([6], [7]). With semiring entries, a semiring matrix can be developed [1], [5], [8]. 

One structure that is a semiring is max-plus algebra. A structure (ℝmax,⊕,⊗)  with ℝmax = ℝ ∪
{−∞} is said to be a max-plus algebra with a maximum ⊕ operation and an addition ⊗ operation. In 

another section, a semiring other than max-plus algebra is min-plus algebra. Min-plus algebra Rmin =
R ∪ {+∞} with minimum (⊕ ′) and addition (⊗) operations with identity elements with respect to ⊕ ′ 
are 𝜀′ = +∞ and 𝑒 = 0. Max-plus algebra and min-plus algebra are isomorphic because of their similar 
structure. It is possible to convert the idea of max-plus algebra into min-plus algebra [5]. The semiring 
element has an inverse to the addition operation so that the determinant of a matrix over the semiring 
can be defined ([3], [6]). The inverse of the semiring matrix can be determined by determining the 
determinant of the semiring matrix. Looking at the characteristics of the semiring, we will specifically 
look at the characteristics of min-plus algebra. It is done because not all semiring properties also apply to 
min-plus algebra. 

As with Group and Ring structures, the commutative characteristic applies to certain Semirings [6], 
[9], [10]. A particular Semiring owns the existence of an inverse element for addition on a Semiring. In 



9545 

 

 
Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 6: 9544-9554, 2024 
DOI: 10.55214/25768484.v8i6.4034 
© 2024 by the authors; licensee Learning Gate 

 

a semiring, the entry on the semiring has the inverse of the + operation so that the determinant of the 
matrix on the semiring can be defined. This study aims to develop the characteristics of the generalized 
inverse matrix over a min-plus algebra. 
 
2. Materials and Methods  

Reducing several properties will form a new algebraic structure. Not all properties of the complete 
structure will also be reduced to the new algebraic structure. The research uses a literature study method 
sourced from books and journal articles. The steps for developing ideas in this study are shown in Figure 
1. 
 

 
Figure 1.  
Procedure for the characteristic study of min-plus algebra. 

 
2.1. Min-Plus Algebra 

The properties of a max-plus algebra can be used to create a min-plus algebra.   
 

Definition 1 

The structure (ℝ𝑚𝑖𝑛,⊕′,⊗)  is said to be a min-plus algebra with ℝ𝑚𝑖𝑛 = ℝ ∪ {+∞}, the binary 

operations ⊕ ′ and ⊗ defined as 𝑎 ⊕ ′ 𝑏 = 𝑚𝑖𝑛 {𝑎, 𝑏} and 𝑎 ⊗ 𝑏 = 𝑎 + 𝑏 for 𝑎, 𝑏 ∈ ℝ𝑚𝑖𝑛. Theorem 1 
provides the algebraic property of min-plus. 
 
Theorem 1 

For an 𝑥, 𝑦, 𝑧 ∈  ℝ𝑚𝑖𝑛 with 𝑒 ≔ 0 and 𝜀′ = +∞ applies 

1. Associative, namely ∀𝑥, 𝑦, 𝑧 ∈ ℝ𝑚𝑖𝑛 , 𝑥 ⊕′ (𝑦 ⊕′ 𝑧) = (𝑥 ⊕′ 𝑦) ⊕ ′𝑧 and 𝑥 ⊗ (𝑦 ⊗ 𝑧) = (𝑥 ⊗
𝑦) ⊗ 𝑧  

2. Commutative, namely ∀𝑥, 𝑦 ∈ ℝ𝑚𝑖𝑛, 𝑥 ⊕ ′𝑦 = 𝑦 ⊕ ′𝑥 and 𝑥 ⊗ 𝑦 = 𝑦 ⊗ 𝑥 

3. Distributive of ⊗ over ⊕ ′, namely ∀𝑥, 𝑦, 𝑥 ∈ ℝ𝑚𝑖𝑛,𝑥 ⊗ (𝑦 ⊕′ 𝑧) = (𝑥 ⊗ 𝑦) ⊕ ′(𝑥 ⊗ 𝑧) 

4. There is a zero element, namely ∀𝑥 ∈ ℝ𝑚𝑖𝑛, 𝑥 ⊕ ′𝜀 = 𝜀 ⊕ ′𝑥 = 𝑥 

5. There is a unit element, namely  ∀𝑥 ∈ ℝ𝑚𝑖𝑛, 𝑥 ⊗ 𝑒 = 𝑒 ⊗ 𝑥 = 𝑥 

6. There is an absorption property by the zero element 𝜀′ towards ⊗,that is 

∀𝑥 ∈ ℝ𝑚𝑖𝑛, 𝑥 ⊗ 𝜀′ = 𝜀′ ⊗ 𝑥 = 𝜀′ 
7. The idempotent property of ⊕ ′, namely ∀𝑥 ∈ ℝ𝑚𝑖𝑛, 𝑥 ⊕ ′𝑥 = 𝑥  
 
Meanwhile, defining the determinant uses permutation characteristics. The definition of permutation is 
given as follows. 
 
 



9546 

 

 
Edelweiss Applied Science and Technology 
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Vol. 8, No. 6: 9544-9554, 2024 
DOI: 10.55214/25768484.v8i6.4034 
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Definition 2 

A permutation matrix is a matrix with exactly one entry (𝑒) and another entry (𝜀′) in each of its 𝑖 −th 

row and 𝑗 −th column. The permutation matrix over the min-plus algebra can be described as 𝑃𝜎 = [𝑝𝑖𝑗]  
with  

𝑝𝑖𝑗 = {
𝑒; 𝑖 = 𝜎(𝑗)

𝜀′; 𝑖 ≠ 𝜎(𝑗)
 

if 𝜎: {1, 2, … , 𝑛} ⟶ {1, 2, … , 𝑛} is a permutation. Thus, 𝑒 appears in the 𝜎(𝑗 − 𝑡ℎ) row in the 𝑗 −th column 

of 𝑃𝜎. 
 
Example 1 

Let 𝜎: {1, 2} → {1, 2} with 𝜎(1) = 2 and 𝜎(2) = 1, then 

𝑝11 = {
𝑒 𝑗𝑖𝑘𝑎 1 = 𝜎(1)

𝜀′ 𝑗𝑖𝑘𝑎 1 ≠ 𝜎(1) 
 , 𝑝11 = 𝜀′ 

𝑝12 = {
𝑒 𝑗𝑖𝑘𝑎 1 = 𝜎(2)

𝜀′ 𝑗𝑖𝑘𝑎 1 ≠ 𝜎(2) 
 , 𝑝12 = 𝑒 

𝑝21 = {
𝑒 𝑗𝑖𝑘𝑎 2 = 𝜎(1)

𝜀′ 𝑗𝑖𝑘𝑎 2 ≠ 𝜎(1) 
 , 𝑝21 = 𝑒 

𝑝21 = {
𝑒 𝑗𝑖𝑘𝑎 2 = 𝜎(2)

𝜀′ 𝑗𝑖𝑘𝑎 2 ≠ 𝜎(2) 
 , 𝑝22 = 𝜀 ′ 

The permutation matrix is [𝜀′ 𝑒
𝑒 𝜀′

]. 

 
Example 2 

Let 𝐴 = [
1 3 −2
3 5 8
4 6 −1

] , 𝑃𝜎 = [
𝜀′ 𝑒 𝜀′
𝜀′ 𝜀′ 𝑒
𝑒 𝜀′ 𝜀′

] we have 

𝐴 ⊗ 𝑃𝜎 = [
1 3 −2
3 5 8
4 6 −1

] ⊗ [
𝜀′ 𝑒 𝜀′
𝜀′ 𝜀′ 𝑒
𝑒 𝜀′ 𝜀′

] = [
−2 1 3
8 3 5

−1 4 6
] 

 

The right-hand multiplication of 𝑃𝜎 creates a permutation of the matrix columns so that the 𝑖 −th 

column of A appears as the 𝜎(𝑖) −th column of 𝐴 ⊗ 𝑃𝜎. The permutation matrix 𝑃𝜎 has an inverse, namely 

𝑃𝜎−1 where 𝑃𝜎−1 is the transpose of 𝑃𝜎 obtained 𝑃𝜎−1 = 𝑃𝜎𝑇 so that 𝑃𝜎 ⊗ 𝑃𝜎−1 = 𝐸. 
If 𝐴 is a matrix over a field, then a single matrix 𝐵 must satisfy the property 𝐴 ⊗ 𝐵 ⊗ 𝐴 = 𝐴. A 

matrix 𝐵 that satisfies this property is called the generalized inverse of matrix 𝐴. In min-plus algebra, 

there is no guarantee that every matrix has a generalized inverse. If 𝐴 has a generalized inverse, then 𝐴 

is considered regular. We will discuss determining whether a matrix 𝐴 is a regular min-plus algebra. 
      A min-plus algebra can be formed based on the characteristics of a max-plus algebra. As an initial 
characteristic, the following theorem is given. 
 
Theorem 2 

Given an idempotent commutative semigroup (𝑆, +). If on S a relation ≥ is defined by  𝑏 ≥ 𝑎 ⟺ 𝑎 +
𝑏 = 𝑏, then the relation  ≤ is a partial order on S. 
Proof : 

Given any 𝑎, 𝑏, 𝑐 ∈ 𝑆 then 

1. Since S is idempotent, then 𝑎 + 𝑎 = 𝑎 ⟺ 𝑎 ≥ 𝑎  



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2. If 𝑏 ≥ 𝑎 and 𝑎 ≥ 𝑏, then 𝑎 + 𝑏 = 𝑏 and 𝑏 + 𝑎 = 𝑎. Since 𝑆 is commutative, then 𝑎 = 𝑏 

3. If 𝑏 > 𝑎 and 𝑎 > 𝑐 then 𝑎 + 𝑏 = 𝑏 and 𝑐 + 𝑎 = 𝑎 then 

𝑏 + 𝑐 = (𝑎 + 𝑏) + 𝑐 = (𝑏 + 𝑎) + 𝑐 = 𝑏 + (𝑎 + 𝑐) = 𝑏 + 𝑎 = 𝑏 

        So, we have  𝑏 > 𝑐. 
        
Definition 3 

In ℝ𝑚𝑖𝑛 , the relation ≥𝑚𝑖𝑛 is defined as 

𝑥 ≥𝑚𝑖𝑛 𝑦 ⟺ 𝑥 ⊕′ 𝑦 = 𝑦   
Theorem 3 

The relation ≥𝑚𝑖𝑛 is a partial order. 
Proof:  

Given 𝑎, 𝑏, 𝑐 ∈  ℝ𝑚𝑖𝑛, then 

1. Since ℝ𝑚𝑖𝑛 is idempotent then 𝑎 ⊕′ 𝑎 = min{𝑎, 𝑎} = 𝑎 therefore 𝑎 ≥𝑚𝑖𝑛 𝑎 

2. If 𝑎 ≥𝑚𝑖𝑛 𝑏 and 𝑏 ≥𝑚𝑖𝑛  𝑎 then 𝑎 ⊕ ′𝑏 = 𝑏 and 𝑏 ⊕ ′𝑎 = 𝑎. Since ℝ𝑚𝑖𝑛 is commutative then 𝑎 = 𝑏 

3. If 𝑎 ≥𝑚𝑖𝑛 𝑏 and 𝑏 ≥𝑚𝑖𝑛  𝑐 then 𝑎 ⊕′ 𝑏 = 𝑏 and 𝑏 ⊕′ 𝑐 = 𝑐 then 

𝑎 ⊕′ 𝑐 = 𝑎 ⊕′ (𝑏 ⊕′ 𝑐) = 𝑎 ⊕′ (𝑏 ⊕′ 𝑐) = (𝑎 ⊕′ 𝑏) ⊕′ 𝑐 = 𝑏 ⊕′ 𝑐 = 𝑐 
       So, 𝑎 ≥𝑚𝑖𝑛  𝑐. 
 
Definition 4 

For ℝ𝑚𝑖𝑛, we use the parsial ordered ≥𝑚𝑖𝑛, that is 𝑎 ≥𝑚𝑖𝑛 𝑏 ⟺ 𝑎 ⊕′ 𝑏 = min(𝑎, 𝑏) = 𝑏. The structure  

(ℝ𝑚𝑖𝑛 , ≤) is a partially ordered set (poset). 
 
Theorem 4 

Let 𝐴 ∈ 𝑀𝑛(ℝ𝑚𝑖𝑛) and supposed 𝐿𝐴: ℝ𝑚𝑖𝑛
𝑛 ⟶ ℝ𝑚𝑖𝑛

𝑛  with 𝐿𝐴(𝑥) = 𝐴 ⊗ 𝑥. We have 𝐴 = 𝑃𝜎 ⊗ 𝐷(𝜆𝑖) for 

a permutation and 𝜆𝑖 > 𝜀 if and only if 𝐿𝐴 injective. 
Proof : 

(⟹) 𝐿𝐴(𝑥) = 𝐿𝐴(𝑥′) such as 𝐴 ⊗ 𝑥 =  𝐴 ⊗ 𝑥′ so 𝑥 = 𝑥′. 
(⟸) It is known that 𝐿𝐴 is injective. For each 𝑖 can be defined 𝐹𝑖 = {𝑗|𝑎𝑗𝑖 > 𝜀} and 𝐺𝑖 = {𝑗|𝑎𝑗𝑘 > 𝜀, 𝑘 ≠

𝑖}. We called 𝐹𝑖 ⊆ 𝐺𝑖; the contradiction assumes that 𝐹𝑖 ⊆ 𝐺𝑖. We will show a contradiction with injective 

𝐿𝐴. 

Let 𝑥 = [𝑥𝑘] with 𝑥𝑘 = {
𝑒 ; 𝑘 ≠ 𝑖
𝜀 ; 𝑘 = 𝑖

 . 

Suppose 𝑏 = 𝐴 ⊗ 𝑥 = ⨂ 𝑎∗𝑘𝑘≠𝑖  with 𝑎∗𝑘 defined the 𝑘 −th column of A. 

Suppose 𝑗 ∈ 𝐹𝑖, then 𝑗 ∈ 𝐺𝑖. Its mean that 𝑘 ≠ 𝑖 for 𝑎𝑗𝑘 > 𝜀. 

In ℝ𝑚𝑖𝑛, we can complete the order relation ≤, namely 𝑎 ≤ 𝑏 if and only if 𝑎 ⊕ ′𝑏 = 𝑎. 

So (ℝ𝑚𝑖𝑛 , ≤) is a poset (partially ordered set). 
 
Definition 5 

A mapping f on a partially ordered set is said to be isotone if for  𝑥 ≤ 𝑦 the result is 𝑓(𝑥) ≤ 𝑓(𝑦). 
 
Example 3 

Given 𝑓:ℝ𝑚𝑖𝑛 ⟶ ℝ𝑚𝑖𝑛 with 𝑓(𝑥) = 𝑥 ⊗ ′7 is an isotone mapping, namely for every 𝑥 ≤ 𝑦  results in 

𝑥 − 7 ≤ 𝑦 − 7 results in 𝑓(𝑥) ≤ 𝑓(𝑦). 
 
Definition 6 

Given (𝐸,≤) is a poset and 𝐴 ⊆ 𝐸. 



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i) For 𝑎 ∈ 𝐴 there is 𝑥 ∈ 𝐴 resulting in 𝑎 ≤ 𝑥 so 𝑎 is called minimum 

ii) For 𝑎 ∈ 𝐴 it is called the minimal element of A if there is 𝑥 ∈ 𝐴 with 𝑎 ≤ 𝑥 then 𝑎 = 𝑥. 
 
Definition 7  

An isotone mapping 𝑓: 𝐷 ⟶ 𝐸 with D, E poset is said to be a residual mapping if for all 𝑏 ∈ 𝐸 then 

{𝑥|𝑏 ≤ 𝑓(𝑥)} has a minimum element denoted 𝑓#(𝑏). The isotone mapping 𝑓#: 𝐸 ⟶ 𝐷 is called residual 
of  f. 
 
Theorem 5 

If 𝑓: 𝐷 → 𝐸 is a residualized mapping, then the equation 𝑓(𝑥) = 𝑏 has a solution if and only if 

𝑓(𝑓# (𝑏)) = 𝑏. 
Proof: 

(⇒) Given 𝑓(𝑥) = 𝑏 has a solution, say 𝑥1. We get 𝑓(𝑥1 ) = 𝑏. Since 𝑓#(𝑏) is a minimal element in 

{𝑥|𝑓(𝑥) ≤ 𝑏}, then 𝑥1 ≤ 𝑓#(𝑏). Since 𝑓 is isotone then 𝑓(𝑥1 ) ≤ 𝑓(𝑓#(𝑏)), according to (*) 𝑓(𝑓#(𝑏)) ≤

𝑏, consequently 𝑏 = 𝑓(𝑥1 ) ≤ 𝑓𝑓# (𝑏) ≤ 𝑏, namely 𝑓(𝑓#(𝑏)) = 𝑏. 

(⇐) Given 𝑓(𝑓^# (𝑏)) = 𝑏, then the equation 𝑓(𝑥) = 𝑏 has a solution, namely 𝑥 = 𝑓# (𝑏). 
 

The function 𝑓 is residualized, because 𝑦 ∈ ℝ𝑚𝑖𝑛 with {𝑥|𝑦 ≤ 𝑥 ⊗′ 7 = 𝑓(𝑥)} is a minimal element, 

namely 𝑥 = 𝑓#(𝑦) = 𝑦 + 7. 
 
Definition 8 

For every 𝑏 ∈ 𝐸 then {𝑥|𝑏 ≤ 𝑓(𝑥)} has a minimal element denoted 𝑓#(𝑏). For 𝑦 ∈ ℝ𝑚𝑖𝑛 

{𝑥|𝑦 ≤ 𝑥 ⊗′ 7 = 𝑓(𝑥)} the minimal element is 𝑥 = 𝑓#(𝑦) = 𝑦 + 7. 
 
Definition 9 

A subsolution of 𝐴 ⊗ 𝑥 = 𝑏 is 𝑥 that satisfies 𝐴 ⊗ 𝑥 ≥ 𝑏, a linear system for obtaining the general 

result of the equation  𝐴 ⊗ 𝑥 = 𝑏. An ordered pair of vectors is defined by 𝑥 ≥ 𝑦 if 𝑥 ⊕′ 𝑦 = 𝑦. 

Since 𝐴 ∈ 𝑅𝑚𝑖𝑛
𝑛×𝑛 and 𝑋 ∈ 𝑅𝑚𝑖𝑛

𝑛×𝑛 then 

𝐴 ⊗ 𝑥 = [

𝑎11 𝑎12

𝑎21 𝑎22

⋯ 𝑎1𝑛

⋯ 𝑎2𝑛

⋮ ⋮
𝑎𝑛1 𝑎𝑛2

⋱ ⋮
⋯ 𝑎𝑛𝑛

] ⊗ [

𝑥1

𝑥2

⋮
𝑥𝑛

] 

     = [

𝑎11 + 𝑥1 ⊕′  𝑎12 + 𝑥2 ⊕′ ⋯⊕′ 𝑎1𝑛 + 𝑥𝑛

𝑎21 + 𝑥1 ⊕′ 𝑎22 + 𝑥2 ⊕′ ⋯⊕′ 𝑎2𝑛 + 𝑥𝑛

⋮
𝑎𝑛1 + 𝑥1 ⊕′  𝑎𝑛2 + 𝑥2 ⊕′ ⋯⊕′ 𝑎𝑛𝑛 + 𝑥𝑛

] 

     =

[
 
 
 
 
⊕ ′𝑎1𝑗 + 𝑥𝑗

⊕ ′𝑎2𝑗 + 𝑥𝑗

⋮
⊕ ′𝑎𝑛𝑗 + 𝑥𝑗]

 
 
 
 

 , 𝑗 = 1,2,… , 𝑛 

with ⊕ ′𝑎1𝑗 + 𝑥𝑗 = 𝑚𝑖𝑛{𝑎11 + 𝑥1,  𝑎12 + 𝑥2, ⋯ , 𝑎1𝑛 + 𝑥𝑛} 



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Form 𝑓𝑗(𝑥𝑗) = [

𝑎1𝑗 + 𝑥1

𝑎2𝑗 + 𝑥2

⋮
𝑎𝑛𝑗 + 𝑥𝑛

] so that 𝐴 ⊗ 𝑥 =

𝑛
⊕′

𝑗 = 1
𝑓𝑗(𝑥𝑗). So, 𝐴 ⊗ 𝑥 =

𝑓1(𝑥1) ⊕′ 𝑓2(𝑥2) ⊕′ ⋯⊕′ 𝑓𝑛(𝑥𝑛). 

For each 𝑗, if 𝑥𝑗ℎ ≤ 𝑥𝑗𝑘 ⟹ 𝑓𝑗(𝑥𝑗ℎ) ≤ 𝑓𝑗(𝑥𝑗𝑘). According Definition 5 and Theorem 5, so {𝑥|𝐴 ⊗ 𝑥 ≥

𝑏} has a minimum element denoted 𝐴#(𝑏). 

Therefore, to determine the solution of the equation 𝐴𝑥 = 𝑏, check wether 𝐴 (𝐴#(𝑏)) = 𝑏. The following 

is a theorem that states this characteristic. 
 
Theorem 6 

If 𝐴 ∈ 𝑅𝑚𝑖𝑛
𝑛×𝑛 and 𝑏 ∈ 𝑅𝑚𝑖𝑛

𝑛 , then the equation 𝐴 ⊗ 𝑥 = 𝑏 has a solution if and only if 𝐴 (𝐴#(𝑏)) = 𝑏. 

In other words, the solution is 𝑥 = 𝐴#(𝑏). 
Proof : 

(⟹) 

It is known that the equation 𝐴 ⊗ 𝑥 = 𝑏 has a solution 𝑥∗, namely 𝐴 ⊗ 𝑥∗ = 𝑏, so that 𝐴 ⊗ 𝑥∗ ≥
𝑏.  

Since 𝐴#𝑏 is the minimal element in {𝑥|𝐴 ⊗ 𝑥 ≥ 𝑏}, then 𝑥∗ ≥ 𝐴#𝑏. It is obtained  

𝐴 ⊗ 𝑥∗ ≤ 𝐴(𝐴#𝑏) 

     ⟺  𝑏 = 𝐴 ⊗ 𝑥∗ ≥ 𝐴(𝐴#𝑏)   ……… . . (∗) 
Furthermore, according to Theorem 5 

         𝐴(𝐴#𝑏) ≥ 𝑏   ………… (∗∗) 

From (∗) and (∗∗) it is obtained 𝐴 (𝐴#(𝑏)) = 𝑏. 

(⟸) 

It is known that 𝐴(𝐴#𝑏) = 𝑏. So the equation 𝐴 ⊗ 𝑥 = 𝑏 has a solution, namely 𝑥 = 𝐴#𝑏. 

Therefore, 𝐴 ⊗ 𝑥 = 𝑏 has a solution 𝑥 = 𝐴#(𝑏). It means that 𝐴 (𝐴#(𝑏)) = 𝑏. 

For example, if 𝐴 ⊗ 𝑥 = 𝑏 has solution 𝑥𝑗, then there is a smallest subsolution 𝐴 ⊗ 𝑥 = 𝑏. 

    𝐴 ⊗ 𝑥 ≥ 𝑏 ⟺
⊕′

𝑗
𝐴𝑖𝑗 ⊗ 𝑥𝑗 ≥ 𝑏𝑖, ∀𝑖 then 

𝐴𝑖1 ⊗ 𝑥1 ⊕′ 𝐴𝑖2 ⊗ 𝑥2 ⊕′ …⊕′ 𝐴𝑖𝑛 ⊗ 𝑥𝑛 ≥ 𝑏𝑖 
𝑖 = 1 ⟹ 𝐴11 ⊗ 𝑥1 ⊕′ 𝐴12 ⊗ 𝑥2 ⊕′ …⊕′ 𝐴1𝑛 ⊗ 𝑥𝑛 ≥ 𝑏1 
𝑖 = 2 ⟹ 𝐴21 ⊗ 𝑥1 ⊕′ 𝐴22 ⊗ 𝑥2 ⊕′ …⊕′ 𝐴2𝑛 ⊗ 𝑥𝑛 ≥ 𝑏2 

⋮ 
𝑖 = 𝑛 ⟹ 𝐴𝑛1 ⊗ 𝑥1 ⊕′ 𝐴𝑛2 ⊗ 𝑥2 ⊕′ …⊕′ 𝐴𝑛𝑛 ⊗ 𝑥𝑛 ≥ 𝑏𝑛 

for 𝑖, min{𝐴𝑖1 +𝑥1, 𝐴𝑖2 + 𝑥2, … , 𝐴𝑖𝑛 + 𝑥𝑛} ≥ 𝑏𝑖. 

For 𝑖, 𝑗, we have 𝐴𝑖𝑗 + 𝑥𝑗 ≥ 𝑏𝑖 , which results in 𝑥𝑗 ≥ 𝑏𝑖 − 𝐴𝑖𝑗 . 

Obtain 𝑥𝑗 ≥ 𝑚𝑎𝑥{𝑏𝑖 − 𝐴𝑖𝑗} for each 𝑖. Next, −𝑥𝑗 ≤ 𝑚𝑖𝑛{−𝑏𝑖 + 𝐴𝑖𝑗} for each 𝑖.  

In other words −𝑥𝑗 ≤ 𝑚𝑖𝑛{−𝑏𝑖 ⊗ 𝐴𝑖𝑗} for each 𝑖. 

So that −𝑥𝑗 = 𝑚𝑖𝑛{−𝑏𝑖 ⊗ 𝐴𝑖𝑗}subsolution of 𝐴 ⊗ 𝑥 = 𝑏 or expressed −𝑥𝑗 = 𝑚𝑖𝑛 {(𝐴𝑖𝑗)
𝑡
⊗ (−𝑏𝑗)}. 

 
Example 4 
Given a system of linear equations over min-plus algebra  



9550 

 

 
Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 6: 9544-9554, 2024 
DOI: 10.55214/25768484.v8i6.4034 
© 2024 by the authors; licensee Learning Gate 

 

[
2 3
4 5

] ⊗ [
𝑥1

𝑥2
] = [

6
7
] 

We get 

−𝑥𝑗 = 𝑚𝑖𝑛{−𝑏𝑖 ⊗ 𝐴𝑖𝑗} 

= [−6 −7] ⊗ [
2 3
4 5

] 

= [−4 ⊕′− 3 −3 ⊕′− 2] 
= [−4 −3] 

So, 𝑥𝑗 = [4 3]. Or, −𝑥𝑗 = 𝑚𝑖𝑛 {(𝐴𝑖𝑗)
𝑡
⊗ (−𝑏𝑗)} = [

2 4
3 5

] ⊗ [
−6
−7

] = [
−4 ⊕′− 3
−3 ⊕′− 2

] = [
−4
3

] 

It can be proven that   
 

[
2 3
4 5

] ⊗ [
𝑥1

𝑥2
] = [

2 3
4 5

] ⊗ [
4
3
] = [

6 ⊕′ 6
8 ⊕′ 8

] = [
6
8
] ≥ [

6
7
] 

 
3. Results and Discussion 

Based on the explanation above, the following properties are obtained. 
 
Definition 7 

A matrix 𝐴 ∈ 𝑀𝑛(𝑅) is said to be invertible over the min-plus algebra if there is a matrix B such that 𝐴 ⊗

𝐵 = 𝐸 with E the identity matrix over the min-plus algebra and is denoted 𝐵 = 𝐴⊗−1
. 

To determine the inverse matrix over min-plus algebra, permutation is required.  
 
Definition 8 

If 𝜆1, 𝜆2, … , 𝜆𝑛 ∈ ℝ𝑚𝑖𝑛, 𝜆𝑖 ≠ 𝜀 then the diagonal matrix is defined as follows. 

𝐷(𝜆𝑖) = [

𝜆1 𝜀
𝜀 𝜆2

⋯ 𝜀
⋮ 𝜀

⋯ ⋯
𝜀 𝜀

⋱ ⋯
⋯ 𝜆𝑛

] 

 
Theorem 7 

Given 𝐴 ∈ 𝑀𝑛(𝑅min). If and only if there is a permutation 𝜎 and values 𝜆𝑖 > 𝜀, 𝑖 ∈ {1,2,… , 𝑛} such that 

𝐴 = 𝑃𝜎 ⊗ 𝐷(𝜆𝑖), then 𝐴 ∈ 𝑀𝑛(𝑅min) has a left inverse. 
 
Proof: 

(⟹) Given 𝐴 ∈ 𝑀𝑛 (𝑅𝑚𝑖𝑛), there exist 𝐵 so that it satisfies the equation 𝐴 ⊗ 𝐵 = 𝐸, meaning  

(1) 𝑚𝑖𝑛𝑘(𝑎𝑖𝑘 + 𝑏𝑖𝑘) = 𝑒 = 0 for every 𝑘 there is 𝑖 so that 𝑎𝑖𝑘 + 𝑏𝑘𝑖 = 𝑒, we have the function 𝑖 = 𝜃(𝑘) 

with 𝑎𝑖𝜃(𝑖) > 𝜀 and 𝑏𝜃(𝑖)𝑖 > 𝜀. 

(2) 𝑚𝑖𝑛𝑘(𝑎𝑖𝑘 + 𝑏𝑘𝑗) = 𝜀′ = ∞ for all 𝑖 ≠ 𝑗 
Based on (2), it is obtained 

(3) 𝑎𝑖𝜃(𝑗) = 𝜀′ for all 𝑖 ≠ 𝑗. 

Since 𝑎𝑖𝜃(𝑖) > 𝜀′ = 𝑎𝑖𝜃(𝑗) for all 𝑖 ≠ 𝑗 then 𝜃 is an injection and permutation function. Meanwhile, 

𝑎𝑖𝜃(𝑖) is a single entry in the 𝜃(𝑖) −th column of 𝐴, which is not 𝜀′. For example, �̂� = 𝑃𝜃 ⊗ 𝐴. The 

𝜃(𝑖) −th row of �̂� is the 𝑖 −th row of 𝐴, which has a larger entry then 𝜀′ in the 𝜃(𝑖) −th column. 

Thus, all larger �̂� diagonal entries become 𝜀′. A has only one non−𝜀′ entry in each column, which is 

also true for �̂�. 



9551 

 

 
Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 6: 9544-9554, 2024 
DOI: 10.55214/25768484.v8i6.4034 
© 2024 by the authors; licensee Learning Gate 

 

So we get 𝑃𝜃 ⊗ 𝐴 = �̂� = 𝐷(𝜆𝑖) with 𝜆𝑖 = 𝑎𝜃−1(𝑖)𝑖 > 𝜀′. 

Suppose, 𝜎 = 𝜃−1, because 𝑃𝜎 ⊗ 𝑃𝜃 = 𝑃𝜃−1 ⊗ 𝑃𝜃 = 𝐸, then 𝐴 = 𝑃𝜎 ⊗ 𝐷(𝜆𝑖). 

So, it is proven that 𝐴 = 𝑃𝜎 ⊗ 𝐷(𝜆𝑖). 

(⟸) Assume 𝐴 = 𝑃𝜎 ⊗ 𝐷(𝜆𝑖) with 𝜆 ∈  𝑅𝑚𝑖𝑛 and 𝜆𝑖 > 𝜀.  

If the statement is true then for example 𝐵 = 𝑃𝜎−1 ⊗ 𝐷(−𝜆𝑖), with −𝜆𝑖 = 𝜆𝑖
⊗−1

. So we have 

𝐴 ⊗ 𝐵 = (𝑃𝜎 ⊗ 𝐷(𝜆𝑖)) ⊗ (𝑃𝜎−1 ⊗ 𝐷(−𝜆𝑖)) 

= 𝑃𝜎 ⊗ (𝐷(𝜆𝑖) ⊗ 𝐷(−𝜆𝑖)) ⊗ 𝑃𝜎−1 

= 𝑃𝜎 ⊗ 𝐸 ⊗ 𝑃𝜎−1 

= 𝑃𝜎 ⊗ 𝑃𝜎−1 

= 𝐸 

And, 𝐴 ⊗ 𝐵 = 𝐸 and 𝐵 is the right inverse of 𝐴. 
From the theorem above, we get the necessary and sufficient conditions for matrix A to be invertible 

over min-plus algebra, namely matrix A is invertible if and only if matrix A is a permuted diagonal matrix 

with 𝐴 = 𝑃𝜎 ⊗ 𝐷(𝜆𝑖).  
The purpose of finding the generalized inverse is to determine the solution of the linear equation 

system 𝐴𝑋𝐵 = 𝐶. A matrix 𝐴 ∈ 𝑅𝑚𝑖𝑛
𝑛×𝑛  has a generalized inverse matrix 𝑋 ∈ 𝑅𝑚𝑖𝑛

𝑛×𝑛 if 𝐴 ⊗ 𝑋 ⊗ 𝐴 = 𝐴. So, 

it can be said that the generalized inverse is the smallest subsolution of the equation 𝐴 ⊗ 𝑋 ⊗ 𝐴 = 𝐴. 

Several steps are required to determine matrix 𝑋 as the generalized inverse of equation 𝐴 ⊗ 𝑋 ⊗ 𝐴 = 𝐴. 
 
Definition 9 

For a matrix 𝐴 ∈ 𝑅𝑚𝑖𝑛
𝑛×𝑛, then the matrix 𝐵 ∈ 𝑅𝑚𝑖𝑛

𝑛×𝑛 is said to be the generalized inverse of the matrix A if 

𝐴 ⊗ 𝐵 ⊗ 𝐴 = 𝐴 is satisfied. 

To determine whether or not there is a matrix 𝐵 that satisfies 𝐴 ⊗ 𝐵 ⊗ 𝐴 = 𝐴, is equivalent to 

determining whether or not there is a solution to the equation 𝐴 ⊗ 𝑋 ⊗ 𝐴 = 𝐴 with 𝐴 ∈ 𝑅𝑚𝑖𝑛
𝑛×𝑛. 

1. Bring the equation 𝐴 ⊗ 𝑋 ⊗ 𝐴 = 𝐴 to the form 𝐴𝑥 = 𝑏. 

2. Determine the matrix 𝑋 
3. Prove that the matrix X is a generalized inverse by substituting it into the equation 

 𝐴 ⊗ 𝑋 ⊗ 𝐴 = 𝐴. 
 

Given 𝐴 ∈ 𝑅𝑚𝑖𝑛
𝑛×𝑛 with operations ⊕ ′ and ⊗. The 𝑖𝑗 −th element in 𝐴 ⊗ 𝑋 ⊗ 𝐴 with 1 ≤ 𝑖, 𝑗 ≤ 𝑛 in 𝐴 ⊗

𝑋 ⊗ 𝐴 is  

[𝐴 ⊗ 𝑋 ⊗ 𝐴]𝑖𝑗 = 𝐴𝑖𝑗 

⟺ [𝐴 ⊗ 𝑋]𝑖𝑙 ⊗ 𝐴𝑙𝑗 = 𝐴𝑖𝑗             1 ≤ 𝑙 ≤ 𝑛 

⟺ 𝐴𝑖𝑘 ⊗ 𝑋𝑘𝑙 ⊗ 𝐴𝑙𝑗 = 𝐴𝑖𝑗             1 ≤ 𝑘, 𝑙 ≤ 𝑛 

⟺ 𝐴𝑖𝑘 ⊗
𝑛

⊕ ′
𝑙 = 1

𝑥𝑘𝑙 ⊗ 𝐴𝑙𝑗 = 𝐴𝑖𝑗        1 ≤ 𝑘 ≤ 𝑛 

⟺
𝑛

⊕ ′
𝑘 = 1

 𝐴𝑖𝑘 ⊗
𝑛

⊕ ′
𝑙 = 1

𝑥𝑘𝑙 ⊗ 𝐴𝑙𝑗 = 𝐴𝑖𝑗 

⟺
𝑛

⊕ ′
𝑘 = 1

𝑛
⊕ ′

𝑙 = 1
𝐴𝑖𝑘 ⊗ 𝑥𝑘𝑙 ⊗ 𝐴𝑙𝑗 = 𝐴𝑖𝑗 

 



9552 

 

 
Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 6: 9544-9554, 2024 
DOI: 10.55214/25768484.v8i6.4034 
© 2024 by the authors; licensee Learning Gate 

 

⟺ 
𝑛
⊕′

𝑘 = 1
[

𝑛
⊕′

𝑙 = 1
(𝐴𝑖𝑘 ⊗ 𝑋𝑘𝑙 ⊗ 𝐴𝑙𝑗)] = 𝐴𝑖𝑗 

So we get 

𝑛
⊕′

𝑖 = 1

𝑛
⊕′

𝑗 = 1
𝑓𝑖𝑗(𝑋𝑘𝑙) = 𝐴𝑖𝑗 . 

The generalized inverse can be found by solving the equation 𝐴 ⊗ 𝑥 = 𝑏 in min-plus algebra. For the 

generalized inverse, denoted 𝐴⊗−1, the form 𝐴 ⊗ 𝑋 ⊗ 𝐴 = 𝐴 is brought to the form 𝐴 ⊗ 𝑥 = 𝑏. For the 

𝑖𝑗 −th element, it is obtained as follows: 

⟺ [𝐴 ⊗ 𝑋 ⊗ 𝐴]𝑖𝑗 = 𝐴𝑖𝑗 

⟺ 𝐴𝑖𝑘 ⊗ 𝑋𝑘𝑙 ⊗ 𝐴𝑙𝑗 = 𝐴𝑖𝑗 

⟺ 𝐴𝑖𝑘 ⊗ 𝑋𝑘𝑙 ⊗ 𝐴𝑙𝑗 ⊗ 𝐴𝑙𝑗
⊗−1 = 𝐴𝑖𝑗 ⊗ 𝐴𝑙𝑗

⊗−1 

⟺ 𝐴𝑖𝑘 ⊗ 𝑋𝑘𝑙 ⊗ 𝐸 = 𝐴𝑖𝑗 − 𝐴𝑙𝑗 

 If suppose 𝑏 = 𝐴𝑖𝑗 − 𝐴𝑙𝑗 then we write   

⟺ 𝐴𝑖𝑘 ⊗ 𝑋𝑘𝑙 = 𝑏 
⟺ 𝐸 ⊗ 𝑋𝑘𝑙 = −𝐴𝑖𝑘 ⊗ 𝑏 

⟺ 𝑋𝑘𝑙 = −𝐴𝑖𝑘 ⊗ 𝑏 

⟺ 𝑋𝑘𝑙 = −[

𝑎11 𝑎12

𝑎21 𝑎22

⋯ 𝑎1𝑘

⋯ 𝑎2𝑘

⋮ ⋮
𝑎𝑖1 𝑎𝑖1

⋱ ⋮
⋯ 𝑎𝑖𝑘

] ⊗ [

𝑏11 𝑏12

𝑏21 𝑏22

⋯ 𝑏1𝑛

⋯ 𝑏2𝑛

⋮ ⋮
𝑏𝑛1 𝑏𝑛2

⋱ ⋮
⋯ 𝑏𝑛𝑛

] 

⟺ 𝑋𝑘𝑙 = [

−𝑎11 −𝑎12

−𝑎21 −𝑎22

⋯ −𝑎1𝑘

⋯ −𝑎2𝑘

⋮ ⋮
−𝑎𝑖1 −𝑎𝑖1

⋱ ⋮
⋯ −𝑎𝑖𝑘

] ⊗ [

𝑏11 𝑏12

𝑏21 𝑏22

⋯ 𝑏1𝑛

⋯ 𝑏2𝑛

⋮ ⋮
𝑏𝑛1 𝑏𝑛2

⋱ ⋮
⋯ 𝑏𝑛𝑛

] 

For 𝑋𝑘𝑙 = [

𝑥11 𝑥12

𝑥21 𝑥22

⋯ 𝑥1𝑙

⋯ 𝑥2𝑙

⋮ ⋮
𝑥𝑘1 𝑥𝑘2

⋱ ⋮
⋯ 𝑥𝑘𝑙

] 

The result,   

𝑥11 = min{−𝑎11 + 𝑏11 , −𝑎12 + 𝑏21, … , −𝑎1𝑘 + 𝑏𝑛1 } 
𝑥12 = min{−𝑎11 + 𝑏12 , −𝑎12 + 𝑏22, … , −𝑎1𝑘 + 𝑏𝑛2 } 

⋮ 
𝑥1𝑙 = min{−𝑎11 + 𝑏1𝑛 , −𝑎12 + 𝑏2𝑛, … , −𝑎1𝑘 + 𝑏𝑛𝑛 } 
𝑥21 = min{−𝑎21 + 𝑏11 , −𝑎22 + 𝑏21, … , −𝑎2𝑘 + 𝑏𝑛1 } 
𝑥22 = min{−𝑎21 + 𝑏12 , −𝑎22 + 𝑏22, … , −𝑎2𝑘 + 𝑏𝑛2 } 

⋮ 
𝑥2𝑙 = min{−𝑎21 + 𝑏1𝑛 , −𝑎22 + 𝑏2𝑛, … , −𝑎2𝑘 + 𝑏𝑛𝑛 } 

⋮ 
𝑥𝑘1 = min{−𝑎𝑖1 + 𝑏11 , −𝑎𝑖2 + 𝑏21, … , −𝑎𝑖𝑘 + 𝑏𝑛1 } 
𝑥𝑘2 = min{−𝑎𝑖1 + 𝑏12 , −𝑎𝑖2 + 𝑏22, … , −𝑎𝑖𝑘 + 𝑏𝑛2 } 

⋮ 
𝑥𝑘𝑙 = min{−𝑎𝑖1 + 𝑏1𝑛 , −𝑎𝑖2 + 𝑏2𝑛, … , −𝑎𝑖𝑘 + 𝑏𝑛𝑛 } 

 
So, it can be written as 



9553 

 

 
Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 6: 9544-9554, 2024 
DOI: 10.55214/25768484.v8i6.4034 
© 2024 by the authors; licensee Learning Gate 

 

𝑋𝑘𝑙 = [

min{−𝑎1𝑘 + 𝑏𝑛1 } min{−𝑎1𝑘 + 𝑏𝑛2 }
min{−𝑎2𝑘 + 𝑏𝑛1 } min{−𝑎2𝑘 + 𝑏𝑛2 }

⋯ min{−𝑎1𝑘 + 𝑏𝑛𝑛 }
⋯ min{−𝑎2𝑘 + 𝑏𝑛𝑛 }

⋮ ⋮
min{−𝑎𝑖𝑘 + 𝑏𝑛1 } min{−𝑎𝑖𝑘 + 𝑏𝑛2 }

⋱ ⋮
⋯ min{−𝑎𝑖𝑘 + 𝑏𝑛𝑛 }

] 

Suppose 𝑏 = 𝐴𝑖𝑗 − 𝐴𝑙𝑗, we have 𝑋𝑘𝑙 =
𝑛

𝑚𝑖𝑛
𝑖 = 1

𝑛
𝑚𝑖𝑛
𝑗 = 1

(−𝑎𝑖𝑘 + 𝑎𝑖𝑗 − 𝑎𝑙𝑗) 

 

From the form 𝑥𝑘𝑙  , we obtain a form that can be expressed 

𝑚
⊕

𝑘 = 1
[

𝑚
⊕

𝑙 = 1
(𝐴𝑖𝑘 + 𝑋𝑘𝑙 + 𝐴𝑙𝑗)] = 𝐴𝑖𝑗 . 

by determining whether or not there is a solution to the equation 𝐴 ⊗ 𝑋 ⊗ 𝐴 = 𝐴 with 𝐴 ∈ ℝ𝑚𝑖𝑛
𝑛×𝑚. 

The 𝑖𝑗 −th element in 𝐴 ⊗ 𝑋 ⊗ 𝐴 is [𝐴 ⊗ 𝑋 ⊗ 𝐴]𝑖𝑗 =
𝑚
⨁′

𝑘 = 1

𝑛
⨁′

𝑙 = 1
(𝐴𝑖𝑘 + 𝑋𝑘𝑙 + 𝐴𝑙𝑗). So, we get the 

equation 
𝑚
⨁′

𝑘 = 1
[

𝑛
⨁′

𝑙 = 1
(𝐴𝑖𝑘 + 𝑋𝑘𝑙 + 𝐴𝑙𝑗)] = 𝐴𝑖𝑗  ……………(3) 

If for each 𝑘, 𝑙 is formed 

𝑓𝑖𝑗(𝑋𝑘𝑙) = 𝐴𝑖𝑘 + 𝑋𝑘𝑙 + 𝐴𝑙𝑗 ……..(4) 
then equation (3) becomes 

𝑛
⨁

𝑖 = 1

𝑚
⨁

𝑖 = 1
𝑓𝑖𝑗(𝑋𝑘𝑙) = 𝐴𝑖𝑗  ………(5) 

It is obtained that 𝑋𝑘𝑙 =
𝑛

𝑚𝑖𝑛
𝑖 = 1

𝑛
𝑚𝑖𝑛
𝑗 = 1

(−𝑎𝑖𝑘 + 𝑎𝑖𝑗 − 𝑎𝑙𝑗) corresponds if substituted into the equation  

𝐴 ⊗ 𝑋 ⊗ 𝐴 = 𝐴.   
  
4. Conclusions 

In this conclusion, the property is obtained that permutation is required determining the inverse 
matrix over min-plus algebra. We obtain several theorems that show the characteristics of the inverse of 
matrices in min-plus algebra, especially generalized matrices. The characterization is obtained by 
considering the characteristics of the solutions linear equations system over min-plus algebra. The 

generalized inverse of the matrix 𝐴 ∈  𝑅𝑚𝑖𝑛
𝑛×𝑛 can be obtained by determining the matrix 𝑋 with entry 

𝑋𝑘𝑙 =
𝑛

𝑚𝑖𝑛
𝑖 = 1

𝑛
𝑚𝑖𝑛
𝑗 = 1

(−𝑎𝑖𝑘 + 𝑎𝑖𝑗 − 𝑎𝑙𝑗) which satisfies 𝐴 ⊗ 𝑋 ⊗ 𝐴 = 𝐴. 

 
Copyright:  
© 2024 by the authors. This article is an open access article distributed under the terms and conditions of 
the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). 
 
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ISSN: 2576-8484 

Vol. 8, No. 6: 9544-9554, 2024 
DOI: 10.55214/25768484.v8i6.4034 
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