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Journal articles on the topic '(max,+) Algebra'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 March 2025

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1

Ranulfo Paiva Barbosa (Sobrinho) and Florentin Smarandache. "Pura Vida Neutrosophic Algebra." Neutrosophic Systems with Applications 9 (September 9, 2023): 101–6. http://dx.doi.org/10.61356/j.nswa.2023.68.

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We introduce Pura Vida Neutrosophic Algebra, an algebraic structure consisting of neutrosophic numbers equipped with two binary operations namely addition and multiplication. The addition can be calculated sometimes with the function min and other times with the max function. The multiplication operation is the usual sum between numbers. Pura Vida Neutrosophic Algebra is an extension of both Tropical Algebra (also known as Min-Plus, or Min-Algebra) and Max-Plus Algebra (also known as Max-algebra). Tropical and Max-Plus algebras are algebraic structures included in semirings and their operations can be used in matrices and vectors. Pura Vida Neutrosophic Algebra is included in Neutrosophic semirings and can be used in Neutrosophic matrices and vectors.

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2

Gavalec, Martin, Zuzana Němcová, and Ján Plavka. "Strong Tolerance and Strong Universality of Interval Eigenvectors in a Max-Łukasiewicz Algebra." Mathematics 8, no. 9 (September 4, 2020): 1504. http://dx.doi.org/10.3390/math8091504.

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The Łukasiewicz conjunction (sometimes also considered to be a logic of absolute comparison), which is used in multivalued logic and in fuzzy set theory, is one of the most important t-norms. In combination with the binary operation ‘maximum’, the Łukasiewicz t-norm forms the basis for the so-called max-Łuk algebra, with applications to the investigation of systems working in discrete steps (discrete events systems; DES, in short). Similar algebras describing the work of DES’s are based on other pairs of operations, such as max-min algebra, max-plus algebra, or max-T algebra (with a given t-norm, T). The investigation of the steady states in a DES leads to the study of the eigenvectors of the transition matrix in the corresponding max-algebra. In real systems, the input values are usually taken to be in some interval. Various types of interval eigenvectors of interval matrices in max-min and max-plus algebras have been described. This paper is oriented to the investigation of strong, strongly tolerable, and strongly universal interval eigenvectors in a max-Łuk algebra. The main method used in this paper is based on max-Ł linear combinations of matrices and vectors. Necessary and sufficient conditions for the recognition of strong, strongly tolerable, and strongly universal eigenvectors have been found. The theoretical results are illustrated by numerical examples.

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3

Chaudhry, Muhammad Anwar, Asfand Fahad, Muhammad Imran Qureshi, and Urwa Riasat. "Some Results about Weak UP-algebras." Journal of Mathematics 2022 (September 16, 2022): 1–6. http://dx.doi.org/10.1155/2022/1206804.

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We introduce a new class of algebras, weak UP-algebras, briefly written as WUP-algebras, as a wider class than the classes of UP-algebras and KU-algebras. We investigate fundamental aspects including maximal elements, branches and the subalgebra consisting of maximal elements of a WUP-algebra. We also study regular congruences on a WUP-algebra as well as the corresponding quotient WUP-algebras. We prove that the congruence generated from the branches of a WUP-algebra is regular and the corresponding quotient algebra χ / q is isomorphic to the WUP-algebra Max χ , the subalgebra of χ consisting of its maximal elements.

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Rudhito, Marcellinus Andy, Sri Wahyuni, Ari Suparwanto, and Frans Susilo. "Matriks atas Aljabar Max-Plus Interval." Jurnal Natur Indonesia 13, no. 2 (November 21, 2012): 94. http://dx.doi.org/10.31258/jnat.13.2.94-99.

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This paper aims to discuss the matrix algebra over interval max-plus algebra (interval matrix) and a method tosimplify the computation of the operation of them. This matrix algebra is an extension of matrix algebra over max-plus algebra and can be used to discuss the matrix algebra over fuzzy number max-plus algebra via its alpha-cut.The finding shows that the set of all interval matrices together with the max-plus scalar multiplication operationand max-plus addition is a semimodule. The set of all square matrices over max-plus algebra together with aninterval of max-plus addition operation and max-plus multiplication operation is a semiring idempotent. As reasoningfor the interval matrix operations can be performed through the corresponding matrix interval, because thatsemimodule set of all interval matrices is isomorphic with semimodule the set of corresponding interval matrix,and the semiring set of all square interval matrices is isomorphic with semiring the set of the correspondingsquare interval matrix.

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5

Butkovič, Peter. "Max-algebra: the linear algebra of combinatorics?" Linear Algebra and its Applications 367 (July 2003): 313–35. http://dx.doi.org/10.1016/s0024-3795(02)00655-9.

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6

Cuninghame-Green, R. A., and P. Butkovič. "Bases in max-algebra." Linear Algebra and its Applications 389 (September 2004): 107–20. http://dx.doi.org/10.1016/j.laa.2004.03.022.

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7

De Schutter, Bart, Ton van den Boom, Jia Xu, and Samira S. Farahani. "Analysis and control of max-plus linear discrete-event systems: An introduction." Discrete Event Dynamic Systems 30, no. 1 (December 2, 2019): 25–54. http://dx.doi.org/10.1007/s10626-019-00294-w.

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AbstractThe objective of this paper is to provide a concise introduction to the max-plus algebra and to max-plus linear discrete-event systems. We present the basic concepts of the max-plus algebra and explain how it can be used to model a specific class of discrete-event systems with synchronization but no concurrency. Such systems are called max-plus linear discrete-event systems because they can be described by a model that is “linear” in the max-plus algebra. We discuss some key properties of the max-plus algebra and indicate how these properties can be used to analyze the behavior of max-plus linear discrete-event systems. Next, some control approaches for max-plus linear discrete-event systems, including residuation-based control and model predictive control, are presented briefly. Finally, we discuss some extensions of the max-plus algebra and of max-plus linear systems.

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8

Suroto, Suroto. "Eigenvalue decomposition of a symmetric matrix over the symmetrized max-plus algebra." Desimal: Jurnal Matematika 4, no. 3 (November 30, 2021): 349–56. http://dx.doi.org/10.24042/djm.v4i3.9959.

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This paper discusses topics in the symmetrized max-plus algebra. In this study, it will be shown the existence of eigenvalue decomposition of a symmetric matrix over symmetrized max-plus algebra. Eigenvalue decomposition is shown by using a function that corresponds to the symmetrized max-plus algebra with conventional algebra. The result obtained is the existence of eigenvalue decomposition of a symmetric matrix over symmetrized max-plus algebra and its application to determine eigenvalues and eigenvectors.

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9

Siswanto, Siswanto. "THE EXISTENCE OF SOLUTION OF GENERALIZED EIGENPROBLEM IN INTERVAL MAX-PLUS ALGEBRA." BAREKENG: Jurnal Ilmu Matematika dan Terapan 17, no. 3 (September 30, 2023): 1341–46. http://dx.doi.org/10.30598/barekengvol17iss3pp1341-1346.

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An eigenproblem of a matrix is where and . Vector and are eigenvector and eigenvalue, respectively. General form of eigenvalue problem is with , . Interval maks-plus algebra is and equipped with a maximum ( and plus operations. The set of matrices which its component elements of is called matrices over interval max-plus algebra and denoted by . Let , eigenproblem in interval max-plus algebra is with and . Vector and are eigenvector and eigenvalue, respectively. In this research, we will discuss the generalization of the eigenproblem in interval max-plus algebra. Especially about the existence of solution of generalized eigenproblem in interval max-plus algebra. Keywords: interval max-plus algebra, generalized eigenproblem, the existence of the solution.

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10

Ariyanti, Gregoria. "A NOTE ON THE SOLUTION OF THE CHARACTERISTIC EQUATION OVER THE SYMMETRIZED MAX-PLUS ALGEBRA." BAREKENG: Jurnal Ilmu Matematika dan Terapan 16, no. 4 (December 15, 2022): 1347–54. http://dx.doi.org/10.30598/barekengvol16iss4pp1347-1354.

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The symmetrized max-plus algebra is an extension of max-plus algebra. One of the problems in the symmetrized max-plus algebra is determining the eigenvalues of a matrix. If the determinant can be defined, the characteristic equation can be formulated as a max-plus algebraic multivariate polynomial equation system. A mathematical tool for solving the problem using operations as in conventional algebra, known as the extended linear complementary problem (ELCP), to determine the solution to the characteristic equation. In this paper, we will describe the use of ELCP in determining the solution to the characteristic equations of matrices over the symmetrized max-plus algebra.

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11

Putri, Zakia Nur Ramadhani, Siswanto Siswanto, and Vika Yugi Kurniawan. "CRAMER’S RULE IN MIN-PLUS ALGEBRA." BAREKENG: Jurnal Ilmu Matematika dan Terapan 18, no. 2 (May 25, 2024): 1147–54. http://dx.doi.org/10.30598/barekengvol18iss2pp1147-1154.

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Cramer’s rule is one of a method for solving a system of linear equations in conventional algebra. The system of linear equation can be solved using Cramer’s rule if . Max-plus algebra is a set where is a set of real numbers, equipped with biner operations and where and . Min-plus Algebra is a set where is a set of real numbers, equipped with biner operations and where and . In max-plus algebra has been formulated Cramer’s rule to solve a system of linear equations. Because max-plus algebra is isomorphic to min-plus algebra, Cramer’s rule can be formulated into min-plus algebra. The purpose of this research is to determine the sufficient conditions for a system of linear equations can be solved using Cramer’s rule. The method used in this research is a literature study that reviews previous research related to min-plus algebra, max-plus algebra, and Cramer’s rule in max-plus algebra. By using the appropriate analogy in max-plus algebra, we can determine the sufficient conditions so that a system of linear equations in min-plus algebra can be solved using Cramer’s rule. Based on the research, the sufficient conditions for a system of linear equations can be solved using Cramer’s rule are for and with the Cramer’s rule is . For an invertible matrix A, Cramer’s rule can be written as .

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12

Qin, Cuifeng, and Zuojun Liu. "Motion Modeling and Control of Lower Limb Exoskeleton Based on Max-Plus Algebra." Discrete Dynamics in Nature and Society 2021 (November 1, 2021): 1–10. http://dx.doi.org/10.1155/2021/8538787.

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Max-plus algebra is a special method to describe the discrete event system. In this paper, it is introduced to describe the motion of lower limb exoskeleton. Based on the max-plus algebra and the timed event graph, the walking process of exoskeleton is modelled. The max-plus algebra approach can describe the logical sequence and safety condition in the walking process, which cannot be achieved via other conventional modelling approaches. The autonomous control of lower limb exoskeleton system is studied via the model based on max-plus algebra. In the end, an FSM (finite state machine) controller embedded with the max-plus algebra model is proposed, and the experiments show ideal speed and gait/phase period control effect, as well as the good safety and stable performance.

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13

Wang, Cailu, Yuanqing Xia, and Yuegang Tao. "Ordered Structures of Polynomials over Max-Plus Algebra." Symmetry 13, no. 7 (June 25, 2021): 1137. http://dx.doi.org/10.3390/sym13071137.

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The ordered structures of polynomial idempotent algebras over max-plus algebra are investigated in this paper. Based on the antisymmetry, the partial orders on the sets of formal polynomials and polynomial functions are introduced to generate two partially ordered idempotent algebras (POIAs). Based on the symmetry, the quotient POIA of formal polynomials is then obtained. The order structure relationships among these three POIAs are described: the POIA of polynomial functions and the POIA of formal polynomials are orderly homomorphic; the POIA of polynomial functions and the quotient POIA of formal polynomials are orderly isomorphic. By using the partial order on formal polynomials, an algebraic method is provided to determine the upper and lower bounds of an equivalence class in the quotient POIA of formal polynomials. The criterion for a formal polynomial to be the minimal element of an equivalence class is derived. Furthermore, it is proven that any equivalence class is either an uncountable set with cardinality of the continuum or a finite set with a single element.

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14

Houssin, L., S. Lahaye, and J. L. Boimond. "TIMETABLE SYNTHESIS USING (MAX,+) ALGEBRA." IFAC Proceedings Volumes 39, no. 3 (2006): 375–80. http://dx.doi.org/10.3182/20060517-3-fr-2903.00201.

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15

Wagneur, Edouard. "Duality in the Max-Algebra." IFAC Proceedings Volumes 31, no. 18 (July 1998): 675–79. http://dx.doi.org/10.1016/s1474-6670(17)42068-4.

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16

Fiedler, Miroslav. "Dominant matrices and max algebra." Linear Algebra and its Applications 434, no. 4 (February 2011): 1189–94. http://dx.doi.org/10.1016/j.laa.2010.10.029.

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17

Molnárová, Monika, and Ján Pribiš. "Matrix period in max-algebra." Discrete Applied Mathematics 103, no. 1-3 (July 2000): 167–75. http://dx.doi.org/10.1016/s0166-218x(99)00242-5.

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18

Semančíková, Blanka. "Orbits in max–min algebra." Linear Algebra and its Applications 414, no. 1 (April 2006): 38–63. http://dx.doi.org/10.1016/j.laa.2005.09.009.

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19

Puquan, Xu, and Xu Xinhe. "An extension on max-algebra *." IFAC Proceedings Volumes 24, no. 14 (June 1991): 34–39. http://dx.doi.org/10.1016/s1474-6670(17)69318-2.

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20

Szabó, Peter. "Goldbach’s conjecture in max-algebra." Computational Management Science 14, no. 1 (November 14, 2016): 81–89. http://dx.doi.org/10.1007/s10287-016-0268-z.

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21

Mufarij Anna Ziaulhaq and Muhamad Zaki Riyanto. "Protokol Otentikasi Menggunakan Konstruksi Matriks Komutatif Atas Matriks Aljabar Max-Plus." Jurnal Fourier 12, no. 2 (October 31, 2023): 51–59. http://dx.doi.org/10.14421/fourier.2023.122.51-59.

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Aljabar max-plus adalah himpunan dilengkapi operasi penjumlahan didefinisikan sebagai operasi maksimum dan operasi perkalian didefinisikan sebagai operasi penjumlahan.Konsep aljabar max-plus diperluas untuk membentuk suatu himpunan matriks atas aljabar max-plus.Lebih lanjut himpunan matriks atas aljabar max-plus merupakan struktur semiring non komutatif.Linde dan Puente [1] telah mengkonstruksi suatu matriks yang bersifat komutatif terhadap operasi perkalian pada matriks atas aljabar max-plus, selanjutnyaMuanalifah dan Sergeev[2] menggunakan konstruksi matriks komutatif tersebut untuk diaplikasikan pada kriptografi, yaitu pada protokol pertukaran kunci Stickel. Artikel ini akan mengembangkan protokol pertukaran kunci tersebut menjadi suatu protokol otentikasi. Fungsi protokol otentikasi yaitu untuk memastikan kebenaran identitas pihak pengirim kepada pihak penerima, agar tidak terjadi pemalsuan data pengirim.Penggunaan matriks komutatif atas aljabar max-plus dimaksudkan untuk meningkatkan keamanan protokol otentikasi dari pihak yang hendak menyadap protokol otentikasi tersebut. The max-plus algebra is a set that completed addition operations defined as maximum operations and multiplication operations defined as addition operations. The concept of max-plus algebra is extended to form a matrix set over max-plus algebra. Furthermore, the set of matrices over max-plus algebra is a noncommutative semiring structure. Linde and Puente [1] has constructed a matrix that is commutative to multiplication operations on matrices over max-plus algebra, then Muanalifah and Sergeev[2] uses the commutative matrix construction to be applied to Stickel key exchange protocol. This article will develop the key exchange protocol into an authentication protocol.The function of the authentication protocol is to ensure the correctness of the identity of the sending party to the receiving party, so that falsification of sender data does not occur. The use of commutative matrices over max-plus algebra is intended to increase the security of authentication protocols from those who want to intercept the authentication protocol.

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22

Valverde-Albacete, Francisco, and Carmen Peláez-Moreno. "The Singular Value Decomposition over Completed Idempotent Semifields." Mathematics 8, no. 9 (September 12, 2020): 1577. http://dx.doi.org/10.3390/math8091577.

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In this paper, we provide a basic technique for Lattice Computing: an analogue of the Singular Value Decomposition for rectangular matrices over complete idempotent semifields (i-SVD). These algebras are already complete lattices and many of their instances—the complete schedule algebra or completed max-plus semifield, the tropical algebra, and the max-times algebra—are useful in a range of applications, e.g., morphological processing. We further the task of eliciting the relation between i-SVD and the extension of Formal Concept Analysis to complete idempotent semifields (K-FCA) started in a prior work. We find out that for a matrix with entries considered in a complete idempotent semifield, the Galois connection at the heart of K-FCA provides two basis of left- and right-singular vectors to choose from, for reconstructing the matrix. These are join-dense or meet-dense sets of object or attribute concepts of the concept lattice created by the connection, and they are almost surely not pairwise orthogonal. We conclude with an attempt analogue of the fundamental theorem of linear algebra that gathers all results and discuss it in the wider setting of matrix factorization.

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Siswanto, Siswanto, and Sahmura Maula Al Maghribi. "Characteristic min-polynomial of a triangular and diagonal strictly double R-astic matrices." F1000Research 13 (July 8, 2024): 771. http://dx.doi.org/10.12688/f1000research.147931.1.

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Background Determinant and characteristic polynomials are important concepts related to square matrices. Due to the absence of additive inverse in max-plus algebra, the determinant of a matrix over max-plus algebra can be represented by a permanent. In addition, there are several types of square matrices over max-plus algebra, including triangular and diagonal strictly double ℝ -astic matrices. A special formula has been devised to determine the permanent and characteristic max-polynomial of those matrices. Another algebraic structure that is isomorphic with max-plus algebra is min-plus algebra. Methods Min-plus algebra is the algebraic structure of triple ( ℝ ε ′ , ⊕ ′ , ⊗ ) . Furthermore, square matrices over min plus algebra are defined by the set of matrices sized n × n , the entries of which are the elements of ℝ ε ′ . Because these two algebraic structures are isomorphic, the permanent and characteristic min-polynomial can also be determined for each square matrix over min-plus algebra, as well as the types of matrices. Results In this paper, we find out the special formulas for determining the permanent and characteristic min-polynomial of the triangular matrix and the diagonal strictly double ℝ -astic matrix. Conclusions We show that the formula for determining the characteristic min-polynomial of the two matrices is the same, for each triangular matrix and strictly double ℝ -astic matrix A , χ A ( x ) = ⨁ r = 0 , 1 , … , n ′ δ n − r ⊗ n r .

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24

Rahayu, Eka Widia, Siswanto Siswanto, and Santoso Budi Wiyono. "MASALAH EIGEN DAN EIGENMODE MATRIKS ATAS ALJABAR MIN-PLUS." BAREKENG: Jurnal Ilmu Matematika dan Terapan 15, no. 4 (December 1, 2021): 659–66. http://dx.doi.org/10.30598/barekengvol15iss4pp659-666.

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Eigen problems and eigenmode are important components related to square matrices. In max-plus algebra, a square matrix can be represented in the form of a graph called a communication graph. The communication graph can be strongly connected graph and a not strongly connected graph. The representation matrix of a strongly connected graph is called an irreducible matrix, while the representation matrix of a graph that is not strongly connected is called a reduced matrix. The purpose of this research is set the steps to determine the eigenvalues and eigenvectors of the irreducible matrix over min-plus algebra and also eigenmode of the regular reduced matrix over min-plus algebra. Min-plus algebra has an ispmorphic structure with max-plus algebra. Therefore, eigen problems and eigenmode matrices over min-plus algebra can be determined based on the theory of eigenvalues, eigenvectors and eigenmode matrices over max-plus algebra. The results of this research obtained steps to determine the eigenvalues and eigenvectors of the irreducible matrix over min-plus algebra and eigenmode algorithm of the regular reduced matrix over min-plus algebra

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25

Wardayani, Ari, and Suroto Suroto. "SEMI MODUL POLINOMIAL FUZZY ATAS ALJABAR MAX-PLUS FUZZY." Jurnal Ilmiah Matematika dan Pendidikan Matematika 3, no. 1 (June 24, 2011): 1. http://dx.doi.org/10.20884/1.jmp.2011.3.1.2968.

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26

Ariyanti, Gregoria, Ana Easti Rahayu Maya Sari, and Christina Manurung. "Notes on matrix inverse over min-plus algebra." Edelweiss Applied Science and Technology 8, no. 6 (December 31, 2024): 9544–54. https://doi.org/10.55214/25768484.v8i6.4034.

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One of the semiring structures is the max-plus algebra, a set with entries 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 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 can be obtained by determining the matrix with entry which satisfies .

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27

Carrillo Flores, Wilber, and Alberto Mariano Rivero Zapata. "Classification of Real Division Algebras." Pesquimat 26, no. 2 (December 30, 2023): 1–10. http://dx.doi.org/10.15381/pesquimat.v26i2.25686.

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This article aims to oﬀer a unifying approach to the basic theory of division algebras by presenting the research of the German-American mathematician Max August Zorn, who classiﬁed alternative division algebras. In section 1 the basic theory of real division algebras is developed. Section 2 presents the Cayley-Dickson Process, which consists of constructing an extension algebra from an algebra provided with a conjugation, similar to the construction of complex numbers from real numbers. In Section 3 presents the classical division algebras R (real), C (complex), H (quaternions) and O (octonions) and mentions some of their applications. In section 4 the main theorem is presented, which establishes that the only (except isomorphism) alternative division algebras are: R, C, H and O (Zorn’s theorem). The classiﬁcation theorems of associative division algebras (Frobenius) and normed division algebras (Hurwitz) are obtained as corollaries of Zorn’s theorem. Finally in section 5 applications of division algebras to Geometry, Number Theory, Classical Physics, Modern Physics, Quantum Mechanics and Cryptography are mentioned.

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Caileanu, Corneliu. "RELIABILITY CALCULUS USING MAX-PLUS ALGEBRA." IFAC Proceedings Volumes 38, no. 1 (2005): 79–82. http://dx.doi.org/10.3182/20050703-6-cz-1902.01815.

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Elsner, L., and P. van den Driessche. "Max-algebra and pairwise comparison matrices." Linear Algebra and its Applications 385 (July 2004): 47–62. http://dx.doi.org/10.1016/s0024-3795(03)00476-2.

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30

Sergeev, Sergei˘. "Fiedler–Pták scaling in max algebra." Linear Algebra and its Applications 439, no. 4 (August 2013): 822–29. http://dx.doi.org/10.1016/j.laa.2011.12.039.

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31

Gavalec, Martin, Ján Plavka, and Hana Tomášková. "Interval eigenproblem in max–min algebra." Linear Algebra and its Applications 440 (January 2014): 24–33. http://dx.doi.org/10.1016/j.laa.2013.10.034.

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32

Müller, Vladimir, and Aljoša Peperko. "On the spectrum in max algebra." Linear Algebra and its Applications 485 (November 2015): 250–66. http://dx.doi.org/10.1016/j.laa.2015.07.013.

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Rosenmann, Amnon, Franz Lehner, and Aljoša Peperko. "Polynomial convolutions in max-plus algebra." Linear Algebra and its Applications 578 (October 2019): 370–401. http://dx.doi.org/10.1016/j.laa.2019.05.020.

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Watanabe, Sennosuke, Akiko Fukuda, Etsuo Segawa, and Iwao Sato. "A walk on max-plus algebra." Linear Algebra and its Applications 598 (August 2020): 29–48. http://dx.doi.org/10.1016/j.laa.2020.03.025.

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35

Menguy, Eric, Jean-Louis Boimond, and Laurent Hardouin. "Feedback Controller Design in Max-Algebra." IFAC Proceedings Volumes 30, no. 6 (May 1997): 1385–90. http://dx.doi.org/10.1016/s1474-6670(17)43555-5.

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Butkovič, P., and R. A. Cuninghame-Green. "On matrix powers in max-algebra." Linear Algebra and its Applications 421, no. 2-3 (March 2007): 370–81. http://dx.doi.org/10.1016/j.laa.2006.09.027.

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Zhao, Ming-wang, Zhong-hua Lu, and Yong-zai Lu. "Parameter identification algorithms in max-algebra." IFAC Proceedings Volumes 24, no. 14 (June 1991): 58–61. http://dx.doi.org/10.1016/s1474-6670(17)69323-6.

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38

Braker, J. G., and G. J. Olsder. "The power algorithm in max algebra." Linear Algebra and its Applications 182 (March 1993): 67–89. http://dx.doi.org/10.1016/0024-3795(93)90492-7.

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39

Bapat, R. B. "Permanents, max algebra and optimal assignment." Linear Algebra and its Applications 226-228 (September 1995): 73–86. http://dx.doi.org/10.1016/0024-3795(95)00304-a.

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Plavka, Ján. "ℓ-Parametric eigenproblem in max-algebra." Discrete Applied Mathematics 150, no. 1-3 (September 2005): 16–28. http://dx.doi.org/10.1016/j.dam.2005.02.017.

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41

Wadu, Mira. "Pengaturan Durasi Traffic Light pada Simpang Empat Kirab Kota Kupang Menggunakan Aljabar Max-Plus." JUTEKS : Jurnal Teknik Sipil 9, no. 2 (October 1, 2024): 136. https://doi.org/10.32511/juteks.v9i2.1346.

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The purpose of this study is to set the duration of traffic lights using max-plus algebra at the Kirab intersection in Kupang City. This study is based on the existing traffic light duration, then a directed graph is formed and modeled in max-plus algebra. Furthermore, the model is analyzed using the power algorithm and the average period of the green light duration for each phase is . In this study, the analysis results were obtained with . Thus, the new green light duration obtained using max-plus algebra for each intersection is 28 seconds, 31 seconds, 23 seconds and 30 seconds.

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Siswanto, Siswanto, and Anggrina Gusmizain. "Determining the Inverse of a Matrix over Min-Plus Algebra." JTAM (Jurnal Teori dan Aplikasi Matematika) 8, no. 1 (January 19, 2024): 244. http://dx.doi.org/10.31764/jtam.v8i1.17432.

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Linear algebra over the semiring R_ε with ⊗ (plus) and ⨁ (maximum) operations which is known as max-plus algebra. One of the isomorphic with this algebra is a min-plus algebra. Min-plus algebra that is the set R_(ε^' )=R∪{ε'}, with ⊗^' (plus) and ⨁' (minimum) operations. Given a matrix whose components are elements of R_(ε^' ) is called min-plus algebra matrices. Any matrix can be connected by an inverse. In conventional algebra, a square matrix is said an invertible matrix if the det⁡〖(A)〗≠0. In contrast to max-plus algebra, a matrix is said to have inverse condition if it meets certain conditions. Some concepts from the max-plus algebra can be transformed to the min-plus, because of their structural similarity. This means that the inverse matrix concept in max-plus can be constructed into a min-plus version. Thus, this study will explain the inverse of a matrix over the min-plus algebra, property of multiplying two invertible matrices, and connection between invertible matrix and linear mapping used the literature study method, with literature sources such as books, journals, articles, and theses. The data analysis technique used in this research is qualitative data analysis technique. Then, this article has a principal result that is matrix A∈R_(ε^')^(n×n) has a right inverse if and only if there are permutations of σ and the value of λ_i<ε', i∈{1,2,3,…,n} such that A=P_σ ⊗^' D(λ_i ) which is the inverse of matrices. Furthermore, if B is the correct inverse that satisfies A⊗^' B=E then B⊗^' A=E and B is uniquely determined by A.

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WALKER, CAROL L., and ELBERT A. WALKER. "A FAMILY OF FINITE DE MORGAN AND KLEENE ALGEBRAS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 20, no. 05 (October 2012): 631–53. http://dx.doi.org/10.1142/s0218488512500298.

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The algebra of truth values for fuzzy sets of type-2 consists of all mappings from the unit interval into itself, with operations certain convolutions of these mappings with respect to pointwise max and min. This algebra generalizes the truth-value algebras of both type-1 and of interval-valued fuzzy sets, and has been studied rather extensively both from a theoretical and applied point of view. This paper addresses the situation when the unit interval is replaced by two finite chains. Most of the basic theory goes through, but there are several special circumstances of interest. These algebras are of interest on two counts, both as special cases of bases for fuzzy theories, and as mathematical entities per se.

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Maharani, Andika Ellena Saufika Hakim, and Ari Suparwanto. "APPLICATION OF SYSTEM MAX-PLUS LINEAR EQUATIONS ON SERIAL MANUFACTURING MACHINE WITH STORAGE UNIT." BAREKENG: Jurnal Ilmu Matematika dan Terapan 16, no. 2 (June 1, 2022): 525–30. http://dx.doi.org/10.30598/barekengvol16iss2pp525-530.

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The set together with the operation maximum (max) denoted as and addition (+) denoted as is called max-plus algebra. Max-plus algebra may be used to apply algebraically a few programs of Discrete Event Systems (DES), certainly one of the examples in the production system. In this study, the application of max-plus algebra in a serial manufacturing machine with a storage unit is discussed. The results of this are the generalization system max-plus-linear equations on a production system that is, in addition, noted the max-plus-linear time-invariant system. From the max-plus-linear time-invariant system, it can be obtained the equation which is then used to determine the beginning time of a production system so the manufacturing machine work periodically. The eigenvector and eigenvalue of the matrix are then used to find the beginning time and the period time of the manufacturing machine. Furthermore, the time when the product leaves the manufacturing machine with the time while the raw material enters the manufacturing machine is given and vice versa are obtained from the max-plus-linear time-invariant system that is can be formed in the equation .

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45

BUTKOVIC, Peter, Hans SCHNEIDER, and Sergei SERGEEV. "Core of a matrix in max algebra and in nonnegative algebra: A survey." Tambov University Reports. Series: Natural and Technical Sciences, no. 127 (2019): 252–71. http://dx.doi.org/10.20310/2686-9667-2019-24-127-252-271.

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This paper presents a light introduction to Perron-Frobenius theory in max algebra and in nonnegative linear algebra, and a survey of results on two cores of a nonnegative matrix. The (usual) core of a nonnegative matrix is defined as ∩ k≥1 span+ (A k ) , that is, intersection of the nonnegative column spans of matrix powers. This object is of importance in the (usual) Perron-Frobenius theory, and it has some applications in ergodic theory. We develop the direct max-algebraic analogue and follow the similarities and differences of both theories.

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46

Maghribi, Sahmura Maula Al, Siswanto Siswanto, and Sutrima Sutrima. "Characteristic Min-Polynomial and Eigen Problem of a Matrix over Min-Plus Algebra." JTAM (Jurnal Teori dan Aplikasi Matematika) 7, no. 4 (October 9, 2023): 1108. http://dx.doi.org/10.31764/jtam.v7i4.16498.

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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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47

Katz, Ricardo D., Hans Schneider, and Sergeı˘ Sergeev. "On commuting matrices in max algebra and in classical nonnegative algebra." Linear Algebra and its Applications 436, no. 2 (January 2012): 276–92. http://dx.doi.org/10.1016/j.laa.2010.08.027.

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Umer, Mubasher, Umar Hayat, and Fazal Abbas. "An Efficient Algorithm for Nontrivial Eigenvectors in Max-Plus Algebra." Symmetry 11, no. 6 (May 30, 2019): 738. http://dx.doi.org/10.3390/sym11060738.

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The eigenproblem for matrices in max-plus algebra describes the steady state of the system, and therefore it has been intensively studied by many authors. In this paper, we propose an algorithm to compute the eigenvalue and the corresponding eigenvectors of a square matrix in an iterative way. The algorithm is extended to compute the nontrivial eigenvectors for Latin squares in max-plus algebra.

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Jones, Daniel. "Matrix roots in the max-plus algebra." Linear Algebra and its Applications 631 (December 2021): 10–34. http://dx.doi.org/10.1016/j.laa.2021.08.008.

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Elsner, Ludwig, and P. van den Driessche. "On the power method in max algebra." Linear Algebra and its Applications 302-303 (December 1999): 17–32. http://dx.doi.org/10.1016/s0024-3795(98)10171-4.

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