Latin Square Komutatif atas Aljabar Max-Plus

Agung, Teosofi Hidayah (2026) Latin Square Komutatif atas Aljabar Max-Plus. Other thesis, Institut Teknologi Sepuluh Nopember.

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Abstract

Aljabar max-plus merupakan struktur aljabar atas ℝmax dengan operasi a ⊕ b = max{a,b} dan a ⊗ b = a + b. Salah satu permasalahan yang penting dalam aljabar max-plus adalah menentukan pasangan matriks yang komutatif terhadap operasi perkalian. Penelitian ini membahas komutativitas dua Latin square terhadap perkalian max-plus. Latin square yang dikaji berordo n dengan himpunan simbol n = {1, 2, ..., n}, sehingga setiap baris dan kolomnya merupakan permutasi dari n. Penelitian ini bertujuan untuk menentukan syarat perlu, syarat cukup, dan prosedur konstruksi pasangan Latin square komutatif. Setiap Latin square direpresentasikan melalui dekomposisi matriks permutasi, yaitu A = ⨁ᵢ₌₁ⁿ i ⊗ P_σᵢᴬ. Melalui representasi tersebut, komutativitas A ⊗ B = B ⊗ A dianalisis berdasarkan komposisi permutasi penyusun A dan B. Hasil penelitian menunjukkan bahwa syarat perlu komutativitas dapat diperoleh dari elemen maksimum, yaitu permutasi simbol maksimum harus saling komutatif. Selanjutnya, diperoleh syarat cukup melalui subgrup abelian dari Sₙ, serta kriteria perlu dan cukup melalui kesamaan himpunan superlevel U_≥vᴬᴮ dan U_≥vᵇᴬ untuk setiap v ∈ {n + 2, n + 3, ..., 2n}. Berdasarkan kriteria tersebut, disusun algoritma untuk mencari pasangan Latin square B yang komutatif dengan suatu Latin square A. Algoritma tersebut memuat tahap alternatif melalui penyusunan ulang permutasi penyusun A ketika permutasi-permutasi tersebut saling komutatif, serta tahap umum yang menggunakan centralizer dari permutasi simbol maksimum, matriks kandidat parsial B*, dan pemeriksaan himpunan superlevel secara bertahap. Penerapan algoritma pada ordo 3, 4, dan 5 menunjukkan bahwa prosedur tersebut dapat digunakan untuk memperoleh kandidat pasangan Latin square komutatif.
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Max-plus algebra is an algebraic structure over ℝmax equipped with the operations a ⊕ b = max{a,b} and a ⊗ b = a + b. One of the important problems in max-plus algebra is determining commutative pairs of matrices under multiplication. This research discusses the commutativity of two Latin squares under max-plus multiplication. The Latin squares studied are of order n with the symbol set n = {1, 2, ..., n}, such that each of their rows and columns is a permutation of n. This research aims to determine the necessary conditions, sufficient conditions, and construction procedures for commutative Latin square pairs. Each Latin square is represented through a permutation matrix decomposition, namely A = ⨁ᵢ₌₁ⁿ i ⊗ P_σᵢᴬ. Through this representation, the commutativity A ⊗ B = B ⊗ A is analyzed based on the composition of the constituent permutations of A and B. The results show that the necessary condition for commutativity can be obtained from the maximum elements, meaning the permutations of the maximum symbol must commute with each other. Furthermore, sufficient conditions are obtained through abelian subgroups of Sₙ, as well as necessary and sufficient criteria through the equality of the superlevel sets U_≥vᴬᴮ and U_≥vᵇᴬ for each v ∈ {n + 2, n + 3, ..., 2n}. Based on these criteria, an algorithm is developed to find a Latin square B that commutes with a given Latin square A. The algorithm contains an alternative stage by rearranging the constituent permutations of A when these permutations are commutative to one another, as well as a general stage utilizing the centralizer of the maximum symbol permutation, a partial candidate matrix B*, and a step-by-step examination of the superlevel sets. The application of the algorithm on orders 3, 4, and 5 demonstrates that the procedure can be used to obtain candidate pairs of commutative Latin squares.

Item Type:	Thesis (Other)
Uncontrolled Keywords:	Aljabar Max-Plus, Latin Square, Matriks Komutatif, Matriks Sirkulan, Permutasi, Semiring Tropikal, Matriks Permutasi, Centralizer, Himpunan Superlevel, Max-Plus Algebra, Latin Square, Commutative Matrices, Circulant Matrices, Permutations, Tropical Semiring, Permutation Matrices, Centralizer, Superlevel Sets
Subjects:	Q Science > QA Mathematics Q Science > QA Mathematics > QA141 Numeracy--Problems, exercises, etc. Q Science > QA Mathematics > QA159 Algebra Q Science > QA Mathematics > QA184 Algebra, Linear
Divisions:	Faculty of Mathematics, Computation, and Data Science > Mathematics > 44201-(S1) Undergraduate Thesis
Depositing User:	Teosofi Hidayah Agung
Date Deposited:	27 Jul 2026 02:07
Last Modified:	27 Jul 2026 02:07
URI:	http://repository.its.ac.id/id/eprint/137658

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