Aljabar max-plus dan sifat-sifatnya

Majid, Abdul (2012) Aljabar max-plus dan sifat-sifatnya. Undergraduate thesis, Universitas Islam Negeri Maulana Malik Ibrahim.

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Abstract

INDONESIA:

Aljabar max-plus yang dinotasikan dengan = (Rmax,... ,... ) merupakan salah satu struktur dalam aljabar yaitu semi-field idempoten. Rmax merupakan himpunan , dimana R merupakan himpunan bilangan real, dengan ε=–∞, sedangkan operasi ... menyatakan maximal dan ... menyatakan penjumlahan normal bilangan real, yang didefinisikan sebagai berikut:

a,b Rmax
a b = max(a,b)
a b = a + b

Aljabar max-plus (Rmax, , ) merupakan semi-ring dengan elemen netral = – dan elemen satuan e = 0, karena untuk setiap a, b, c Rmax berlaku sifat-sifat berikut:
i. (Rmax, ) membentuk semi-grup komutatif dengan elemen identitas , karena (Rmax, ) memiliki sifat assosiatif, komutatif terhadap operasi .
ii. (Rmax, ) membentuk grup abelian dengan elemen identitas e, dan memiliki elemen netral yang bersifat menyerap terhadap operasi , karena (Rmax, ) memiliki sifat assosiatif, komutatif, terdapat elemen identitas, dan elemen invers terhadap operasi .
iii. (Rmax, , ) membentuk semi-ring, karena berdasarkan sifat-sifat di atas maka (Rmax, ) membentuk semi-grup komutatif dengan elemen identitas, (Rmax, ) membentuk grup abelian dengan elemen identitas e, dan memiliki elemen netral yang bersifat menyerap terhadap operasi , dan (Rmax, , ) memiliki sifat distributif operasi terhadap operasi .

Semi-ring Rmax merupakan semi-ring komutatif jika operasi bersifat komutatif dan merupakan semi-ring idempoten jika operasi bersifat idempoten, dan semi-ring komutatif Rmax merupakan semi-field jika setiap elemen tak netralnya mempunyai invers terhadap operasi . Maka, terlihat bahwa (Rmax, , ) merupakan semi-field idempoten. Maka disarankan kepada peneliti selanjutnya untuk membahas tentang aljabar max-plus pada matrik, pada fungsi skalar, pada masalah nilai eigen dan vektor eigen, dan lain-lain. Aljabar max-plus memiliki peranan yang sangat banyak dalam menyelesaikan persoalan di beberapa bidang seperti teori graf, fuzzy, kombinatorika, teori sistem, teori antrian dan proses stokastik. Karena penelitian ini adalah aljabar max-plus, maka bisa diteliti pula tentang aljabar min-plus.

ENGLISH:

Max-plus algebra are denoted by = (Rmax, , ) is one of the algebraic structure of idempotent semi-field. Rmax is the set , where R is the set of real numbers, with
= – , while the operation stated maximum and normal addition of real numbers, which are defined as follows:

a,b Rmax
a b = max(a,b)
a b = a + b

Max-plus algebra (Rmax, , ) is a semi-ring with neutral element = – and identity element e = 0, since for every a, b, c Rmax apply the following properties:
i. (Rmax, ) form a commutative semi-group with identity element , because (Rmax, ) has the properties of associative, and commutative operation on .
ii. (Rmax, ) form abelian group with identity element e, and has a neutral element that are absorbed to the operation , because (Rmax, ) has the properties of associative, commutative, there is the identity element and inverse elements of the operation .
iii. (Rmax, , ) form a semi-ring, because based on the properties of the above then (Rmax, ) form a commutative semi-group with identity element , (Rmax, ) form abelian group with identity element e, and has neutral element that are absorbed to the operation , and (Rmax, , ) has a distributive nature of the operations to the operation .

Semi-ring Rmax is a commutative semi-ring if the operation hold on commutative and idempotent semi-ring if the operation hold on idempotent, and commutative semi-ring Rmax is the semi-field if every element of neutrality did not have the inverse of the operation . Thus, it appears that (Rmax, , ) is an idempotent semi-field.

It is advisable to research further to discuss about the max-plus algebra on the matrix, the scalar function, the problem of eigenvalues and eigenvectors, and others. Max-plus algebra has a role very much in solving problems in several fields such as graph theory, fuzzy, kombinatorika, systems theory, queuing theory and stochastic processes. Because of this research is the max-plus algebra, so it can be observed also on the min-plus algebra.

Item Type: Thesis (Undergraduate)
Supervisor: Jamhuri, Mohammad and Nashichuddin, Achmad
Keywords: Semi-grup; Semi-ring; Semi-field; Aljabar Max-plus; Max-plus Algebra
Subjects: 01 MATHEMATICAL SCIENCES > 0101 Pure Mathematics > 010101 Algebra and Number Theory
Departement: Fakultas Sains dan Teknologi > Jurusan Matematika
Depositing User: M. Muzakir
Date Deposited: 24 May 2017 05:29
Last Modified: 24 May 2017 05:29
URI: http://etheses.uin-malang.ac.id/id/eprint/6720

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