Pelabelan Lokal Titik Graf Hasil Diagram Lattice Subgrup Zn

Ikhsanul Halikin

Abstract


Abstract

A group is a system that contains a set and a binary operation satisfying four axioms, i.e., the set is closed under binary operation, associative, has an identity element, and each element has an inverse. Since the group is essentially a set and the set itself has subsets, so if the binary operation is applied to its subsets then it satisfies the group's four axioms, the subsets with the binary operation are called subgroups. The group and subgroups further form a partial ordering relation. Partial ordering relation is a relation that has reflexive, antisymmetric, and transitive properties. Since the connection of subgroups of a group is partial ordering relation, it can be drawn a lattice diagram. The set of integers modulo n, , is a group under addition modulo n. If the subgroups of are represented as vertex and relations that is connecting two subgroups are represented as edgean , then a graph is obtained. Furthermore, the vertex in this graph can be labeled by their subgroup elements. In this research, we get the result about the characteristic of the lattice diagram of and the existence of vertex local labeling.

Abstrak

Grup merupakan sistem yang memuat sebuah himpunan dan operasi biner yang memenuhi 4 aksioma, yaitu operasi pada himpunannya bersifat tertutup, assosiatif, memiliki elemen identitas, dan setiap elemennya memiliki invers. Grup pada dasarnya adalah himpunan dan himpunan itu memiliki himpunan bagian. Jika operasi tersebut diberlakukan pada himpunan bagiannya dan memenuhi 4 aksioma grup maka himpunan bagian dan operasi tersebut disebut subgrup. Grup dan subgrup ini selanjutnya membentuk suatu relasi pengurutan parsial. Relasi pengurutan parsial adalah suatu relasi yang memiliki sifat refleksif, antisimetris, dan transitif. Oleh karenanya, relasi subgrup-subgrup dari suatu grup ini dapat digambar diagram latticenya. Himpunan bilangan bulat modulo n, , merupakan grup terhadap operasi penjumlahan modulo n. Jika subgrup pada direpresentasikan sebagai titik dan relasi yang menghubungkan dua buah subgrupnya direpresentasikan sebagai sisi, maka diperoleh suatu graf. Titik-titik pada graf ini dapat dilabeli berdasarkan elemen-elemen subgrupnya. Pada penelitian ini diperoleh hasil kajian mengenai karakteristik diagram lattice subgrup dan eksistensi pelabelan lokal titiknya.


Keywords


Lattice Diagram; Subgroup; Vertex Local Labelling.

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References


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DOI: https://doi.org/10.24256/jpmipa.v6i1.409

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