# Characterization of an algebraic structure and its fuzzifications

Akram, Muhammad (2017) Characterization of an algebraic structure and its fuzzifications. PhD thesis, Universiti Tun Hussein Onn Malaysia.

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## Abstract

Semigroups are very important in many areas of mathematics and play a very significant role in the development of mathematics. The concept of Γ-semigroup was given as a generalization of semigroup and ternary semigroup. Fuzzy set and its extensions are equally important and have a lot of applications in every field of life. Some very important extensions of algebraic structures are their fuzzifications and intuitionistic fuzzifications. In this research, some work on different aspects of an algebraic structure and its fuzzifications are given. Here the concept of biΓ-ternary semigroup is given as a generalization of ternary semigroup and Γ-semigroup. The notions of biΓ-ternary subsemigroup, biΓ-ternary left (right, lateral, bi and quasi) ideals of biΓ-ternary semigroup are defined and characterized. Some properties of these ideals and relationships between them are discussed. Moreover, the regular biΓ-ternary semigroup is defined and characterized in terms of its ideals. Furthermore, the concept of intuitionistic N-fuzzy set as a new extension of fuzzy set and intuitionistic fuzzy set is defined by considering the negative meanings of information. The notions of level set and characteristic functions of intuitionistic N-fuzzy set is also defined here. As the applications of intuitionistic N-fuzzy set, the notion of this set is applied to the ideal theory of some algebraic structures such as semigroup, Γ-semigroup and the biΓ-ternary semigroup to get the corresponding intuitionistic N-fuzzy structures and substructures and characterized them. Finally, the concept of fuzzy soft set and bipolar fuzzy soft set is applied to the ideals of Γ-semigroups and biΓ-ternary semigroups. Some special operations, OR, AND, extended union, extended intersection, restricted union and restricted intersection between the fuzzy soft and bipolar fuzzy soft ideals of Γ-semigroup and biΓ-ternary semigroup are discussed. It is also proved that these operations also included same type fuzzy soft and bipolar fuzzy soft ideals of Γ-semigroup and biΓ-ternary semigroup.

Item Type: Thesis (PhD) Q Science > QA Mathematics > QA150 Algebra Mr. Mohammad Shaifulrip Ithnin 02 Jul 2018 03:24 02 Jul 2018 03:24 http://eprints.uthm.edu.my/id/eprint/10009 View Download Statistic