A Study of Cubic Sets in Semigroups and Generalized Structures

Abstract

It is common knowledge that common models with their limited boundaries of truth and falsehood are not sufficient to detect the reality so there is a need to discover other systems which are able to address the daily life problems. In every branch of science problems arise which abound with uncertainties and impaction. Some of these problems are related to human life, some others are subjective while others are objective and classical methods are not sufficient to solve such problems because they cannot handle various ambiguities involved. To overcome this problem, Zadeh (Zadeh., 1965) introduced the concept of a fuzzy set which provides a useful mathematical tool for describing the behavior of systems that are either too complex or ill-defined to admit precise mathematical analysis by classical methods. The literature in fuzzy set theory is rapidly expanding and application of this concept can be seen in a variety of disciplines such as artificial intelligence, computer science, control engineering, expert systems, operating research, management science, and robotics. Atanassov (Atanassov,1986), defined the notions of intuitionistic fuzzy sets as the generalization of fuzzy sets. Atanassov and many others applied the concept of an intuitionistic fuzzy set to algebra, topological space, knowledge engineering, natural language and neural network etc. Cubic sets are the generalizations of fuzzy sets and intuitionistic fuzzy sets, in which there are two representations, one is used for the degree of membership and other is used for the degree of non-membership. Membership function is handle in the form of interval while non-membership is handled through ordinary fuzzy set (Jun et al., 2010 ). Hyperstructure theory was introduced in 1934, when Marty (Marty, 1934 ) defined hypergroups, began to analyze their properties and applied them to groups. In the following decades and nowadays, a number of different hyperstructures are widely studied from the theoretical point of view. Nowadays, hyperstructures have a lot of applications to several domains of mathematics and computer science and they are studied in many countries of the world. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. This thesis consists of eight chapters. In chapter one, we present some basic definitions and results which are directly used in our work. Here we discussed semigroups, LA-semigroups, LA-semihypergroups, fuzzy sets, interval valued fuzzy sets and cubic sets. In chapter two, we introduce a new concept of v H -LA-semigroups with examples. In addition to this we show that every LA-semihypergroup is an v H -LA-semigroup and each LA-semigroup endowed with an equivalence relation can induce an v H -LA-semigroup. We explore isomorphism theorems with the help of regular relations, v H -LA-subsemigroups and ideals, hyperorder on v H -LA-semigroups and direct product of v H -LA-semigroups. In chapter three, we introduce the concept of a generalized cubic set and defined the concept of generalized cubic subsemigroups (ideals) of semigroups and investigate some related properties. Specifically, we introduced the concept of ( , q ) Γ Γ Δ ∈ ∈ ∨ -cubic ideal, ( , q ) Γ Γ Δ ∈ ∈ ∨ -cubic quasi-ideal, ( , q ) Γ Γ Δ ∈ ∈ ∨ -cubic bi-ideal and ( , q ) Γ Γ Δ ∈ ∈ ∨ -cubic prime/semiprime ideal in semigroups. Chapter four, deals with the study of cubic sets in non-associative algebraic structure, namely LA-semigroups. LA-semigroups are the generalization of the well-known associative structure, namely commutative semigroups. Here we define some basic properties of the cubic sets in LA-semigroups. Further we explore some useful characterizations of regular and intra-regular LA-semigroups by using the idea of cubic sets. In chapter five, we define ( , q ) Γ Γ Δ ∈ ∈ ∨ -cubic sub LA-semigroups, ( , q ) Γ Γ Δ ∈ ∈ ∨ -cubic ideals, ( , q ) Γ Γ Δ ∈ ∈ ∨ -cubic generalized bi-ideals, iv ( , q ) Γ Γ Δ ∈ ∈ ∨ -cubic bi-ideals, ( , q ) Γ Γ Δ ∈ ∈ ∨ -cubic interior-ideals and ( , q ) Γ Γ Δ ∈ ∈ ∨ -cubic quasi-ideals of an LA-semigroup and give some interesting results We mainly focus on the intra-regular LA-semigroups in term of newly defined generalized cubic ideals and give some useful characterizations. In chapter six, we initiate a study of cubic sets in left almost semihypergroups. By using the concept of cubic sets, we introduce the notion of cubic sub LA-semihypergroups (hyperideals and bi-hyperideals) and discuss some basic results on cubic sets in LA-semihypergroups. At the end we discuss some properties concerning the image and preimage of cubic hyperideals. In chapter seven, we define different types of generalized cubic hyperideals in LA-semihypergroups and we present some results on images and preimages of ( , q ) Γ Γ Δ ∈ ∈ ∨ -cubic hyperideals of LA-semihypergroups. At the end we give some characterizations of regular LA-semihypergroups in terms of ( , ) k ∈∈∨q -cubic hyperideals and ( , q ) Γ Γ Δ ∈ ∈ ∨ -cubic (resp., left, right, two sided, bi, generalized bi, interior, quasi)-hyperideals of LA-semihypergroups. In chapter eight, we define the idea of cubic ideals, cubic relations, cubic regular relations, cubic Rees relation of an v H -LA-semigroup and investigate some helpful conclusion on it. We define the notion of generalized cubic v H -LA-subsemigroups, generalized cubic v H -ideals of v H -LA-semigroups and chat about some of their essential properties. At the end we talk about the direct product of v H -LA-Semigroups, direct product of n - v H -LA-semigroups and direct product of the generalized n -cubic sets of n - v H -LA-semigroups in terms of cubic sets and generalized cubic sets.