Algebraic Study of Soft Lattice Theory and Its Applications to Distributed Computing System

Algebraic Study of Soft Lattice Theory and Its Applications to Distributed Computing System

Chapter One

Aim and Objectives

The aim of this research is to develop algebraic theorems on soft lattice defined via conjunction and disjunction, and present its application to distributed computing system.

The research objectives are to:

redefine the concepts of conjunction and disjunction as binary operations on soft sets and present their properties;
define soft lattice in terms of the redefined conjunction and disjunction withillustrations;
introduce soft Boolean algebra and obtain some results;
extend the concept of soft lattice to distributed, modular and isomorphic soft lattices;
apply soft lattice theory to distributed computing

CHAPTER TWO

LITERATURE REVIEW

The origin of soft set theory could be traced to the work of Pawlak (1999) titled Hard and Soft Set in Proceeding of the International Workshop on rough sets and knowledge discovery at Banff. His notion of soft sets is a unified view of classical, rough and fuzzy sets. This motivated Molodtsov (1999) titled soft set theory: first result. Therein, the basic notions of the theory of soft sets and some of its possible applications were presented. For positive motivation, the work discussed some future problems with regards to the theory.

In order to solidify the theory of soft set, Maji et al. (2002), defined some basic terms of the theory such as equality of two soft sets, subset and super set of a soft set, complement of a soft set, null soft set, and absolute soft set with examples. Binary operations like AND, OR, union and intersection were also defined. De Morgan’s laws and a number of results are verified in soft set theory context.

Chen et al. (2005) focused their discussion on the parameterization reduction of soft sets and its applications. First they pointed out that the results of soft set reductions offered in Maji et al. (2002) can be restructured. Therein they observed that the algorithms used to first compute the choice value to select the optimal objects for the decision problems can be redefined and they illustrated this with an example. Finally, they proposed parameterization reduction of soft sets and compared it with the concept of attributes reduction in rough sets theory. Mushrif et al. (2006) studied the texture classification via soft set theory.

Aktas and Cagman (2007) extended the basic properties of soft sets in Maji et al. (2002) and compared the soft sets to the related concept of fuzzy sets and rough sets. They then gave a definition of soft groups, and derived their basic properties.

Herawan et al. (2010) gave an alternative approach for attribute reduction in multi-valued information system under soft set theory. The work emphasized that based on the notion of multi-soft sets and AND operation, attribute re- duction can be defined. It is shown that the results obtained are equivalent with Pawlak’s rough reduction. Also, Xu et al. (2010) introduced the notion of vague soft set which is an extension of soft set. Therein the basic properties of vague soft sets were presented and discussed.

Li (2010) defined the soft lattices, soft sub-lattices, discussed its properties, and studied the relation between the soft lattices and the fuzzy soft set. Also, Babitha and Sunil (2010), introduced the concept of soft set relations as a soft subset of the Cartesian product of the soft sets and many related concepts such as equivalent soft set relation, partition, composition and function are discussed. Nagarajan and Meenambigai (2011) extended the study of soft lattices by using soft set theory. The notions of soft lattices, soft distributive lattices, soft modular lattices, soft lattice ideals, soft lattice homomorphisms are introduced and several related properties are investigated.

Ozturk and Inan (2011) illustrated the interconnections between the various operations in soft set and defined the notion of restricted symmetric difference of soft sets and investigated its properties. However, Manemaran (2011) fur- ther discussed fuzzy soft sets algebraic structures and defined fuzzy soft group. Operations on fuzzy soft groups and some related results were proved. Further- more, definitions of fuzzy soft functions and fuzzy soft homomorphism were also defined. Finally, the theorems on homomorphic image and homomorphic pre image were discussed in details.

Ghosh et al. (2011) also extended soft set to ring theory and fuzzy soft ideal theory. Also, Atagun and Sezgin (2011) studied soft subrings and soft ideals of a ring. Moreover, the concept of soft subfields of a field and soft sub module of a left R-module were defined. Some related properties about soft substructures of rings, fields and modules are investigated and illustrated by some examples.

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Sezgin and Atagun (2011) proved that certain De Morgan’s law holds in soft set theory with respect to different operations on soft sets. Soft set is being currently extended to intuitionistic fuzzy soft sets and the concept of intuition- istic fuzzy soft sets to semi group theory. The notion of intuitionistic fuzzy soft ideals over a semi group is introduced with their basic properties investigated (Zhou et al., 2011). Also, some lattice structures of the set of all intuitionistic fuzzy soft ideals of a semi group were derived.

Yang and Guo (2011) studied the concept of anti-reflexive kernel, symmetric kernel, reflexive closure, and symmetric closure of a soft set relation. Finally, soft set relation mapping and inverse soft set relation mappings were proposed and some related properties were discussed. Also, in the same year, (Ge and Yan, 2011) further investigated the operational rules given by (Maji et al., 2002 and Ali et al., 2009) to obtain some necessary and sufficient conditions that made corresponding operational rules to hold.

Marudai and Rajendran (2011) studied (Molodtsov, 1999) notion of soft set and fuzzy soft set considering the fact that the parameters are mostly fuzzy hedges or fuzzy parameters. Also study the notion of fuzzy soft lattice on groups, homomorphic image, preimage of fuzzy soft lattice, arbitrary family of fuzzy soft lattice and fuzzy normal soft lattice using T-norms.

Dauda and Ibrahim (2013) developed partial ordering in soft set context. In the same year, Kurt (2013) introduced and investigated the basic concepts of soft set theory and soft group theory. Some examples for soft groups and soft subgroups are also studied.

Akram and Feng (2013) introduced the concept of soft Lie subalgebras (respec-tively, soft Lie ideals) and state some of their fundamental properties. Also the concept of soft intersection Lie subalgebras (respectively, soft Lie ideals) are introduced and some related properties are investigated. Soft set theory has potential applications in many different fields which include the smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability theory, and measurement theory (Molodtsov, 1999).

In Molodtsov (1999) applications of soft set to stability and regularization, game theory and operational research were discussed in details. Extension of soft set theory to real analysis and its applications were also presented.

CHAPTER THREE

FUNDAMENTALS OF SOFT SET THEORY

In this chapter, fundamentals of soft set theory, operations and its various algebraic structures are presented with redefined concept of conjunction and disjunction as binary operations.

Soft Set Theory

In this section, some basic definitions of soft theory and some suitable examples are given. Let U be a universal set and E be the set of all possible parameters under consideration with respect to U . Let the power set of U (i.e., the set of all subsets of U ) be denoted by P (U ) and A is a subset of the parameters, E (A⊆ E). The parameters are attributes, characteristics or properties associ- ated with the objects in U . Then we have the following: definitions.

Definition 3.1.1 A pair (F, E) is called a soft set over U if and only if F is a mapping of E into the set of all subsets of the set U . That is, a soft set is a parametrized family of subsets of the set U . For all e ∈ E, F (e) is considered as the set of e−approximate elements of the soft set (F, E).

CHAPTER FOUR

SOFT BOOLEAN ALGEBRA

Redefined Concept of Conjunction (∧) and Disjunction (∨) in Soft SetTheory

Based on the difficulties we encountered in the Definition 3.1.8 and 3.1.9 we hereby redefine the Concept of Conjunction (∧) and Disjunction (∨) as follows:

Definition 4.1.1

Given any two soft sets (F, A) and (G, B) over a common universe U, the disjunction of (F, A) and (G, B) denoted by (F, A) ∨ (G, B) is defined as

(F, A) ∨ (G, B) = (H, A ∪ B), where H(α) =F (α) ∪ G(α), ∀α ∈ A ∪ B.

Definition 4.1.2

Given any two soft sets (F, A) and (G, B) over a common universe U, the conjunction of (F, A) and (G, B) denoted by (F, A) ∧ (G, B) is defined as (F, A) ∧ (G, B) = (P, A ∩ B), where P (α) =F (α) ∩ G(α), ∀ α ∈ A ∩ B.

CHAPTER FIVE

SOFT LATTICE THEORY AND APPLICATION

Soft Lattice Theory

There are two standard ways of defining lattice in classical setting: viz., based on algebraic structure or based on the notion of order. In this chapter, we present soft lattice theory based on algebraic structure and on the notion of order with its related application to distributed system.

Definition 5.1.1

Let (Γ, E) be a soft set. Let A, B, C ⊆ E such that (F, A) , (G, B) and (H, C) are all defined. Then (Γ, E) together with the binary operations ∨ and ∧ is called soft lattice if the following axioms are satisfied:

CHAPTER SIX

SUMMARY, CONCLUSION AND RECOMMENDATIONS

Summary

This thesis presents a critical study of soft set theory, its algebraic structure and application in soft lattice distributed computing. It is made up of six chapters as shown summarily below.

Chapter one covers the general introduction of the thesis, which includes the historical background of the study, motivation and justification, and the aim and objectives of the study.

A critical literature survey of the fundamentals of soft set theory, in respect to its origin and general concept, application point of view and the algebraic point of view are presented in chapter two.

Chapter three presents a critical study of the fundamentals of soft set theory, its operation and various algebraic structures. We also redefined the concept of conjunction and disjunction as an algebraic operators on soft set and used in chapter four to propose soft Boolean algebra.

In chapter four, a perception named soft Boolean algebra is introduced and some related results are established.

Chapter five discusses soft lattice and some of its algebraic properties are introduced. Some important classes of lattices using the concept of soft sets theory are presented and some results are established. It also presents some applications of soft lattice theory in distributed computing system.

Conclusion

In this thesis, fundamentals of soft set theory, algebraic structures of soft set, soft Boolean algebras and soft lattice theory are presented. We introduced some important classes of lattices using the concept of soft sets theory and presented their properties; and some new results were obtained. We discussed the various applications of soft lattice theory to solve problems in distributed computing system.

The following new results are the main contributions of this thesis:

Redefined Concept of Conjunction (AND) and Disjunction (OR)

Recommendations

Soft set theory is a relatively new area in mathematics and very applicable in real life decision making problems that involve uncertainties. The notions of soft lattice implication algebras and soft lattice implication sub-algebras could be interesting for further studies. We recommend that the application of soft lattice be extended to graph decomposition, cryptography and analysis of partial order traces for temporal logic predicate.

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