## ABSTRACT

In this thesis, we crisply present the fundamentals of soft set theory to emphasize

that soft set has enough developed basic supporting tools through which

various algebraic structures in theoretical point of view could be developed.

The concepts of conjunction and disjunction are redefined as binary operations

on soft sets and their properties are presented. A perception named soft

Boolean algebra is introduced where some related results were established. It

is shown that if SB is a collection of all soft sets under a common universe

U, then (SB;^;_; ;; ~U ) is a Boolean algebra. For any two soft sets (F;A),

(G;B) 2 (SB), domination, idempotents, absorption and complement laws

are satisfied, where ; and ~U are unique. We define soft lattice in terms of

the redefined conjunction and disjunction and present some examples. Upper

bound and least upper bound, lower bound and greatest lower bound were

defined in soft set context. Soft lattice is redefine in terms of supremum and

infimum and it is shown that the two definitions are equivalent. Given any soft

semilattice (;E), where (e1) (e2) if and only if (e1) ^ (e2) = (e1),

8 e1; e2 2 E, we show that ((;E);) is an ordered soft set in which every

pair of elements has greatest lower bound. The idea of soft lattice is extended

to distributed soft lattice, modular soft lattice and isomorphic soft lattice and

their properties are presented with some related results. We established that

if (;E) is an ordered soft set and A;B E, such that : (F;A) ! (G;B)

is defined by (F(e1)) = fF(e2) 2 (F;A) : F(e1) F(e2); 8F(e1) 2 (F;A)g,

then (F;A) is isomorphic to the range of ordered by containment . Finally,

some applications of soft lattice theory to distributed computing system are

presented where it is shown that, a predicate is linear if and only if it is meetclosed.

If B is a linear predicate with the efficient advancement property, then

there exists an efficient algorithm to determine the least consistent cut that

satisfy B (if any). We presented an algorithm to detect a linear predicate of a

consistent cut and showed that the slice of a distributed computing is uniquely

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defined for all predicate.

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## TABLE OF CONTENTS

Flyleaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Certification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

CHAPTER ONE

GENERAL INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Statement of Problem . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Aim and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

CHAPTER TWO

LITERATURE REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

CHAPTER THREE

FUNDAMENTALS OF SOFT SET THEORY . . . . . . . . . . . . . . . . . . . . 16

3.1 Soft Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Soft Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Properties of Soft Set Operations . . . . . . . . . . . . . . . . . 20

3.4 Algebraic Structures via Soft Sets . . . . . . . . . . . . . . . . . 31

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CHAPTER FOUR

SOFT BOOLEAN ALGEBRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 Redefined Concept of Conjunction (^) and Disjunction (_) in

Soft Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Properties of the Redefined Conjunction (^)and Disjunction (_) 36

4.3 Concept of Soft Boolean Algebra . . . . . . . . . . . . . . . . . 41

4.4 Some Results on Soft Boolean Algebra . . . . . . . . . . . . . . 46

CHAPTER FIVE

SOFT LATTICE THEORY AND APPLICATION . . . . . . . . . . . . . . . 57

5.1 Soft Lattice Theory . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Some Theorems on Soft Lattice Theory . . . . . . . . . . . . . . 65

5.3 Hasse Diagram for Soft Lattice . . . . . . . . . . . . . . . . . . 69

5.4 Isomorphic Soft Lattices and Soft Sublattices . . . . . . . . . . . 71

5.5 Distributive and Modular Soft Lattices . . . . . . . . . . . . . . 79

5.6 Soft Lattice Distributed Computing . . . . . . . . . . . . . . . . 90

5.6.1 Detecting global predicate . . . . . . . . . . . . . . . . . 91

5.6.2 Slicing distributed computing . . . . . . . . . . . . . . . 95

CHAPTER SIX

SUMMARY, CONCLUSION AND RECOMMENDATIONS. . . . 101

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . 109

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

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## CHAPTER ONE

GENERAL INTRODUCTION

1.1 Introduction

Most of our real life problems in engineering, social and medical science, economics,

environment etc., involve imprecise data and their solutions involve

the use of mathematical principles based on uncertainty and imprecision. To

handle such uncertainties, a number of theories have been proposed. According

to (Molodtsov, 1999) some of these are probability theory, fuzzy sets theory,

intuitionistic fuzzy sets, interval mathematics and rough sets, etc. All these

theories, however, are associated with an inherent limitation, which is the

inadequacy of the parameterization tool associated with these theories.

Fuzzy set was developed by (Zadeh, 1965), in an attempt to deal with the

problems of uncertainties. This theory has been found to be appropriate to

some extent.

Let A be a subset of set X, A called indicator function, is defined as

A =

8><

>: 1

;

if x

2

A;

0; if x =2 A:

Obviously there is one-to-one correspondence between a set and its indicator

function.

Let U be a universe. A fuzzy set X over U is a set defined by a function X

representing a mapping

X : U ! [0; 1]:

Here, X is called the membership function of X, and the value X(u) is called

1

the grade of membership of u 2 U, and represents the degree of u belonging

to the fuzzy set X. Thus, a fuzzy set X over U can be represented as follows:

X = f(X(u)=u) : u 2 U; X(u) 2 [0; 1]g

One of the deficiencies of fuzzy set is how to set the membership function

(Molodtsov, 1999).

Given the various observations on some of the existing tools for solving uncertainty

problems, soft set theory emerged to soften these limitations.

Definition1.1.1: Soft set (Molodtsov, 1999)

Let U be a universe set and let E be a set of parameters (each parameter could

be a word or a sentence). Let P(U) denote the power set of U.

A pair (F; E) is called a soft set over a given universe set U if and only if

F is a mapping of a set of parameters E into the power set of U. That is,

F : E ! P(U). Clearly, a soft set over U is a parameterized family of

subsets of a given universe U. Also, for any e 2 E, F (e) is considered as the

set of e–approximate elements of the soft set (F;E).

Soft sets could be regarded as neighbourhood systems, and they are special

cases of context-dependent fuzzy sets (Molodtsov, 1999). In soft set theory the

problem of setting the membership function, among other related problems,

simply does not arise. This makes the theory very convenient and easy to

apply in practice.

Example 1.1.1

(i) Let (X, ) be a topological space, that is, X is a set and is a toplogy (a

family of subsets of X called the open sets of X). Then, the family of neighborhoods

T(x) of point x, where T (x) = fV 2 j x 2 V g, may be considered

2

as the soft set (T (x) ; ).

(ii) Let A be a fuzzy set and A be the membership function of the fuzzy set

A, that is, A is a mapping of U into [0; 1], let F () = fx 2 U j A(x)

g; 2 [0; 1] be a family of level sets for function A. If the family F is

known, A (x) can be found by means of the definition:

A (x) = Sup

2 [0; 1] ;

x 2 F()

. Hence every fuzzy set A may be considered as the soft set (F; [0; 1]):

(iii) Let U = fC1; C2; C3; C4; C5; C6; C7; C8; C9; C10g be the set of Cars

under consideration, E be a set of parameters defined as follows:

E = {e1 = expensive, e2 = beautiful, e3 = manual gear, e4 = cheap, e5 =

automatic gear, e6 = in good repair, e7 = in bad repair}. Then the soft set

(F;E) describes the attractiveness of the Cars under consideration.

Li (2010) used the concept of soft sets to defined soft lattice, gave some properties

of soft lattice and discussed the relationship between soft lattice and

fuzzy sets. The notion of soft lattice, soft sublattice, complete soft lattice,

modular soft lattice, distributive soft lattice and soft chain were studied using

the conventional conjunction and disjunction (Karaaslan et al., 2012).

However, in this thesis we redefine the concepts of conjunction and disjunction

and thereby define soft lattice via the redefined notions and obtain results.

The concept of soft Boolean algebra is proposed with some theorems. The

application of soft lattice to distributed computing system is introduced.

3

1.2 Statement of Problem

The concept of soft sets proposed by (Molodtsov, 1999) received much attentions

in research domain due to its resourcefulness in engineering, control

system, computer science and real life situations. However, the idea of soft

lattice theory in an algebraic perspective and its application to distributed

computing system have not been studied.

1.3 Justification

Modern set theory formulated by George Cantor is fundamental for the whole

of mathematics. The issues associated with the notion of set are the concept

of vagueness and uncertainties. Mathematics requires that all mathematical

notions including set must be exact. This vagueness and uncertainties or the

representation of imperfect knowledge has been a serious problem for so long a

time for logicians and mathematicians. As a result the concept of approximate

solution is introduced. Such concepts include the notion of soft set theory,

where an approximate solution is provided.

1.4 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:

(i) redefine the concepts of conjunction and disjunction as binary operations

on soft sets and present their properties;

4

(ii) define soft lattice in terms of the redefined conjunction and disjunction

with illustrations;

(iii) introduce soft Boolean algebra and obtain some results;

(iv) extend the concept of soft lattice to distributed, modular and isomorphic

soft lattices;

(v) apply soft lattice theory to distributed computing system.

1.5 Methodology

We used the existing literature on development and fundamentals of soft set

theory to obtain new results. Various researches on lattice theory including its

applications were studied, as guiding principles in extending soft set theory to

lattice theory, called soft lattice.

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