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
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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
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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.
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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;
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(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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