ALGEBRAICSTUDYOFSOFTLATTICETHEORY ANDITSAPPLICATIONSTODISTRIBUTED COMPUTINGSYSTEM
ALGEBRAICSTUDYOFSOFTLATTICETHEORY ANDITSAPPLICATIONSTODISTRIBUTED COMPUTINGSYSTEM
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Date
2016-03
Authors
YUSUF, AHMEDOMEIZA
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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)∈ (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
infimumanditisshownthatthetwodefinitionsareequivalent. Givenanysoft
semilattice (Γ,E), where Γ(e1)⊆ Γ(e2) if and only if Γ(e1)∧ Γ(e2) = Γ(e1),
∀ e1,e2∈ E, we show that ((Γ,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 (Γ,E) is an ordered soft set and A,B⊆ E, such that θ : (F,A)→ (G,B)
is defined by θ(F(e1)) ={F(e2)∈ (F,A) : F(e1)⊆ F(e2),∀F(e1)∈ (F,A)},
then (F,A) is isomorphic to the range of θ 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
consistentcutandshowedthatthesliceofadistributedcomputingisuniquely defined for all predicate.
Description
A THESIS SUBMITTED TO THE
SCHOOL OF POST GRADUATE STUDIES,
AHMADU BELLO UNIVERSITY, ZARIA-NIGERIA,
IN PARTIAL FULFILMENT OF THE REQUIREMENTS
FOR THE AWARD OF DOCTOR OF PHILOSOPHY
IN MATHEMATICS
DEPARTMENT OF MATHEMATICS
AHMADU BELLO UNIVERSITY, ZARIA,
NIGERIA
Keywords
ALGEBRAICSTUDY,, SOFT LATTICE THEORY,, APPLICATIONS,, DISTRIBUTED COMPUTING SYSTEM,