25-05-2012, 03:43 PM
Rough set theory applied to (fuzzy) ideal theory
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Abstract
We use covers of the universal set to de-ne approximation operators on the power set of the given set. In Section 1, we
determine basic properties of the upper approximation operator and show how it can be used to give algebraic structural
properties of certain subsets. We de-ne a particular cover on the set of ideals of a commutative ring with identity in such a
way that both the concepts of the (fuzzy) prime spectrum of a ring and rough set theory can simultaneously.
Introduction
In 1982, Pawlak introduced the concept of a rough
set [18]. This concept is fundamental to the examination
of granularity in knowledge. It is a concept which
has many applications in data analysis. The idea is
to approximate a subset of a universal set by a lower
approximation and an upper approximation in the following
manner. A partition of the universe is given.
The lower approximation is the union of those members
of the partition contained in the given subset and
the upper approximation is the union of those
members of the partition which have a nonempty
intersection with the given subset. It is well known
that a partition induces an equivalence relation on
a set and vice versa. The properties of rough setscan thus be examined via either partitions or equivalence
relations. The members of the partition (or
equivalence classes) can be formally described by
unary set-theoretic operators [27], or by successor
functions for upper approximation spaces [7,8]. This
axiomatic approach allows not only for a wide range
of areas in mathematics to fall under this approach,
but also a wide range of areas to be used to describe
rough sets. Some examples are topology, (fuzzy)
abstract algebra, (fuzzy) directed graphs, (fuzzy) --
nite state machines, modal logic, interval structures
[7,14,15,17,19,27–29]. One may generalize the use
of partitions or equivalence relations to that of covers
or relations [17,20,22,24,25,29].
Upper and lower approximations dened by
covers
Let V be nonempty set and let P(V) denote the
power set of V. Let s be a function of P(V) into itself.
We are interested in the following conditions on
s since they are the ones that hold for upper approximation
operators de-ned via an equivalence relation:
For further research projects, one might examine
Theorem 2.11 and Corollary 2.12 for L=[0; 1] since
L in this case is the lattice most often used and for
which most of the applications exist. Also, we know
that the closed subsets of FP in the Zariski topology
are exactly theFP{I}, where I ∈FI. Because of the
importance of the FP{I} ∪{I} in the above development,
their topological properties may be of interest.
The ultimate goal is to further develop the ideas initiated
here and apply them to the study of fuzzy intersection
equations. A possible start is to examine the
examples of fuzzy intersection equations given and
studied in [13].