نظریه مجموعه راف اعمال شده برای نظریه ایده آل (فازی)

We use covers of the universal set to define 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 define 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 be brought to bear on the study of (fuzzy) ideals of a ring.

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 sets can 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]. In this paper, 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. Section 1 lays the ground work for our main results which appear in Section 2. In Section 2, 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 [6], and rough set theory can simultaneously be brought to bear on the study of (fuzzy) ideals of a ring. The notion of the (fuzzy) prime spectrum of a ring generalizes that of aDne varieties, where the study of polynomial equations occurs. The notion of a fuzzy subset is of course due to Zadeh [30], and a fuzzy substructure of an algebraic structure is due to Rosenfeld [21].