Abstract
The formal content of near set theory can be summarised in terms of three concepts: a perceptual system, a nearness relation and a near set. Perceptual systems and different forms of nearness relations have been already successfully related to important mathematical structures (e.g., approach spaces) and described in the frameworks of general topology and category theory. However, since near sets actually do not form any regular structure, there is lack of similar results about the concept of a near set. The main goal of this paper is to fill this gap and provide a mathematical basis for near sets on a similar abstract level as it has been already done for nearness relations. However, we take in the paper a down-to-earth approach; that is, instead of seeking the richest mathematical structures which express our intuitions (call it up-to-sky approach), we present in the paper the simplest category theoretic structures which, however, are rich enough to convey our ideas. Thus, although many times we actually deal with sheaves, we speak about them as pre-sheaves. The main reason is that a sheaf (in contrast to a pre-sheaf) embodies the idea of gluing (in the very similar way like manifolds, which are obtainable by gluing open subsets of Euclidean space), which is irrelevant to the study of near sets and rough sets. Furthermore, the concept a pre-sheaf is much simpler than a sheaf and should be easily “digested” by readers who are not very familiar with category theory.
Highlights
In these days of “soft mathematics”—or, according to its more sophisticated name, “soft computing paradigm”— intuitive and simple concepts of set and set membership have been extended to more complex mathematical objects such as fuzzy sets (e.g., [1]) or rough sets (e.g., [2])
Our idea is expressible by a concept of a pre-sheaf, that is a contravariant functor from a topological space to the category of sets. (It is worth a mention that here or there we deal with richer structures than pre-sheaves, that is, sheaves; a sheaf— to a pre-sheaf—embodies the idea of gluing, which is irrelevant to the study of near sets and rough sets, and would make definitions quite complex.) Firstly, a perceptual system is a simple example of a pre-sheaf, which is a sub-pre-sheaf of the most elementary sheaf considered in the literature
A lower approximation operator may be naturally converted into a pre-sheaf, whereas a near set would be an analogue of an upper approximation operator; near set theory may be considered as a kind of pre-sheafification of rough set theory
Summary
In these days of “soft mathematics”—or, according to its more sophisticated name, “soft computing paradigm”— intuitive and simple concepts of set and set membership have been extended to more complex mathematical objects such as fuzzy sets (e.g., [1]) or rough sets (e.g., [2]). Pre-sheaves may be viewed as a well-defined methodological perspective from which near set theory deals with data tables and concepts derivable from them. Under this view, a lower approximation operator may be naturally converted into a pre-sheaf, whereas a near set would be an analogue of an upper approximation operator; near set theory may be considered as a kind of pre-sheafification of rough set theory
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