Pointless Form of Rough Sets

Lecture Notes in Computer Science, 2010

The central notion of a rough set is indiscernibility based on equivalence relation. Since equivalence relation shows strong bondage in an equivalence class, it forms a Galois connection and the difference between upper and lower approximations is lost. We here introduce two different equivalence relations, the one for upper approximation, and the other for lower approximation, and construct composite approximation operator consisting of different equivalence relations. We show that a collection of fixed points with respect to the operator is a lattice, and that there exists a representation theorem for that construction. 1. Introduction This paper is written to make a difference between topological space and rough set theory [1, 2] clear in a term of lattice theory [3, 4]. Rough set provides a method for data analysis, based on the notion of indiscernibility which is defined by equivalence relation [5, 6]. Since equivalence classes can be analogously used as open sets, similar notions used in topological space can be defined. Upper and lower approximations in a rough set theory correspond to closure and internal set, respectively [7, 8]. On one hand, closure and internal set are defined under the constraint of a topological space (i.e., closed with respect to finite intersection and to any union). On the other hand, upper and lower approximation can be defined independent of such a kind of constraint. The essential difference between operations in a topological space and approximations in a rough set is the relationship between an element and a set (open set or equivalence class) containing the element. Any elements in an equivalence class have the same equivalence class, different from the case of topological space.

In this paper, we intend to study a connection between rough sets and lattice theory. We introduce the concepts of upper and lower rough ideals (filters) in a lattice. Then, we offer some of their properties with regard to prime ideals (filters), the set of all fixed points, compact elements, and homomorphisms.

2011

This paper deals with rough set approach on lattice theory. We represent the lattices for rough sets determined by an equivalence relation. Without any loss of generality, we have defined the rough set as a pair of sets (lower approximation set, upper approximation set) and then we showed that the collection of all rough sets of an approximations by an equivalence relation form a lattice by some order relation. In this paper we are able to deal with information sources in a set-theoretic manner. We also given an integrated approach to form lattices by choice function and lattice structure in rough set theory. The simple notion of this paper is to show the lattice structure in rough set theory by using indiscernible equivalence relation. Some important results are also proved. Finally, some examples are considered to illustrate the paper. c ©2011 World Academic Press, UK. All rights reserved.

Mathematics and Statistics, 2023

Rough sets are extensions of classical sets characterized by vagueness and imprecision. The main idea of rough set theory is to use incomplete information to approximate the concept of imprecision or uncertainty, or to treat ambiguous phenomena and problems based on observation and measurement. In Pawlak rough set model, equivalence relations are a key concept, and equivalence classes are the foundations for lower and upper approximations. Developing an algebraic structure for rough sets will allow us to study set theoretic properties in detail. Several researchers studied rough sets from an algebraic perspective and a number of structures have been developed in recent years, including rough semigroups, rough groups, rough rings, rough modules, and rough vector spaces. The purpose of this study is to demonstrate the usefulness of rough set theory in group theory. There have been several papers investigating the roughness in algebraic structures by substituting an algebraic structure for the universe set. In this paper, rough groups are defined using upper and lower approximations of rough sets from a finite universe instead of considering the whole universe. Here we have considered a finite universe Λ along with a relation χ which classifies the universe into equivalence classes. We have identified all rough sets with respect to this relation. The upper and lower approximated sets have been taken separately and these form a rough group equivalence relation (χ rog) and it partitions the group (2 Λ , △) into equivalence classes. In this paper, the rough group approximation space (2 Λ , χ rog) has been defined along with upper and lower approximations and properties of subsets of 2 Λ with respect to rough group equivalence relations have been illustrated.

Formalized Mathematics, 2009

Rough sets, developed by Pawlak, are an important tool to describe a situation of incomplete or partially unknown information. One of the algebraic models deals with the pair of the upper and the lower approximation. Although usually the tolerance or the equivalence relation is taken into account when considering a rough set, here we rather concentrate on the model with the pair of two definable sets, hence we are close to the notion of an interval set. In this article, the lattices of rough sets and intervals are formalized. This paper, being essentially the continuation of [6], is also a step towards the formalization of the algebraic theory of rough sets, as in [7] or [13].