A classification study of rough sets generalization

In the development of rough set theory, many different interpretations and formulations have been proposed and studied. One can classify the studies of rough sets into algebraic and constructive approaches. While algebraic studies focus on the axiomatization of rough set algebras, the constructive studies concern with the construction of rough set algebras from other well known mathematical concepts and structures. The constructive approaches are particularly useful in the real applications of rough set theory. The main objective of this thesis to provide a systematic review existing works on constructive approaches and to present some additional results. Both constructive and algebraic approaches are first discussed with respect to the classical rough set model. In particular, three equivalent constructive definitions of rough set approximation operators are examined. They are the element based, the equivalence class based, and the subsystem based definitions. Based on the element based and subsystem based definitions, generalized rough set models are reviewed and summarized. One can extend the element based definition by using any binary relations instead of equivalence relations in the classical rough set model. Many classes of rough set models can be established based on the properties of binary relations. The subsystem based definition can be extended in the set-theoretical setting, which leads to rough set models based on Pawlak approximation space, topological space, and closure system. Finally, the connections between the algebraic studies, relation based, and subsystem based formulations are established

On the Lattice of Intervals and Rough Sets

Rough sets, developed by Pawlak [6], 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 [3], is also a step towards the formalization of the algebraic theory of rough sets, as in [4] or [9].Grabowski Adam - Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok, PolandJastrzębska Magdalena - Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok, PolandGrzegorz Bancerek. Complete lattices. Formalized Mathematics, 2(5):719-725, 1991.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Adam Grabowski. Basic properties of rough sets and rough membership function. Formalized Mathematics, 12(1):21-28, 2004.Amin Mousavi and Parviz Jabedar-Maralani. Relative sets and rough sets. Int. J. Appl. Math. Comput. Sci., 11(3):637-653, 2001.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Z. Pawlak. Rough sets. International Journal of Parallel Programming, 11:341-356, 1982, doi:10.1007/BF01001956.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Y. Y. Yao. Interval-set algebra for qualitative knowledge representation. Proc. 5-th Int. Conf. Computing and Information, pages 370-375, 1993.Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215-222, 1990

Representation of Nelson Algebras by Rough Sets Determined by Quasiorders

In this paper, we show that every quasiorder $R$ induces a Nelson algebra $\mathbb{RS}$ such that the underlying rough set lattice $RS$ is algebraic. We note that $\mathbb{RS}$ is a three-valued {\L}ukasiewicz algebra if and only if $R$ is an equivalence. Our main result says that if $\mathbb{A}$ is a Nelson algebra defined on an algebraic lattice, then there exists a set $U$ and a quasiorder $R$ on $U$ such that $\mathbb{A} \cong \mathbb{RS}$.Comment: 16 page

Covering rough sets based on neighborhoods: An approach without using neighborhoods

Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as neighborhood have been applied to covering rough sets. In this paper, we further investigate the covering rough sets based on neighborhoods by approximation operations. We show that the upper approximation based on neighborhoods can be defined equivalently without using neighborhoods. To analyze the coverings themselves, we introduce unary and composition operations on coverings. A notion of homomorphismis provided to relate two covering approximation spaces. We also examine the properties of approximations preserved by the operations and homomorphisms, respectively.Comment: 13 pages; to appear in International Journal of Approximate Reasonin