On the Completion of Rough Sets System Determined by Arbitrary Binary Relations

Abstract

In this paper, a solution is given to the problem proposed by Järvinen in [8]. A smallest completion of the rough sets system determined by an arbitrary binary relation is given. This completion, in the case of a quasi order, coincides with the rough sets system which is a Nelson algebra. Further, the algebraic properties of this completion has been studied.is a completion of (R * , ≤).In 1987, Gehrke and Walker [6] showed that the structure of the rough sets system determined by an equivalence relation R on U is isomorphic to 2 I × 3 J , where I = {R(x) | |R(x)| = 1} and J = {R(x) | |R(x)| > 1}. 2 I is the set of maps from I to a two element chain and 3 J is the set of maps from J to a three element chain. For reflexive relations R on U , the rough sets system determined by R can be order embed in (2 I × 3 J , ≤). Proposition 3.2. [8] If R is a reflexive relation on a set U , then (2 I ×3 J , ≤) is a completion of (R * , ≤), where I = {R(x) | |R(x)| = 1} and J = {R(x) | |R(x)| > 1}.

Rough sets determined by tolerances

“…In the case of rough sets induced by arbitrary binary relations, we knew quite a little about their structure. Practically only the results presented in [9] about the completion DM(RS) were known. In this work, we have extended this knowledge in case of a reflexive relation by showing that DM(RS) forms a paraorthomodular lattice.…”

“…We denote the Dedekind-MacNeille completion of RS by DM(RS). Umadevi [9] has proved that for any binary relation R on U ,…”

“…Umadevi [9] mentioned without proof that for any binary relation, DM(RS) forms a pseudo-Kleene algebra. For the sake of completeness, we write the following lemma.…”

“…However, it is known that for an arbitrary tolerance, the ordered set RS is not necessarily a lattice; see [8], for instance. In [9], D. Umadevi presented the Dedekind-MacNeille completion of RS for arbitrary binary relations. In this work, we denote this completion by DM(RS).…”

“…Note that Kleene algebras are distributive pseudo-Kleene algebras. Umadevi noted in [9] that DM(RS) forms a pseudo-Kleene algebra. In the literature [10,11] can be found several studies in which approximation operators are defined by using certain combinations of two or more equivalence relations, meaning that concepts are described by using many granulation spaces.…”