A Method of Representing Rough Sets System Determined by Quasi Orders

Nagarajan, E. K. R.; Umadevi, D.

doi:10.1007/s11083-011-9245-x

Cited by 9 publications

(6 citation statements)

References 22 publications

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“…For the definitions of some algebras not given here like regular double Stone algebra, Kleene algebra, Nelson algebra, etc., the readers are asked to refer the literature [5,11,13,14].…”

Section: Theorem 23 [5]mentioning

confidence: 99%

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

Umadevi¹

2015

Fundamenta Informaticae

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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}.

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“…For the definitions of some algebras not given here like regular double Stone algebra, Kleene algebra, Nelson algebra, etc., the readers are asked to refer the literature [5,11,13,14].…”

Section: Theorem 23 [5]mentioning

confidence: 99%

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

Umadevi¹

2015

Fundamenta Informaticae

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“…Not only they are useful in data-mining (cf. [5]), but in this case the construction of rough set systems assumes an unexpected amazing meaning (see [7]) (on the topic, see also [14]). In fact, if P is a partial order upper bounded by a set M of maximal elements (always if it is finite), then for all m ∈ M, ↑ m is a singleton definable set.…”

Section: And 83); the Fact That The Prime Ideals Of A P−algebra Lie mentioning

confidence: 99%

Rough Sets and Other Mathematics: Ten Research Programs

Pagliani¹

2015

Springer Proceedings in Mathematics &Amp; Statistics

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Since its inception, interesting connections between Rough Set Theory and different mathematical and logical topics have been investigated. This paper is a survey of some less known although highly interesting connections, which extend from Rough Set Theory to other mathematical and logical fields. The survey is primarily thought of as a guide for new directions to be explored.Keywords Rough sets · Algebraic logic · Topology Information from Data and Information as MetaphorAs is well known, the starting point of Rough Set Theory is an indiscernible space U, E , where U is a set and E ⊆ U × U is an equivalence relation such that x, y ∈ E states that items x and y take exactly the same attribute-values according to an evaluation recorded in an Information System.Given any relational structure U, R , with R ⊆ U × U , and X ⊆ U , the set R(X ) = {y : ∃x ∈ X ( x, y) ∈ R)} will by named the R-neighborhood of X. If X = {a} we shall write R(a). Thus, by means of E−neighborhoods, from any indiscernibility space the following operators are defined on ℘ (U ):is called an approximation space. Any equivalence class modulo E is a neighborhood E(a) for some a ∈ U , and represents a "basic property," that is, a unique array of attribute-values, hence a subset of U definable by means of the given evaluation. Moreover, forany X , (l E)(X ) and P. Pagliani

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“…Rough sets induced by quasiorders are in the focus of current interest; see [11][12][13]16], for example. Let us denote by ℘(U ) the power set of U , that is, the set of all subsets of U .…”

Section: Rough Setsmentioning

confidence: 99%

Monteiro Spaces and Rough Sets Determined by Quasiorder Relations: Models for Nelson algebras

Järvinen¹,

Radeleczki

2014

Fundamenta Informaticae

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The theory of rough sets provides a widely used modern tool, and in particular, rough sets induced by quasiorders are in the focus of the current interest, because they are strongly interrelated with the applications of preference relations and intuitionistic logic. In this paper, a structural characterisation of rough sets induced by quasiorders is given. These rough sets form Nelson algebras defined on algebraic lattices. We prove that any Nelson algebra can be represented as a subalgebra of an algebra defined on rough sets induced by a suitable quasiorder. We also show that Monteiro spaces, rough sets induced by quasiorders and Nelson algebras defined on T 0 -spaces that are Alexandrov topologies can be considered as equivalent structures, because they determine each other up to isomorphism.

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A Method of Representing Rough Sets System Determined by Quasi Orders

Cited by 9 publications

References 22 publications

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

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

Rough Sets and Other Mathematics: Ten Research Programs

Monteiro Spaces and Rough Sets Determined by Quasiorder Relations: Models for Nelson algebras

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