Representation of Nelson algebras by rough sets determined by quasiorders

Järvinen, Jouni; Radeleczki, Sándor

doi:10.1007/s00012-011-0149-9

Cited by 33 publications

(42 citation statements)

References 15 publications

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“…Since ℘(U ) and ℘(U ) are algebraic and completely distributive lattices, R * is an algebraic and completely distributive lattice as shown in [10]. Also in [11], it is proved that R * is a Nelson algebra. Figure 2…”

Section: Umadevi / On the Completion Of Rough Sets System Determinmentioning

confidence: 91%

“…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%

“…Järvinen and his co-authors [8,10,11] have studied the ordered structure of the rough sets system determined by arbitrary binary relations. Following were their conclusions regarding the structure of rough sets system determined by various relations:…”

Section: Completions Of Rough Sets Systemmentioning

confidence: 99%

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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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Section: Umadevi / On the Completion Of Rough Sets System Determinmentioning

confidence: 91%

“…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%

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

Umadevi¹

2015

Fundamenta Informaticae

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“…Note that in [9], Jouni Järvinen and Sándor Radeleczki considered rough sets determined by quasiorders (reflexive and transitive binary relations), and showed that they form Nelson algebras.…”

Section: Lukasiewicz-moisil Algebrasmentioning

confidence: 99%

Hidden modalities in algebras with negation and implication

Kondo¹,

Järvinen²,

Mattila³

et al. 2013

Math. Appl.

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Abstract. Lukasiewicz 3-valued logic may be seen as a logic with hidden truthfunctional modalities defined by ♦A := ¬A → A and A := ¬(A → ¬A). It is known that axioms (K), (T), (B), (D), (S4), (S5) are provable for these modalities, and rule (RN) is admissible. We show that, if analogously defined modalities are adopted in Lukasiewicz 4-valued logic, then (K), (T), (D), (B) are provable, and (RN) is admissible. In addition, we show that in the canonical n-valued LukasiewiczMoisil algebras Ln, identities corresponding to (K), (T), and (D) hold for all n ≥ 3 and 1 = 1. We define analogous operations in residuated lattices and show that residuated lattices determine modal systems in which axioms (K) and (D) are provable and 1 = 1 holds. Involutive residuated lattices satisfy also the identity corresponding to (T). We also show that involutive residuated lattices do not satisfy identities corresponding to (S4) nor (S5). Finally, we show that in Heyting algebras, and thus in intuitionistic logic, ♦ and are equal, and they correspond to the double negation ¬¬.

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“…The algebraic structure of generalized rough approximations and rough sets were mostly studied by Jarvinen in [8,9]. Recently, he has investigated about the algebraic structure of the rough sets system determined by a quasi order(reflexive and transitive) [10,11]. With this motivation, in this paper we study about a different algebraic structure on the rough sets system determined by a quasi order.…”

Section: Introductionmentioning

confidence: 99%

Algebra of Rough Sets based on Quasi Order

Nagarajan¹,

Umadevi²

2013

Fundamenta Informaticae

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In this paper, we give a characterization theorem for rough sets based on quasi order. We obtain an algebra on the rough sets system determined by a quasi order which is the generalization of the algebra of rough sets system determined by an equivalence relation given in [1]. The properties of this algebra are abstracted at various levels and define new class of algebras. Further we give a representation theorem for the new class of algebras.

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Representation of Nelson algebras by rough sets determined by quasiorders

Cited by 33 publications

References 15 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

Hidden modalities in algebras with negation and implication

Algebra of Rough Sets based on Quasi Order

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