A certain conception of the calculus of rough sets.

Bonikowski, Zbigniew

doi:10.1305/ndjfl/1093634405

Cited by 48 publications

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“…By defining minimal upper sample and selection function respectively, Bonikowski and Zhang et al [3,25] successfully defined intersection and union operations on rough sets. However, proper complement cannot be defined for every rough set.…”

Section: Classic Rough Sets Algebramentioning

confidence: 99%

“…In fact, except the fundamental sets, only pseudo complement can be defined for each rough set. In this way, rough sets formed a complete atomic Stone algebra [3]. In addition, taking advantage of relative pseudo complement operation between two rough sets, Pagliani [12] proved that rough sets system constituted a Heyting algebra under disjoint representation and was isomorphic to a semi-simple Nelson algebra.…”

Section: Classic Rough Sets Algebramentioning

confidence: 99%

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A Triple Structure of Rough Sets Based on Selection Function

2015

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Different from rough sets in Pawlak's sense, which is a binary approximation operations based structure, in this paper, we propose a new rough equivalence relation based on triple approximation operations induced by selection function. The same as traditional rough sets research, we consider the algebra issue of new rough sets system and construct lattice structure in an algebraic fashion. An isomorphic relationship is studied between proposed rough sets algebra structure and that in Pawlak's sense. In addition, some examples are shown and studied in order to indicate the effectiveness of new equivalence relation in distinguishing and describing subsets of universe. Besides, we also study restriction of selection congruence on subalgebra built by covering of universe. Finally, conclusion on the axioms of middle approximation operation are shown to clarify properties of new structure.

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Section: Classic Rough Sets Algebramentioning

confidence: 99%

Section: Classic Rough Sets Algebramentioning

confidence: 99%

A Triple Structure of Rough Sets Based on Selection Function

2015

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“…R * (A) and R * (A) are made of unions of equivalence classes, hence depend on the partition chosen, hence on the observer. Bonikowski in Bonikowski (1992) has proved that, given two subsets A and B of S there exists another subset C of S such that…”

Section: The Algebraic Structure Of Approximation Spacesmentioning

confidence: 99%

Algebraic models of deviant modal operators based on de Morgan and Kleene lattices

Cattaneo

Ciucci

Dubois

2011

Information Sciences

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Abstract. An algebraic model of a kind of modal extension of de Morgan logic is described under the name MDS5 algebra. The main properties of this algebra can be summarized as follows: (1) it is based on a de Morgan lattice, rather than a Boolean algebra; (2) a modal necessity operator that satisfies the axioms N , K, T , and 5 (and as a consequence also B and 4) of modal logic is introduced; it allows one to introduce a modal possibility by the usual combination of necessity operation and de Morgan negation; (3) the necessity operator satisfies a distributivity principle over joins. The latter property cannot be meaningfully added to the standard Boolean algebraic models of S5 modal logic, since in this Boolean context both modalities collapse in the identity mapping. The consistency of this algebraic model is proved, showing that usual fuzzy set theory on a universe U can be equipped with a MDS5 structure that satisfies all the above points (1)- (3), without the trivialization of the modalities to the identity mapping. Further, the relationship between this new algebra and Heyting-Wajsberg algebras is investigated. Finally, the question of the role of these deviant modalities, as opposed to the usual nondistributive ones, in the scope of knowledge representation and approximation spaces is discussed.

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“…In some cases, only a fragment of logical studies connected with various rough set models is indicated in the literature [28,2,37,4,7,20,11,5,30,15,14,38,24,40,16,17,22] mostly from algebraic perspective. We shall work in the domain of covering-based rough sets.…”

Section: Introductionmentioning

confidence: 99%

Covering-based rough sets and modal logics. Part I

Chakraborty

2016

International Journal of Approximate Reasoning

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A certain conception of the calculus of rough sets.

Cited by 48 publications

References 0 publications

A Triple Structure of Rough Sets Based on Selection Function

A Triple Structure of Rough Sets Based on Selection Function

Algebraic models of deviant modal operators based on de Morgan and Kleene lattices

Covering-based rough sets and modal logics. Part I

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