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Categories and Algebras from Rough Sets : New Facets

More A. K., Banerjee M.

Fundamenta Informaticae

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2016

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Vol. 148, nr 1/2

173--190

EN

Rough sets are investigated from the viewpoint of topos theory. Two categories RSC and ROUGH of rough sets and a subcategory ξ-RSC are focussed upon. It is shown that RSC and ROUGH are equivalent. Generalizations RSC(C ) and ξ-RSC(C ) are proposed over an arbitrary topos C. RSC(C ) is shown to be a quasitopos, while ξ-RSC(C ) forms a topos in the special case when C is Boolean. An example of RSC(C ) is given, through which one is able to define monoid actions on rough sets. Next, the algebra of strong subobjects of an object in RSC is studied using the notion of relative rough complementation. A class of contrapositionally complemented 'c.V.c' lattices is obtained as a result, from the object class of RSC. Moreover, it is shown that such a class can also be obtained if the construction is generalized over an arbitrary Boolean algebra.

2

Algebras of Definable and Rough Sets in Quasi Order-based Approximation Spaces

Kumar A., Banerjee M.

Fundamenta Informaticae

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2015

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Vol. 141, nr 1

37--55

EN

A pair of approximation operators, based on the notion of granules in generalized approximation spaces, was studied in an earlier work by the authors. In this article, we investigate algebraic structures formed by the definable sets and also by the rough sets determined by this pair of approximation operators. The definable sets are open sets of an Alexandrov topological space, and form a completely distributive lattice in which the set of completely join irreducible elements is join dense. The collection of rough sets also forms a similar structure. Representation results for such classes of completely distributive lattices as well as Heyting algebras in terms of definable and rough sets are obtained. Further, two unary operators on rough sets are considered, making the latter constitute a structure that is named a ‘rough lattice’. Representation results for rough lattices are proved.

3

Rough Sets : Some Foundational Issues

Chakraborty M.K., Banerjee M.

Fundamenta Informaticae

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2013

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Vol. 127, nr 1-4

1--15

EN

The article analyses prevalent definitions of rough sets from the foundational and mathematical perspectives. In particular, the issue of language dependency in the definitions, and implications of the definitions on the issue of vagueness are discussed in detail.

4

Rough Dialogue and Implication Lattices

Chakraborty M.K., Banerjee M.

Fundamenta Informaticae

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2007

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Vol. 75, nr 1-4

123-139

EN

A dialogue is an `activity' by a pair of agents to arrive at some kind of understanding over a concept/belief/piece of information etc. represented by a subset (the extension) in some universe of discourse. The universe is partitioned into two different sets of granules (equivalence classes) representing the perceptions of the agents. So, there are two approximation spaces at the beginning. A third approximation space arises out of superimposition of the two partitions. A dialogue is a finite process of gradual enhancement of the two base subsets of the agents, in their `common' approximation space. Through this process, various kinds of overlap may emerge between the two final subsets. A first introduction of the idea of a dialogue in rough context was made in [6]. This paper further develops the notion and focusses upon the study of the above-mentioned overlaps in a systematic manner. Given two sets A and B in an approximation space, there are nine possible inclusion relations among the sets lo(A), A, up(A), lo(B), B and up(B) where lo and up denote the lower and upper approximation operators respectively. There are five resulting equivalence classes and the quotient set forms a lattice by implication ordering. That is, of the nine relations, only five are independent and they form an implication or entailment lattice. Starting with this basic lattice other implication lattices are formed. Relationship of these lattices with the various overlap conditions between the final pair of sets arrived at after a dialogue is studied. Finally, examples are given, one of which relates dialogues in rough context with rough belief revision [3] - in a line similar to the approach of [5].

5

Logic for Rough Truth

Banerjee M.

Fundamenta Informaticae

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2006

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Vol. 71, nr 2,3

139-151

EN

Pawlak had proposed the notion of rough truth in 1987 [16]. The article takes a fresh look at this ``soft'' truth, and presents a formal system LR, that is shown to be sound and complete with respect to a semantics determined by this notion. LR is based on the modal logic S5. Notable is the rough consequence relation defining LR (a first version introduced in [9]), and rough consistency (also introduced in [9]), used to prove the completeness result. The former is defined in order to be able to derive roughly true propositions from roughly true premisses in an information system. The motivation for the latter stems from the observation that a proposition and its negation may well be roughly true together. A characterization of LR-consequence shows that the paraconsistent discussive logic J of Ja\'skowski is equivalent to LR. So, LR, developed from a totally independent angle, viz. that of rough set theory, gives an alternative formulation to this well-studied logic. It is further observed that pre-rough logic [3] and 3-valued ukasiewicz logic are embeddable into LR.