Representing Regular Pseudocomplemented Kleene Algebras by Tolerance-Based Rough sets

Järvinen, Jouni; Radeleczki, Sándor

doi:10.1017/s1446788717000283

Cited by 15 publications

(7 citation statements)

References 20 publications

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“…On the other hand, some three-valued algebraic structures have been represented as rough sets generated by one covering [57,59,58,4,37]. Our results are more general, since many-valued algebraic structures correspond to sequences of rough sets determined by a sequence of coverings.…”

Section: Tab 11: Structures and Operation Between Orthopairsmentioning

confidence: 85%

Sequences of Refinements of Rough Sets: Logical and Algebraic Aspects

Boffa

Gerla

2020

Lecture Notes in Computer Science

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In this thesis, a generalization of the classical Rough set theory [81] is developed considering the so-called sequences of orthopairs that we define in [16] as special sequences of rough sets.Mainly, our aim is to introduce some operations between sequences of orthopairs, and to discover how to generate them starting from the operations concerning standard rough sets (defined in [29]). Also, we prove several representation theorems representing the class of finite centered Kleene algebras with the interpolation property [28], and some classes of finite residuated lattices (more precisely, we consider Nelson algebras [86], Nelson lattices [21], IUML-algebras [69] and Kleene lattice with implication [24]) as sequences of orthopairs. I am deeply thankful to my family: my parents, my sister Rosa, my brother Angelo and his girlfriend Roberta, for their unconditional love and spiritual support in all my life. v vii "I magination is the Discovering Faculty, pre-eminently. It is that which penetrates into the unseen worlds around us, the worlds of Science. It is that which feels & discovers what is, the real which we see not, which exists not for our senses. Those who have learned to walk on the threshold of the unknown worlds, by means of what are commonly termed par excellence the exact sciences, may then with the fair white wings of Imagination hope to soar further into the unexplored amidst which we live.

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Section: Tab 11: Structures and Operation Between Orthopairsmentioning

confidence: 85%

Sequences of Refinements of Rough Sets: Logical and Algebraic Aspects

Boffa

Gerla

2020

Lecture Notes in Computer Science

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“…In addition, we proved [9, Theorem 4.8] that RS (T ) is a completely distributive lattice if and only if T is induced by an irredundant covering. We also showed in [10] that if T is a tolerance induced by an irredundant covering, then RS (T ) forms a regular double pseudocomplemented lattice such that for any (A, B) ∈ RS (T ),…”

Section: Introductionmentioning

confidence: 89%

The structure of multigranular rough sets

Järvinen,

Radeleczki

2020

Preprint

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“…It is proved in [6] that when T is a tolerance induced by an irredundant covering of U , RS (T ) is a regular double p-algebra. Recall from [8], for example, that an algebra (L, ∨, ∧, * , + , 0, 1) is called a double p-algebra if (L, ∨, ∧, 0, 1) is a bounded lattice such that * is the pseudocomplement operation and + is the dual pseudocomplement operation on L. Note that this means that for all a ∈ A Boolean lattice is a bounded distributive lattice L such that each element a ∈ L has a complement a ′ which satisfies a ∧ a ′ = 0 and a ∨ a ′ = 1.…”

Section: Further Properties Of Rs (E T )mentioning

confidence: 99%

“…Obviously, each equivalence E on U is such that its equivalence classes U/E form an irredundant covering of U and that the "tolerance" induced by U/E is E. Tolerances induced by an irredundant covering of U play an important role in Section 4. It is known (see [6,7]) that if T is a tolerance induced by an irredundant covering, then this covering is {T (x) | T (x) is a block}. Lemma 2.13.…”

Section: Tolerances Induced By Coveringsmentioning

confidence: 99%

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Defining rough sets using tolerances compatible with an equivalence

Järvinen

Kovács

Radeleczki

2019

Information Sciences

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We consider tolerances T compatible with an equivalence E on U , meaning that the relational product E •T is included in T . We present the essential properties of E-compatible tolerances and study rough approximations defined by such E and T . We consider rough set pairs (XE, X T ), where the lower approximation XE is defined as is customary in rough set theory, but X T allows more elements to be possibly in X than X E . Motivating examples of E-compatible tolerances are given, and the essential lattice-theoretical properties of the ordered set of rough sets {(XE , X T ) | X ⊆ U } are established.

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Representing Regular Pseudocomplemented Kleene Algebras by Tolerance-Based Rough sets

Abstract: We show that any regular pseudocomplemented Kleene algebra defined on an algebraic lattice is isomorphic to a rough set Kleene algebra determined by a tolerance induced by an irredundant covering.

Cited by 15 publications

References 20 publications

Sequences of Refinements of Rough Sets: Logical and Algebraic Aspects

Sequences of Refinements of Rough Sets: Logical and Algebraic Aspects

The structure of multigranular rough sets

Defining rough sets using tolerances compatible with an equivalence

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