Generalized Ideals and Co-granular Rough Sets
03 Jul 2017-pp 23-42
TL;DR: This research will be of relevance for a number of logico-algebraic approaches to rough sets that proceed from point-wise definitions of approximations and also for using alternative approximation in spatial mereological contexts involving actual contact relations.
Abstract: Lattice-theoretic ideals have been used to define and generate non granular rough approximations over general approximation spaces over the last few years by few authors. The goal of these studies, in relation based rough sets, have been to obtain nice properties comparable to those of classical rough approximations. In this research paper, these ideas are generalized in a severe way by the present author and associated semantic features are investigated by her. Granules are used in the construction of approximations in implicit ways and so a concept of co-granularity is introduced. Knowledge interpretation associable with the approaches is also investigated. This research will be of relevance for a number of logico-algebraic approaches to rough sets that proceed from point-wise definitions of approximations and also for using alternative approximations in spatial mereological contexts involving actual contact relations. The antichain based semantics invented in earlier papers by the present author also applies to the contexts considered.
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TL;DR: This paper defines several measurements to compare the granularity of neighborhood granulations, and generates “OR” and “AND” decision rules based on multigranulation fusion strategies that are employed to make decisions in the presence of disease diagnosis problems.
Abstract: Multigranulation rough set over two universes provides a new perspective to combine multiple granulation knowledge in a multigranulation space in practical reality. Note that there are always non-essential neighborhood granulations, which would affect the efficiency and quality of decision making. Therefore, selecting valuable granulations and reducing worthless ones are necessary for the application of multigranulation rough set in decision process. In this paper, we first define several measurements to compare the granularity of neighborhood granulations, using which the granulation selection with multigranulation rough set is characterized. Then, the selection algorithms in the multigranulation space are developed. Third, we generate “OR” and “AND” decision rules based on multigranulation fusion strategies. As an application, these decision rules are employed to make decisions in the presence of disease diagnosis problems. In the end, the effectiveness and efficiency of the proposed algorithms are examined with numerical experiments on selective data sets.
19 citations
29 Jun 2020
TL;DR: This expository paper is intended to explain basic aspects of granular computing from a critical perspective, their range of applications and provide directions relative to general rough sets and related formal approaches to vagueness.
Abstract: A number of nonequivalent perspectives on granular computing are known in the literature, and many are in states of continuous development. Further related concepts of granules and granulations may be incompatible in many senses. This expository paper is intended to explain basic aspects of these from a critical perspective, their range of applications and provide directions relative to general rough sets and related formal approaches to vagueness. General granular principles related to knowledge are also mentioned.
9 citations
Cites background from "Generalized Ideals and Co-granular ..."
...In the present author’s classification, a rough approximation operator may be granular (in the axiomatic sense), co-granular, pointwise, abstract or empirical [31]....
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Posted Content•
TL;DR: Higher order versions of granular operator spaces and variants are presented uniformly as partial algebraic systems for generalizing Skowron-Polkowski style of rough mereology and further algorithms grounded in mereological nearness, suited for decision-making in human-machine interaction contexts, are proposed.
Abstract: Granular operator spaces and variants had been introduced and used in theoretical investigations on the foundations of general rough sets by the present author over the last few years. In this research, higher order versions of these are presented uniformly as partial algebraic systems. They are also adapted for practical applications when the data is representable by data table-like structures according to a minimalist schema for avoiding contamination. Issues relating to valuations used in information systems or tables are also addressed. The concept of contamination introduced and studied by the present author across a number of her papers, concerns mixing up of information across semantic domains (or domains of discourse). Rough inclusion functions (\textsf{RIF}s), variants, and numeric functions often have a direct or indirect role in contaminating algorithms. Some solutions that seek to replace or avoid them have been proposed and investigated by the present author in some of her earlier papers. Because multiple kinds of solution are of interest to the contamination problem, granular generalizations of RIFs are proposed, and investigated. Interesting representation results are proved and a core algebraic strategy for generalizing Skowron-Polkowski style of rough mereology (though for a very different purpose) is formulated. A number of examples have been added to illustrate key parts of the proposal in higher order variants of granular operator spaces. Further algorithms grounded in mereological nearness, suited for decision-making in human-machine interaction contexts, are proposed by the present author. Applications of granular \textsf{RIF}s to partial/soft solutions of the inverse problem are also invented in this paper.
5 citations
Cites background from "Generalized Ideals and Co-granular ..."
...The reader may refer to [40, 12] for more on the following definition....
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01 Jan 2018
TL;DR: In this research chapter, dualities and representations of various kinds associated with the semantics of rough sets are explained, critically reviewed, new proofs have been proposed, open problems are specified and new directions are suggested.
Abstract: In this research chapter, dualities and representations of various kinds associated with the semantics of rough sets are explained, critically reviewed, new proofs have been proposed, open problems are specified and new directions are suggested. Some recent duality results in the literature are also adapted for use in rough contexts. New results are also proved on granular connections between generalized rough and L-fuzzy sets by the present author. Philosophical aspects of the concepts have also been considered by her in this research chapter.
3 citations
01 Jan 2022
1 citations
References
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01 Jan 2011
TL;DR: Rough Mereology as mentioned in this paper is a tool for reasoning under uncertainty, formulated in terms of parts by Lesniewski, and borrows from Fuzzy Set Theory and Rough Set Theory ideas of the containment to a degree.
Abstract: The monograph offers a view on Rough Mereology, a tool for reasoning under uncertainty, which goes back to Mereology, formulated in terms of parts by Lesniewski, and borrows from Fuzzy Set Theory and Rough Set Theory ideas of the containment to a degree. The result is a theory based on the notion of a part to a degree. One can invoke here a formula Rough: Rough Mereology : Mereology = Fuzzy Set Theory : Set Theory. As with Mereology, Rough Mereology finds important applications in problems of Spatial Reasoning, illustrated in this monograph with examples from Behavioral Robotics. Due to its involvement with concepts, Rough Mereology offers new approaches to Granular Computing, Classifier and Decision Synthesis, Logics for Information Systems, and are--formulation of well--known ideas of Neural Networks and Many Agent Systems. All these approaches are discussed in this monograph. To make the exposition self--contained, underlying notions of Set Theory, Topology, and Deductive and Reductive Reasoning with emphasis on Rough and Fuzzy Set Theories along with a thorough exposition of Mereology both in Lesniewski and Whitehead--Leonard--Goodman--Clarke versions are discussed at length. It is hoped that the monograph offers researchers in various areas of Artificial Intelligence a new tool to deal with analysis of relations among concepts.
77 citations
TL;DR: The concepts of upper and lower rough ideals (filters) in a lattice are introduced and some of their properties with regard to prime ideals, the set of all fixed points, compact elements, and homomorphisms are offered.
77 citations
"Generalized Ideals and Co-granular ..." refers background in this paper
...Concepts of rough ideals have also been studied by different authors in specific algebras (see for example [12,13])- these studies involve the use of rough concepts within algebras....
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TL;DR: The three types of Yao's lower and upper approximations of any set with respect to any similarity relation are investigated and a comparison between these types is given.
73 citations
"Generalized Ideals and Co-granular ..." refers background in this paper
...Set theoretic generalizations of the approach in [4,5,6] are proposed in this section by the present author....
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...Among the latter class, few new approximations have been studied in [4,5,6] over general approximation spaces of the form (X,R) with X being a set and R being at least a reflexive relation....
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...The significance of the obtained results and potential application contexts are not explored in the three papers mentioned [4,5,6] in sufficient detail and many open problems remain hidden....
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...In the relational approximation contexts of [4,5,6], subsets of generalized zeros are generalized zeros....
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...The approximations in [6] are more general than the ones introduced and studied in [4,5]....
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01 Jan 1971
TL;DR: In this paper, a theory of pseudo-complements for posets (partially ordered sets) is developed, where the concepts of ideal and semi-ideal are introduced and a few results about them are obtained.
Abstract: In this paper a theory of pseudo-complements is developed for posets (partially ordered sets). The concepts of ideal and semi-ideal are introduced for posets and a few results about them are obtained. These results together with known results about pseudo-complements in distributive lattices lead to the main results. It is proved that if in a pseudo-complemented semilattice or dual semilattice every element is normal, then it is a Boolean algebra. Using this result new proofs for two known theorems are obtained. The existence of maximal ideals in posets is established and it is shown that the dual ideal of dense elements of a poset with 0 is the product of all the maximal dual ideals. Already, there exists a theory of pseudo-complements for lattices. Frink [5 ] has obtained a generalisation of the theory for semilattices. In this paper we extend some of the results of Frink [5] and Balachandran [1] to posets (partially ordered sets). We obtain these extensions by using the concept of semi-ideal, which we define in ?2. This paper consists of three sections. In ?1 we summarise some known results which we use in later sections. ?2 deals with some of the properties of semi-ideals and ideals in posets. Our definition of poset ideal is different from that introduced by Frink [4]; however in a lattice our definition is equivalent to the usual definition. Using the results obtained in ?2, we develop a theory of pseudo-complements for posets in ?3. 1. Preliminaries. We shall denote the ordering relation in a poset by ieI ai and rliET ai respectively. When A is finite, say, A = { a,, a2, * , an }, the lattice-sum and the lattice-product of the as are denoted by al+a2+ * . . +a. and a,*. a. respectively. The Received by the editors February 3, 1970. AMS 1969 subject classifications. Primary 0620; Secondary 0630, 0635.
44 citations
"Generalized Ideals and Co-granular ..." refers background in this paper
...The original motivations for the approach relate to the strategies for generalizing the concept of lattice ideal to partially ordered sets (see [19,20,21])....
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01 Jan 2008
TL;DR: In this article, some methods for generating topologies are obtained using binary relations and the relationship between these methods are discussed, and several examples are given to indicate counter connections, and a quasi-discrete topology from a symmetric relation instead of an equivalence relation is obtained.
Abstract: Relation on a set is a simple mathematical model to which many real-life data can be connected. In fact, topological structures are generalized methods for measuring similarity and dissimilarity between objects in the uni- verses. In this work, some methods for generating topologies are obtained using binary relations. The relationship between these methods are discussed. We also investigate some properties of these topologies. Moreover, we obtain a quasi-discrete topology from a symmetric relation instead of an equivalence relation. Finally, several examples are given to indicate counter connections.
42 citations
"Generalized Ideals and Co-granular ..." refers background in this paper
...Set theoretic generalizations of the approach in [4,5,6] are proposed in this section by the present author....
[...]
...Among the latter class, few new approximations have been studied in [4,5,6] over general approximation spaces of the form (X,R) with X being a set and R being at least a reflexive relation....
[...]
...The significance of the obtained results and potential application contexts are not explored in the three papers mentioned [4,5,6] in sufficient detail and many open problems remain hidden....
[...]
...In the relational approximation contexts of [4,5,6], subsets of generalized zeros are generalized zeros....
[...]
...The approximations in [6] are more general than the ones introduced and studied in [4,5]....
[...]