Generalized Ideals and Co-granular Rough Sets

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.

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.

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.

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.

Abstract: In relational approach to general rough sets, ideas of directed relations are supplemented with additional conditions for multiple algebraic approaches in this research paper. The relations are also specialized to representations of general parthood that are upper-directed, reflexive and antisymmetric for a better behaved groupoidal semantics over the set of roughly equivalent objects by the first author. Another distinct algebraic semantics over the set of approximations, and a new knowledge interpretation are also invented in this research by her. Because of minimal conditions imposed on the relations, neighborhood granulations are used in the construction of all approximations (granular and pointwise). Necessary and sufficient conditions for the lattice of local upper approximations to be completely distributive are proved by the second author. These results are related to formal concept analysis. Applications to student centered learning and decision making are also outlined.

Abstract: The theory of rough sets is an extension of set theory with two additional unary set-theoretic operators defined based on a binary relation on the universe. These two operators are related to the modal operators in modal logics. By exploring the relationship between rough sets and modal logics, this paper proposes and examines a number of extended rough set models. By the properties satisfied by a binary relation, such as serial, reflexive, symmetric, transitive, and Euclidean, various classes of algebraic rough set models can be derived. They correspond to different modal logic systems. With respect to graded and probabilistic modal logics, graded and probabilistic rough set models are also discussed.

Abstract: In many familiar classes of algebraic structures kernels of congruence relations are uniquely specified by the inverse images q~-l(0)= {x [ q~(x)= 0} of a specified constant 0. On the one hand, q~-l(0) is nothing else but the 0-class of the kernel congruence of q~, on the other hand q~-~(0) can be axiomatized intrinsically, namely q l (0 ) is an ideal (in rings, Boolean algebras, or more generally in Heyt ing algebras), a normal subgroup, resp. normal subloop (in groups, resp. loops) or a filter (in Implicat ion algebras or Boolean algebras again, where 0 is replaced by the unit). In this paper we investigate common features of all the above structures by using a general notion of " ideal" , which makes sense in all universal algebras having a constant 0 and which specializes to the familar concepts of ideal, normal subgroup or filter in each of the algebras quoted above. In all universal algebras the 0-classes of congruence relations are easily seen to be ideals, but we shall require that conversely each ideal is the 0-class of a unique congruence relation. Such algebras, or ra ther classes of algebras with this proper ty will be called "classes with ideal determined congruences" or shortly ideal determined. In Part 1, after presenting the precise definitions, we shall show that the ideal determined varieties are characterized by a Mal 'cev condition, which turns out to be a combination of Fichtner 's condition for 0-regularity together with a ternary te rm r(x, y, z) which is a weakened form of Mal 'cev 's permutabil i ty term. From a result of Hagem ann it follows that ideal-determined varieties have modular congruence lattices, so the theory of commutators becomes readily available. In

Abstract: While studying rough equality within the framework of the modal system S 5, an algebraic structure called rough algebra [1], came up. Its features were abstracted to yield a topological quasi-Boolean algebra (tqBa). In this paper, it is observed that rough algebra is more structured than a tqBa. Thus, enriching the tqBa with additional axioms, two more structures, viz. pre-rough algebra and rough algebra, are denned. Representation theorems of these algebras are also obtained. Further, the corresponding logical systems L 1 L 2 are proposed and eventually, L 2 is proved to be sound and complete with respect to a rough set semantics.

Abstract: This paper is the second part of the paper [2]. Both of themare in the field of region-based (or Whitehedian) theory of space, which is an important subfield of Qualitative Spatial Reasoning (QSR). The paper can be considered also as an application of abstract algebra and topology to some problems arising and motivated in Theoretical Computer Science and QSR. In [2], different axiomatizations for region-based theory of space were given. The most general one was introduced under the name "Contact Algebra". In this paper some categories defined in the language of contact algebras are introduced. It is shown that they are equivalent to the category of all semiregular T$_0$-spaces and their continuous maps and to its full subcategories having as objects all regular (respectively, completely regular; compact; locally compact) Hausdorff spaces. An algorithm for a direct construction of all, up to homeomorphism, finite semiregular T$_0$-spaces of rank n is found. An example of an RCC model which has no regular Hausdorff representation space is presented. The main method of investigation in both parts is a lattice-theoretic generalization of methods and constructions from the theory of proximity spaces. Proximity models for various kinds of contact algebras are given here. In this way, the paper can be regarded as a full realization of the proximity approach to the region-based theory of space.

Abstract: [i]A Geometry of Approximation[/i] addresses Rough Set Theory, a field of interdisciplinary research first proposed by Zdzislaw Pawlak in 1982, and focuses mainly on its logic-algebraic interpretation. The theory is embedded in a broader perspective that includes logical and mathematical methodologies pertaining to the theory, as well as related epistemological issues. Any mathematical technique that is introduced in the book is preceded by logical and epistemological explanations. Intuitive justifications are also provided, insofar as possible, so that the general perspective is not lost. – Such an approach endows the present treatise with a unique character. Due to this uniqueness in the treatment of the subject, the book will be useful to researchers, graduate and pre-graduate students from various disciplines, such as computer science, mathematics and philosophy. It features an impressive number of examples supported by about 40 tables and 230 figures. The comprehensive index of concepts turns the book into a sort of encyclopaedia for researchers from a number of fields. – [i]A Geometry of Approximation[/i] links many areas of academic pursuit without losing track of its focal point, Rough Sets. – Table of contents : Preface. - Glossary of terms. - Introduction. - 1. A Mathematics of Perception. - 2. The Logico-algebraic Theory of Rough Sets. - 3. The Modal Logic of Rough Sets. - 4. A Relational Approach to Rough Sets. - 5. A Dialogical Approach. - Index. - Bibliography. M.-M. V.