775 research outputs found

    Stone-type representations and dualities for varieties of bisemilattices

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    In this article we will focus our attention on the variety of distributive bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and involutive bisemilattices. After extending Balbes' representation theorem to bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn duality and introduce the categories of 2spaces and 2spaces⋆^{\star}. The categories of 2spaces and 2spaces⋆^{\star} will play with respect to the categories of distributive bisemilattices and De Morgan bisemilattices, respectively, a role analogous to the category of Stone spaces with respect to the category of Boolean algebras. Actually, the aim of this work is to show that these categories are, in fact, dually equivalent

    Related Study of Soft Set and Its Application A Review

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    Abstract In the present paper some literature related to soft sets are collected. The literature is motivated by Molodsov

    Matroidal approaches to rough sets via closure operators

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    AbstractThis paper studies rough sets from the operator-oriented view by matroidal approaches. We firstly investigate some kinds of closure operators and conclude that the Pawlak upper approximation operator is just a topological and matroidal closure operator. Then we characterize the Pawlak upper approximation operator in terms of the closure operator in Pawlak matroids, which are first defined in this paper, and are generalized to fundamental matroids when partitions are generalized to coverings. A new covering-based rough set model is then proposed based on fundamental matroids and properties of this model are studied. Lastly, we refer to the abstract approximation space, whose original definition is modified to get a one-to-one correspondence between closure systems (operators) and concrete models of abstract approximation spaces. We finally examine the relations of four kinds of abstract approximation spaces, which correspond exactly to the relations of closure systems

    A comparison of two types of rough sets induced by coverings

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    Rough set theory is an important technique in knowledge discovery in databases. In covering-based rough sets, many types of rough set models were established in recent years. In this paper, we compare the covering-based rough sets defined by Zhu with ones defined by Xu and Zhang. We further explore the properties and structures of these types of rough set models. We also consider the reduction of coverings. Finally, the axiomatic systems for the lower and upper approximations defined by Xu and Zhang are constructed

    The Relation Between Rough Sets And Fuzzy Sets Via Topological Spaces

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    Abstract: Theories of rough sets and fuzzy sets are related and complementary methodologies to handle uncertainty of vagueness and coarseness, respectively. They are generalizations of classical set theory for modeling vagueness and uncertainty. A fundamental question concerning both theories is their connections and differences. There have been many studies on this topic. Topology is a branch of mathematics, whose ideas exist not only in almost all branches of mathematics but also in many real life applications. The topological structure on an abstract set is used as the base, which used to extract knowledge from data. In this paper: topological structure is used to study the relation between rough sets and fuzzy sets. Membership function is used to convert from rough set to fuzzy set and vice versa. This conversion will achieve the advantages of two theories. Some examples and theories are introduced to indicate the importance of using general binary relations in the construction of rough set concepts, and indicate the relation between rough sets and fuzzy sets according to the topological spaces

    Tameness in generalized metric structures

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    We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and Zambrano on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections to classes of fuzzy structures, and structures on sheaves
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