1,343 research outputs found

    Quantitative Concept Analysis

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    Formal Concept Analysis (FCA) begins from a context, given as a binary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that the first set consists of all objects that satisfy all attributes in the second, and vice versa. Many applications, though, provide contexts with quantitative information, telling not just whether an object satisfies an attribute, but also quantifying this satisfaction. Contexts in this form arise as rating matrices in recommender systems, as occurrence matrices in text analysis, as pixel intensity matrices in digital image processing, etc. Such applications have attracted a lot of attention, and several numeric extensions of FCA have been proposed. We propose the framework of proximity sets (proxets), which subsume partially ordered sets (posets) as well as metric spaces. One feature of this approach is that it extracts from quantified contexts quantified concepts, and thus allows full use of the available information. Another feature is that the categorical approach allows analyzing any universal properties that the classical FCA and the new versions may have, and thus provides structural guidance for aligning and combining the approaches.Comment: 16 pages, 3 figures, ICFCA 201

    Bombay hypertopologies

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    [EN] Recently it was shown that, in a metric space, the upper Wijsman convergence can be topologized with the introduction of a new far-miss topology. The resulting Wijsman topology is a mixture of the ball topology and the proximal ball topology. It leads easily to the generalized or g-Wijsman topology on the hyperspace of any topological space with a compatible LO-proximity and a cobase (i.e. a family of closed subsets which is closed under finite unions and which contains all singletons). Further generalization involving a topological space with two compatible LO-proximities and a cobase results in a new hypertopology which we call the Bombay topology. The generalized locally finite Bombay topology includes the known hypertopologies as special cases and moreover it gives birth to many new hypertopologies. We show how it facilitates comparison of any two hypertopologies by proving one simple result of which most of the existing results are easy consequences.Di Maio, G.; Meccariello, E.; Naimpally, S. (2003). Bombay hypertopologies. Applied General Topology. 4(2):421-444. doi:10.4995/agt.2003.2042.SWORD4214444

    List of communications

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    The lattice of quasi-uniformities

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    Includes bibliographical references.Over the last thirty years much progress has been made in the investigation of the lattice of uniformities on a set X. In particular, Pelant, Reiterman, Rodl and Simon have published several articles concerning anti-atoms and complements in this lattice. The aim of this dissertation is to begin a similar investigation into the lattice of quasi-uniformities θ(X) on a set X. It starts off with a summary of results obtained for the lattice of topologies on X, which, having been studied in great detail in the past, is intended as an example as to what may be achieved with θ(X). An exposit ion of the lattice of uniformities is then given. We conclude by commencing an investigation into the lattice of quasi-uniformities on X . Where possible, results obtained for the lattice of uniformities are generalized to θ(X), and some original results for θ(X) are also presented
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