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The comparison of sets of objects is a research topic with applications in diverse fields such as computer science, biology and psychology. Since the introduction of the Jaccard index, many techniques have been proposed. This paper aims at extending an existing framework of comparison indices for sets. Firstly, the novel indices account for similarities between elements, rather than identity of elements as is the case for existing techniques. As a result, a richer framework of comparison indices is obtained. The use of fuzzy quantifiers in this framework is shown. Secondly, the machinery for sets is extended to the case of multisets, which results in two classes of comparison indices. The first class considers each element instance as a separate element, while the second class considers groups of elements instances as an atomic entity. The number of instances is then a property of this group, that is taken into account when calculating similarity between element groups. [ABSTRACT FROM AUTHOR]

Traditional databases can manage only crisp information, a limitation that also holds for geographic information systems and spatial databases. In this paper, we present a technique based on triangulated irregular networks (or TINs for short) and fuzzy set theory to model imprecise or uncertain regions. A fuzzy region is represented by a Extended TIN, which allows for an associated value for each point of the region in the presented approach to be considered; this associated value will be a membership grade. As is common in fuzzy set theory, membership grades can indicate a degree of "belonging to the set"; in our approach these are the degree to which every crisp location belongs to the fuzzy region (membership grades in fuzzy set theory can have other interpretations7 as well, but these are not needed for the modelling of fuzzy regions). While modelling a fuzzy region as described provides a more accurate model of a real world situation, it does require many operators from the geographic realm to be extended and also new operators (mainly from the fuzzy realm) to be added at the object level. In this paper, from the GIS realm, the calculation of the surface area and the minimum bounding rectangle for fuzzy regions are considered; from the fuzzy realm the calculation of the α-cut is considered. Other operations (i.e. convex hull of a fuzzy region, distance between two fuzzy regions, ...) are still under development. [ABSTRACT FROM AUTHOR]