In this paper, a solution is given to the problem proposed by J¨arvinen in [8]. A smallest completion of the rough sets system determined by an arbitrary binary relation is given. This completion, in the case of a quasi order, coincides with the rough sets system which is a Nelson algebra. Further, the algebraic properties of this completion has been studied.
In this paper the context of independent sets $J^{p}_{L}$ is assigned to the complete lattice (P(M),⊆) of all subsets of a non-empty set M. Some properties of this context, especially the irreducibility and the span, are investigated.
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This paper generalizes the notion of symmetrical neighbourhoods, which have been used to define connectivity in the case of sets, to the wider framework of complete lattices having a sup-generating family. Two versions (weak and strong) of the notion of a symmetrical dilation are introduced, and they are applied to the generation of ``connected components'' from the so-called ``geodesic dilations''. It turns out that any ``climbing'' ``weakly symmetrical'' extensive dilation induces a ``geodesic'' connectivity. When the lattice is the one of subsets of a metric space, the connectivities which are obtained in this way may coincide with the usual ones under some conditions, which are clarified. The abstract theory can be applied to grey-level and colour images, without any assumption of translation-invariance of operators.
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Classically, connectivity is a topological notion for sets, often introduced by means of arcs. An algebraic definition, called connection, has been proposed by Serra to extend the notion of connectivity to complete sup-generated lattices. A connection turns out to be characterized by a family of openings parameterized by the sup-generators, which partition each element of the lattice into maximal components. Starting from a first connection, several others may be constructed; e.g., by applying dilations. The present paper applies this theory to numerical functions. Every connection leads to segmenting the support of the function under study into regions. Inside each region, the function is r-continuous, for a modulus of continuity r given a priori, and characteristic of the connection. However, the segmentation is not unique, and may be particularized by other considerations (self-duality, large or low number of point components, etc.). These variants are introduced by means of examples for three different connections: flat zone connections, jump connections, and smooth path connections. They turn out to provide remarkable segmentations, depending only on a few parameters. In the last section, some morphological filters are described, based on flat zone connections, namely openings by reconstruction, flattenings and levelings.
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Mathematical morphology is a geometric approach in image processing and analysis with a strong mathematical flavor. Originally, it was developed as a powerful tool for shape analysis in binary and, later, grey-scale images. But it was soon recognized that the underlying ideas could be extended naturally to a much wider class of mathematical objects, namely complete lattices. This paper presents, in a bird's eye view, the foundations of mathematical morphology, or more precisely, the theory of morphological operators on complete lattices.
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