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1

On the Completion of Rough Sets System Determined by Arbitrary Binary Relations

Umadevi D.

Fundamenta Informaticae

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2015

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Vol. 137, nr 3

413--424

EN

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.

2

Decomposition of Relations and Concept Lattices

Berghammer R., Winter M.

Fundamenta Informaticae

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2013

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Vol. 126, nr 1

37--82

EN

We introduce the decomposition of an arbitrary relation into a sequential composition of three relations, viz. of a mapping with a partial order and then the transpose of a mapping. After presenting some basic properties, we investigate the specific classes of junkfree, irreducible and minimal decompositions and show that for all relations a minimal decomposition exists. We also study decompositions with regard to DedekindMacNeille completions and concept lattices. These constructions are closely related to decompositions of relations. In our setting the fundamental theorem of concept lattices states that concept lattices are minimal-complete decompositions and all such decompositions are isomorphic. As a further main result we prove that the cutDedekindMacNeille completion of the order that belongs to the minimal decomposition of a relation is isomorphic to the concept lattice of that relation. Instead of considering binary relations on sets, we will work point-free within the general framework of allegories. This complement-free approach implies that the results of the paper can be applied to all models of these algebraic structures, including, for instance, lattice-valued fuzzy relations.

3

On a complete lattice of retracts of a free monoid generated by three elements

Foryś W.

Opuscula Mathematica

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2008

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Vol. 28, no. 2

123-127

EN

We prove that the family of retracts of a free monoid generated by three elements, partially ordered with respect to the inclusion, is a complete lattice.

4

Geodesy and connectivity in lattices

Ronse Ch., Serra J.

Fundamenta Informaticae

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2001

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Vol. 46, nr 4

349-395

EN

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.

5

Fundamenta morphologicae mathematicae

Goutsias J., Heijmans H.J.A.M.

Fundamenta Informaticae

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2000

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Vol. 41, Nr 1,2

1-31

EN

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.

6

Connections for sets and functions

Serra J.

Fundamenta Informaticae

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2000

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Vol. 41, Nr 1,2

147-186

EN

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.