Abstract
Over many different kinds of cryptomorphisms equivalent with closure systems (and so with closure operators), we focus here on implication relations and the related overhanging relations, as introduced and axiomatized in a previous paper (Domenach and Leclerc, Math Soc Sci 47(3):349–366, 2004). In relation with data analysis motivations, we particularize the axioms on overhanging relations in order to account for some types of closure systems, such as nested or distributive ones. We also examine the lattice structure of overhanging relations, which is isomorphic to the lattice of closure systems, and derived structures for particular sets of overhangings.
Résumé
Parmi bien des types de structures liés par cryptomorphisme aux systèmes de fermeture (et donc aux fermetures), nous considérons ici plus particulièrement les relations d’implication et les relations d’emboitement qui leur sont liées, telles qu’introduites et caractérisées axiomatiquement dans un article précédent (Domenach and Leclerc, Math Soc Sci 47(3):349–366, 2004). En relation avec des motivations issues de l’analyse des données, nous établissons des systèmes d’axiomes particuliers pour les relations d’emboitement associées à des systèmes de fermeture particuliers, comme les familles de parties totalement ordonnées ou distributives. Nous abordons aussi l’étude du treillis des relations d’emboitement, qui est isomorphe à celui des systèmes de fermeture, ainsi que des structures qui en résultent sur les ensembles particuliers d’emboitements.
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Domenach, F., Leclerc, B. The structure of the overhanging relations associated with some types of closure systems. Ann Math Artif Intell 49, 137–149 (2007). https://doi.org/10.1007/s10472-007-9061-6
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DOI: https://doi.org/10.1007/s10472-007-9061-6