# Structures latticielles, correspondances de Galois contraintes et classification symbolique

Abstract : This thesis is in the field of the latticial analysis of data in the situation, very general, where objects of various nature are described by variables of various types; one makes simply the assumption (realistic) according to which each variable takes his values in a lattice. The problems of processing of such data (extraction of knowledge) often amount seeking to obtain Moore families of a particular type, for example arborescent, and thus to impose structural constraints. Within this framework, we study initially particular Moore families, hierarchies, of which we characterize the implicational canonical basis. With this intention, we introduce a new type of binary relations on subsets of a set, called (\em overhanging relations). We put them in a one-to-one correspondence with unspecified Moore families, establish their bond with one of the arrow relations, and reconsider their properties in the hierarchical case, where they initially appeared. In one second part, we are interested in the Galois connection associated with a binary table (to which data of the type indicated above can always be brought back). We examine the constraints then to be imposed on a binary table so that closed sets obtained belong to prescribed Moore families, or of desired type. We then obtained some binary relations called (\em biclosed). Given two closure spaces $(E, \varphi)$ and $(E', \varphi')$, a relation is biclosed if any line of its matric representation corresponds to a closed set by $\varphi$, and any column with a closed set by $\varphi'$. We establish an isomorphism between the set of biclosed relations and that of Galois connections between the two lattices of closed sets induced by $\varphi$ and $\varphi'$. In the finite case, we deduce some effective algorithms for the adjustment of a Galois connection to an unspecified mapping between two lattices, or for the calculation of the join of two polarities. In a third part, we apply the preceding results to the study of the introduction of classifying constraints to a data table. We reconsider various uses of Galois connections (or the couples residuated / residual mappings) in models and methods of classification. Those are revisited in the optics of a unified frame based on bicloseds, and, by taking of account the results of the first part, differents ways are traced for definition of new methods. These parts are preceded by a synthesis on lattices and Galois connections.
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Document type :
Theses
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https://tel.archives-ouvertes.fr/tel-00003403
Contributor : Florent Domenach <>
Submitted on : Thursday, September 25, 2003 - 9:20:46 AM
Last modification on : Tuesday, January 19, 2021 - 11:08:07 AM
Long-term archiving on: : Wednesday, September 12, 2012 - 10:30:53 AM

### Identifiers

• HAL Id : tel-00003403, version 1

### Citation

Florent Adrien Domenach. Structures latticielles, correspondances de Galois contraintes et classification symbolique. Autre [cs.OH]. Université Panthéon-Sorbonne - Paris I, 2002. Français. ⟨tel-00003403⟩

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