Using the the dependently-typed programming language Agda, we formalise orders, with attention to the theory of Galois Connections, and showcase it by formalising a few results of the category of algebraic contexts with relational homomorphisms presented by Jipsen (2012) and Moshier (2013). We aim to exhibit an internal theory of Galois Connections and Closure operators where the ambient space need not have a notion of meets (intersections), which are the usual medium in presenting antisymmetry of partial orders. Instead we consider `symmetric quotients' as being the relational counterpart of propositional calculus' primitive connective: equivalence. We argue that it as a more natural primitive than meet --- especially its connection with certain proof heuristics regarding posets. Moreover, not only do we constrain ourselves to an unconventional set of primitive operators, but in fact we discard the familiar setting of relations and sets in favour of the more general setting of ordered categories with converse (OCCs) --- in fact, a large portion does not require identities and so semigroupoids may be used instead. / Thesis / Master of Science (MSc)

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In this thesis, we develop a relational localisation theory for topological algebras, i.e., a theory that studies local approximations of a topological algebra’s relational counterpart. In order to provide an appropriate framework for our considerations, we first introduce a general Galois theory between continuous functions and closed relations on an arbitrary topological space. Subsequently to this rather foundational discussion, we establish the desired localisation theory comprising the identification of suitable subsets, the description of local structures, and the retrieval of global information from local data. Among other results, we show that the restriction process with respect to a sufficiently large collection of local approximations of a Hausdorff topological algebra extends to a categorical equivalence between the topological quasivariety generated by the examined structure and the one generated by its localisation. Furthermore, we present methods for exploring topological algebras possessing certain operational compactness properties. Finally, we explain the developed concepts and obtained results in the particular context of three important classes of topological algebras by analysing the local structure of essentially unary topological algebras, topological lattices, and topological modules of compact Hausdorff topological rings.

We introduce a special set of relations on a finite set, called clausal relations. A restricted version of the Galois connection between polymorphisms and invariants, called Pol-CInv, is studied, where the invariant relations are clausal relations. Clones arising from this Galois connection, so-called C-clones, are investigated. Finally, we show that clausal relations meet a sufficient condition that is known to ensure polynomial time solvability of the corresponding CSP.

In this thesis, we develop a relational localisation theory for topological algebras, i.e., a theory that studies local approximations of a topological algebra’s relational counterpart. In order to provide an appropriate framework for our considerations, we first introduce a general Galois theory between continuous functions and closed relations on an arbitrary topological space. Subsequently to this rather foundational discussion, we establish the desired localisation theory comprising the identification of suitable subsets, the description of local structures, and the retrieval of global information from local data. Among other results, we show that the restriction process with respect to a sufficiently large collection of local approximations of a Hausdorff topological algebra extends to a categorical equivalence between the topological quasivariety generated by the examined structure and the one generated by its localisation. Furthermore, we present methods for exploring topological algebras possessing certain operational compactness properties. Finally, we explain the developed concepts and obtained results in the particular context of three important classes of topological algebras by analysing the local structure of essentially unary topological algebras, topological lattices, and topological modules of compact Hausdorff topological rings.

We introduce a special set of relations on a finite set, called clausal relations. A restricted version of the Galois connection between polymorphisms and invariants, called Pol-CInv, is studied, where the invariant relations are clausal relations. Clones arising from this Galois connection, so-called C-clones, are investigated. Finally, we show that clausal relations meet a sufficient condition that is known to ensure polynomial time solvability of the corresponding CSP.

L'extraction de règles d'association et de bi-clusters sont deux techniques de fouille de données complémentaires majeures, notamment pour l'intégration de connaissances. Ces techniques sont utilisées dans de nombreux domaines, mais aucune approche permettant de les unifier n'a été proposée. Hors, réaliser ces extractions indépendamment pose les problèmes des ressources nécessaires (mémoire, temps d'exécution et accès aux données) et de l'unification des résultats. Nous proposons une approche originale pour extraire différentes catégories de modèles de connaissances tout en utilisant un minimum de ressources. Cette approche est basée sur la théorie des ensembles fermés et utilise une nouvelle structure de données pour extraire des représentations conceptuelles minimales de règles d'association, bi-clusters et règles de classification. Ces modèles étendent les règles d'association et de classification et les bi-clusters classiques, les listes d'objets supportant chaque modèle et les relations hiérarchiques entre modèles étant également extraits. Cette approche a été appliquée pour l'analyse de données d'interaction protéomiques entre le virus VIH-1 et l'homme. L'analyse de ces interactions entre espèces est un défi majeur récent en bio-informatique. Plusieurs bases de données intégrant des informations hétérogènes sur les interactions et des connaissances biologiques sur les protéines ont été construites. Les résultats expérimentaux montrent que l'approche proposée peut traiter efficacement ces bases de données et que les modèles conceptuels extraits peuvent aider à la compréhension et à l'analyse de la nature des relations entre les protéines interagissant. / Knowledge pattern extraction is one of the major topics in the data mining and background knowledge integration domains. Out of several data mining techniques, association rule mining and bi-clustering are two major complementary tasks for these topics. These tasks gained much importance in many domains in recent years. However, no approach was proposed to perform them in one process. This poses the problems of resources required (memory, execution times and data accesses) to perform independent extractions and of the unification of the different results. We propose an original approach for extracting different categories of knowledge patterns while using minimum resources. This approach is based on the frequent closed patterns theoretical framework and uses a novel suffix-tree based data structure to extract conceptual minimal representations of association rules, bi-clusters and classification rules. These patterns extend the classical frameworks of association and classification rules, and bi-clusters as data objects supporting each pattern and hierarchical relationships between patterns are also extracted. This approach was applied to the analysis of HIV-1 and human protein-protein interaction data. Analyzing such inter-species protein interactions is a recent major challenge in computational biology. Databases integrating heterogeneous interaction information and biological background knowledge on proteins have been constructed. Experimental results show that the proposed approach can efficiently process these databases and that extracted conceptual patterns can help the understanding and analysis of the nature of relationships between interacting proteins.

In this paper, we generalize the notions of polymorphisms and invariant relations to arbitrary categories. This leads us to a Galois connection that coincides with the classical case from universal algebra if the underlying category is the category of sets, but remains applicable no matter how the category is changed. In analogy to the situation in universal algebra, we characterize the Galois closed classes by local closures of clones of operations and local closures of what we will introduce as clones of (generalized) relations. Since the approach is built on purely category-theoretic properties, we will also discuss the dualization of our notions.

In this paper, we generalize the notions of polymorphisms and invariant relations to arbitrary categories. This leads us to a Galois connection that coincides with the classical case from universal algebra if the underlying category is the category of sets, but remains applicable no matter how the category is changed. In analogy to the situation in universal algebra, we characterize the Galois closed classes by local closures of clones of operations and local closures of what we will introduce as clones of (generalized) relations. Since the approach is built on purely category-theoretic properties, we will also discuss the dualization of our notions.