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.