Positive extension and completion problems for a class of structured matrices

A class of matrices, defined by a displacement rank property, is introduced. Completion and extension problems are studied for matrices in this class, under certain positivity constraints. The extension problem is reduced to a standard interpolation problem for Schur matrix valued functions.     

A simple construction of representable relation algebras with non‐representable completions

We give a simple new construction of representable relation algebras with non‐representable completions. Using variations on our construction, we show that the elementary closure of the class of completely representable relation algebras is not finitely axiomatizable (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Profinite completions of some groups acting on trees

We investigate the profinite completions of a certain family of groups acting on trees. It turns out that for some of the groups considered, the completions coincide with the closures of the groups in the full group of tree automorphisms. However, we introduce an infinite series of groups for which that is not so, and describe the kernels of natural homomorphisms of the profinite completions onto the aforementioned closures of respective groups.     

Free Distributive Completions of Partial Complete Lattices

The free distributive completion of a partial complete lattice is the complete lattice that it is freely generated by the partial complete lattice in the most distributive way. This can be described as being a universal solution in the sense of universal algebra. Free distributive completions generalize the constructions of tensor products and of free completely distributive complete lattices over partially ordered sets.     

Eigenvalue completions by affine varieties

In this paper we provide new necessary and sufficient conditions for a general class of eigenvalue completion problems.

     

Subcategories of Filter Tower Spaces

The concept of a convergence tower space, or equivalently, a convergence approach space is formulated here in the context of a Cauchy setting in order to include a completion theory. Subcategories of filter tower spaces are defined in terms of axioms involving a general t-norm, T, in order to include a broad range of spaces. A T-regular sequence for a filter tower space is defined and, moreover, it is shown that the category of T-regular objects is a bireflective subcategory of all filter tower spaces. A completion theory for subcategories of filter tower spaces is given.     

Minimal comparability completions of arbitrary graphs

A transitive orientation of an undirected graph is an assignment of directions to its edges so that these directed edges represent a transitive relation between the vertices of the graph. Not every graph has a transitive orientation, but every graph can be turned into a graph that has a transitive orientation, by adding edges. We study the problem of adding an inclusion minimal set of edges to an arbitrary graph so that the resulting graph is transitively orientable. We show that this problem can be solved in polynomial time, and we give a surprisingly simple algorithm for it. We use a vertex incremental approach in this algorithm, and we also give a more general result that describes graph classes Π for which Π completion of arbitrary graphs can be achieved through such a vertex incremental approach.     

A calculus of direct limits of probability spaces

In this paper, several results on probability spaces have been extended to direct limits of such. The calculus includes convergence sequences, conditional expectations, completeness questions and extensions of probability spaces.