Klaus Keimel

TCS Journal 2012 Journal Article

Extending algebraic operations to D -completions

• Klaus Keimel
• Jimmie D. Lawson

In this article, we show how separately continuous algebraic operations on T 0 -spaces and the laws that they satisfy, both identities and inequalities, can be extended to the D -completion, that is, the universal monotone convergence space completion. Indeed we show that the operations can be extended to the lattice of closed sets, but in this case it is only the linear identities that admit extension. Via the Scott topology, the theory is shown to be applicable to dcpo-completions of posets. We also explore connections with the construction of free algebras in the context of monotone convergence spaces.

TCS Journal 2006 Journal Article

Projective topology on bifinite domains and applications

• Samy Abbes
• Klaus Keimel

We revisit extension results from continuous valuations to Radon measures for bifinite domains. In the framework of bifinite domains, the Prokhorov theorem (existence of projective limits of Radon measures) appears as a natural tool, and helps building a bridge between Measure theory and Domain theory. The study we present also fills a gap in the literature concerning the coincidence between projective and Lawson topology for bifinite domains. Motivated by probabilistic considerations, we study the extension of measures in order to define Borel measures on the space of maximal elements of a bifinite domain.

TCS Journal 2004 Journal Article

The probabilistic powerdomain for stably compact spaces

• Mauricio Alvarez-Manilla
• Achim Jung
• Klaus Keimel

This paper reviews the one-to-one correspondence between stably compact spaces (a topological concept covering most classes of semantic domains) and compact ordered Hausdorff spaces. The correspondence is extended to certain classes of real-valued functions on these spaces. This is the basis for transferring methods and results from functional analysis to the non-Hausdorff setting. As an application of this, the Riesz Representation Theorem is used for a straightforward proof of the (known) fact that every valuation on a stably compact space extends uniquely to a Radon measure on the Borel algebra of the corresponding compact Hausdorff space. The view of valuations and measures as certain linear functionals on function spaces suggests considering a weak topology for the space of all valuations. If these are restricted to the probabilistic or sub-probabilistic case, then another stably compact space is obtained. The corresponding compact ordered space can be viewed as the set of (probability or sub-probability) measures together with their natural weak topology.

TCS Journal 2000 Journal Article

Foreword

• Klaus Keimel
• Michael Mislove
• Constantine Tsinakis