Mathematical Structures in Computer Science

Paper

Domain theoretic characterisations of quasi-metric completeness in terms of formal balls

SALVADOR ROMAGUERAa1 and OSCAR VALEROa2

a1 Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, 46071 Valencia, Spain Email: sromague@mat.upv.es

a2 Departamento de Ciencias Matemáticas e Informática, Universidad de las Islas Baleares, 07122 Palma de Mallorca, Baleares, Spain Email: o.valero@uib.es

Abstract

We characterise those quasi-metric spaces (X, d) whose poset BX of formal balls satisfies the condition

\begin{linenomath}\begin{equation}
\text{for every }
(x,r),(y,s)\in \mathbf{B}X,\ (x,r)\ll (y,s)\Leftrightarrow
d(x,y)<r-s.
\tag{$*$}
\end{equation}\end{linenomath}

(*)

From this characterisation, we then deduce that a quasi-metric space (X, d) is Smyth-complete if and only if BX is a dcpo satisfying condition (*). We also give characterisations in terms of formal balls for sequentially Yoneda complete quasi-metric spaces and for Yoneda complete T1 quasi-metric spaces. Finally, we discuss several properties of the Heckmann quasi-metric on the formal balls of any quasi-metric space.

(Received June 25 2009)

(Revised December 15 2009)

(Online publication April 07 2010)

Footnotes

The authors are grateful for the support of the Spanish Ministry of Science and Innovation, grant MTM2009-12872-C02-01, and for the support of Generalitat Valenciana, grant ACOMP2009/005.