Computational Effects in Topological Domain Theory

AbstractThis paper contributes towards establishing the category QCB, of topological quotients of countably based spaces, and its subcategory TP, of topological predomains, as a flexible framework for denotational semantics of programming languages. In particular, we show that both categories have free algebras for arbitrary countable parametrised equational theories, and are thus, following ideas of Plotkin and Power, able to model a wide range of computational effects. Furthermore, we give an explicit construction of the free algebras

Extending Algebraic Operations to D-Completions

In this article we show how separately continuous algebraic operations on T0-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. © 2009 Elsevier B.V. All rights reserved

D-completions and the d-topology

In this article we give a general categorical construction via reflection functors for various completions of T0-spaces subordinate to sobrification, with a particular emphasis on what we call the D-completion, a type of directed completion introduced by Wyler [O. Wyler, Dedekind complete posets and Scott topologies, in: B. Banaschewski, R.-E. Hoffmann (Eds.), Continuous Lattices Proceedings, Bremen 1979, in: Lecture Notes in Mathematics, vol. 871, Springer Verlag, 1981, pp. 384-389]. A key result is that all completions of a certain type are universal, hence unique (up to homeomorphism). We give a direct definition of the D-completion and develop its theory by introducing a variant of the Scott topology, which we call the d-topology. For partially ordered sets the D-completion turns out to be a natural dcpo-completion that generalizes the rounded ideal completion. In the latter part of the paper we consider settings in which the D-completion agrees with the sobrification respectively the closed ideal completion. © 2008 Elsevier B.V. All rights reserved

Extending algebraic operations to D -completions

In this article, we show how separately continuous algebraic operations on T0-spaces and the laws that they satisfy, both identities and inequalities, can be extended to theD-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. © 2011 Elsevier B.V. All rights reserved