Extensions of valuations
Jean Goubault-Larrecq · Mathematical Structures in Computer Science · 2005
Continuous valuations have been proposed by several authors as a way of modelling probabilistic non-determinism in programming language semantics. Let is algebraic, where every continuous valuation is the sup of a directed family of simple valuations based on finite elements. We exhibit another class of spaces in which every continuous valuation is quasi-simple, the so-called finitarily coherent spaces – in this case there is a largest extension to the Alexandroff topology. In general, the extension to the Alexandroff topology is not unique, unless, for example, the original valuation is bicontinuous. We also show that other natural spaces of valuations, namely those of discrete valuations and point-continuous valuations, can be characterised by similar extension theorems.