Abstract
We investigate the Eilenberg-Moore algebras of the extended probabilistic powerdomain monad Vw over the category TOP0 of T0 topological spaces and continuous maps. We prove that every Vw-algebra in our setting is a weakly locally convex sober topological cone, and that a map is the structure map of a Vw-algebra if and only if it is continuous and sends every continuous valuation to its unique barycentre. Conversely, for locally linear sober cones—a strong form of local convexity—, the mere existence of barycentres entails that the barycentre map is the structure map of a Vw-algebra; moreover the algebra morphisms are exactly the linear continuous maps in that case.We also examine the algebras of two related monads, the simple valuation monad Vf and the point-continuous valuation monad Vp. In TOP0 their algebras are fully characterised as weakly locally convex topological cones and weakly locally convex sober topological cones, respectively. In both cases, the algebra morphisms are continuous linear maps between the corresponding algebras.
Highlights
The probabilistic powerdomain construction on directed complete partially ordered sets was introduced by Jones and Plotkin and employed to give semantics to programming languages with probabilistic features [14, 13]
Real numbers to Scott-open subsets of the dcpo. Jones proved that this construction is a monad on the category of dcpos and Scott-continuous functions. She proved that this monad can be restricted to the full subcategory of continuous domains, the algebras of this monad in the category of continuous domains are the continuous abstract probabilistic domains, and the algebra homomorphisms are continuous linear maps
Kirch [20] generalised Jones and Plotkin’s probabilistic powerdomain by stipulating that a valuation might take values that are not finite. He showed that this construction is again a monad that can be restricted to the category of continuous domains, and the algebras of this monad in the category of continuous domains are the continuous d-cones, a notion well investigated in [25]
Summary
The probabilistic powerdomain construction on directed complete partially ordered sets (dcpo for short) was introduced by Jones and Plotkin and employed to give semantics to programming languages with probabilistic features [14, 13]. They showed that the extended probabilistic powerdomain construction is a monad over the category of T0 topological spaces and considered its algebras in related categories in the same paper, leaving a conjecture that the algebras of this monad on the category of stably compact spaces and continuous functions are the stably compact locally convex topological cones Restricting this monad to the category of compact ordered spaces (compact pospaces) and continuous monotone maps, Keimel located the algebras of this monad to be the compact convex ordered sets embeddable in locally convex ordered topological vector spaces [16]. Whenever R+ is treated as a topological space, we mean that it is equipped with the Scott topology until stated otherwise
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