Uniqueness of directed complete posets based on Scott closed set lattices

In analogy to a result due to Drake and Thron about topological spaces, this paper studies the dcpos (directed complete posets) which are fully determined, among all dcpos, by their lattices of all Scott-closed subsets (such dcpos will be called $C_{\sigma}$-unique). We introduce the notions of down-linear element and quasicontinuous element in dcpos, and use them to prove that dcpos of certain classes, including all quasicontinuous dcpos as well as Johnstone's and Kou's examples, are $C_{\sigma}$-unique. As a consequence, $C_{\sigma}$-unique dcpos with their Scott topologies need not be bounded sober.Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1607.0357

Domains and Probability Measures: A Topological Retrospective

Domain theory has seen success as a semantic model for high-level programming languages, having devised a range of constructs to support various effects that arise in programming. One of the most interesting - and problematic - is probabilistic choice, which traditionally has been modeled using a domain-theoretic rendering of sub-probability measures as valuations. In this talk, I will place the domain-theoretic approach in context, by showing how it relates to the more traditional approaches such as functional analysis and set theory. In particular, we show how the topologies that arise in the classic approaches relate to the domain-theoretic rendering. We also describe some recent developments that extend the domain approach to stochastic process theory

Quasiexact posets and the moderate meet-continuity

The study of weak domains and quasicontinuous domains leads to the consideration of two types generalizations of domains. In the current paper, we define the weak way-below relation between two nonempty subsets of a poset and quasiexact posets. We prove some connections among quasiexact posets, quasicontinuous domains and weak domains. Furthermore, we introduce the weak way-below finitely determined topology and study its links to Scott topology and the weak way-below topology first considered by Mushburn. It is also proved that a dcpo is a domain if it is quasiexact and moderately meet continuous with the weak way-below relation weakly increasing

A Radon-Nikod\'ym Theorem for Valuations

We enquire under which conditions, given two $\sigma$-finite, $\omega$-continuous valuations $\nu$ and $\mu$, $\nu$ has density with respect to $\mu$. The answer is that $\nu$ has to be absolutely continuous with respect to $\mu$, plus a certain Hahn decomposition property, which happens to be always true for measures.Comment: 22 pages, 2 figure

Free dcpo-algebras via directed spaces

Directed spaces are natural topological extensions of dcpos in domain theory and form a cartesian closed category. We will show that the D-completion of free algebras over a Scott space $\Sigma L$, on the context of directed spaces, are exactly the free dcpo-algebras over dcpo $L$, which reveals the close connection between directed powerspaces and powerdomains. By this result, we provide a topological representation of upper, lower and convex powerdomains of dcpos uniformly.Comment: 18 page

One-step Closure, Ideal Convergence and Monotone Determined Space

Monotone determined spaces are natural topological extensions of dcpo. Its main purpose is to build an extended framework for domain theory. In this paper, we study the one-step closure and ideal convergence on monotone determined space. Then we also introduce the equivalent characterizations of c-spaces and locally hypercompact space. The main results are:1.Every c-space has one-step closure and every locally hypercompact space has weak one-step closure;2.A monotone determined space has one-step closure if and only if it is d-meet continuous and has weak one-step closure. 3.IS-convergence(resp. IGS-convergence) is topological iff X is a c-space (resp. locally hypercompact space); 4.If X is a d-meet continuous space, then the following three conditions are equivalent to each other: (i) X is c-space; (ii) The net (xj ) ISL-converges to x iff (xj ) I-converges to x with respect to Lawson topology; (iii) The net (xj ) IGSL-converges to x iff (xj ) I-converges to x with respect to Lawson topology.Comment: 14page