arXiv subjects

Zhenchao Lyu

Publications and source records attributed to Zhenchao Lyu.

12 recordsLinked to original sources

arXiv2026

Finite-valuation approximable structures: a solution to the Jung--Tix problem of probabilistic powerdomains

We introduce the category $ω{\bf FVA}$ of finite-valuation approximable domains, a full subcategory of continuous domains contained in the category of pointed countably based FS-domains. We prove that $ω{\bf FVA}$ is Cartesian closed and closed under both the subprobabilistic and probabilistic valuation powerdomains. Hence the valuation monads $\mathcal{V}_{\leq 1}$ and $\mathcal{V}_1$ restrict to $ω{\bf FVA}$, yielding a positive answer to the category-existence form of the Jung--Tix problem, a long-standing open problem in domain theory since the 1990s. In particular, we develop a new factorization-approximation framework for constructing objects from a given class of known objects or structures. Applying this method to the class of subprobabilistic powerdomains over finite posets, we construct the class $ω{\bf FVA}$ and show that it is closed under Scott-continuous retracts, finite products, function spaces, $\mathcal{V}_{\leq 1}$ and $\mathcal{V}_1$ monads.

cs.LO↗

arXiv2026

The reflective hull of the two-element chain in DCPO: properness, maximal Γ-faithfulness, and an internal reflection formula

Let $\DCPO$ be the category of dcpos and Scott-continuous maps, and let $\Rtwo$ be the reflective hull of the two-element chain $2$. We prove that $\Rtwo$ is the least non-discrete proper reflective full subcategory of $\DCPO$. It is closed under limits and Skula-closed sub-dcpos and contains every sober dcpo. We show that $\Rtwo$ is the maximal $Γ$-faithful full subcategory of $\DCPO$ that strictly contains weakly dominated dcpos. Finally, we give the concrete construction of $2$-reflection internally by a transfinite iteration and a criterion for reflective full subcategories of $\DCPO$, analogous to the Keimel--Lawson conditions for $T_0$ topological spaces.

math.GN↗

arXiv2026

FS-domains are not always RB-domains

We prove that Lawson's planar closed-disk domain is not an RB-domain. This domain is the dcpo of all closed disks in the Euclidean plane, together with the whole plane as bottom, ordered by reverse inclusion. Since this domain is an FS-domain, it gives a concrete example of an FS-domain which is not an RB-domain, answering negatively the long-standing open problem in domain theory of whether FS-domains and RB-domains are identical.

math.GN↗

arXiv2026

Characterizing finite posets whose probabilistic powerdomain are RB-domains

We classify the finite posets whose probabilistic powerdomain is an RB-domain. For a finite nonempty poset $P$, let $\Vone(P)$ be the probability powerdomain of $P$, which is the probability simplex ordered by the stochastic order. We prove that $\Vone(P)$ is an RB-domain if and only if $P$ has a least element and the undirected Hasse graph of $P$ is a tree. Consequently, the probabilistic powerdomain does not preserve RB-domains; the four-point diamond gives a finite counterexample. The proof separates two obstructions. First, if $P$ has no least element, then the face of probability measures supported on the minimal points must be fixed pointwise by every deflation below the identity. Secondly, once a least element exists, the Hasse graph is connected, and a cycle in it makes the local stochastic cone non-simplicial. A Euclidean finite-step cone argument then rules out the finite-valued monotone approximations supplied by the RB property.

math.CO↗

arXiv2026

Monotone determined spaces via $\mathbb{C}$-generated spaces

The category of monotone determined spaces is an extended topological framework for dcpos in domain theory. We first show that monotone determined spaces are exactly the spaces generated by one-point convergence spaces, and then naturally form a convenient Cartesian closed category of $\mathbb{C}$-generated spaces. We then show that monotone determined spaces are not always compact Hausdorff generated, answering the question raised by Ingo Battenfeld in 2013. Moreover, we generalize the notion of monotone determined spaces by introducing $\mathcal{C}$-determined spaces and showing that categories of $\mathcal{C}$-determined spaces correspond to coreflective subcategories of topological spaces. This yields a uniform construction of several convenient categories determined by directed, chain and monotone sequential convergence classes. We finally discuss the relationships among them, including the categories generated by continuous spaces, quasicontinuous spaces and Scott spaces of dcpos.

math.GN↗

arXiv2025

The Scott space of lattice of closed subsets with supremum operator as a topological semilattice

We present several equivalent conditions of the continuity of the supremum function from the square of the Scott space of $C(X)$ to itself under mild assumptions, where $C(X)$ denotes the lattice of closed subsets of a $\mathbf{T_0}$ topological space. We also show that a $\mathbf{T_0}$ space is quasicontinuous (quasialgebraic) iff the lattice of its closed subsets is a quasicontinuous (quasialgebraic) domain by using $n$-approximation. Furthermore, we provide a necessary condition for when a topological space possesses a Scott completion. This allows us to give more examples which do not have Scott completions.

math.GN↗

arXiv2023

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 $ΣL$, 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.

cs.LO↗

arXiv2022

Continuity and core compactness of topological spaces

We investigate two approximation relations on a T0 topological space, the n-approximation, and the d-approximation, which are generalizations of the way-below relation on a dcpo. Different kinds of continuous spaces are defined by the two approximations and are all shown to be directed spaces. We show that the continuity of a directed space is very similar to the continuity of a dcpo in many aspects, which indicates that the notion of directed spaces is a suitable topological extension of dcpos.The main results are: (1) A topological space is continuous iff it is a retract of an algebraic space;(2) a directed space X is core compact iff for any directed space Y, the topological product is equal to the categorical product in DTop of X and Y respectively;(3) a directed space is continuous (resp., algebraic, quasicontinuous, quasialgebraic) iff the lattice of its closed subsets is continuous (resp., algebraic, quasicontinuous, quasialgebraic).

math.GN↗

arXiv2022

The Directed Probabilistic Powerspace

Probabilistic powerdomain in domain theory plays an important role in modeling the semantics of nondeterministic functional programming languages with probabilistic choice. In this paper, we extend the notion of powerdomain to directed spaces, which is equivalent to the notion of the T0 monotone-determined space [4]. We construct the probabilistic powerspace of the directed space, which is defined as a free directed space-cone. In addition, the relationships between our construction and classical probabilistic powerdomain are studied.

math.GN↗

arXiv2021

Two topologies on the lattice of Scott closed subsets

For a poset $P$, let $σ(P)$ and $Γ(P)$ respectively denote the lattice of its Scott open subsets and Scott closed subsets ordered by inclusion, and set $ΣP=(P,σ(P))$. In this paper, we discuss the lower Vietoris topology and the Scott topology on $Γ(P)$ and give some sufficient conditions to make the two topologies equal. We built an adjunction between $σ(P)$ and $σ(Γ(P))$ and proved that $ΣP$ is core-compact iff $ΣΓ(P)$ is core-compact iff $ΣΓ(P)$ is sober, locally compact and $σ(Γ(P))=\upsilon(Γ(P))$ (the lower Vietoris topology). This answers a question in [17]. Brecht and Kawai [2] asked whether the consonance of a topological space $X$ implies the consonance of its lower powerspace, we give a partial answer to this question at the last part of this paper.

math.GN↗

arXiv2019

Core-compactness of Smyth powerspaces

We prove that the Smyth powerspace Q(X) of a topological space X is core-compact if and only if X is locally compact. As a straightforward consequence we obtain that the Smyth powerspace construction does not preserve core-compactness generally.

math.GN↗

arXiv2019

Domain-complete and LCS-complete spaces

We study $G_δ$ subspaces of continuous dcpos, which we call domain-complete spaces, and $G_δ$ subspaces of locally compact sober spaces, which we call LCS-complete spaces. Those include all locally compact sober spaces-in particular, all continuous dcpos-, all topologically complete spaces in the sense of Čech, and all quasi-Polish spaces-in particular, all Polish spaces. We show that LCS-complete spaces are sober, Wilker, compactly Choquet-complete, completely Baire, and $\odot$-consonant-in particular, consonant; that the countably-based LCS-complete (resp., domain-complete) spaces are the quasi-Polish spaces exactly; and that the metrizable LCS-complete (resp., domain-complete) spaces are the completely metrizable spaces. We include two applications: on LCS-complete spaces, all continuous valuations extend to measures, and sublinear previsions form a space homeomorphic to the convex Hoare powerdomain of the space of continuous valuations.

math.GN↗