arXiv subjects

Hui Kou

Publications and source records attributed to Hui Kou.

14 recordsLinked to original sources

arXiv2026

A Natural Fuzzy Order on Fuzzy Numbers

This paper introduces a natural fuzzy order on fuzzy numbers that extends the natural orders on real numbers and interval numbers. We investigate its completeness properties and show that the space of uniformly bounded fuzzy numbers is conically complete and conically cocomplete, and that it is complete if and only if the underlying continuous t-norm is the G\"odel t-norm. Moreover, it is proved that this space constitutes a $[0,1]$-enriched domain if and only if the underlying continuous t-norm satisfies the (S) condition. These results provide a foundation for ordering fuzzy numbers.

math.CT↗

arXiv2026

The reflective hull of the two-element chain in DCPO: properness, maximal \Gamma-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 $\Gamma$-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

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

We introduce the category $\omega{\bf FVA}$ of finite-valuation approximable domains, a full subcategory of continuous domains contained in the category of pointed countably based FS-domains. We prove that $\omega{\bf FVA}$ 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 $\omega{\bf FVA}$, 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 $\omega{\bf FVA}$ 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

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

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↗

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 $\Sigma L$, 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.

cs.LO↗

arXiv2022

$\mathcal{S}_X$-convergence and locally hypercompact spaces

In this paper, we give a topological version of Scott convergence theorem for locally hypercompact spaces. We introduce the notion of $\mathcal{S}^*_X$-convergence on a $T_0$ topological space $X$, and define the notion of finitely approximated spaces. Monotone determined spaces are natural topological extensions of dcpos. The main results are: (1) A monotone determined space $X$ is a locally hypercompact space iff $\mathcal{S}^*_X$-convergence is topological. (2) For a $T_0$ space $X$, $\mathcal{S}^*_X$-convergence is topological iff $X$ is a finitely approximating space. (3) If the Lawson topology on a monotone determined space $X$ is compact, then $X$ is a dcpo endowed with the Scott topology.

math.GN↗

arXiv2022

Free algebras over directed spaces

Directed spaces are natural topological extensions of dcpos in domain theory and form a cartesian closed category. In order to model nondeterministic semantics, the power structures over directed spaces were defined through the form of free algebras. We show that free algebras over any directed space exist by the Adjoint Functor Theorem. An c-space (resp.\ b-space) can be characterized as a continuous (resp.\ algebraic) directed space. We show that continuous spaces are just all retracts of algebraic spaces by means of topological ideals, which are generalizations of the rounded ideals. Moreover, by categorical methods, we show that the carrier spaces of free algebras over continuous (resp.\ algebraic) spaces are still continuous (resp.\ algebraic) spaces.

math.CT↗

arXiv2022

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↗

arXiv2022

Power structures of directed spaces

Powerdomains in domain theory plays an important role in modeling the semantics of nondeterministic functional programming languages.\ In this paper,\ we extend the notion of powerdomain to the category of directed spaces,\ which is equivalent to the notion of the\ $T_0$\ monotone-determined space\ \cite{EN2009}.\ We define the notion of upper,\ lower and convex powerspace of a directed space by the way of free algebras.\ We show that the upper,\ lower and convex powerspace over any directed space exist and give their concrete structures.\ Generally,\ the upper,\ lower and convex powerspaces of a directed spaces are different from the upper,\ lower and convex powerdomains of a dcpos endowed with the Scott topology and the observationally-induced upper and lower powerspaces introduced by Battenfeld and Sch\"{o}der in 2015. Keywords: powerdomain,\ directed lower powerspace of directed spaces,\ directed upper powerspace of directed spaces,\ directed convex powerspace of directed spaces,\ observationally-induced lower powerspace,\ observationally-induced lower powerspace

math.GN↗

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 $\sigma(P)$ and $\Gamma(P)$ respectively denote the lattice of its Scott open subsets and Scott closed subsets ordered by inclusion, and set $\Sigma P=(P,\sigma(P))$. In this paper, we discuss the lower Vietoris topology and the Scott topology on $\Gamma(P)$ and give some sufficient conditions to make the two topologies equal. We built an adjunction between $\sigma(P)$ and $\sigma(\Gamma(P))$ and proved that $\Sigma P$ is core-compact iff $\Sigma\Gamma(P)$ is core-compact iff $\Sigma\Gamma(P)$ is sober, locally compact and $\sigma(\Gamma(P))=\upsilon(\Gamma(P))$ (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↗