arXiv subjects

Qingguo Li

Publications and source records attributed to Qingguo Li.

At least 37 records · Page 2Linked to original sources

arXiv2024

Representations of domains via closure spaces in the quantale-valued setting

With a commutative unital quantale $L$ as the truth value table, this study focuses on the representations of $L$-domains by means of $L$-closure spaces. First, the notions of interpolative generalized $L$-closure spaces and directed closed sets are introduced. It is proved that in an interpolative generalized $L$-closure space (resp., $L$-closure space), the collection of directed closed sets with respect to the inclusion $L$-order forms a continuous $L$-dcpo (resp., an algebraic $L$-dcpo). Conversely, it is shown that every continuous $L$-dcpo (resp., algebraic $L$-dcpo) can be reconstructed by an interpolative generalized $L$-closure space (resp., $L$-closure space). Second, when $L$ is integral, the notion of dense subspaces of generalized $L$-closure spaces is introduced. By means of dense subspaces, an alternative representation for algebraic $L$-dcpos is given. Moreover, the concept of $L$-approximable relations between interpolative generalized $L$-closure spaces is introduced. Consequently, a categorical equivalence between the category of interpolative generalized $L$-closure spaces (resp., $L$-closure spaces) with $L$-approximable relations and that of continuous $L$-dcpos (resp., algebraic $L$-dcpos) with Scott continuous mappings is established.

math.GN↗

arXiv2023

Xi-Zhao Model Preserves WD Spaces

For a $T_1$ space $X$, Zhao and Xi constructed a dcpo model $\hat{P}$, where $P$ is a bounded complete algebraic poset model of $X$. In this paper, we formulate the closed WD subsets of the maximal point space $\mathrm{Max}(\hat{P})$ and the Scott space $Σ\hat{P}$, and then prove that $X$ is a WD space if and only if $Σ\hat{P}$ is a WD space. It is also shown that the sobrification $X^s$ (resp., the well-filtered reflection $X^w$) of $X$ can be embedded into the sobrification $\hat{P}^s$ (resp., the well-filtered reflection $\hat{P}^w$) of $\hat{P}$ as a saturated subspace. Finally, we introduce two new concepts ``${\rm H}$-model spaces" and ``weak ${\rm H}$-model spaces", and provide a general framework to prove that such $T_1$ spaces can be preserved by Xi-Zhao dcpo models.

math.GN↗

arXiv2023

GFDC: A Granule Fusion Density-Based Clustering with Evidential Reasoning

Currently, density-based clustering algorithms are widely applied because they can detect clusters with arbitrary shapes. However, they perform poorly in measuring global density, determining reasonable cluster centers or structures, assigning samples accurately and handling data with large density differences among clusters. To overcome their drawbacks, this paper proposes a granule fusion density-based clustering with evidential reasoning (GFDC). Both local and global densities of samples are measured by a sparse degree metric first. Then information granules are generated in high-density and low-density regions, assisting in processing clusters with significant density differences. Further, three novel granule fusion strategies are utilized to combine granules into stable cluster structures, helping to detect clusters with arbitrary shapes. Finally, by an assignment method developed from Dempster-Shafer theory, unstable samples are assigned. After using GFDC, a reasonable clustering result and some identified outliers can be obtained. The experimental results on extensive datasets demonstrate the effectiveness of GFDC.

cs.LG↗

arXiv2023

Three-way Imbalanced Learning based on Fuzzy Twin SVM

Three-way decision (3WD) is a powerful tool for granular computing to deal with uncertain data, commonly used in information systems, decision-making, and medical care. Three-way decision gets much research in traditional rough set models. However, three-way decision is rarely combined with the currently popular field of machine learning to expand its research. In this paper, three-way decision is connected with SVM, a standard binary classification model in machine learning, for solving imbalanced classification problems that SVM needs to improve. A new three-way fuzzy membership function and a new fuzzy twin support vector machine with three-way membership (TWFTSVM) are proposed. The new three-way fuzzy membership function is defined to increase the certainty of uncertain data in both input space and feature space, which assigns higher fuzzy membership to minority samples compared with majority samples. To evaluate the effectiveness of the proposed model, comparative experiments are designed for forty-seven different datasets with varying imbalance ratios. In addition, datasets with different imbalance ratios are derived from the same dataset to further assess the proposed model's performance. The results show that the proposed model significantly outperforms other traditional SVM-based methods.

cs.LG↗

arXiv2023

The core compactly generated topology

M. Escardó et al. asked whether the core compactly generated topology of a sober space is again sober and the sobrification of a core compactly generated space again core compactly generated. In this note, we answer the problem by displaying a counterexample, which reveals that the core compactly generated spaces are not closed under sobrifications. Meantime, we obtain that the core compactly generated spaces are closed under ω-well-filterifications and D-completions. Furthermore, we find that the core compactly generated topology of the Smyth power space of a well-filtered space coincides with the Scott topology. Finally, we provide a characterization for the core-compactness of core compactly generated spaces.

math.GN↗

arXiv2023

The $d^{*}$-space

In this paper, we introduce the concept of $d^{\ast}$-spaces. We find that strong $d$-spaces are $d^{\ast}$-spaces, but the converse does not hold. We give a characterization for a topological space to be a $d^{\ast}$-space. We prove that the retract of a $d^{\ast}$-space is a $d^{\ast}$-space. We obtain the result that for any $T_{0}$ space $X$ and $Y$, if the function space $TOP(X,Y)$ endowed with the Isbell topology is a $d^{\ast}$-space, then $Y$ is a $d^{\ast}$-space. We also show that for any $T_{0}$ space $X$, if the Smyth power space $Q_{v}(X)$ is a $d^{\ast}$-space, then $X$ is a $d^{\ast}$-space. Meanwhile, we give a counterexample to illustrate that conversely, for a $d^{\ast}$-space $X$, the Smyth power space $Q_{v}(X)$ may not be a $d^{\ast}$-space.

math.GN↗

arXiv2023

Weakly meet $s_{Z}$-continuity and $δ_{Z}$-continuity

Based on the concept of weakly meet $s_{Z}$-continuouity put forward by Xu and Luo in \cite{qzm}, we further prove that if the subset system $Z$ satisfies certain conditions, a poset is $s_{Z}$-continuous if and only if it is weakly meet $s_{Z}$-continuous and $s_{Z}$-quasicontinuous, which improves a related result given by Ruan and Xu in \cite{sz}. Meanwhile, we provide a characterization for the poset to be weakly meet $s_{Z}$-continuous, that is, a poset with a lower hereditary $Z$-Scott topology is weakly meet $s_{Z}$-continuous if and only if it is locally weakly meet $s_{Z}$-continuous. In addition, we introduce a monad on the new category $\mathbf{POSET_δ}$ and characterize its $Eilenberg$-$Moore$ algebras concretely.

math.GN↗

arXiv2023

On $k$-ranks of topological spaces

In this paper, the concepts of $K$-subset systems and $k$-well-filtered spaces are introduced, which provide another uniform approach to $d$-spaces, $s$-well-filtered spaces (i.e., $\mathcal{U}_{S}$-admissibility) and well-filtered spaces. We prove that the $k$-well-filtered reflection of any $T_{0}$ space exists. Meanwhile, we propose the definition of $k$-rank, which is an ordinal that measures how many steps from a $T_{0}$ space to a $k$-well-filtered space. Moreover, we derive that for any ordinal $α$, there exists a $T_{0}$ space whose $k$-rank equals to $α$. One immediate corollary is that for any ordinal $α$, there exists a $T_{0}$ space whose $d$-rank (respectively, $wf$-rank) equals to $α$.

math.GN↗

arXiv2023

One-step closure, weak one-step closure and meet continuity

This paper studies the weak one-step closure and one-step closure properties concerning the structure of Scott closures. We deduce that every quasicontinuous domain has weak one-step closure and show that a quasicontinuous poset need not have weak one-step closure. We also constructed a non-continuous poset with one-step closure, which gives a negative answer to an open problem posed by Zou et al.. Finally, we investigate the relationship between weak one-step closure property and one-step closure property and prove that a poset has one-step closure if and only if it is meet continuous and has weak one-step closure.

math.GN↗

arXiv2023

A Hofmann-Mislove theorem for $c$-well-filtered spaces

The Hofmann-Mislove theorem states that in a sober space, the nonempty Scott open filters of its open set lattice correspond bijectively to its compacts saturated sets. In this paper, the concept of $c$-well-filtered spaces is introduced. We show that a retract of a $c$-well-filtered space is $c$-well-filtered and a locally Lindelöf and $c$-well-filtered $P$-space is countably sober. In particular, we obtain a Hofmann-Mislove theorem for $c$-well-filtered spaces.

math.GN↗

arXiv2022

Not every countable complete lattice is sober

The study of the sobriety of Scott spaces has got an relative long history in domain theory. Lawson and Hoffmann independently proved that the Scott space of every continuous directed complete poset (usually called domain) is sober. Johnstone constructed the first directed complete poset whose Scott space is non-sober. Not long after, Isbell gave a complete lattice with non-sober Scott space. Based on Isbell's example, Xu, Xi and Zhao showed that there is even a complete Heyting algebra whose Scott space is non-sober. Achim Jung then asked whether every countable complete lattice has a sober Scott space. Let $ΣP$ be the Scott space of poset $P$. In this paper, we first prove that the topology of the product space $ΣP\times ΣQ$ coincides with the Scott topology on the product poset $P\times Q$ if the set $Id(P)$ and $Id(Q)$ of all non-trivial ideals of posets $P$ and $Q$ are both countable. Based on this result, we deduce that a directed complete poset $P$ has a sober Scott space, if $Id(P)$ is countable and the space $ΣP$ is coherent and well-filtered. Thus a complete lattice $L$ with $Id(L)$ countable has a sober Scott space. Making use the obtained results, we then construct a countable complete lattice whose Scott space is non-sober and thus give a negative answer to Jung's problem.

math.GN↗

arXiv2022

Categorical Representations of Continuous Domains and Continuous L-Domains Based on Closure Spaces

Closure space has proven to be a useful tool to restructure lattices and various order structures.This paper aims to provide a novel approach to characterizing some important kinds of continuous domains by means of closure spaces. By introducing an additional map into a given closure space, the notion of F-augmented generalized closure space is presented. It is shown that F-augmented generalized closure spaces generate exactly continuous domains. Moreover, the notion of approximable mapping is identified to represent Scott-continuous functions between continuous domains. These results produce a category equivalent to that of continuous domains with Scottcontinuous functions. At the same time, two subclasses of F-augmented generalized closure spaces are considered which are representations of continuous L-domains and continuous bounded complete domains, respectively.

cs.LO↗

arXiv2021

On some open problems concerning strong d-spaces and super H-sober spaces

In this paper, we highlight some open problems stated by Xu and Zhao. In particular, we focus on strong $d$-spaces and answer two open problems concerning strong $d$-spaces. One is about the product space of an arbitrary family of strong $d$-spaces. The other concerns the reflectivity of the category $\mathbf{S}$-$\mathbf{Top}_{d}$ of strong $d$-spaces. Meantime, for an irreducible subset system ($R$-subset system for short) $\mathrm{H}$ and a $T_{0}$ space $X$, we prove that $\mathrm{H}$ naturally satisfies property $M$. Based on this, we deduce that $X$ is super $\mathrm{H}$-sober iff $X$ is $\mathrm{H}$-sober and $\mathrm{H}$ satisfies property $Q$. Additionally, we give positive answers to some questions posed by Xu in \cite{Xiaoquan21}. Furthermore, we obtain that the finite product of hyper-sober spaces is hyper-sober.

math.GN↗

arXiv2021

D-completion, well-filterification and sobrification

In this paper, we obtain some sufficient conditions for the D-completion of a T0 space to be the well-filterification of this space, the well-filterification of a T0 space to be the sobrification of this space and the D-completion of a T0 space to be the sobrification, respectively. Moreover, we give an example to show that a tapered closed set may be neither the closure of a directed set nor the closed KF-set, respectively. Because the tapered closed set is a closed WD-set, the example also gives a negative answer to a problem proposed by Xu. Meantime, a new direct characterization of the D-completion of a T0 space is given by using the notion of pre-c-compact elements.

math.GN↗

arXiv2020

The categorical equivalence between disjunctive sequent calculi and algebraic L-domains

This paper establishes a purely syntactic representation for the category of algebraic L-domains with Scott-continuous functions as morphisms. The central tool used here is the notion of logical states, which builds a bridge between disjunctive sequent calculi and algebraic L-domains. To capture Scott-continuous functions between algebraic L-domains, the notion of consequence relations between disjunctive sequent calculi is also introduced. It is shown that the category of disjunctive sequent calculi with consequence relations as morphisms is categorical equivalent to that of algebraic L-domains with Scott-continuous functions as morphisms.

cs.LO↗

arXiv2019

Continuous Domains in Formal Concept Analysis

Formal Concept Analysis has proven to be an effective method of restructuring complete lattices and various algebraic domains. In this paper, the notions of attribute continuous formal context and continuous formal concept are introduced by considering a selection F of fnite subsets of attributes. Our decision of a selection F relies on a kind of generalized interior operators. It is shown that the set of continuous formal concepts forms a continuous domain, and every continuous domain can be obtained in this way. Moreover, an notion of F-morphism is also identified to produce a category equivalent to that of continuous domains with Scott-continuous functions. This paper also consider the representations of various subclasses of continuous domains such as algebraic domains, bounded complete domains and stably continuous semilattices. These results explore the fundamental idea of domain theory in Formal Concept Analysis from a categorical viewpoint.

cs.LO↗

arXiv2018

A characterization of the consistent Hoare powerdomains over dcpos

It has been shown that for a dcpo P, the Scott closure of Γ_c(P) in Γ(P) is a consistent Hoare powerdomain of P, where Γ_c(P) is the family of nonempty, consistent and Scott closed subsets of P, and Γ(P) is the collection of all nonempty Scott closed subsets of P. In this paper, by introducing the notion of a \vee-existing set, we present a direct characterization of the consistent Hoare powerdomain: the set of all \vee-existing Scott closed subsets of a dcpo P is exactly the consistent Hoare powerdomain of P. We also introduce the concept of an F-Scott closed set over each dcpo-\vee-semilattice. We prove that the Scott closed set lattice of a dcpo P is isomorphic to the family of all F-Scott closed sets of P's consistent Hoare powerdomain.

cs.LO↗

arXiv2018

Domains via approximation operators

In this paper, we tailor-make new approximation operators inspired by rough set theory and specially suited for domain theory. Our approximation operators offer a fresh perspective to existing concepts and results in domain theory, but also reveal ways to establishing novel domain-theoretic results. For instance, (1) the well-known interpolation property of the way-below relation on a continuous poset is equivalent to the idempotence of a certain set-operator; (2) the continuity of a poset can be characterized by the coincidence of the Scott closure operator and the upper approximation operator induced by the way below relation; (3) meet-continuity can be established from a certain property of the topological closure operator. Additionally, we show how, to each approximating relation, an associated order-compatible topology can be defined in such a way that for the case of a continuous poset the topology associated to the way-below relation is exactly the Scott topology. A preliminary investigation is carried out on this new topology.

cs.LO↗