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Dexue Zhang

Publications and source records attributed to Dexue Zhang.

At least 19 recordsLinked to original sources
arXiv2026

d-Spectral Bitopological Spaces

We introduce and study the category of \emph{d-spectral spaces}, a bitopological analogue of the classical spectral spaces of Stone and Hochster. A d-spectral space is a compact, d-sober bitopological space such that both open set lattices are coherent frames, where d-sobriety is the bitopological notion of sobriety due to Jung and Moshier. We show that the category of spectral spaces embeds into the category of d-spectral spaces as a simultaneously reflective and coreflective full subcategory. Moreover, we prove that d-spectral spaces are precisely the spectra of d-lattices. Key to this result is the d-lattice of compact open sets associated to a d-spectral space and the spectrum construction for d-lattices. We also show that the patch space of a d-spectral space is d-Boolean and that the de Groot dual of a d-spectral space is again d-spectral, mirroring the corresponding classical properties of spectral spaces. Our results demonstrate that d-spectral spaces form a natural and well-behaved bitopological extension of the spectral space framework.

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arXiv2025

d-Boolean algebras and their bitopological representation

We present a Stone duality for bitopological spaces in analogy to the duality between Stone spaces and Boolean algebras, in the same vein as the duality between d-sober bitopological spaces and spatial d-frames established by Jung and Moshier. Precisely, we introduce the notion of d-Boolean algebras and prove that the category of such algebras is dually equivalent to the category of compact and zero-dimensional bitopological spaces satisfying the T0 separation axiom.

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arXiv2024

The bounded ideal monad on the category of quasi-metric spaces and its algebras

The notion of bounded ideals is introduced for quasi-metric spaces. Such ideals give rise to a monad, the bounded ideal monad, on the category of quasi-metric spaces and non-expansive maps. Algebras of this monad are metric version of local dcpos of Mislove. It is shown that an algebra of the bounded ideal monad is a standard quasi-metric space of which the formal balls form a local dcpo; and that a continuous algebra is a standard quasi-metric space of which the formal balls form a local domain.

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arXiv2024

A Boolean-valued space approach to separation axioms and sobriety of bitopological spaces

This paper presents a study of separation axioms and sobriety of bitopological spaces from the point of view of fuzzy topology via identifying bitopological spaces with topological spaces valued in the Boolean algebra of four elements. A system of separation axioms is proposed making use of Boolean-valued specialization order of bitopological spaces; The relationship between d-sobriety of bitopological spaces proposed by Jung and Moshier and sobriety of fuzzy topological spaces is studied; A Hofmann-Mislove theorem for bitopological spaces is established.

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arXiv2024

Introductory notes on real-enriched categories

Real-enriched categories are categories with real numbers as enrichment. Precisely, a real-enriched category is a category enriched over the commutative and unital quantale composed of the unit interval and a continuous t-norm. These notes present a brief introduction to such categories, focusing on the presheaf monad and its submonads in the category of real-enriched categories.

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arXiv2023

Smyth complete real-enriched categories

This paper investigates Smyth completeness of categories enriched over a quantale obtained by equipping the unit interval of real numbers with a continuous t-norm. A real-enriched category is Smyth-complete if each of its forward Cauchy nets has a unique limit in the open ball topology of its symmetrization. It is demonstrated that Smyth completeness can be characterized as a categorical property and as a real-valued topological property. Explicitly, it is shown that a real-enriched category is Smyth complete if and only if it is separated and all of its ideals are representable, if and only if its Alexandroff real-valued topology is sober.

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arXiv2022

A Hofmann-Mislove theorem for approach spaces

The Hofmann-Mislove theorem says that the ordered set of open filters of the open-set lattice of a sober topological space is isomorphic to the ordered set of compact saturated sets (ordered by reverse inclusion) of that space. This paper concerns a metric analogy of this result. To this end, the notion of compact functions of approach spaces is introduced. Such functions are an analog of compact subsets in the enriched context. It is shown that for a sober approach space $X$, the metric space of proper open $[0,\infty]$-filters of the metric space of upper regular functions of $X$ is isomorphic to the opposite of the metric space of inhabited and saturated compact functions of $X$, establishing a Hofmann-Mislove theorem for approach spaces.

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arXiv2022

Formal balls of ${\sf Q}$-categories

The construction of the formal ball model for metric spaces due to Edalat and Heckmann was generalized to ${\sf Q}$-categories by Kostanek and Waszkiewicz. This paper concerns the influence of the structure of the quantale ${\sf Q}$ on the connection between Yoneda completeness of ${\sf Q}$-categories and directed completeness of their sets of formal balls. In the case that ${\sf Q}$ is the interval $[0,1]$ equipped with a continuous t-norm $\&$, it is shown that in order that Yoneda completeness of each ${\sf Q}$-category be equivalent to directed completeness of its set of formal balls, a necessary and sufficient condition is that the t-norm $\&$ is Archimedean.

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arXiv2021

Continuous [0,1]-lattices and injective [0,1]-approach spaces

In 1972, Dana Scott proved a fundamental result on the connection between order and topology which says that injective $T_0$ spaces are precisely continuous lattices endowed with Scott topology. This paper investigates whether this is true in an enriched context, where the enrichment is the quantale obtained by equipping the interval $[0,1]$ with a continuous t-norm. It is shown that for each continuous t-norm, the specialization $[0,1]$-order of a separated and injective $[0,1]$-approach space $X$ is a continuous $[0,1]$-lattice and the $[0,1]$-approach structure of $X$ coincides with the Scott $[0,1]$-approach structure of its specialization $[0,1]$-order; but, unlike in the classical situation, the converse fails in general.

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arXiv2021

Sober topological spaces valued in a quantale

The notion of sobriety is extended to the realm of topological spaces valued in a commutative and unital quantale, via an adjunction between a category of quantale modules and the category of quantale-valued topological spaces. Relations between such sober spaces and quantale-valued domains based on flat ideals are investigated.

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arXiv2020

The saturated prefilter monad

This paper considers some extensions of the notion of filter to the quantale-valued context, including saturated prefilter, $\top$-filter and bounded saturated prefilter. The question is whether these constructions give rise to monads on the category of sets. It is shown that the answer depends on the structure of the quantale. Specifically, if the quantale is the unit interval equipped with a continuous t-norm, then these constructions give rise to monads if and only if the implication operator corresponding to that t-norm is continuous at each point off the diagonal.

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arXiv2020

Quantale-valued dissimilarity

Inspired by the theory of apartness relations of Scott, we establish a positive theory of dissimilarity valued in an involutive quantale $\mathsf{Q}$ without the aid of negation. It is demonstrated that a set equipped with a $\mathsf{Q}$-valued dissimilarity is precisely a symmetric category enriched in a subquantaloid of the quantaloid of back diagonals of $\mathsf{Q}$. Interactions between $\mathsf{Q}$-valued dissimilarities and $\mathsf{Q}$-valued similarities (which are equivalent to $\mathsf{Q}$-valued equalities in the sense of H{ö}hle--Kubiak) are investigated with the help of lax functors. In particular, it is shown that similarities and dissimilarities are interdefinable if $\mathsf{Q}$ is a Girard quantale with a hermitian and cyclic dualizing element.

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arXiv2020

Completely distributive enriched categories are not always continuous

In contrast to the fact that every completely distributive lattice is necessarily continuous in the sense of Scott, it is shown that complete distributivity of a category enriched over the closed category obtained by endowing the unit interval with a continuous t-norm does not imply its continuity in general. Necessary and sufficient conditions for the implication are presented.

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arXiv2018

Scott approach distance on metric spaces

The notion of Scott distance between points and subsets in a metric space, a metric analogy of the Scott topology on an ordered set, is introduced, making a metric space into an approach space. Basic properties of Scott distance are investigated, including its topological coreflection and its relation to injective $T_0$ approach spaces. It is proved that the topological coreflection of the Scott distance is sandwiched between the $d$-Scott topology and the generalized Scott topology; and that every injective $T_0$ approach space is a cocomplete and continuous metric space equipped with its Scott distance.

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arXiv2018

A comparative study of ideals in fuzzy orders

This paper presents a comparative study of three kinds of ideals in fuzzy order theory: forward Cauchy ideals (generated by forward Cauchy nets), flat ideals and irreducible ideals, including their role in connecting fuzzy order with fuzzy topology.

math.GM↗
arXiv2018

Flat ideals in the unit interval with the canonical fuzzy order

A characterization of flat ideals in the unit interval with the canonical fuzzy order is obtained with the help of the ordinal sum decomposition of continuous t-norms. This characterization will be useful in the study of topological and domain theoretic properties of fuzzy orders.

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