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A categorical equivalence between Q-domains and interpolative generalized Q-closure spaces

IF 3.2 1区数学 Q2 COMPUTER SCIENCE, THEORY & METHODS

Fuzzy Sets and Systems Pub Date : 2025-01-16 DOI:10.1016/j.fss.2025.109280

Guojun Wu , Wei Yao , Qingguo Li

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引用次数: 0

Abstract

With a commutative unital quantale

Q

as the truth value table, this study focuses on representations of

Q

-domains by means of generalized

Q

-closure spaces. First, the notions of interpolative generalized

Q

-closure spaces and directed closed sets are introduced. It is proved that for an interpolative generalized

Q

-closure space (resp., a

Q

-closure space), the collection of directed closed sets with respect to the inclusion

Q

-order forms a continuous

Q

-dcpo (resp., an algebraic

Q

-dcpo). Conversely, it is shown that every continuous

Q

-dcpo (resp., algebraic

Q

-dcpo) can be reconstructed from an interpolative generalized

Q

-closure space (resp., a

Q

-closure space). Second, the notion of dense subspaces of generalized

Q

-closure spaces is provided. By means of dense subspaces, an alternative representation for algebraic

Q

-dcpos is given. Furthermore, the notion of approximable

Q

-relations between interpolative generalized

Q

-closure spaces is introduced. Consequently, a categorical equivalence is established between the category of interpolative generalized

Q

-closure spaces (resp.,

Q

-closure spaces) with approximable

Q

-relations and that of continuous

Q

-dcpos (resp., algebraic

Q

-dcpos) with Scott continuous mappings.

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Q 域与内插广义 Q 封闭空间之间的等价性

本文以可交换的单位量子Q为真值表，研究了用广义Q闭包空间表示Q域的方法。首先，引入了插值广义q闭空间和有向闭集的概念。证明了对于一个插值广义q闭包空间。（一个q闭空间），关于包含q阶的有向闭集的集合形成了一个连续的Q-dcpo（相对于Q-dcpo）。，代数Q-dcpo)。相反地，我们发现每一个连续的Q-dcpo (resp。，代数Q-dcpo)可以由插值广义q -闭包空间重构。（一个q闭包空间）。其次，给出了广义q闭包空间的密子空间的概念。利用密集子空间，给出了代数Q-dcpos的另一种表示形式。进一步，引入了插值广义q闭空间间的近似q关系的概念。因此，在插值广义q闭空间的范畴之间建立了一个范畴等价。，具有近似q关系的q -闭包空间)和连续Q-dcpos的q -闭包空间。，具有Scott连续映射的代数Q-dcpos。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Fuzzy Sets and Systems 数学-计算机：理论方法

CiteScore

6.50

自引率

17.90%

发文量

321

审稿时长

6.1 months

期刊介绍： Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies. In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.