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The Categorical Equivalence Between Domains and Interpolative Generalized Closure Spaces

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Abstract

Closure space has been proven to be a useful tool to restructure lattices and various order structures. This paper aims to provide an approach to characterizing domains by means of closure spaces. The notion of an interpolative generalized closure space is presented and shown to generate exactly domains, and the notion of an approximable mapping between interpolative generalized closure spaces is identified to represent Scott continuous functions between domains. These produce a category equivalent to that of domains with Scott continuous functions. Meanwhile, some important subclasses of domains are discussed, such as algebraic domains, L-domains, bounded-complete domains, and continuous lattices. Conditions are presented which, when fulfilled by an interpolative generalized closure space, make the generated domain fulfill some restrictive conditions.

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Acknowledgements

We would like to give thanks to the anonymous referees for their comments and suggestions, which are of great help to improve this paper. This paper was supported by the National Natural Science Foundation of China (12231007) and Shandong Provincial Natural Science Foundation (ZR2022MA022).

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

    Longchun Wang

  2. School of Mathematics, Hunan University, Changsha, 410082, Hunan, China

    Qingguo Li

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  1. Longchun Wang
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Correspondence to Qingguo Li.

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Wang, L., Li, Q. The Categorical Equivalence Between Domains and Interpolative Generalized Closure Spaces. Stud Logica 111, 187–215 (2023). https://doi.org/10.1007/s11225-022-10024-3

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