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Suitable sets for paratopological groups

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

A subset S of a paratopological group G is a suitable set for G, if S is a discrete subspace of G, \(S\cup \{e\}\) is closed, and the subgroup \(\langle S\rangle \) of G generated by S is dense in G. Suitable sets in topological groups were studied by many authors. The aim of the present paper is to provide a start-up for a general investigation of suitable sets for paratopological groups, looking to what extent we can (by proving propositions) or cannot (by constructing examples) generalize to paratopological groups results which hold for topological groups, and to pose a few challenging questions for possible future research. We shall discuss when paratopological groups of different classes have suitable sets. Namely, we consider paratopological groups (in particular, countable) satisfying different separation axioms, paratopological groups which are compact-like spaces, and saturated (in particular, precompact) paratopological groups. Also we consider when a property of a group to have a suitable set is preserved with respect to (open or dense) subgroups, products and extensions.

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Notes

  1. We clarified these details via a personal communication with Guran.

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Acknowledgements

The authors thank to Igor Guran for consultations and to Taras Banakh for sharing with them the book [9] and other help. Moreover, the authors wish to thank the referee for carefully reading preliminary version of this paper and providing many valuable suggestions.

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  1. School of mathematics and statistics, Minnan Normal University, 363000, Zhangzhou, People’s Republic of China

    Fucai Lin & Tingting Shi

  2. Fujian Key Laboratory of Granular Computing and Application, Minnan Normal University, Zhangzhou, 363000, People’s Republic of China

    Fucai Lin

  3. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, Ukraine

    Alex Ravsky

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  1. Fucai Lin
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The first author is supported by the Key Program of the Natural Science Foundation of Fujian Province (no: 2020J02043), the NSFC (no. 11571158), the lab of Granular Computing, the Institute of Meteorological Big Data-Digital Fujian and Fujian Key Laboratory of Data Science and Statistics.

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Lin, F., Ravsky, A. & Shi, T. Suitable sets for paratopological groups. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 115, 183 (2021). https://doi.org/10.1007/s13398-021-01129-w

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