Atomicity of Boolean algebras and vector lattices in terms of order convergence
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- by Antonio Avilés, Eugene Bilokopytov and Vladimir G. Troitsky;
- Proc. Amer. Math. Soc. 152 (2024), 3275-3287
- DOI: https://doi.org/10.1090/proc/16855
- Published electronically: June 18, 2024
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Abstract:
We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic.References
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Bibliographic Information
- Antonio Avilés
- Affiliation: Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain
- ORCID: 0000-0003-0291-3113
- Email: avileslo@um.es
- Eugene Bilokopytov
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Canada
- MR Author ID: 1057766
- ORCID: 0000-0001-7075-886X
- Email: bilokopy@ualberta.ca
- Vladimir G. Troitsky
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Canada
- MR Author ID: 341818
- Email: troitsky@ualberta.ca
- Received by editor(s): November 20, 2023
- Received by editor(s) in revised form: March 5, 2024, and March 9, 2024
- Published electronically: June 18, 2024
- Additional Notes: The first author was supported by project PID2021-122126NB-C32 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” and by Fundación Séneca - Agencia de Ciencia y Tecnología de la Región de Murcia (21955/PI/22). The second author was supported by Pacific Institute for the Mathematical Sciences. The third author was supported by Natural Sciences and Engineering Research Council of Canada.
- Communicated by: Stephen Dilworth
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3275-3287
- MSC (2020): Primary 06E10, 06F20, 46A40; Secondary 06E05, 46A19, 54A20
- DOI: https://doi.org/10.1090/proc/16855
- MathSciNet review: 4767262