Quantale-valued Cauchy tower spaces and completeness

Abstract

[EN] Generalizing the concept of a probabilistic Cauchy space, we introduce quantale-valued Cauchy tower spaces. These spaces encompass quantale-valued metric spaces, quantale-valued uniform (convergence) tower spaces and quantale-valued convergence tower groups. For special choices of the quantale, classical and probabilistic metric spaces are covered and probabilistic and approach Cauchy spaces arise. We also study completeness and completion in this setting and establish a connection to the Cauchy completeness of a quantale-valued metric space.Jäger, G.; Ahsanullah, TMG. (2021). Quantale-valued Cauchy tower spaces and completeness. Applied General Topology. 22(2):461-481. https://doi.org/10.4995/agt.2021.15610OJS461481222J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1989.T. M. G. Ahsanullah and G. Jäger, Probabilistic uniform convergence spaces redefined, Acta Math. Hungarica 146 (2015), 376-390. https://doi.org/10.1007/s10474-015-0525-6T. M. G. Ahsanullah and G. Jäger, Probabilistic uniformization and probabilistic metrization of probabilistic convergence groups, Math Slovaca 67 (2017), 985-1000. https://doi.org/10.1515/ms-2017-0027P. Brock and D. C. Kent, Approach spaces, limit tower spaces, and probabilistic convergence spaces, Appl. Cat. Structures 5 (1997), 99-110. https://doi.org/10.1023/A:1008633124960H. R. Fischer, Limesräume, Math. Ann. 137 (1959), 269-303. https://doi.org/10.1007/BF01360965R. C. Flagg, Completeness in continuity spaces, in: Category Theory 1991, CMS Conf. Proc. 13 (1992), 183-199.R. C. Flagg, Quantales and continuity spaces, Algebra Univers. 37 (1997), 257-276. https://doi.org/10.1007/s000120050018L. C. Florescu, Probabilistic convergence structures, Aequationes Math. 38 (1989), 123-145. https://doi.org/10.1007/BF01839999G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott, Continuous lattices and domains, Cambridge University Press, 2003. https://doi.org/10.1017/CBO9780511542725D. Hofmann and C. D. Reis, Probabilistic metric spaces as enriched categories, Fuzzy Sets and Systems 210 (2013), 1-21. https://doi.org/10.1016/j.fss.2012.05.005U. Höhle, Commutative, residuated l-monoids, in: Non-classical logics and their applications to fuzzy subsets (U. Höhle, E. P. Klement, eds.), Kluwer, Dordrecht 1995, pp. 53-106. https://doi.org/10.1007/978-94-011-0215-5_5G. Jäger, A convergence theory for probabilistic metric spaces, Quaest. Math. 38 (2015), 587-599. https://doi.org/10.2989/16073606.2014.981734G. Jäger and T. M. G. Ahsanullah, Probabilistic limit groups under a $t$-norm, Topology Proceedings 44 (2014), 59-74.G. Jäger and T. M. G. Ahsanullah, Characterization of quantale-valued metric spaces and quantale-valued partial metric spaces by convergence, Applied Gen. Topology 19, no. 1 (2018), 129-144. https://doi.org/10.4995/agt.2018.7849G. Jäger, Quantale-valued uniform convergence towers for quantale-valued metric spaces, Hacettepe J. Math. Stat. 48, no. 5 (2019), 1443-1453. https://doi.org/10.15672/HJMS.2018.585G. Jäger, The Wijsman structure of a quantale-valued metric space, Iranian J. Fuzzy Systems 17, no. 1 (2020), 171-184.H. H. Keller, Die Limes-Uniformisierbarkeit der Limesräume, Math. Ann. 176 (1968), 334-341. https://doi.org/10.1007/BF02052894D. C. Kent and G. D. Richardson, Completions of probabilistic Cauchy spaces, Math. Japonica 48, no. 3 (1998), 399-407.F. W. Lawvere, Metric spaces, generalized logic, and closed categories, Rendiconti del Seminario Matematico e Fisico di Milano 43 (1973), 135-166. Reprinted in: Reprints in Theory and Applications of Categories} 1 (2002), 1-37. https://doi.org/10.1007/BF02924844R. Lowen, Index Analysis, Springer, London, Heidelberg, New York, Dordrecht 2015. https://doi.org/10.1007/978-1-4471-6485-2R. Lowen and Y. J. Lee, Approach theory in merotopic, Cauchy and convergence spaces. I, Acta Math. Hungarica 83, no. 3 (1999), 189-207. https://doi.org/10.1023/A:1006717022079R. Lowen and Y. J. Lee, Approach theory in merotopic, Cauchy and convergence spaces. II, Acta Math. Hungarica 83, no. 3 (1999), 209-229. https://doi.org/10.1023/A:1006769006149R. Lowen and B. Windels, On the quantification of uniform properties, Comment. Math. Univ. Carolin. 38, no. 4 (1997), 749-759.R. Lowen and B. Windels, Approach groups, Rocky Mountain J. Math. 30, no. 3 (2000), 1057-1073. https://doi.org/10.1216/rmjm/1021477259J. Minkler, G. Minkler and G. Richardson, Subcategories of filter tower spaces, Appl. Categ. Structures 9 (2001), 369-379. https://doi.org/10.1023/A:1011226611840H. Nusser, A generalization of probabilistic uniform spaces, Appl. Categ. Structures 10 (2002), 81-98. https://doi.org/10.1023/A:1013375301613H. Nusser, Completion of probabilistic uniform limit spaces, Quaest. Math. 26 (2003), 125-140. https://doi.org/10.2989/16073600309486049G. Preuß, Seminuniform convergence spaces, Math. Japonica 41, no. 3 (1995), 465-491.G. Preuß, Foundations of topology. An approach to convenient topology, Kluwer Academic Publishers, Dordrecht, 2002. https://doi.org/10.1007/978-94-010-0489-3Q. Pu and D. Zhang, Preordered sets valued in a GL-monoid, Fuzzy Sets and Systems 187 (2012), 1-32. https://doi.org/10.1016/j.fss.2011.06.012G. N. Raney, A subdirect-union representation for completely distributive complete lattices, Proc. Amer. Math. Soc. 4 (1953), 518-512. https://doi.org/10.1090/S0002-9939-1953-0058568-4E. E. Reed, Completion of uniform convergence spaces, Math. Ann. 194 (1971), 83-108. https://doi.org/10.1007/BF01362537G. D. Richardson and D. C. Kent, Probabilistic convergence spaces, J. Austral. Math. Soc. (Series A) 61 (1996), 400-420. https://doi.org/10.1017/S1446788700000483B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland, New York, 1983