Characterization of quantale-valued metric spaces and quantale-valued partial metric spaces by convergence

Gunther Jäger, T. M. G. Ahsanullah


We identify two categories of quantale-valued convergence tower spaces that are isomorphic to the categories of quantale-valued metric spaces and quantale-valued partial metric spaces, respectively. As an application we state a quantale-valued metrization theorem for quantale-valued convergence tower groups.


L-metric space; L-partial metric space; L-convergence tower space; L-convergence tower group; metrization

Subject classification

54A20; 54A40; 54E35; 54E70; 54E99.

Full Text:



S. Abramsky and A. Jung, Domain Theory, in: S. Abramsky, D. M. Gabby, T. S. E. Maibaum (Eds.), Handbook of Logic and Computer Science, Vol. 3, Claredon Press, Oxford 1994.

J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1989.

T. M. G. Ahsanullah and G. Jäger, On approach limit groups and their uniformization, Int. J. Contemp. Math. Sciences 9 (2014), 195-213.

T. M. G. Ahsanullah and G. Jäger, Probabilistic uniformization and probabilistic metrization of probabilistic convergence groups, Math. Slovaca 67 (2017), 985-1000.

T. M. G. Ahsanullah and G. Jäger, Stratified LMN-convergence tower groups and their stratified LMN-uniform convergence tower structures, Fuzzy Sets and Systems 330 (2018), 105-123.

R. Beattie and H.-P. Butzmann, Convergence Structures and Applications to Functional Analysis, Springer Science & Business Media, 2002.

P. Brock, Probabilistic convergence spaces and generalized metric spaces, Int. J. Math. and Math. Sci. 21 (1998), 439-452.

P. Brock and D. C. Kent, Approach spaces, limit tower spaces, and probabilistic convergence spaces, Appl. Cat. Structures 5 (1997), 99-110.

R. C. Flagg, Quantales and continuity spaces, Algebra Univers. 37 (1997), 257-276.

G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott,Continuous lattices and domains, Cambridge University Press 2003

U. Höhle, M-valued sets and sheaves over integral cl-monoids, in: S.E. Rodabaugh, E.P. Klement and U. Höhle (eds.), Applications of category theory to fuzzy subsets, Kluwer, Boston 1992, 33-72.

U. Höhle, Presheaves over GL-monoids, in: U. Höhle and E.P. Klement (eds.), Non-classical logics and their applications to fuzzy subsets, Kluwer, Boston 1995, 127-157.

U. Höhle, Commutative, residuated l-monoids, in: Non-classical logics and their applications to fuzzy subsets (U. Höhle, E. P. Klement, eds.), Kluwer, Dordrecht 1995, 53-106.

G. Jäger, A convergence theory for probabilistic metric spaces, Quaestiones Math. 38 (2015), 587-599.

G. Jäger, Stratified LMN-convergence tower spaces, Fuzzy Sets and Systems 282 (2016), 62-73.

G. Jäger and T. M. G. Ahsanullah, Probabilistic limit groups under a t-norm, Topology Proc. 44 (2014), 59-74.

R. Kopperman, S. Matthews and H. Pajoohesh, Partial metrizability in value quantales, Applied General Topology 5 (2004), 115-127.

H.-J. Kowalsky, Limesraume und Komplettierung, Math. Nachrichten 12 (1954), 301-340.

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.

R. Lowen, Approach spaces. The missing link in the topology-uniformity-metric triad, Claredon Press, Oxford 1997.

S. G. Matthews, Metric domains for completeness, PhD thesis, University of Warwick, 1985.

S. G. Matthews, Partial metric topology, Annals of the New York Academy of Sciences 728 (1994), 183-197.

G. Preuss, Theory of topological structures, D. ReidelPublishing Company, Dor-drecht/Boston/Lancaster/Tokyo 1988.

G. Preuss, Foundations of topology. An approach to convenient topology, Kluwer Aca-demic Publishers, Dordrecht 2002.

G. N. Raney, A subdirect-union representation for completely distributive complete lattices, Proc. Amer. Math. Soc. 4 (1953), 518-512.

G. D. Richardson and D. C. Kent, Probabilistic convergence spaces, J. Austral. Math. Soc. (Series A) 61 (1996), 400-420.

K. I. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Mathe-matics 234, Longman, Burnt Mill, Harlow 1990.

B. Schweizer and A. Sklar, Probabilistic metric spaces, North Holland, New York, 1983.

R. M. Tardiff, Topologies for probabilistic metric spaces, Pacific J. Math. 65 (1976), 233-251.

J. Wu and Y. Yue, Formal balls in fuzzy partial metric spaces, Iranian J. Fuzzy Systems 14 (2017), 155-164.

Yue, Separated ∆+-valued equivalences as probabilistic partial metric spaces, Journalof Intelligent & Fuzzy Systems 28 (2015), 2715–2724.

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1. Quantale-Valued Generalizations of Approach Groups
T. M. G. Ahsanullah, Gunther Jäger
New Mathematics and Natural Computation  vol: 15  issue: 01  first page: 1  year: 2019  
doi: 10.1142/S1793005719500017

Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147