Document Type : Research Paper

**Authors**

^{1}
University of Applied Sciences Stralsund, Stralsund, Germany

^{2}
School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

**Abstract**

We show, for a commutative and integral quantale, that the recently introduced category of $\top$-uniform convergence spaces is a topological category which possesses natural function spaces, making it Cartesian closed. Furthermore, as two important examples for $\top$-uniform convergence spaces, we study probabilistic uniform spaces and quantale-valued metric spaces. The underlying $\top$-convergence spaces are also described and it is shown that in the case of a probabilistic uniform space this $\top$-convergence is the convergence of a fuzzy topology with conical neighbourhood filters. Finally it is shown that the category of $\top$-uniform convergence spaces can be embedded into the category of stratified lattice-valued uniform convergence spaces as a reflective subcategory.

**Keywords**

[1] J. Adámek, H. Herrlich, G. E. Strecker, Abstract and concrete categories, Wiley, New York, 1989.

[2] R. Bĕlohlávek, Fuzzy relation systems, foundation and principles, Klumer Academic/Plenum Publishers, New York, Boston, Dordrecht, London, Moscow, 2002.

[3] C. H. Cook, H. R. Fischer, Uniform convergence structures, Mathematische Annalen, 173 (1967), 290-306.

[4] A. Craig, G. Jäger, A common framework for lattice-valued uniform spaces and probabilistic uniform limit spaces, Fuzzy Sets and Systems, 160(9) (2009), 1177-1203.

[5] J. Fang, Stratified L-ordered convergence structures, Fuzzy Sets and Systems, 161 (2010), 2130-2149.

[6] J. Fang, Y. Yue, ⊤-diagonal conditions and continuous extension theorem, Fuzzy Sets and Systems, 321 (2017), 73-89.

[7] R. C. Flagg, Quantales and continuity spaces, Algebra Universalis, 37 (1997), 257-276.

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

[9] J. Gutierrez-Garcia, On stratified L-valued filters induced by ⊤-filters, Fuzzy Sets and Systems, 157 (2006), 813-819.

[10] J. He, H. Lai, L. Shen, Towards probabilistic partial metric spaces: Diagonals between distance distributions, Fuzzy Sets and Systems, 370 (2019), 99-119.

[11] U. Höhle, Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Mathematica, 38 (1982), 289-323.

[12] U. Höhle, Commutative, residuated l-monoids, in: U. H¨ohle, E. P. Klement, (Eds.), Non-classical logics and their applications to fuzzy subsets, Kluwer, Dordrecht, (1995), 53-106.

[13] U. Höhle, A. Šostak, Axiomatic foundations of fixed-basis fuzzy topology, in: U. Höhle, S. E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, 3, Kluwer Academic Publishers, Dordrecht, (1999), 123-272.

[14] D. Hofmann, G. J. Seal, W. Tholen, Monoidal topology, Cambridge University Press, Cambridge, 2014.

[15] G. Jäger, M. Burton, Stratified L-uniform convergence spaces, Quaestiones Mathematicae, 28 (2005), 11-36.

[16] H. Lai, D. Zhang, Fuzzy topological spaces with conical neighborhood systems, Fuzzy Sets and Systems, 330 (2018), 87-104.

[17] 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.

[18] R. S. Lee, The category of uniform convergence spaces is Cartesian closed, Bulletin of the Australian Mathematical Society, 15 (1976), 461-465.

[19] L. Li, Q. Jin, K. Hu, Lattice-valued convergence associated with CNS spaces, Fuzzy Sets and Systems, 370 (2019), 91-98.

[20] G. Preuss, Foundations of topology: An approach to convenient topology, Kluwer Academic Publishers, Dordrecht, 2002.

[21] Q. Pu, D. Zhang, Preordered sets valued in a GL-monoid, Fuzzy Sets and Systems, 187 (2012), 1-32.

[22] G. N. Raney, A subdirect-union representation for completely distributive complete lattices, Proceedings American Mathematical Society, 4 (1953), 518-512.

[23] L. Reid, G. Richardson, Connecting ⊤ and lattice-valued convergences, Iranian Journal of Fuzzy Systems, 15(4), (2018), 151-169.

[24] L. Reid, G. Richardson, Lattice-valued spaces: ⊤-completions, Fuzzy Sets and Systems, 369 (2019), 1-19.

[25] B. Schweizer, A. Sklar, Probabilistic metric spaces, North Holland, New York, 1983.

[26] O. Wyler, Filter space monads, regularity, completions, TOPO 1972 - General Topology and its Applications, Lecture notes in Mathematics, 378, Springer-Verlag, Berlin - Heidelberg - New York, (1974), 591-637.

[27] Q. Yu, J. Fang, The category of ⊤-convergence spaces and its Cartesian-closedness, Iranian Journal of Fuzzy Systems, 14(3) (2017), 121-138.

[28] Y. Yue, J. Fang, Completeness in probabilistic quasi-uniform spaces, Fuzzy Sets and Systems, 370 (2019), 34-62.

[29] Y. Yue, J. Fang, The ⊤-filter monad and its applications, Fuzzy Sets and Systems, 382 (2020), 79-97.

[30] D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems, 158 (2007), 349-366.

March and April 2022

Pages 133-149