Monoidal closedness of the category of $\top$-semiuniform convergence spaces

Yıl 2022, Cilt: 51 Sayı: 5, 1348 - 1370, 01.10.2022

Lin Zhang Bin Pang

https://doi.org/10.15672/hujms.1065246

Cited By: 7

Öz

Lattice-valued semiuniform convergence structures are important mathematical structures in the theory of lattice-valued topology. Choosing a complete residuated lattice $L$ as the lattice background, we introduce a new type of lattice-valued filters using the tensor and implication operations on $L$, which is called $\top$-filters. By means of $\top$-filters, we propose the concept of $\top$-semiuniform convergence structures as a new lattice-valued counterpart of semiuniform convergence structures. Different from the usual discussions on lattice-valued semiuniform convergence structures, we show that the category of $\top$-semiuniform convergence spaces is a topological and monoidal closed category when $L$ is a complete residuated lattice without any other requirements.

Anahtar Kelimeler

T-semiuniform convergence, T-filter, monoidal closedness, residuated lattice

Destekleyen Kurum

Natural Science Foundation of China

Proje Numarası

12071033，11701122

Kaynakça

• [1] J. Adámek, H. Herrlich and G.E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1990.
• [2] R. Bělohlávek, Fuzzy Relational Systems, Foundation and Principles, Kluwer Academic, Plenum Publishers, New York, Boston, Dordrecht, London, Moscow, 2002.
• [3] P. Eklund, J. Gutiérrez García, U. Höhle and J. Kortelainen, Semigroups in Complete Lattice, Quantales, Modules and Related Topics, Spring International Publishing AG, Gewerbestrasse, Switzerland, 2018.
• [4] J.M. Fang, Relationships between L-ordered convergence structures and strong L-topologies, Fuzzy Sets and Systems, 161, 2923-2944, 2010.
• [5] J.M. Fang, Lattice-valued preuniform convergence spaces, Fuzzy Sets and Systems, 251, 52-70, 2014.
• [6] J.M. Fang and Z. Fang, Monoidal closedness of the category of stratified L-semiuniform convergence spaces, Fuzzy Sets and Systems, 425, 83-99, 2021.
• [7] P.V. Flores, R.N. Mohapatra and G. Richardson, Lattice-valued spaces: fuzzy convergence, Fuzzy Sets and Systems 157, 2706-2714, 2006.
• [8] P.V. Flores and G.D. Richardson, Lattice-valued convergence: diagonal axioms, Fuzzy Sets and Systems, 159(19), 2520-2528, 2008.
• [9] U. Höhle, Probabilistic topologies induces by L-fuzzy uniformities, Manuscr. Math. 38, 289-323, 1982.
• [10] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18, 145-174, 1967.
• [11] G. Jäger, A category of L-fuzzy convergence spaces, Quaest. Math. 24, 501-517, 2001.
• [12] G. Jäger, Stratified $LMN$-convergence tower spaces, Fuzzy Sets and Systems, 282, 62-73, 2016.
• [13] G. Jäger and M.H. Burton, Stratified L-uniform convergence spaces, Quaest. Math. 28, 11-36, 2015.
• [14] G. Jäger and Y.L. Yue, $\top$-uniform convergence spaces, Iran. J. Fuzzy Syst. https://dx.doi.org/10.22111/ijfs.2021.6528, 2022.
• [15] L. Li and Q. Jin, On stratified L-convergence spaces: pretopological axioms and diagonal axioms, Fuzzy Sets and Systems, 204, 40-52, 2012.
• [16] L. Li and Q. Jin, p-Topologicalness and p-regularity for lattice-valued convergence spaces, Fuzzy Sets and Systems, 238, 26-45, 2014.
• [17] L. Li, Q. Jin and K. Hu, On stratified L-convergence spaces: Fischer’s diagonal axiom, Fuzzy Sets and Systems, 267, 31-40, 2015.
• [18] G. Preuss, Foundations of Topology-An Approach to Convenient Topology, Kluwer Academic Publishers, Dordrecht, 2002.
• [19] B. Pang, Degrees of separation properties in stratified L-generalized convergence spaces using residual implication, Filomat, 31(20), 6293-6305, 2017.
• [20] B. Pang, Stratified L-ordered filter spaces, Quaest. Math. 40 (5), 661-678, 2017.
• [21] B. Pang, Categorical properties of L-fuzzifying convergence spaces, Filomat 32 (11), 4021-4036, 2018.
• [22] B. Pang, Convenient properties of stratified L-convergence tower spaces, Filomat 33 (15), 4811-4825, 2019.
• [23] B. Pang, Convergence structures in M-fuzzifying convex spaces, Quaest. Math. 43 (11), 1541-1561, 2020.
• [24] Z.Y. Xiu, Q.H. Li and B. Pang, Fuzzy convergence structures in the framework of L-convex spaces, Iran. J. Fuzzy Syst. 17(4), 139-150, 2020.
• [25] W. Yao, On many-valued stratified L-fuzzy convergence spaces, Fuzzy Sets and Systems, 159, 2503-2519, 2008.
• [26] W. Yao, On L-fuzzifying convergence spaces, Iran. J. Fuzzy Syst. 6(1), 63-80, 2009.
• [27] Q. Yu and J.M. Fang, The category of $\top$-convergence spaces and its Cartesian closedness, Iran. J. Fuzzy Syst. 14(3), 121-138, 2017.
• [28] L. Zhang and B. Pang, Strong L-concave structures and L-convergence structures, J. Nonlinear Convex Anal. 21, 2759-2769, 2020.
• [29] L. Zhang and B. Pang, The category of residuated lattice valued filter spaces, Quaest. Math., https://doi.org/10.2989/16073606.2021.1973140, 2021.
• [30] L. Zhang, J.M. Fang and W.J. Wang, Monoidal closedness of L-generalized convergence spaces, Iran. J. Fuzzy Syst. 16(5), 139-153, 2019.