As a natural extension of probabilistic uniform spaces, we further introduce and study$L$-Čech uniform spaces, and also present their connection with $L$-Čech closure spaces.
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez:In this paper, we first give a concrete example to illustrate that the construction of$L$-Čech closure operators given in [27, Theorem 3.1] fails to hold when $L$ is a continuous residuated lattice.
📊
Temel Bulgular & Veri:By strengthening $L$ to be completely distributive, we propose two new kinds of operators, which are respectively called generalized closure operators and generalized$L$-closure operators, and establish the categorical relation between them.
💡
Sonuç & Katkı:As a natural extension of probabilistic uniform spaces, we further introduce and study$L$-Čech uniform spaces, and also present their connection with $L$-Čech closure spaces.
In this paper, we first give a concrete example to illustrate that the construction of$L$-Čech closure operators given in [27, Theorem 3.1] fails to hold when $L$ is a continuous residuated lattice. By strengthening $L$ to be completely distributive, we propose two new kinds of operators, which are respectively called generalized closure operators and generalized$L$-closure operators, and establish the categorical relation between them. As an application, we prove the existence of a Galois connection between the category of stratified generalized$L$-closure spaces and the category of stratified$L$-convex spaces. As a natural extension of probabilistic uniform spaces, we further introduce and study$L$-Čech uniform spaces, and also present their connection with $L$-Čech closure spaces.
J. Adámek, H. Herrlich and G.E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1990.
B. Chacko, Fuzzy closure operator induced by a fuzzy pseudo metric, Int. J. Fuzzy Math. Syst. 2, 345–350, 2012.
E. Čech, Topological Spaces, Interscience Publishers, John Wiley and Sons, New York, 1966.
P. Cordero, M. Enciso, A. Mora and V. Vychodil, Parameterized simplification logic I: reasoning with implications and classes of closure operators, Int. J. Gen. Syst. 49, 724–746, 2020.
B. De Baets, Analytical Solution Methods for Fuzzy Relational Equations, Springer US, Boston, MA, 2000.
D. Dikranjan and E. Giuli, Closure operators induced by topological epireflections, COIL Math. Sot. J. Bolyai. 41, 233–246, 1983.
M. Diker, Ş. Dost and A.A. Uğur, Interior and closure operators on texture spaces-I: Basic concepts and Čech closure operators, Fuzzy Sets Syst. 161, 935–953, 2009.
M. Diker, Ş. Dost and A.A. Uğur, Interior and closure operators on texture spaces-II: Dikranjan-Giuli closure operators and Hutton algebras, Fuzzy Sets Syst. 161, 954– 972, 2010.
Y. Dong and F.-G. Shi, L-cones and L-dual cones, Fuzzy Sets Syst. 469, 108626, 2023.
G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003.
P. Hájek, Metamathematics of fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.
O.A. Heubo-Kwegna and J.B. Nganou, Closure operations on MV-algebras, Fuzzy Sets Syst. 418, 139–152, 2021.
B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal. Appl. 58, 559–571, 1977.
U. Höhle, Probabilistic uniformization of fuzzy topologies, Fuzzy Sets Syst. 1, 311–332, 1978.
V. Istrăţescu, An Introduction to the Theory of Probabilistic Metric Spaces, with Applications, Ed. Tehnică Bucureşti, 1974.
T. Kuhr and V. Vychodil, Fuzzy logic programming reduced to reasoning with attribute implications, Fuzzy Sets Syst. 262, 1–20, 2015.
R. Lowen, Fuzzy uniform spaces, J. Math. Anal. Appl. 82, 370–385, 1981.
E.H. Moore, Introduction to a Form of General Analysis, Vol. 2, Yale University Press, 1910.
N. Madrid, J. Medina, M. Ojeda-Aciego and I. Perfilieva, Toward the use of fuzzy relations in the definition of mathematical morphology operators, J. Fuzzy Set Valued Anal. 2016, 87–98, 2016.
M. Ojeda-Hernández, I.P. Cabrera, P. Cordero and E. Muñoz-Velasco, Fuzzy closure structures as formal concepts, Fuzzy Sets Syst. 463, 108458, 2023.
M. Ojeda-Hernández, I.P. Cabrera, P. Cordero and E. Muñoz-Velasco, Fuzzy closure structures as formal concepts II, Fuzzy Sets Syst. 473, 108734, 2023.
B. Pang and F.-G. Shi, Subcategories of the category of L-convex spaces, Fuzzy Sets Syst. 313, 61–74, 2017.
B. Pang and F.-G. Shi, Strong inclusion orders between L-subsets and its applications in L-convex spaces, Quaes. Math. 41, 1021–1043, 2018.
J. Poelmans, S.O. Kuznetsov, D.I. Ignatov and G. Dedene, Formal concept analysis in knowledge processing: a survey on models and techniques, Expert Syst. Appl. 40, 6601–6623, 2013.
J. Qiao, A note on the specialization order of L-co-topological space, Fuzzy Syst. Math. 29, 37–43, 2015. (in Chinese).
J. Qiao, On the reflective and coreflective subcategory of stratified L-Čech closure spaces, Fuzzy Sets Syst. 390, 105–117, 2020.
V. Chvátal, Antimatroids, Betweenness, Convexity, Research Trends in Combinatorial Optimization, 57–64, 2009.
C. Shen and F.-G. Shi, Characterizations of L-convex spaces via domain theory, Fuzzy Sets Syst. 380, 44–63, 2020.
F.-G. Shi, Theory of $L_{\beta}$-nested sets and Lα-nested sets and its applications, Fuzzy Syst. Math. 4, 65–72, 1995 (in Chinese).
M. Van De Vel, Theory of Convex Structures, North-Holland, Amsterdam, 1993.
G.J. Wang, Theory of topological molecular lattices, Fuzzy Sets Syst. 47, 351–376, 1992.
M. Ward and R.P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc. 1939.
S.N. Shukla and S. Tiwari, Ultra-pseudo metric spaces and their characterization with completely regular topological spaces, Topol. Appl. 353, 108955, 2024.