Type-2 lattice-valued preorders: A common framework of lattice-valued preorders and various kinds of metrics

[1] R. A. Aliev, W. Pedrycz, B. G. Guirimov, R. R. Aliev, U. Ilhan, M. Babagil, S. Mammadli, Type-2 fuzzy nueral
networks with fuzzy clustering and differential evolution optimization, Information Sciences, 181(9) (2011), 1591-1608.

[2] R. Bˇelohl´avek, Fuzzy relational systems: Foundations and principles, Kluwer Academic Publishers, Plenum Publishers, New York, 2002.

[3] R. Bˇelohl´avek, J. W. Dauben, G. J. Klir, Fuzzy logic and mathematics: A historical perspective, Oxford University
Press, Cary, 2017.

[4] I. P. Cabrera, P. Cordero, F. Garc´ıa-Pardo, M. Ojeda-Aciego, B. De Baets, Galois connections between a fuzzy
preordered structure and a general fuzzy structure, IEEE Transactions on Fuzzy Systems, 26(3) (2018), 1274-1287.

[5] V. V. Chistyakov, Metric modulars and their application, Doklady Mathematics, 73 (2006), 32-35.

[6] V. V. Chistyakov, Modular metric spaces, I: Basic concepts, Nonlinear Analysis: Theory, Methods and Applications,
72(1) (2010), 1-14.

[7] B. I. Choi, F. C. H. Rhee, Interval type-2 fuzzy membership function generation methods for pattern recognition,
Information Sciences, 179(13) (2009), 2102-2122.

[8] G. De Cooman, E. E. Kerre, Order norms on bounded partially ordered sets, Journal of Fuzzy Mathematics, 2(2)
(1994), 281-310.

[9] M. Demirci, On many-valued partitions and many-valued equivalence relations, International Journal of Uncertainty,
Fuzziness and Knowledge-Based Systems, 11(02) (2003), 235-253.

[10] M. Demirci, Representations of the extensions of many-valued equivalence relations, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 11(03) (2003), 319-341.

[11] D. Dutta, M. Sen, Type-2 fuzzy equivalence relation on a groupoid under balanced and semibalanced maps, Journal
of Informatics and Mathematical Sciences, 10 (2018), 133-151.

[12] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64(3) (1994), 395-399.

[13] M. Gondran, M. Minoux, Graphs, dioids and semirings: New models and algorithms, Springer Science and Business
Media, 2008.

[14] J. Goubault-Larrecq, Non-Hausdorff topology and domain theory, Cambridge University Press, Cambridge, 2013.

[15] V. Gregori, A. L´opez-Crevill´en, S. Morillas, A. Sapena, On convergence in fuzzy metric spaces, Topology and its
Applications, 156(18) (2009), 3002-3006.

[16] V. Gregori, J. J. Mi˜nana, S. Morillas, A. Sapena, Characterizing a class of completable fuzzy metric spaces, Topology
and its Applications, 203 (2016), 3-11.

[17] V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets and Systems, 115(3) (2000), 485-489.

[18] O. Grigorenko, J. J. Mi˜nana, A. Sostak, O. Valero, ˇ On t-conorm based fuzzy (pseudo) metrics, Axioms, 9(3) (2020), 78.U. H¨ohle, Classification of subsheaves over GL-algebras, Logic Colloquium, 98 (2000), 238-261.

[20] B. Q. Hu, C. Y. Wang, On type-2 fuzzy relations and interval-valued type-2 fuzzy sets, Fuzzy Sets and Systems,
236 (2014), 1-32.

[21] G. J¨ager, W. Yao, Quantale-valued gauge spaces, Iranian Journal of Fuzzy Systems, 15(1) (2018), 103-122.

[22] N. N. Karnik, J. M. Mendel, Operations on type-2 fuzzy sets, Fuzzy Sets and Systems, 122(2) (2001), 327-348.

[23] G. M. Kelly, Basic concepts of enriched category theory, In: London Mathematical Society Lecture Notes Series,
64, Cambridge University Press, Cambridge, 1982.

[24] E. P. Klement, R. Mesiar, E. Pap, Triangular norms, Kluwer Academic Publishers, Dordrecht, 2000.

[25] I. Kramosil, J. Mich´alek, Fuzzy metrics and statistical metric spaces, Kybernetika, 11(5) (1975), 336-344.

[26] M. Kr¨otzsch, Generalized ultrametric spaces in quantitative domain theory, Theoretical Computer Science, 368(1-2)
(2006), 30-49.

[27] H. Lai, D. Zhang, Complete and directed complete Ω-categories, Theoretical Computer Science, 388(1-3) (2007),
1-25.

[28] F. W. Lawvere, Metric spaces, generalized logic, and closed categories, Rendiconti del Seminario Mat´ematico e
Fisico di Milano, 43 (1973), 135-166.

[29] F. Liu, J. Lu, G. Zhang, Unsupervised heterogeneous domain adaptation via shared fuzzy equivalence relations,
IEEE Transactions on Fuzzy Systems, 26(6) (2018), 3555-3568.

[30] S. G. Matthews, Partial metric topology, Annals of the New York Academy of Sciences, 728(1) (1994), 183-197.

[31] K. Menger, Statistical metrics, Proceedings of the National Academy of Sciences of the United States of America,
28(12) (1942), 535-537.

[32] J. Moˇckoˇr, P. Hurtik, Lattice-valued F-transforms and similarity relations, Fuzzy Sets and Systems, 342 (2018),
67-89.

[33] N. N. Morsi, On fuzzy pseudo-normed vector spaces, Fuzzy Sets and Systems, 27(3) (1988), 351-372.

[34] I. L. Reilly, P. V. Subrahmanyam, M. K. Vamanamurthy, Cauchy sequences in quasi-pseudo-metric spaces, Monatshefte f¨ur Mathematik, 93 (1982), 127-140.

[35] B. Schweizer, A. Sklar, Statistical metric spaces, Pacific Journal of Mathematics, 10(1) (1960), 313-334.

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

[37] M. Sen, D. Dutta, A. Deshpande, Type-2 fuzzy G-tolerance relation and its properties, International Journal of
Analysis and Applications, 15(2) (2017), 172-178.

[38] C. Shen, Y. Shi, F. G. Shi, H. Andradi, Characterizations of pointwise pseudo-metrics via pointwise closed-ball
systems, IEEE Transactions on Fuzzy Systems, 30(5) (2021), 1212-1223.

[39] S. Shukla, D. Gopal, W. Sintunavarat, A new class of fuzzy contractive mappings and fixed point theorems, Fuzzy
Sets and Systems, 350 (2018), 85-94.

[40] D. W. W. W. Tan, A simplified type-2 fuzzy logic controller for real-time control, ISA Transactions, 45(4) (2006),
503-516.

[41] M. L. J. Van De Vel, Theory of convex structures, North-Holland, Elsevier Science Publishers B.V., Amsterdam,
1993.

[42] C. Y. Wang, Type-2 fuzzy rough sets based on extended t-norms, Information Sciences, 305 (2015), 165-183.

[43] J. Z. Xiao, X. H. Zhu, H. Zhou, On the topological structure of KM fuzzy metric spaces and normed spaces, IEEE
Transactions on Fuzzy Systems, 28(8) (2019), 1575-1584.

[44] W. Yao, Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy directed complete posets, Fuzzy Sets and
Systems, 161(7) (2010), 973-987.

[45] W. Yao, Y. She, L. X. Lu, Metric-based L-fuzzy rough sets: Approximation operators and definable sets, KnowledgeBased Systems, 163 (2019), 91-102.

[46] W. Yao, F. G. Shi, Quantitative domains via fuzzy sets: Part II: Fuzzy Scott topology on fuzzy directed-complete
posets, Fuzzy Sets and Systems, 173(1) (2011), 60-80.

[47] W. Yao, G. Zhang, C. J. Zhou, Real-valued hemimetric-based fuzzy rough sets and an application to contour extraction of digital surfaces, Fuzzy Sets and Systems, 459 (2023), 201-219.