Degrees of compatible $L$-subsets and compatible mappings

Shi, Fu-Gui ; Sun, Yan

Kybernetika, Tome 60 (2024) no. 2, pp. 172-196

Abstract (VO)

Based on a completely distributive lattice $L$, degrees of compatible $L$-subsets and compatible mappings are introduced in an $L$-approximation space and their characterizations are given by four kinds of cut sets of $L$-subsets and $L$-equivalences, respectively. Besides, some characterizations of compatible mappings and compatible degrees of mappings are given by compatible $L$-subsets and compatible degrees of $L$-subsets. Finally, the notion of complete $L$-sublattices is introduced and it is shown that the product of complete $L$-sublattices is still a complete $L$-sublattice and the compatible degree of an $L$-subset is a complete $L$-sublattice.

Détail

DOI : 10.14736/kyb-2024-2-0172

Classification : 06B75, 06D10, 06D72
Keywords: $L$-approximation spaces; compatible $L$-subsets; compatible mappings; complete $L$-sublattices

Bibliographie

[1] Ajmal, N., Thomas, K. V.: Fuzzy lattices. Inf. Sci. 79 (1994), 271-291. | DOI | MR

[2] Bandler, W., Kohout, L. J.: Semantics of implication operators and fuzzy relational products. Int. J. Man-Machine Studies 12 (1980), 89-116. | DOI | MR

[3] Bělohlávek, R.: Fuzzy Relational Systems. Foundations and Principles. Kluwer Academic Publishers, New York 2002.

[4] Bělohlávek, R., Dvořák, J., Outrata, J.: Fast factorization by similarity in formal concept analysis of data with fuzzy attributes. J. Comput. Syst. Sci. 73 (2007), 6, 1012-1022. | DOI | MR

[5] Berry, V., Nicolas, F.: Improved parameterized complexity of the maximum agreement subtree and maximum compatible tree problems. IEEE/ACM Trans. Comput. Biol. Bioinform. 3 (2006), 3, 289-302. | DOI

[6] Birkhoff, G.: Lattice Theory. AMS, 1948. | MR | Zbl

[7] Demirci, M.: Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, Part I: Fuzzy functions and their applications. Int. J. Gen. Syst. 32 (2003), 123-155. | DOI | MR

[8] Dong, Y. Y., Shi, F.-G.: $L$-fuzzy sub-effect algebras. Mathematics 9 (2021), 14, 1596. | DOI

[9] Fang, J. M.: Residuated Lattices and Fuzzy Sets. Science Press (in Chinese), 2012.

[10] Ganapathy, G., Warnow, T. J.: Approximating the complement of the maximum compatible subset of leaves of trees. Lecture Notes Comput. Sci. 2462 (2002), 122-134. | DOI | MR

[11] Ganter, B., Wille, R.: Formal Concept Analysis-Mathematical Foundations. Springer, 1999. | DOI | MR

[12] Gao, Y., Pang, B.: Subcategories of the category of $\top$-convergence spaces. Hacet. J. Math. Stat. 53 (2024), 1, 88-106. | DOI | MR

[13] Gégény, D., Radeleczki, S.: Rough L-fuzzy sets: Their representation and related structures. Int. J. Approx. Reason. 142 (2022), 1-12. | DOI | MR

[14] Goguen, J. A.: $L$-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 145-174. | DOI | MR | Zbl

[15] Goguen, J. A.: Logic of inexact concepts. Synthese 19 (1968), 325-373. | DOI

[16] Gottwald, S.: Treatise on Many-Valued Logics. Research Studies Press, Baldock 2001. | MR

[17] Huang, H. L., Shi, F.-G.: $L$-fuzzy numbers and their properties. Inf. Sci. 178 (2008), 1141-1151. | DOI | MR

[18] Klawonn, F.: Fuzzy points, fuzzy relations and fuzzy functions. In: Discovering World with Fuzzy Logic (Novak, Perfilieva, eds.), Physica-Verlag, 2000, pp. 431-453. | MR

[19] Klawonn, F., Castro, J. L.: Similarity in fuzzy reasoning. Mathware Soft Comput. 2 (1995), 197-228. | MR

[20] Konecny, J., Krupka, M.: Complete relations on fuzzy complete lattices. Fuzzy Sets Syst. 320 (2017), 1, 64-80. | DOI | MR

[21] Li, H. Y., Shi, F.-G.: Degrees of fuzzy compactness in $L$-fuzzy topological spaces. Fuzzy Sets Syst., 161 (2010) 988-1001. | DOI | MR

[22] Li, J., Shi, F.-G.: $L$-fuzzy convexity induced by $L$-convex fuzzy sublattice degree. Iran. J. Fuzzy Syst. 14 (2017), 5, 83-102. | MR

[23] Liang, C. Y., Shi, F.-G.: Degree of continuity for mappings of ($L$,$M$)-fuzzy topological spaces. J. Intell. Fuzzy Syst. 27 (2014), 2665-2677. | DOI | MR

[24] Pang, B.: Bases and subbases in ($L$,$M$)-fuzzy convex spaces. J. Comput. Appl. Math. 39 (2020), 41. | DOI | MR

[25] Pang, B.: Convergence structures in $M$-fuzzifying convex spaces. Quaest. Math. 43 (2020), 11, 1541-1561. | DOI | MR

[26] Pang, B.: Quantale-valued convex structures as lax algebras. Fuzzy Sets Syst.473 (2023), 108737. | DOI | MR

[27] Pang, B.: Fuzzy convexities via overlap functions. IEEE T. fuzzy Syst. 31 (2023), 4, 1071-1082. | DOI

[28] Pawlak, Z.: Rough sets and fuzzy sets. Fuzzy Sets Syst. 17 (1985), 99-102. | DOI | MR

[29] Shen, Y. T., Xiong, Y. J., Xia, W., Soatto, S.: Towards backward-compatible representation learning. In: Proc. IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2020, pp. 6368-6377.

[30] Shi, F.-G.: Theory of $L_\alpha$-nest sets and $L_\beta$-nest sets and their applications. Fuzzy Syst. Math. 4 (1995), 65-72. | DOI | MR

[31] Shi, F.-G.: $L$-fuzzy relation and $L$-fuzzy subgroup. J. Fuzzy Math. 8 (2000), 491-499. | MR

[32] Shi, F.-G., Xin, X.: $L$-fuzzy subgroup degrees and $L$-fuzzy normal subgroup degrees. J. Adv. Res. Pure Math. 3 (2011), 92-108. | DOI | MR

[33] Shi, Y., Pang, B., Baets, B. De: Fuzzy structures induced by fuzzy betweenness relations. Fuzzy Sets Syst. 466 (2023), 108443. | DOI | MR

[34] Sun, Y., Mi, J. S., Chen, J. K., Liu, W.: A new fuzzy multi-attribute group decision-making method with generalized maximal consistent block and its application in emergency management. Knowl. Based Syst. 215 (2021), 106594. | DOI

[35] Sun, Y., Shi, F.-G.: Representations of L-fuzzy rough approximation operators. Inf. Sci. 645 (2023), 119324. | DOI

[36] Tepav\u{c}ević, A., Trajkovski, D.: $L$-fuzzy lattices: an introduction. Fuzzy Sets Syst. 123 (2001), 209-216. | DOI | MR

[37] Wang, G. J.: Theory of topological molecular lattices. Fuzzy Sets Syst. 47 (1992), 351-376. | DOI | MR

[38] Xiu, Z. Y., Li, Q.-H., Pang, B.: Fuzzy convergence structures in the framework of L-convex spaces. Iran. J. Fuzzy Syst. 17 (2020), 4, 139-150. | MR

[39] Yao, Y. Y.: Three-way granular computing, rough sets, and formal concept analysis. Int. J. Approx. Reason. 116 (2020), 106-125. | DOI | MR

[40] Ying, M.: A new approach for fuzzy topology (I). Fuzzy Sets Syst. 39 (3) (1991), 303-321. | DOI | MR

[41] Yuan, B., Wu, W.: Fuzzy ideals on a distributive lattice. Fuzzy Sets Syst. 35 (1990), 231-240. | DOI | MR

[42] Zadeh, L. A.: Fuzzy sets. Inform. Control. 8 (1965), 338-353. | DOI | MR | Zbl

[43] Zadeh, L. A.: Similarity relations and fuzzy orderings. Inf. Sci. 3 (1971) 177-200. | DOI | MR

[44] Zadeh, L. A.: The concept of a linguistic variable and its application to approximation reasoning I. Inf. Sci. 8 (1975), 3, 199-251. | DOI | MR

[45] Zhang, L., Pang, B.: Monoidal closedness of the category of $L$-semiuniform convergence spaces. Hacet. J. Math. Stat. 51 (2022), 5, 1348-1370. | DOI | MR

[46] Pang, L. Zhang andB.: Convergence structures in $(L,M)$-fuzzy convex spaces. Filomat 37 (2023), 9, 2859-2877. | DOI | MR

[47] Zhang, L., Pang, B., Li, W.: Subcategories of the category of stratified $(L,M)$-semiuniform convergence tower spaces. Iran. J. Fuzzy Syst. 20 (2023), 4, 179-192. | MR

[48] Zhang, L., Pang, B.: A new approach to lattice-valued convergence groups via $\top$-filters. Fuzzy Sets Syst. 455 (2023), 198-221. | DOI | MR

[49] Zhao, F. F., Pang, B.: Equivalence among $L$-closure (interior) operators, $L$-closure (interior) systems and $L$-enclosed (internal) relations. Filomat 36 (2022), 3, 979-1003. | DOI | MR