Korean J. Math. Vol. 32 No. 4 (2024) pp.801-811
DOI: https://doi.org/10.11568/kjm.2024.32.4.801

Two types of algebraic structures based on generalized residuated lattices

Article Sidebar

PDF
Published: Dec 30, 2024
Keywords:
Generalized residuated lattices, bi-preorders, bi-interior( bi-closure) operators, bi-interior(bi-closure) systems, left(right) rough sets
Mathematics Subject Classification:
03E72, 54A40, 54B10

Main Article Content

Jin-Won Park
Department of Mathematics Education, Je-Ju National University, Jeju 63243, Republic of Korea
Young-Hee Kim
Ingenium College of Liberal Arts, Kwangwoon University, Seoul 01897, Republic of Korea
https://orcid.org/0000-0003-3430-5315

Abstract

In this paper, we introduce two types of left and right algebraic structures. We investigate the relations between bi-interior(bi-closure) operators and bi-interior(bi-closure) systems. We explore how a bi-preordered space leads to the formation of right and left rough sets.



Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.

Supporting Agencies

Kwangwoon University

References

[1] R. Bělohlávek, "Fuzzy Relational Systems," Kluwer Academic Publishers, New York, 2002. Available: https://www.kybernetika.cz/content/2003/1/119/paper.pdf Google Scholar

[2] R. Bělohlávek, "Fuzzy closure operator," J. Math. Anal. Appl., vol. 262, pp. 473–486, 2001. Google Scholar

[3] L. Biacino and G. Gerla, "Closure systems and L-subalgebras," Inf. Sci., vol. 33, pp. 181–195, 1984. Google Scholar

[4] L. Biacino and G. Gerla, "An extension principle for closure operators," J. Math. Anal. Appl., vol. 198, pp. 1–24, 1996. Google Scholar

[5] L. Guo, G. Q. Zhang, and Q. Li, "Fuzzy closure systems on L-ordered sets," Math. Log. Quart., vol. 57(3), pp. 281–291, 2011. DOI: https://doi.org/10.1002/malq.201010007 Google Scholar

[6] J. Fang and Y. Yue, "L-fuzzy closure systems," Fuzzy Sets Syst., vol. 161, pp. 1242–1252, 2010. DOI: https://doi.org/10.1016/j.fss.2009.10.002 Google Scholar

[7] G. Georgescu and A. Popescue, "Non-commutative Galois connections," Soft Comput., vol. 7, pp. 458–467, 2003. Google Scholar

[8] P. Hájek, "Metamathematics of Fuzzy Logic," Kluwer Academic Publishers, Dordrecht, 1998. Google Scholar

[9] J. M. Ko and Y. C. Kim, "Bi-closure systems and bi-closure operators on generalized residuated lattices," J. Intell. Fuzzy Syst., vol. 36, pp. 2631–2643, 2019. DOI: https://doi.org/10.3233/JIFS-18493 Google Scholar

[10] J. M. Ko and Y. C. Kim, "Various operations and right completeness in generalized residuated lattices," J. Intell. Fuzzy Syst., vol. 40, pp. 149–164, 2021. DOI: https://doi.org/10.3233/JIFS-190424 Google Scholar

[11] Z. Pawlak, "Rough sets," Internat. J. Comput. Inform. Sci., vol. 11, pp. 341–356, 1982. Google Scholar

[12] Z. Pawlak, "Rough sets: Theoretical Aspects of Reasoning about Data," Kluwer Academic Publishers, Dordrecht, 1991. Google Scholar

[13] J. Qiao and B. Q. Hu, "Granular variable precision L-fuzzy rough sets based on residuated lattices," Fuzzy Sets Syst., vol. 336, pp. 148–166, 2018. Google Scholar

[14] C. J. Mulvey, "Quantales," Suppl. Rend. Circ. Mat. Palermo Ser. II, vol. 12, pp. 99–104, 1986. Google Scholar

[15] E. Turunen, "Mathematics Behind Fuzzy Logic," Springer-Verlag, 1999. Google Scholar

[16] C. Y. Wang, "Topological characterizations of generalized fuzzy rough sets," Fuzzy Sets Syst., vol. 312, pp. 109–125, 2017. DOI: https://doi.org/10.1016/j.fss.2016.02.005 Google Scholar

[17] C. Y. Wang, "Topological structures of L-fuzzy rough sets and similarity sets of L-fuzzy relations," Int. J. Approx. Reason., vol. 83, pp. 160–175, 2017. DOI: https://doi.org/10.1016/j.ijar.2017.01.002 Google Scholar

[18] M. Ward and R. P. Dilworth, "Residuated lattices," Trans. Amer. Math. Soc., vol. 45, pp. 335–354, 1939. Google Scholar