Abstract
In this paper, the concept of the fuzzy filter of IL-algebra is introduced. This concept generalizes the notion of the fuzzy filter of BL-algebra, MTL-algebra, and other allied algebraic structures. Three kinds of prime filters and fuzzy prime filters of IL-algebra are defined, and a few interesting properties are obtained. It is proved that quotient algebra formed with the help of a fuzzy filter corresponding to IL-algebra is an IL-algebra. Isomorphism theorems for filters and fuzzy filters of IL-algebra are established.
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References
Borzooei RA, Shoar SK, Ameri R (2012) Some types of filters in MTL-algebras. Fuzzy Sets Syst 187:92–102. https://doi.org/10.1016/j.fss.2011.09.001
Chakraborty MK, Sen J (1998) MV-algebra embedded in a CL-algebra. Int J Approx Reason 18:217–229. https://doi.org/10.1016/S0888-613X(98)00007-3
Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: an algebraic glimpse at substructural logics, vol 151. Elsevier, Amsterdam
Gasse BV, Deschrijver G, Cornelis C, Kerre EE (2010) Filters of residuated lattices and triangle algebras. Inf Sci 180(16):3006–3020. https://doi.org/10.1016/j.ins.2010.04.010
Girard J-Y (1987) Linear logic. Theoretical Computer Science 50(1):1–101. https://doi.org/10.1016/0304-3975(87)90045-4
Hájek P (1998) Metamathematics of fuzzy logic. Kluwer Academic Publishers, Dordrecht. https://doi.org/10.1007/978-94-011-5300-3
Höhle U (1996) MV-algebra valued filter theory. Quaest Math 19(1–2):23–46. https://doi.org/10.1080/16073606.1996.9631824
Islam S, Sanyal A, Sen J (2020) Filter theory of IL-algebras. Jr Cal Math Soc 16(2):113–126
Jun YB, Xu Y, Zhang XH (2005) Fuzzy filters of MTL-algebras. Inform Sci 175(1–2):120–138. https://doi.org/10.1016/j.ins.2004.11.004
Kadji A, Lele C, Tonga M (2017) Fuzzy prime and maximal filters of residuated lattices. Soft Comput 21:1913–1922. https://doi.org/10.1007/s00500-016-2113-2
Kihara H, Ono H (2008) Algebraic characterizations of variable separation properties. Rep Math Log 43:43–63
Kondo M, Dudek WA (2008) Filter theory of BL algebras. Soft Comput 12:419–423. https://doi.org/10.1007/s00500-007-0178-7
Kondo M, Turunen E (2012) Prime Filters on Residuated Lattices. In: (2012) IEEE 42nd international symposium on multiple-valued logic. Victoria, BC, Canada pp. 89–91. https://doi.org/10.1109/ISMVL.2012.40
Lianzhen L, Kaitai L (2005) Fuzzy implicative and Boolean filters of \(R_0\) algebras. Inform Sci 171(1–3):61–71. https://doi.org/10.1016/j.ins.2004.03.017
Lianzhen L, Kaitai L (2005) Fuzzy Boolean and positive implicative filters of BL-algebras. Fuzzy Sets Syst 152(2):333–348. https://doi.org/10.1016/j.fss.2004.10.005
Liu L, Li K (2005) Fuzzy filters of BL-algebras. Inform Sci 173:141–154. https://doi.org/10.1016/j.ins.2004.07.009
Liu Y, Qin K, Xu Y (2011) Fuzzy prime filters of lattice implication algebras. Fuzzy Inform Eng 3(3):235–246. https://doi.org/10.1007/s12543-011-0080-y
Ma ZM (2014) Lattices of (generalized) fuzzy filters in residuated lattices. J Intell Fuzzy Syst 27(5):2281–2287. https://doi.org/10.3233/IFS-141191
Ma X, Zhan J (2008) On \((\in ,\in ,\vee q)\)-fuzzy filters of BL-algebras. J Syst Sci Complex 21:144–158. https://doi.org/10.1007/s11424-008-9073-2
Ma X, Zhan J, Jun Y (2009) On \((\in ,\in ,\vee q)\)-Fuzzy Filters of \(R0\)-algebras. Math Log Q 55(5):493–508. https://doi.org/10.1002/malq.200810022
Motamed S, Torkzadeh L (2017) A new class of BL-algebras. Soft Comput 21(3):687–698. https://doi.org/10.1007/s00500-016-2043-z
Rachůnek J, Šalounová D (2008) Fuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras. Inform Sci 178(17):3474–3481. https://doi.org/10.1016/j.ins.2008.05.005
Rasiowa H (1974) An algebraic approach to non-classical logics. North Holland, Amsterdam
Saeid AB, Motamed S (2014) A new filter in BL-algebras. J Intell Fuzzy Syst 27(6):2949–2957. https://doi.org/10.3233/IFS-141254
Sen J (2001) Some embeddings in linear logic and related issues. Ph.D. Thesis, University of Calcutta, India
Troelstra AS (1992) Lectures on linear logic. No. 29, Center for the Study of Language and Information, Stanford
Turunen E (1999) BL-algebras of basic fuzzy logic. Mathw Soft Comput 6:49–61
Turunen E (2001) Boolean deductive system of BL-algebras. Arch Math Log 40:467–473. https://doi.org/10.1007/s001530100088
Wang W, Xin X (2011) On fuzzy filters of Heyting-algebras. Discret Contin Dyn Syst Ser S 4(6):1611–1619. https://doi.org/10.3934/dcdss.2011.4.1611
Wang W, Xin X (2011) On fuzzy filters of pseudo BL-algebras. Fuzzy Sets Syst 162(1):27–38. https://doi.org/10.1016/j.fss.2010.09.006
Yin Y, Zhan J (2010) New types of fuzzy filters of BL-algebras. Comput Math Appl 60(7):2115–2125. https://doi.org/10.1016/j.camwa.2010.07.054
Zhang J (2009) Fuzzy prime Boolean filters and their operations in IMTL-algebras. Fuzzy Inf Eng 1:401–419. https://doi.org/10.1007/s12543-009-0031-z
Zhang J, Zhou H (2006) Fuzzy filters on the residuated lattices. New Math Nat Comput 2(1):11–28. https://doi.org/10.1142/S1793005706000373
Zhao F, Zhu X (2011) Fuzzy prime filters in lattice effect algebra. In: International Conference on Multimedia Technology, ICMT pp. 5869–5871. https://doi.org/10.1109/ICMT.2011.6002647
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The authors would like to express their sincere thanks to the reviewers for their valuable suggestions and comments.
Funding
The research of the last two authors is funded by the Department of Higher Education, Government of West Bengal, India [Project number- 257(Sanc)/ST/P/S&T/16G-45/2017 dated 25.03.2018].
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Islam, S., Sanyal, A. & Sen, J. Fuzzy filters of IL-algebras. Soft Comput 26, 7017–7027 (2022). https://doi.org/10.1007/s00500-022-06985-1
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DOI: https://doi.org/10.1007/s00500-022-06985-1