Abstract
We introduce the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. By using this new idea, we introduce the notions of interval valued \((\in,\in\!\vee\,q)\) -fuzzy filters of pseudo BL-algebras and investigate some of their related properties. Some characterization theorems of these generalized interval valued fuzzy filters are derived. The relationship among these generalized interval valued fuzzy filters of pseudo BL-algebras is considered. Finally, we consider the concept of implication-based interval valued fuzzy implicative filters of pseudo BL-algebras, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Bhakat SK (2000). \((\in,\in\!\vee\,q)\) -fuzzy normal, quasinormal and maximal subgroupsFuzzy Sets Syst 112: 299–312
Bhakat SK and Das P (1996). \((\in,\in\!\vee\,q)\)-fuzzy subgroupsFuzzy Sets Syst 80: 359–368
Biswas R (1994). Rosenfeld’s fuzzy subgroups with interval valued membership functions. Fuzzy Sets Syst 63: 87–90
Chang CC (1958). Algebraic analysis of many valued logic. Trans Am Math Soc 88: 467–490
Curry HB (1965). Foundation of mathematical logics. McGraw-Hill, New York
Davvaz B (2006). \((\in,\in\!\vee\,q)\)-fuzzy subnear-rings and idealsSoft Comput 10: 206–211
Davvaz B and Corsini P (2007). Redefined fuzzy \(H_v\) -submodules and many valued implicationsInform Sci 177: 865–875
Deschrijver G (2007). Arithmetric operators in interval-valued fuzzy theory. Inform Sci 177: 2906–2924
Di Nola A, Georgescu G and Iorgulescu A (2000). Boolean products of BL-algebras. J Math Anal Appl 251: 106–131
Di Nola A, Georgescu G, Iorgulescu A (2002) Pseudo BL-algebras: Part I,II. Mult Valued Log 8:673–714, 715–750
Di Nola A and Leustean L (2003). Compact representations of BL-algebras. Arch Math Log 42: 737–761
Dvurečenskij A (2001a). States on pseudo MV-algebras. Stud Log. 68: 301–327
Dvurečenskij A (2001b). On pseudo MV-algebras. Soft Comput 5: 347–354
Dvurečenskij A (2007). Every linear pseudo BL-algebra admits a state. Soft Comput 11: 495–501
Flonder P, Georgescu G and Iorgulescu A (2001). Pseudo-t-norms and pseudo BL-algebras. Soft Comput 5: 355–371
Georgescu G (2004). Bosbach states on fuzzy structures. Soft Comput 8: 217–230
Georgescu G and Iorgulescu A (2001). Pseudo MV-algebras. Mult Valued Log 6: 95–135
Georgescu G and Leustean L (2002). Some classes of pseudo BL-algebras. J Aust Math Soc 73: 127–153
Hájek P (1998). Metamathematics of fuzzy logic. Kluwer Academic Press, Dordrecht
Iorgulescu A (2004). Iséki algebras. Connection with BL-algebras.. Soft Comput 8: 449–463
Janowitz MV and Johnshon CS (1969). A note on Brouwerian and Glivenko semigroups. J Lond Math Soc 2: 733–736
Ma X, Zhan J and Xu Y (2007). Generalized fuzzy filters of BL-algebras. Appl Math A J Chin Universities Ser B 22(4): 490–496
Nemitz WC (1965). Implicative semilattices. Trans AMS Math Soc 117: 123–142
Pu PM and Liu YM (1980). Fuzzy topology I: Neighourhood structure of a fuzzy point and Moore-Smith convergence. J Math Anal Appl 76: 571–599
Rachunek J (2002). Prime spectra of a non-commutative generalizations of MV-algebras. Algebra Universalis 48: 151–169
Rachunek J (2002). A non-commutative generalizations of MV-algebras. Czechoslov Math J 52: 255–273
Turunen E (2001). Boolean deductive systems of BL-algebras. Arch Math Log 40: 467–473
Turunen E and Sessa S (2001). Local BL-algebras. Mult Valued Log 6: 229–249
Yuan XH, Zhang C and Ren YH (2003). Generalized fuzzy groups and many valued applications. Fuzzy Sets Syst 138: 205–211
Zadeh LA (1965). Fuzzy sets. Inform Control 8: 338–353
Zadeh LA (1975). The concept of a linguistic variable and its application to approximate reason I. Inform Sci 8: 199–249
Zadeh LA (2005). Toward a generalized theory of uncertainty (GTU)-an outine. Inform Sci 172: 1–40
Zeng W and Li H (2006). Relationship between similarity measure and intropy of interval valued fuzzy set. Fuzzy Sets Syst 157: 1477–1484
Zhan J, Davvaz B and Shum KP (2008). A new view of fuzzy hypernear-rings. Inform Sci 178: 425–438
Zhang XH and Li WH (2006). On pseudo BL-algebras and BCC-algebras. Soft Comput 10: 941–952
Zhang XH, Qin K and Dudek WA (2007). Ultra LI-ideals in lattice implication algebras and MTL-algebras. Czechoslov Math J 57(132): 591–605
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhan, J., Dudek, W.A. & Jun, Y.B. Interval valued \((\in,\in\!\vee\,q)\) -fuzzy filters of pseudo BL-algebras. Soft Comput 13, 13–21 (2009). https://doi.org/10.1007/s00500-008-0288-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-008-0288-x