The lattices of 𝔏-fuzzy state filters in state residuated lattices

References

[1] Bahls, P.—Cole, J.—Jipsen, P.—Tsinakis, C.: Cancellative residuated lattices, Algebra Universalis 50 (2003), 83–106.10.1007/s00012-003-1822-4Search in Google Scholar

[2] Blount, K.—Tsinakis, C.: The structure of residuated lattices, Internat. J. Algebra Comput. 13 (2003), 437–461.10.1142/S0218196703001511Search in Google Scholar

[3] Blyth, T. S.: Lattices and Ordered Algebraic Structures, Springer, London, 2005.Search in Google Scholar

[4] Chang, C. C.: Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 467–490.10.1090/S0002-9947-1958-0094302-9Search in Google Scholar

[5] Ciungu, L. C.: Bosbach and Riečan states on residuated lattices, J. Appl. Funct. Anal. 2 (2008), 175–188.Search in Google Scholar

[6] Ciungu, L.C.—Dvurečenskij, A.—Hyčko, M.: State BL-algebras, Soft Comput. 15 (2011), 619–634.10.1007/s00500-010-0571-5Search in Google Scholar

[7] Dvurečenskij, A.—Rachůnek, J.—Šalounová, D.: State operators on generalizations of fuzzy structures, Fuzzy Sets and Systems 187 (2012), 58–76.10.1016/j.fss.2011.05.023Search in Google Scholar

[8] Esteva, F.—Godo, L.: Monoidal t-norm-based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems 124 (2001), 271–288.10.1016/S0165-0114(01)00098-7Search in Google Scholar

[9] Flaminio, T.—Montagna, F.: MV-algebras with internal states and probabilistic fuzzy logic, Internat. J. Approx. Reason 50 (2009), 138–152.10.1016/j.ijar.2008.07.006Search in Google Scholar

[10] Galatos, N.—Jipsen, P.—Kowalski, T.—Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier, Tokyo, 2007.Search in Google Scholar

[11] Georgescu, G.: Bosbach states on fuzzy structures, Soft Comput. 8 (2004), 217–230.10.1007/s00500-003-0266-2Search in Google Scholar

[12] H ájek, P.: Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998.10.1007/978-94-011-5300-3Search in Google Scholar

[13] He, P. F.—Xin, X. L.—Yang, Y. W.: On state residuated lattices, Soft Comput. 19 (2015), 2083–2094.10.1007/s00500-015-1620-xSearch in Google Scholar

[14] Kowalski, T.—Ono, H.: Residuated Lattices: An Algebraic Glimpse at Logic without Contraction. Studies in Logic, Elsevier, 2001.Search in Google Scholar

[15] Jipsen, P.—Tsinakis, C.: A survey of residuated lattices. In: Ordered Algebraic Structures, J. Martinez, Eds., Kluwer, Dordrecht, 2002, pp. 19–56.10.1007/978-1-4757-3627-4_3Search in Google Scholar

[16] Jun, Y. B.—Xu, Y.—Zhang, X. H.: Fuzzy filters of MTL-algebras, Inform Sci. 175 (2005), 120–138.10.1016/j.ins.2004.11.004Search in Google Scholar

[17] Liu, L. Z.: On the existence of states on MTL-algebras, Inform. Sci. 220 (2013), 559–567.10.1016/j.ins.2012.07.046Search in Google Scholar

[18] Liu, L. Z.—Li, K. T.: Fuzzy filters of BL-algebras, Inform Sci. 173 (2005), 141–154.10.1016/j.ins.2004.07.009Search in Google Scholar

[19] Liu, L. Z.—Li, K. T.: Boolean filters and positive implicative filters of residuated lattices, Inform Sci. 177 (2007), 5725–5738.10.1016/j.ins.2007.07.014Search in Google Scholar

[20] Liu, L. Z.—Li, K. T.: Fuzzy Boolean and positive implicative filters of BL-algebras, Fuzzy Sets and Systems 152 (2005), 333–348.10.1016/j.fss.2004.10.005Search in Google Scholar

[21] Liu, L. Z.—Zhang, X. Y.: States onR₀-algebras, Soft Comput. 12 (2008), 1099–1104.10.1007/s00500-008-0276-1Search in Google Scholar

[22] Ma, Z. M.: Two types of MTL-L-filters in residuated lattices, J. Intell. Fuzzy Syst. 27 (2014), 681–689.10.3233/IFS-131026Search in Google Scholar

[23] Ma, Z. M.: Lattices of (generalized) fuzzy filters in residuated lattices, J. Intell. Fuzzy Syst. 27 (2014), 2281–2287.10.3233/IFS-141191Search in Google Scholar

[24] Ma, Z. M.—Hu, B. Q.: Characterizations and new subclasses ofI-filters in residuated lattices, Fuzzy Sets and Systems 247 (2014), 92–107.10.1016/j.fss.2013.11.009Search in Google Scholar

[25] Ma, Z. M.—Yang, W.: Hybrid generalized Bosbach and Riečan states on non-commutative residuated lattices, Int. J. Gen. Syst 45 (2016), 1–23.10.1080/03081079.2015.1110579Search in Google Scholar

[26] Mertanen, J.—Turunen, E.: States on semi-divisible generalized residuated lattices reduce to states on MV-algebras, Fuzzy Sets and Systems 159 (2008), 3051–3064.10.1016/j.fss.2008.01.036Search in Google Scholar

[27] Kondo, M.—Dudek, W. A.: Filter theory of BL-algebras, Soft Comput. 12 (2008), 419–423.10.1007/s00500-007-0178-7Search in Google Scholar

[28] Mundici, D.: Averaging the truth-value in Łukasiewicz sentential logic, Studia Logica 55 (1995), 113–127.10.1007/BF01053035Search in Google Scholar

[29] Rasouli, S.—Davvaz, B.: Rough filters based on residuated lattices, Knowledge and Information Systems 58 (2019), 399–424.10.1007/s10115-018-1219-5Search in Google Scholar

[30] Riečan, B.: On the probability on BL-algebras, Acta Math. Nitra 4 (2000), 3–13.10.1007/s005000050082Search in Google Scholar

[31] Turunen, E.—Mertanen, J.: States on semi-divisible residuated lattices, Soft Comput. 12 (2008), 353–357.10.1007/s00500-007-0182-ySearch in Google Scholar

[32] Ward, M.—Dilworth, P. R.: Residuated lattice, Trans. Amer. Math. Soc. 45 (1939), 335–354.10.1007/978-1-4899-3558-8_32Search in Google Scholar

[33] Wang, W.—Xin, X. L.: On fuzzy filters of pseudo BL-algebras, Fuzzy Sets and Systems 162 (2011), 27–38.10.1016/j.fss.2010.09.006Search in Google Scholar

[34] Wang, Z. D.—Fang, J. X.: Onv-filters and normalv-filters of a residuated lattice with a weakvt-operator, Inform Sci. 178 (2008), 3465–3473.10.1016/j.ins.2008.04.003Search in Google Scholar

[35] Zhang, X. H.—Zhou, H. J.—Mao, X. Y.: IMTL(MV)-filters and fuzzy IMTL(MV)-filters of residuated lattices, J. Intell. Fuzzy Syst. 26 (2014), 589–596.10.3233/IFS-120752Search in Google Scholar

[36] Zhu, Y. Q.—Xu, Y.: On filter theory of residuated lattices, Inform. Sci. 180 (2010), 3614–3632.10.1016/j.ins.2010.05.034Search in Google Scholar