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\(n\)-contractive BL-logics. (English) Zbl 1266.03041
Summary: In the field of many-valued logics, Hájek’s Basic Logic BL was introduced in [P. Hájek, Metamathematics of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)]. In this paper we study four families of \(n\)-contractive (i.e., satisfying the axiom \({\varphi^n\rightarrow\varphi^{n+1}}\), for some \({n\in\mathbb{N}^+}\)) axiomatic extensions of BL and their corresponding varieties: \(\mathrm{BL}^{n}\), \(\mathrm{SBL}^{n}\), \(\mathrm{BL}_{n}\) and \(\mathrm{SBL}_{n}\). Concerning \(\mathrm{BL}^{n}\) we have that every \(\mathrm{BL}^{n}\)-chain is isomorphic to an ordinal sum of MV-chains of at most \(n + 1\) elements, whilst every \(\mathrm{BL}_{n}\)-chain is isomorphic to an ordinal sum of \(\mathrm{MV}_{n}\)-chains (for \(\mathrm{SBL}^{n}\) and \(\mathrm{SBL}_{n}\) a similar property holds, with the difference that the first component must be the two-element Boolean algebra); all these varieties are locally finite. After a preliminary section, we study generic and \(k\)-generic algebras, completeness and computational complexity results, amalgamation and interpolation properties. Finally, we analyze the first-order versions of these logics, from the point of view of completeness and arithmetical complexity.

MSC:
03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness
03G25 Other algebras related to logic
06B20 Varieties of lattices
06D35 MV-algebras
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[1] Aglianò P., Ferreirim I., Montagna F.: Basic Hoops: an Algebraic Study of Continuous t-norms. Studia Logica 87(1), 73–98 (2007). doi: 10.1007/s11225-007-9078-1 · Zbl 1127.03049
[2] Aglianò P., Montagna F.: Varieties of BL-algebras I: general properties. J. Pure Appl. Algebra 181(2–3), 105–129 (2003). doi: 10.1016/S0022-4049(02)00329-8 · Zbl 1034.06009
[3] Aguzzoli, S., Bova, S., Marra, V.: Applications of finite duality to locally finite varieties of BL-algebras. In: Artemov, S., Nerode, A. (eds.), Logical Foundations of Computer Science–International Symposium, LFCS 2009, Deerfield Beach, FL, USA, January 3–6, 2009. Proceedings, Lecture Notes in Computer Science, vol. 5407, pp. 1–15. Springer, Berlin (2009). doi: 10.1007/978-3-540-92687-0_1 · Zbl 1211.03098
[4] Bianchi M., Montagna F.: Supersound many-valued logics and Dedekind-MacNeille completions. Arch. Math. Log. 48(8), 719–736 (2009). doi: 10.1007/s00153-009-0145-3 · Zbl 1185.03040
[5] Blok W., Ferreirim I.: On the structure of hoops. Algebra Universalis 43(2–3), 233–257 (2000). doi: 10.1007/s000120050156 · Zbl 1012.06016
[6] Blok, W., Pigozzi, D.: Algebraizable logics. No. 396 in memoirs of The American Mathematical Society. Am. Math. Soc. (1989); ISBN:0-8218-2459-7. Available on http://orion.math.iastate.edu/dpigozzi/ · Zbl 0664.03042
[7] Burris, S., Sankappanavar, H.P.: A course in Universal Algebra, Graduate Texts in Mathematics, vol. 78. Springer, New York (1981). An updated and revised electronic edition is available on http://www.math.uwaterloo.ca/\(\sim\)snburris/htdocs/ualg.html · Zbl 0478.08001
[8] Busaniche M., Cabrer L.: Canonicity in subvarieties of BL-algebras. Algebra Universalis 62(4), 375–397 (2009). doi: 10.1007/s00012-010-0055-6 · Zbl 1200.03049
[9] Ciabattoni A., Esteva F., Godo L.: T-norm based logics with n-contraction. Neural Netw. World 16(5), 453–495 (2008). doi: 10.1093/jigpal/jzn014
[10] Cignoli, R., D’Ottaviano, I., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning, Trends in Logic, vol. 7. Kluwer Academic Publishers, Berlin (1999); ISBN:9780792360094
[11] Cintula, P.: From fuzzy logic to fuzzy mathematics. Ph.D. thesis, FNSPE CTU, Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering–Czech Technical University in Prague, Trojanova 13, 120 00 Prague 2, Czech Republic (2004). Available on http://www2.cs.cas.cz/\(\sim\)cintula/thesis.pdf
[12] Cintula P., Esteva F., Gispert J., Godo L., Montagna F., Noguera C.: Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Ann. Pure Appl. Log. 160(1), 53–81 (2009). doi: 10.1016/j.apal.2009.01.012 · Zbl 1168.03052
[13] Cintula P., Hájek P.: Triangular norm predicate fuzzy logics. Fuzzy Sets Syst. 161(3), 311–346 (2010). doi: 10.1016/j.fss.2009.09.006 · Zbl 1200.03020
[14] Davey B.A., Priestley H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002). doi: 10.2277/0521784514 ISBN:9780521784511 · Zbl 1002.06001
[15] di Nola A., Lettieri A.: One chain generated varieties of MV-algebras. J. Algebra 225(2), 667–697 (2000). doi: 10.1006/jabr.1999.8136 · Zbl 0949.06004
[16] Esteva F., Godo L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst. 124(3), 271–288 (2001). doi: 10.1016/S0165-0114(01)00098-7 · Zbl 0994.03017
[17] Ferreirim, I.: On varieties and quasivarieties of hoops and their reducts. Ph.D. thesis, University of Illinois at Chicago, Chicago (1992)
[18] Galatos N., Jipsen P., Kowalski T., Ono H.: Residuated Lattices: an Algebraic Glimpse at Substructural Logics, Studies in Logic and The Foundations of Mathematics, vol. 151. Elsevier, Amsterdam (2007) ISBN:978-0-444-52141-5 · Zbl 1171.03001
[19] Grigolia, R.: Algebraic analysis of Łukasiewicz-tarski n-valued logical systems. In: Selected Papers on Łukasiewicz Sentencial Calculi, pp. 81–91. Polish Academy of Science, Ossolineum (1977)
[20] Hájek P.: Metamathematics of Fuzzy Logic, Trends in Logic, vol. 4, paperback edn. Kluwer Academic Publishers, Berlin (1998) ISBN:9781402003707
[21] Horčík R., Noguera C., Petrík M.: On n-contractive fuzzy logics. Math. Log. Quart. 53(3), 268–288 (2007). doi: 10.1002/malq.200610044 · Zbl 1125.03017
[22] Labuschagne, C., van Alten, C.: On the MacNeille completion of MTL-chains. In: Proceedings of the Ninth International Conference on Intelligent Technologies, October 7–9, 2008. Samui, Thailand (2008)
[23] MacNeille, H.M.: Partially ordered sets. Trans. Amer. Math. Soc. 42(3), 416–460 (1937). Available on http://www.jstor.org/stable/1989739 · Zbl 0017.33904
[24] Metcalfe G., Montagna F.: Substructural fuzzy logics. J. Symbolic Log. 72(3), 834–864 (2007). doi: 10.2178/jsl/1191333844 · Zbl 1139.03017
[25] Montagna F.: Interpolation and Beth’s property in propositional many-valued logics: a semantic investigation. Ann. Pure. Appl. Log. 141(1–2), 148–179 (2006). doi: 10.1016/j.apal.2005.11.001 · Zbl 1094.03011
[26] Montagna F., Noguera C.: Arithmetical complexity of first-order predicate fuzzy logics over distinguished semantics. J. Log. Comput. 20(2), 399–424 (2010). doi: 10.1093/logcom/exp052 · Zbl 1198.03032
[27] Montagna F., Noguera C., Horčík R.: On weakly cancellative fuzzy logics. J. Log. Comput. 16(4), 423–450 (2006). doi: 10.1093/logcom/exl002 · Zbl 1113.03021
[28] van Alten, C.J.: Preservation theorems for MTL-Chains. Log. J. IGPL (2010). doi: 10.1093/jigpal/jzp088 · Zbl 1253.03098
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