Abstract
In this short paper we will discuss on saturated and κ-saturated models of many-valued (t-norm based fuzzy) logics. Using these peculiar structures we show a representation theorem à la Di Nola for several classes of algebras including MV, Gödel, product, BL, NM and WNM-algebras. Then, still using (κ)-saturated algebras, we finally show that some relevant subclasses of algebras related to many-valued logics also enjoy the joint embedding property and the amalgamation property.
References
[BM11] BIANCHI, M.-MONTAGNA, F.: n-contractive BL-logics, Arch. Math. Logic 50 (2011), 257-285, doi:10.1007/s00153-010-0213-8. 10.1007/s00153-010-0213-8Suche in Google Scholar
[BS81] BURRIS, S.-SANKAPPANAVAR, H. P.: A Course in Universal Algebra, Springer, Berlin-Heidelberg-New York, 1981. A revised and free edition is available on http://www.math.uwaterloo.ca/snburris/htdocs/UALG/univ-algebra2012.pdf. 10.1007/978-1-4613-8130-3Suche in Google Scholar
[BS06] BELL, J. L.-SLOMSON, A. B.: Models and Ultraproducts: An Introduction, Dover Publications, Mineola, NY, 2006. Suche in Google Scholar
[CDM00] CIGNOLI, R.-D’OTTAVIANO, I.-MUNDICI, D.: Algebraic Foundations of Many-Valued Reasoning. Trends in Logic, Vol. 7, Kluwer Academic Publishers, Dordrecht, 2000. 10.1007/978-94-015-9480-6_2Suche in Google Scholar
[CEG09] 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. Logic 160 (2009), 53-81. doi:10.1016/j.apal.2009.01.012. 10.1016/j.apal.2009.01.012Suche in Google Scholar
[CHN11a] CINTULA, P.-HÁJEK, P.-NOGUERA, C.: Handbook of Mathematical Fuzzy Logic, Vol. 1. College Publications, 2011. Suche in Google Scholar
[CHN11b] CINTULA, P.-HÁJEK, P.-NOGUERA, C.: Handbook of Mathematical Fuzzy Logic, Vol. 2. College Publications, 2011. [CK90] CHANG, C. C.-KEISLER, H. J.: Model Theory, North Holland, Amsterdam, 1990. Suche in Google Scholar
[CMM11] CORTONESI, T.-MARCHIONI, E.-MONTAGNA, F.: Quantifier elimination and other model-theoretic properties of BL-algebras, Notre Dame J. Form. Log. 52 (2011), 339-379. doi:10.1215/00294527-1499336. 10.1215/00294527-1499336Suche in Google Scholar
[CN10] CINTULA, P.-NOGUERA, C.: Implicational (semilinear) logics I: a new hierarchy, Arch. Math. Logic 49 (2010), 417-446. doi:10.1007/s00153-010-0178-7. 10.1007/s00153-010-0178-7Suche in Google Scholar
[DN91] DI NOLA, A.: Representation and reticulation by quotients of MV-algebras, Ric. Mat. 40 (1991), 291-297. Suche in Google Scholar
[DNLS10] DI NOLA, A.-LENZI, G.-SPADA, L.: Representation of MV-algebras by regular ultrapowers of [0, 1], Arch. Math. Logic 49 (2010), 491-500. doi:10.1007/s00153-010-0182-y. 10.1007/s00153-010-0182-ySuche in Google Scholar
[EG01] ESTEVA, F.-GODO, L.: Monoidal t-norm based logic: Towards a logic for leftcontinuous t-norms, Fuzzy Sets Syst. 124 (2001), 271-288. doi:10.1016/S0165-0114(01)00098-7. 10.1016/S0165-0114(01)00098-7Suche in Google Scholar
[Fla08] FLAMINIO, T.: Strong non-standard completeness for fuzzy logics, Soft Cmput. 12 (2008), 321-333. doi:10.1007/s00500-007-0184-9. 10.1007/s00500-007-0184-9Suche in Google Scholar
[FMS62] FRAYNE, T. E.-MOREL, A. C.-SCOTT, D.: Reduced direct products, Fund. Math. 51 (1962), 195-228 (Abstract: Notices Amer. Math. Soc. 5 (1958), p. 674). Suche in Google Scholar
[GJKO07] GALATOS, N.-JIPSEN, P.-KOWALSKI, T.-ONO, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Stud. Logic Found. Math. 151, Elsevier, Amsterdam, 2007. Suche in Google Scholar
[Haj98] HÁJEK, P.: Metamathematics of Fuzzy Logic. Trends in Logic, Vol. 4, Kluwer Academic Publishers, Amsterdam, 1998. 10.1007/978-94-011-5300-3Suche in Google Scholar
[Kei61] KEISLER, H. J.: Ultraproducts and elementary classes, Indag. Math. 23 (1961), 477-495 (Nederl. Akad. Wetensch. Proc. Ser. A 64). 10.1016/S1385-7258(61)50048-0Suche in Google Scholar
[Kei64] KEISLER, H. J.: Good ideals in fields of sets, Ann. of Math. (2) 79 (1964), 338-359. 10.2307/1970549Suche in Google Scholar
[Kei10] KEISLER, H. J.: The ultraproduct construction. In: Ultrafilters across Mathematics. Contemp. Math. 530, Amer. Math. Soc., Providence, RI, 2010, pp. 163-179. doi:10.1090/conm/530/10444; (Preprint: http://www.math.wisc.edu/\keisler/ultraproducts-web-final.pdf). 10.1090/conm/530/10444Suche in Google Scholar
[KO01] KOWALSKI, T.-ONO, H.: Residuated lattices: an algebraic glimpse at logic without contraction. Tech. report, Japan Advanced Institute of Science and Technology, 2001. Suche in Google Scholar
[Kun72] KUNEN, K.: Ultrafilters and independent sets, Trans. Amer. Mathematical Soc. 172 (1972), 299-306. doi:10.1090/S0002-9947-1972-0314619-7. 10.1090/S0002-9947-1972-0314619-7Suche in Google Scholar
[LS77] LACAVA, F.-SAELI, D.: Sul model-completamento della teoria delle Ł-catene, Boll. Unione Mat. Ital. A (5) 14 (1977), 107-110. Suche in Google Scholar
[Mar12] MARCHIONI, E.: Amalgamation through quantifier elimination for varieties of commutative residuated lattices, Arch. Math. Logic 51 (2012), 15-34. doi:10.1007/s00153-011-0251-x. 10.1007/s00153-011-0251-xSuche in Google Scholar
[Mon05] MONTAGNA, F.: Generating the variety of BL-algebras, Soft Comput. 9 (2005), 869-874. doi:10.1007/s00500-004-0450-z. 10.1007/s00500-004-0450-zSuche in Google Scholar
[Mon11] MONTAGNA, F.: Completeness with respect to a chain and universal models in fuzzy logic, Arch. Math. Logic 50 (2011), 161-183. doi:10.1007/s00153-010-0207-6. 10.1007/s00153-010-0207-6Suche in Google Scholar
[MV62] MORLEY, M.-VAUGHT, R.: Homogeneous universal models, Math. Scand. 11 (1962), 37-57. 10.7146/math.scand.a-10648Suche in Google Scholar
[Neg69] NEGREPONTIS, S.: The Stone space of the saturated Boolean algebras, Trans. Amer. Math. Soc. 141 (1969), 515-527. doi:10.1090/S0002-9947-1969-0248057-2. 10.1090/S0002-9947-1969-0248057-2Suche in Google Scholar
[NEG08] NOGUERA, C.-ESTEVA, F.-GISPERT, J.: On triangular norm based axiomatic extensions of the weak nilpotent minimum logic, MLQ Math. Log. Q. 54 (2008), 387-409. doi:10.1002/malq.200710054. 10.1002/malq.200710054Suche in Google Scholar
[Yas74] YASUHARA, M.: The amalgamation property, the universal-homogeneous models, and the generic models, Math. Scand. 34 (1974), 5-36.10.7146/math.scand.a-11501Suche in Google Scholar
© 2015
Artikel in diesem Heft
- Editorial. Many-Valued Logic ’12
- Antonio Di Nola
- On the Modal μ-Calculus Over Finite Symmetric Graphs
- A Note on Saturated Models for Many-Valued Logics
- ℍ-Perfect Pseudo MV-Algebras and Their Representations
- Some Notes on Elimination Properties for The Theory of Riesz MV-Chains
- The Riesz Hull of a Semisimple MV-Algebra
- The Failure of The Amalgamation Property for Semilinear Varieties of Residuated Lattices
- The Chinese Remainder Theorem for Strongly Semisimple MV-Algebras and Lattice-Groups
- Algebraically Closed Abelian l-Groups
- Possibilistic and Probabilistic Logic under Coherence: Default Reasoning and System P
- Functional Many-Valued Relations
Artikel in diesem Heft
- Editorial. Many-Valued Logic ’12
- Antonio Di Nola
- On the Modal μ-Calculus Over Finite Symmetric Graphs
- A Note on Saturated Models for Many-Valued Logics
- ℍ-Perfect Pseudo MV-Algebras and Their Representations
- Some Notes on Elimination Properties for The Theory of Riesz MV-Chains
- The Riesz Hull of a Semisimple MV-Algebra
- The Failure of The Amalgamation Property for Semilinear Varieties of Residuated Lattices
- The Chinese Remainder Theorem for Strongly Semisimple MV-Algebras and Lattice-Groups
- Algebraically Closed Abelian l-Groups
- Possibilistic and Probabilistic Logic under Coherence: Default Reasoning and System P
- Functional Many-Valued Relations