zbMATH — the first resource for mathematics

Another proof of the completeness of the Łukasiewicz axioms and of the extensions of Di Nola’s theorem. (English) Zbl 1320.06010
Using Farkas’ lemma the authors embed every finite partial subalgebra of an MV-chain into the standard rational MV-algebra. This easily yields a new proof of Chang’s completeness theorem for Łukasiewicz logic. This method was also used, for a more general completeness theorem, in the paper by R. Cignoli and D. Mundici [Mult.-Valued Log. 6, No. 1-2, 89-94 (2001; Zbl 1020.03019)]. Combining these techniques with the main result of the paper by M. Botur [Fuzzy Sets Syst. 178, No. 1, 24-37 (2011; Zbl 1252.03145)], the authors then provide a new proof of Di Nola’s representation theorem, without using model-theoretic techniques. Similar results for zero-cancellative partially ordered monoids were obtained in the paper by J. Paseka [Math. Slovaca 64, No. 3, 777-788 (2014; Zbl 1337.06008)].

06D35 MV-algebras
03B50 Many-valued logic
Full Text: DOI arXiv
[1] Blok, W.; Ferreirim, I., On the structure of hoops, Algebra Universalis, 43, 233-257, (2000) · Zbl 1012.06016
[2] Blok, W., van Alten, C.J.: On the finite embeddability property for residuated ordered groupoids. Trans. Amer. Math. Soc. 357, 4141-4157 (2005) · Zbl 1083.06013
[3] Botur, M., A non-associative generalization of Hájeks BL-algebras, Fuzzy Sets and Systems, 178, 24-37, (2011) · Zbl 1252.03145
[4] Chang, C.C., Algebraic analysis of many valued logics, Trans. Amer. Math. Soc., 88, 467-490, (1958) · Zbl 0084.00704
[5] Chang, C.C., A new proof of the completeness of the łukasiewicz axioms, Trans. Amer. Math. Soc., 93, 74-80, (1959) · Zbl 0093.01104
[6] Chang, C.C., Keisler, H.J.: Model Theory. Elsevier (1973) · Zbl 0084.00704
[7] Cignoli, R.L.O., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer (2000) · Zbl 0937.06009
[8] Cignoli, R.L.O., Mundici, D.: On Partial isomorphisms on totally ordered abelian groups and Hájek’s completeness theorem for basic logic. Multiple valued Logic, (Gordon and Breach), Special issue dedicated to the memory of Grigore Moisil, 6, 89-94 (2001) · Zbl 1020.03019
[9] Dziobiak, W., Kravchenko, A.V., Wojciechowski P.: Equivalents for a quasivariety to be generated by a single algebraic structure. Studia Logica 91, 113-123 (2009) · Zbl 1167.08004
[10] Gispert, J., Mundici, D.: MV-algebras: a variety for magnitudes with archimedean units. Algebra Universalis 53, 7-43 (2005) · Zbl 1093.06010
[11] Hahn, H.: Über die nichtarchimedischen Größensysteme. Sitzungsber. d. Akademie d. Wiss. Wien, Math.-Naturw. Klasse 116, 601-655 (1907) · JFM 38.0501.01
[12] Di Nola, A.: Representation and reticulation by quotients of MV-algebras. Ric. Mat. XL, 291-297 (1991) · Zbl 0767.06013
[13] Di Nola, A., Lenzi, G., Spada, L.: Representation of MV-algebras by regular ultrapowers of [0, 1]. Arch. Math. Log. 49, 491-500 (2010) · Zbl 1196.06005
[14] Farkas, G., Über die theorie der einfachen ungleichungen, J. Reine Angew. Math., 124, 1-27, (1902) · JFM 32.0169.02
[15] Keisler H.J.: A survey of ultraproducts. In: Proc. Internat. Congr. Logic, Methodology and Philosophy of Science, pp. 112-126. North-Holland, Amsterdam (1965)
[16] Mundici, D.: Advanced Łukasiewicz calculus and MV-algebras. Trends in Logic, vol. 35 Springer, New York, (2011) · Zbl 1235.03002
[17] Paseka, J.: Representations of zero-cancellative pomonoids. Math. Slovaca 64, 777-788 (2014) · Zbl 1337.06008
[18] Schrijver, A.: Theory of linear and integer programming. Wiley-Interscience series in discrete mathematics and optimization, John Wiley & sons (1998)
[19] Wojciechowski, P.J., Embeddings of totally ordered MV-algebras of bounded cardinality, Fund. Math., 203, 57-63, (2009) · Zbl 1174.06016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.