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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)].

MSC:
06D35 MV-algebras
03B50 Many-valued logic
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References:
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