Algebra universalis

, Volume 73, Issue 3–4, pp 277–290 | Cite as

Another proof of the completeness of the Łukasiewicz axioms and of the extensions of Di Nola’s Theorem

  • Michal Botur
  • Jan Paseka


The main aim of this paper is twofold. Firstly, to present a new method based on Farkas’ Lemma for the rational numbers, showing how to embed any finite partial subalgebra of a linearly ordered MV-algebra into \({\mathbb{Q}\cap[0, 1]}\). and then to establish a new proof of the completeness of the Łukasiewicz axioms based on this method. Secondly, to present a purely algebraic proof of Di Nola’s Representation Theorem for MV-algebras and to extend his results to the restriction of the standard MV-algebra on the rational numbers.

Key words and phrases

MV-algebra ultraproduct Di Nola’s Representation Theorem Farkas’ Lemma 

2010 Mathematics Subject Classification

Primary: 06D35 Secondary: 03B50 


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Faculty of SciencePalacký University OlomoucOlomoucCzech Republic
  2. 2.Department of Mathematics and Statistics, Faculty of ScienceMasaryk UniversityBrnoCzech Republic

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