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

Abstract

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.