Abstract
The infinite-valued logic of Łukasiewicz was originally defined by means of an infinite-valued matrix. Łukasiewicz took special forms of negation and implication as basic connectives and proposed an axiom system that he conjectured would be sufficient to derive the valid formulas of the logic; this was eventually verified by M. Wajsberg. The algebraic counterparts of this logic have become know as Wajsberg algebras. In this paper we show that a Wajsberg algebra is complete and atomic (as a lattice) if and only if it is a direct product of finite Wajsberg chains. The classical characterization of complete and atomic Boolean algebras as fields of sets is a particular case of this result.
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This research was partially supported by the Consejo Nacional Investigaciones Científicas y Técnicas de la República Argentina (CONICET).
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Cignoli, R. Complete and atomic algebras of the infinite valued Łukasiewicz logic. Stud Logica 50, 375–384 (1991). https://doi.org/10.1007/BF00370678
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DOI: https://doi.org/10.1007/BF00370678