A common generalization for MV-algebras and Łukasiewicz–Moisil algebras

Abstract

We introduce the notion of n-nuanced MV-algebra by performing a Łukasiewicz–Moisil nuancing construction on top of MV-algebras. These structures extend both MV-algebras and Łukasiewicz–Moisil algebras, thus unifying two important types of structures in the algebra of logic. On a logical level, n-nuanced MV-algebras amalgamate two distinct approaches to many valuedness: that of the infinitely valued Łukasiewicz logic, more related in spirit to the fuzzy approach, and that of Moisil n-nuanced logic, which is more concerned with nuances of truth rather than truth degree. We study n-nuanced MV-algebras mainly from the algebraic and categorical points of view, and also consider some basic model-theoretic aspects. The relationship with a suitable notion of n-nuanced ordered group via an extension of the Γ construction is also analyzed.

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Correspondence to Andrei Popescu.

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Georgescu, G., Popescu, A. A common generalization for MV-algebras and Łukasiewicz–Moisil algebras. Arch. Math. Logic 45, 947–981 (2006). https://doi.org/10.1007/s00153-006-0020-4

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Keywords

  • n-Nuanced MV-algebra
  • Łukasiewicz–Moisil algebra
  • n-Nuanced ordered group

Mathematics Subject Classification (2000)

  • 06D35
  • 03G20
  • 06F15