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.
Similar content being viewed by others
References
Belluce L.P. (1986) Semisimple algebras of infinite valued logic and bold fuzzy set theory. Can. J. Math. 38, 1356–1379
Belluce L.P., Di Nola A., Lettieri A. (1993) Local MV-algebras. Redincoti Circolo Matematico di Palermo 42, 347–361
Boicescu V., Filipoiu A., Georgescu G., Rudeanu S. (1991) Łukasiewicz–Moisil algebras. North-Holland, Amsterdam
Burris S., Sankappanavar H.P. (1981) A course in universal algebra Graduate Texts in Mathematics No 78. Springer, Berlin Heidelberg New York
Chang C.C. (1958) Algebraic analysis of many valued logics. Trans. Am. Math. Soc. 88, 467–490
Chang C.C., Keisler H.J. (1973) Model theory. North Holland, Amsterdam
Cignoli R., D’Ottaviano I., Mundici D. (2000) Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht, p. 7
Cignoli R. (1982) Proper n-valued Łukasiewicz algebras as S-algebras of Łukasiewicz n-valued propositional calculi. Studia Logica 41, 3–16
Di Nola A. (1993) MV-algebras in the treatment of uncertainty. In: Löwen P., Roubens E. (eds) Proceedings of the International IFSA Congress, Bruxelles, 1991. Kluwer, Dordrecht, pp. 123–131
Georgescu G., Vraciu C. (1970) On the characterization of centered Łukasiwicz algebras. J. Algebra 16, 486–495
Grigolia R.S.: Algebraic analysis of Łukasiewicz–Tarski logical systems. In: Wojcicki R., Malinkowski G. (eds.) Selected Papers on Łukasiewicz Sentential Calculi, Osolineum, pp. 81–92. Wroclav (1977)
Iorgulescu A. (1998) Connections between MV n algebras and n-valued Lukasiewicz–Moisil algebras Part I. Discrete Math. 181, 155–177
Iorgulescu A. (1999) Connections between MV n algebras and n-valued Lukasiewicz–Moisil algebras Part II. Discrete Math. 202, 113–134
Iorgulescu A. (2000) Connections between MV n algebras and n-valued Lukasiewicz–Moisil algebras Part IV. J. Univ. Comput. Sci. 6(I): 139–154
Łukasiewicz J. (1920) On three-valued logic (Polish). Ruch Filozoficzny 5, 160–171
Lukasiewicz J., Tarski A. (1930) Untersuchungen uber den Aussagenkalkul. C. R. Séances Soc. Sci. Lettres Varsovie 23, 30–50
Mac Lane S. (1971) Categories for the Working Mathematician. Springer, Berlin Heidelberg New York
Moisil, Gr.C.: Notes sur les logique non-chrysippiennes. Ann. Sci. Univ. Jassy 27, 86–98, 176–185, 233–243 (1941)
Moisil Gr.C. (1963) Le algebre di Lukasiewicz. An. Univ. C.I. Parhon, Acta Logica 6, 97–135
Moisil Gr.C. (1965) Old and New Essays on Non-classical Logics (Romanian). Ştiinţifică, Bucharest
Moisil Gr.C. (1972) Essais sur les logiques non-chrysippiennes. Academiei, Bucharest
Moisil Gr.C. (1975) Lectures on the Logic of Fuzzy Reasoning. Ştiinţifică, Bucharest
Monk J.D. (1976) Mathematical Logic. Springer, Berlin Heidelberg New York
Mundici D. (1986) Interpretation of AFC*-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63
Mundici D. (1988) Free products in the category of abelian l-groups with strong unit. J. Algebra 113, 81–109
Pawlak Z. (1991) Rough Sets. Kluwer, Dordrecht
Post E. (1921) Introduction to a general theory of elementary propositions. Am. J. Math. 43, 163–185
Rasiowa H. An Algebraic Approach to Non-classical Logic. North-Holland, Amsterdam, Polish Scientific Publ., Warszawa (1974)
Rosenbloom P. (1942) Post algebras. Postulates and general theory. Am. J. Math. 64, 163–183
Zadeh L. (1965) Fuzzy sets. Inf. Control 8, 338–353
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-006-0020-4