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ALGEBRAIC GEOMETRY FOR MV-ALGEBRAS

Published online by Cambridge University Press:  12 December 2014

LAWRENCE P. BELLUCE ,
ANTONIO DI NOLA  and
GIACOMO LENZI
LAWRENCE P. BELLUCE
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF BRITISH COLUMBIA VANCOUVER, B. C., CANADAE-mail:[email protected]
ANTONIO DI NOLA
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF SALERNO 84084 FISCIANO (SA), ITALYE-mail:[email protected]:[email protected]
GIACOMO LENZI
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF SALERNO 84084 FISCIANO (SA), ITALYE-mail:[email protected]:[email protected]
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Abstract

In this paper we try to apply universal algebraic geometry to MV algebras, that is, we study “MV algebraic sets” given by zeros of MV polynomials, and their “coordinate MV algebras”. We also relate algebraic and geometric objects with theories and models taken in Łukasiewicz many valued logic with constants. In particular we focus on the structure of MV polynomials and MV polynomial functions on a given MV algebra.

Keywords

Algebraic GeometryLukasiewicz LogicMV-algebras
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 4 , December 2014 , pp. 1061 - 1091
Copyright
Copyright © The Association for Symbolic Logic 2014 

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References

REFERENCES

Baumslag, G., Myasnikov, A., and Remeslennikov, V., Algebraic geometry over groups I: Algebraic sets and ideal theory. Journal of Algebra, vol. 219 (1999), pp. 16–79.Google Scholar
Belluce, L. P., Di Nola, A., and Lenzi, G., On generalizing the Nullstellensatz and McNaughton’s Theorem for MV algebras, submitted.Google Scholar
Burris, Stan and Sankapannavar, H. P., A course in universal algebra, Springer-Verlag, New York, 1981.Google Scholar
Cignoli, R., D’Ottaviano, I. and Mundici, D., Algebraic foundations of many valued reasoning, Kluwer, Dordrecht, 2000.CrossRefGoogle Scholar
Daniyarova, E., Myasnikov, A., and Remeslennikov, V., Unification theorems in algebraic geometry. Aspects of infinite groups, World Scientific, Singapore, 2008, pp. 80–111.CrossRefGoogle Scholar
Daniyarova, E., Universal algebraic geometry (Russian). Doklady Akademii Nauk, vol. 439 (2011), pp. 730–732.Google Scholar
Dubuc, E. and Poveda, Y., Representation theory of MV algebras. Annals of Pure and Applied Logic, vol. 161 (2010), pp. 1024–1046.Google Scholar
Kharlampovich, O. and Myasnikov, A., Elementary theory of free non-abelian groups. Journal of Algebra, vol. 302 (2006), pp. 451–552.CrossRefGoogle Scholar
Komori, Y., Completeness of two theories on ordered abelian groups and embedding relations. Nagoya Mathematical Journal, vol. 77 (1980), pp. 33–39.CrossRefGoogle Scholar
Lacava, F., Alcune proprietà delle L-algebre e delle L-algebre esistenzialmente chiuse. Bollettino U.M.I., vol. 16-A (1979), pp. 360–366.Google Scholar
Lacava, F., Sulle classi delle L-algebre e degli l-gruppi abeliani algebricamente chiusi. Bollettino U.M.I., vol. 1-B (1987), pp. 703–712.Google Scholar
Lawvere, F. W., Comments on the development of topos theory. Development of mathematics: 1950-2000, volume II, Birkhauser, Basel, 2000, pp. 715–734.Google Scholar
MacLane, S. and Moerdijk, I., Sheaves in geometry and logic, Springer, Berlin, 1992.Google Scholar
Marra, V. and Spada, L., The dual adjunction between MV-algebras and Tychonoff spaces. Studia Logica, vol. 100 (2012), pp. 253–278.Google Scholar
Mundici, D., Advanced Lukasiewicz calculus and MV-algebras, Springer, Berlin, 2011.Google Scholar
Myasnikov, A. and Remeslennikov, V., Algebraic geometry over groups II: Logical foundations. Journal of Algebra, vol. 234 (2000), pp. 225–276.Google Scholar
Plotkin, B. I., Some concepts of algebraic geometry in universal algebra (Russian), Algebra i Analiz, vol. 9 (1997), pp. 224–248.Google Scholar
Plotkin, B. I., Seven lectures on the universal algebraic geometry, arXiv:math/0204245v1, 2002, preprint.Google Scholar
Plotkin, B. I., Algebras with the same (algebraic) geometry (Russian). Trudy Matematicheskogo Instituta Imeni V. A. Steklova, vol. 242 (2003), pp. 176–207.Google Scholar
Sela, Z., Diophantine geometry over groups VI: The elementary theory of a free group. Geometric and Functional Analysis, vol. 16 (2006), pp. 707–730.Google Scholar
Zadeh, L., Fuzzy sets. Information and Control, vol. 8 (1965), pp. 338–353.CrossRefGoogle Scholar