Abstract
Riesz MV-algebras are a variety of algebras strongly connected to Riesz spaces. In this short article we investigate some elimination properties of the first-order theory RMV of linearly ordered Riesz MV-algebras and show that RMV admits elimination of quantifiers and uniform elimination of imaginary elements. In the process, we also prove several other results such as modelcompleteness, o-minimality, definability of Skolem functions, and a version of the Di Nola Representation Theorem for Riesz MV-algebras.
References
[1] BAAZ, M.-VEITH, H.: Quantifier elimination in fuzzy logic. In: Computer Science Logic. Lecture Notes in Comput. Sci., Springer, Berlin-Heidelberg, 1999, pp. 399-414.Search in Google Scholar
[2] BEDE, B.-DI NOLA, A.: Elementary calculus in Riesz MV-algebras, Internat. J. Approx. Reason. 36 (2004), 129-149.10.1016/j.ijar.2003.09.003Search in Google Scholar
[3] BURRIS, S.-SANKAPPANAVAR, H.: A Course in Universal Algebra, The Millennium Edition, 2012. http://www.math.uwaterloo.ca/snburris/htdocs/ualg.html.Search in Google Scholar
[4] CHANG, C.-KEISLER, H. J.: Model Theory, North-Holland Publishing Company, Amsterdam, 1973.Search in Google Scholar
[5] CIGNOLI, R.-D’OTTAVIANO, I. M. L.-MUNDICI, D.: Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht, 1999.10.1007/978-94-015-9480-6Search in Google Scholar
[6] DI NOLA, A.: Representation and reticulation by quotients of MV-algebras, Ric. Mat. XL (1991), 291-297.Search in Google Scholar
[7] DI NOLA, A.-LETTIERI, A.: Coproduct MV-algebras, nonstandard reals, and Riesz spaces, J. Algebra 185 (1996), 605-620.10.1006/jabr.1996.0342Search in Google Scholar
[8] DI NOLA, A.-LEUSTEAN, I.: Łukasiewicz logic and Riesz spaces, Soft Comput. 18 (2014), 2349-2363.10.1007/s00500-014-1348-zSearch in Google Scholar
[9] DI NOLA, A.-LEUSTEAN, I.: Łukasiewicz Logic and MV-Algebras. In: Handbook of Mathematical Fuzzy Logic, Vol. II (P. Cintula, P. Hájek, C. Noguera, eds.). College Publications, 2011.Search in Google Scholar
[10] DI NOLA, A.-LEUSTEAN, I.-FLONDOR, P.: MV-modules, J. Algebra 267 (2003), 21-40.10.1016/S0021-8693(03)00332-6Search in Google Scholar
[11] VAN DEN DRIES, L.: Theories of definable Skolem functions, J. Symbolic Logic 49 (1984), 625-629.10.2307/2274194Search in Google Scholar
[12] VAN DEN DRIES, L.: Tame Topology and O-minimal Structures, Cambridge University Press, Cambridge, 1998.10.1017/CBO9780511525919Search in Google Scholar
[13] FUCHS, L.: Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963.Search in Google Scholar
[14] HODGES, W.: Model theory. Encyclopedia Math. Appl. 42, Cambridge University Press, Cambridge, 1993.Search in Google Scholar
[15] LABUSCHAGNE, C. C. A.-VAN ALTEN, C. J.: On the variety of Riesz spaces, Indag.Math. (N.S.) 18 (2007), 61-68.10.1016/S0019-3577(07)80006-1Search in Google Scholar
[16] LENZI, G.-MARCHIONI, E.: An algebraic characterization of o-minimal and weakly o-minimal MV-chains, J. Pure Appl. Algebra 218 (2014), 90-100.10.1016/j.jpaa.2013.04.014Search in Google Scholar
[17] LUXEMBURG, W. A. J.-ZAANEN, A. C.: Riesz spaces I, North-Holland, Amsterdam, 1971.Search in Google Scholar
[18] MARCHIONI, E.: Amalgamation through quantifier elimination for varieties of commutative residuated lattices, Arch. Math. Logic 51 (2012), 15-34.10.1007/s00153-011-0251-xSearch in Google Scholar
[19] MARKER, D.: Model theory. An Introduction. Grad. Texts in Math. 217, Springer-Verlag, New York, 2002.Search in Google Scholar
[20] MUNDICI, D.: Interpretation of AF C∗-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), 15-63. 10.1016/0022-1236(86)90015-7Search in Google Scholar
© 2015
Articles in the same Issue
- Editorial. Many-Valued Logic ’12
- Antonio Di Nola
- On the Modal μ-Calculus Over Finite Symmetric Graphs
- A Note on Saturated Models for Many-Valued Logics
- ℍ-Perfect Pseudo MV-Algebras and Their Representations
- Some Notes on Elimination Properties for The Theory of Riesz MV-Chains
- The Riesz Hull of a Semisimple MV-Algebra
- The Failure of The Amalgamation Property for Semilinear Varieties of Residuated Lattices
- The Chinese Remainder Theorem for Strongly Semisimple MV-Algebras and Lattice-Groups
- Algebraically Closed Abelian l-Groups
- Possibilistic and Probabilistic Logic under Coherence: Default Reasoning and System P
- Functional Many-Valued Relations
Articles in the same Issue
- Editorial. Many-Valued Logic ’12
- Antonio Di Nola
- On the Modal μ-Calculus Over Finite Symmetric Graphs
- A Note on Saturated Models for Many-Valued Logics
- ℍ-Perfect Pseudo MV-Algebras and Their Representations
- Some Notes on Elimination Properties for The Theory of Riesz MV-Chains
- The Riesz Hull of a Semisimple MV-Algebra
- The Failure of The Amalgamation Property for Semilinear Varieties of Residuated Lattices
- The Chinese Remainder Theorem for Strongly Semisimple MV-Algebras and Lattice-Groups
- Algebraically Closed Abelian l-Groups
- Possibilistic and Probabilistic Logic under Coherence: Default Reasoning and System P
- Functional Many-Valued Relations