Algebraically Closed Abelian l-Groups
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Wolfgang Rump
Abstract
Every semifield of non-zero characteristic is either a field of prime characteristic or a semifield of characteristic 1. Semifields of characteristic 1 are equivalent to abelian lattice-ordered groups. It is proved that such a semifield A is algebraically closed if and only if the pure equations xn = a and certain quadratic equations are solvable in A. Using a sheaf representation for z-projectable abelian l-groups on the co-Zariski space of minimal primes, a sheaf-theoretic characterization of algebraic closedness in characteristic 1 is obtained. Concerning the solvability of quadratic equations, the criterion consists in a topological condition for the base space. The results are built upon an analysis of rational functions in characteristic 1. While polynomials satisfy the “fundamental theorem of algebra”, the multiplicative structure of rational functions is determined by means of “divisors”, extracted from the additive structure of A modulo parallelogram identities.
References
[1] ANDERSON, M.-FEIL, T.: Lattice-ordered Groups, D. Reidel Publishing Co., Dordrecht, 1988.10.1007/978-94-009-2871-8Search in Google Scholar
[2] BIGARD, A.-KEIMEL, K.-WOLFENSTEIN, S.: Groupes et anneaux réticulés. Lecture Notes in Math. 608, Springer-Verlag, Berlin-New York, 1977.10.1007/BFb0067004Search in Google Scholar
[3] CONNES, A.-CONSANI, C.: Characteristic one, entropy and the absolute point. In: Noncommutative Geometry, Arithmetic, and Related Topics. The Twenty-First Meeting of the Japan-U.S. Mathematics Institute, Baltimore 2009, Johns Hopkins Univ. Press, Baltimore, MD, 2011, pp. 75-139.Search in Google Scholar
[4] CUNINGHAME-GREEN, R. A.-MEIJER, P. F. J.: An algebra for piecewise-linear minimax problems, Discrete Appl. Math. 2 (1980), 267-294.10.1016/0166-218X(80)90025-6Search in Google Scholar
[5] DI NOLA, A.-FERRAIOLI, A. R.-LENZI, G.: Algebraically closed MV-algebras and their sheaf representation, Ann. Pure Appl. Logic 164 (2013), 349-355.10.1016/j.apal.2012.10.017Search in Google Scholar
[6] DUBUC, E. J.-POVEDA, Y. A.: Representation theory of MV-algebras, Ann. Pure Appl. Logic 161 (2010), 1024-1046.10.1016/j.apal.2009.12.006Search in Google Scholar
[7] EINSIEDLER, M.-KAPRANOV, M.-LIND, D.: Non-Archimedean amoebas and tropical varieties, J. Reine Angew. Math. 601 (2006), 139-157.Search in Google Scholar
[8] FILIPOIU, A.-GEORGESCU, G.: Compact and Pierce representations of MV-algebras, Rev. Roumaine Math. Pures Appl. 40 (1995), 599-618.Search in Google Scholar
[9] GLASS, A. M. W.: Partially Ordered Groups. Ser. Algebra 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1999.10.1142/3811Search in Google Scholar
[10] GROTHENDIECK, A.-DIEUDONNÉ, J. A.: Eléments de Géométrie Algébrique, Springer-Verlag, Berlin-Heidelberg-New York, 1971.Search in Google Scholar
[11] KEIMEL, K.: The Representation of Lattice-ordered Groups and Rings by Sections in Sheaves. Lectures on the Applications of Sheaves to Ring Theory. Lecture Notes in Math. 248, Berlin, 1971, pp. 1-98.10.1007/BFb0058563Search in Google Scholar
[12] KNOX, M. L.-MCGOVERN, W.WM.: Feebly projectable l-groups, Algebra Universalis 62 (2009), 91-11210.1007/s00012-010-0041-zSearch in Google Scholar
[13] KOLOKOLTSOV, V. N.-MASLOV, V. P.: Idempotent Analysis and Its Applications (Moscow, 1994. Translated by V. E. Nazaikinskii, with an appendix by Pierre Del Moral). Mathematics and Its Applications 401, Kluwer Academic Publishers Group, Dordrecht, 1997.10.1007/978-94-015-8901-7_1Search in Google Scholar
[14] LACAVA, F.: On classes of algebraically closed Ł-algebras and abelian l-groups, Boll. Unione Mat. Ital. B (7) 1 (1987), 703-712 (Italian).Search in Google Scholar
[15] LACAVA, F.: Quasilocal Łukasiewicz algebras, Boll. Unione Mat. Ital. B (7) 11 (1997), 961-972 (Italian).Search in Google Scholar
[16] LITVINOV, G. L.: The Maslov dequantization, idempotent and tropical mathematics: a very brief introduction. In: Idempotent Mathematics and Mathematical Physics. Contemp. Math. 377, Amer. Math. Soc., Providence, RI, 2005, pp. 1-17.Search in Google Scholar
[17] MCGOVERN, W. WM.: Neat rings, J. Pure Appl. Algebra 205 (2006), 243-265.10.1016/j.jpaa.2005.07.012Search in Google Scholar
[18] MIKHALKIN, G.: Tropical geometry and its applications. In: International Congress of Mathematicians, Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 827-852.10.4171/022-2/40Search in Google Scholar
[19] MILNE, J. S.: Algebraic Number Theory, Course Notes (revised version), 2012. http://www.jmilne.org/math/CourseNotes/ANT.pdf.Search in Google Scholar
[20] MUNDICI, D.: Interpretation of AFC∗ algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), 15-63.10.1016/0022-1236(86)90015-7Search in Google Scholar
[21] RUMP, W.: Abelian l-groups and a characterization of the maximal spectrum of a Prüfer domain, J. Pure Appl. Algebra 218 (2014), 2204-2017.10.1016/j.jpaa.2014.03.011Search in Google Scholar
[22] RUMP, W.-YANG, Y. C.: Jaffard-Ohm correspondence and Hochster duality, Bull. Lond. Math. Soc. 40 (2008), 263-273.10.1112/blms/bdn006Search in Google Scholar
[23] SPEYER, D.-STURMFELS, B.: Tropical mathematics, 2004. arXiv:math/0408099v1.Search in Google Scholar
[24] STURMFELS, B.: A combinatorial introduction to tropical geometry, Berlin, 2007, http://math.berkeley.edu/∼bernd/tropical/sec1.pdfSearch in Google Scholar
[25] WEINERT, J. J.-WIEGANDT, R.: On the structure of semifields and lattice-ordered groups, Period. Math. Hungar. 32 (1996), 129-147. 10.1007/BF01879738Search 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
