On approximation of locally compact groups by finite algebraic systems
Authors:
L. Yu. Glebsky and E. I. Gordon
Journal:
Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 21-28
MSC (2000):
Primary 26E35, 03H05; Secondary 28E05, 42A38
DOI:
https://doi.org/10.1090/S1079-6762-04-00126-X
Published electronically:
March 30, 2004
MathSciNet review:
2048428
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Abstract: We discuss the approximability of locally compact groups by finite semigroups and finite quasigroups (latin squares). We show that if a locally compact group $G$ is approximable by finite semigroups, then it is approximable by finite groups, and thus many important groups are not approximable by finite semigroups. This result implies, in particular, the impossibility to simulate the field of reals in computers by finite associative rings. We show that a locally compact group is approximable by finite quasigroups iff it is unimodular.
- S. Albeverio, E. I. Gordon, and A. Yu. Khrennikov, Finite-dimensional approximations of operators in the Hilbert spaces of functions on locally compact abelian groups, Acta Appl. Math. 64 (2000), no. 1, 33–73. MR 1828556, DOI 10.1023/A:1006457731833
- M. A. Alekseev, L. Yu. Glebskiĭ, and E. I. Gordon, On approximations of groups, group actions and Hopf algebras, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 256 (1999), no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 3, 224–262, 268 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 107 (2001), no. 5, 4305–4332. MR 1708567, DOI 10.1023/A:1012485910692
- Trevor Evans, Some connections between residual finiteness, finite embeddability and the word problem, J. London Math. Soc. (2) 1 (1969), 399–403. MR 249344, DOI 10.1112/jlms/s2-1.1.399
- Trevor Evans, Word problems, Bull. Amer. Math. Soc. 84 (1978), no. 5, 789–802. MR 498063, DOI 10.1090/S0002-9904-1978-14516-9
GC L. Yu. Glebsky, Carlos J. Rubio, Latin squares, partial latin squares and its generalized quotients, preprint math.CO/0303356, http://xxx.lanl.gov/, submitted to Combinatoric and Graphs.
- E. I. Gordon, Nonstandard methods in commutative harmonic analysis, Translations of Mathematical Monographs, vol. 164, American Mathematical Society, Providence, RI, 1997. Translated from the Russian manuscript by H. H. McFaden. MR 1449873, DOI 10.1090/mmono/164
- Evgenii I. Gordon and Olga A. Rezvova, On hyperfinite approximations of the field R, Reuniting the antipodes—constructive and nonstandard views of the continuum (Venice, 1999) Synthese Lib., vol. 306, Kluwer Acad. Publ., Dordrecht, 2001, pp. 93–102. MR 1895385
- M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 2, 109–197. MR 1694588, DOI 10.1007/PL00011162
- A. J. W. Hilton, Outlines of Latin squares, Combinatorial design theory, North-Holland Math. Stud., vol. 149, North-Holland, Amsterdam, 1987, pp. 225–241. MR 920647, DOI 10.1016/S0304-0208(08)72889-5
- John von Neumann, Invariant measures, American Mathematical Society, Providence, RI, 1999. MR 1744399
- Manfred Wolff and Peter A. Loeb (eds.), Nonstandard analysis for the working mathematician, Mathematics and its Applications, vol. 510, Kluwer Academic Publishers, Dordrecht, 2000. MR 1790871, DOI 10.1007/978-94-011-4168-0
- O. Chein, H. O. Pflugfelder, and J. D. H. Smith (eds.), Quasigroups and loops: theory and applications, Sigma Series in Pure Mathematics, vol. 8, Heldermann Verlag, Berlin, 1990. MR 1125806
- Michael A. Arbib (ed.), Algebraic theory of machines, languages, and semigroups, Academic Press, New York-London, 1968. With a major contribution by Kenneth Krohn and John L. Rhodes. MR 0232875
- Herbert John Ryser, Combinatorial mathematics, The Carus Mathematical Monographs, No. 14, Mathematical Association of America; distributed by John Wiley and Sons, Inc., New York, 1963. MR 0150048, DOI 10.5948/UPO9781614440147
- A. M. Vershik and E. I. Gordon, Groups that are locally embeddable in the class of finite groups, Algebra i Analiz 9 (1997), no. 1, 71–97 (Russian); English transl., St. Petersburg Math. J. 9 (1998), no. 1, 49–67. MR 1458419
AGKh S. Albeverio, E. Gordon, A. Khrennikov, Finite dimensional approximations of operators in the spaces of functions on locally compact abelian groups, Acta Applicandae Mathematicae 64(1), 33–73, October 2000.
AGG M. A. Alekseev, L. Yu. Glebskii, E. I. Gordon, On approximations of groups, group actions and Hopf algebras, Representation Theory, Dynamical Systems, Combinatorial and Algebraic Methods. III, A. M. Vershik editor, Russian Academy of Sciences, St. Petersburg Branch of Steklov Mathematical Institute, Zapiski Nauchn. Seminarov POMI 256 (1999), 224–262; English transl., Journal of Mathematical Sciences, 107, No. 5 (2001), 4305–4332.
Ev1 T. Evans, Some connection between residual finiteness, finite embeddability and the word problem, J. Lond. Math. Soc. (2), 1 (1969), 399–403.
Ev2 T. Evans, Word problems, Bull. American Math. Soc., 84, No. 5 (1978), 789–802.
GC L. Yu. Glebsky, Carlos J. Rubio, Latin squares, partial latin squares and its generalized quotients, preprint math.CO/0303356, http://xxx.lanl.gov/, submitted to Combinatoric and Graphs.
Gor E. Gordon, Nonstandard Methods in Commutative Harmonic Analysis, AMS, Providence, Rhode Island, 1997.
Resv E. I. Gordon, O. A. Rezvova, On hyperfinite approximations of the field $\mathbb {R}$, Reuniting the Antipodes—Constructive and Nonstandard Views of the Continuum, Proceedings of the Symposium in San Servolo/Venice, Italy, May 17–20, 2000. B. Ulrich, H. Ossvald and P. Schuster, editors. Synthése Library, volume 306, Kluwer Academic Publishers, Dordrecht, 2001.
Gr M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. 1 (1999), 109–197.
Hil A. J. W. Hilton, Outlines of latin squares, Ann. Discrete Math. 34 (1987), 225–242.
Neum J. von Neumann, Invariant Measures, AMS, Providence, RI, 1998.
Loeb Nonstandard Analysis for the Working Mathematicians, P. A. Loeb and M. P. H. Wolff, editors. Mathematics and Applications, volume 510, Kluwer Academic Publishers, Dordrecht/Boston/London, 2000.
quas Quasigroups and Loops. Theory and Applications, O. Chein, H. O. Pfulgfelder and J. D. H. Smith, editors. Sigma Series in Pure Mathematica, volume 8, Heldermann Verlag, Berlin, 1990.
RT J. Rhodes, B. Tilson, Theorems on local structure of finite semigroups, Algebraic theory of machines, languages and semigroups, M. A. Arbib, ed., Acad. Press, New York & London, 1968.
Ryser H. J. Ryser, Combinatorial Mathematics, The Carus Mathematical Monographs, 15, The Mathematical Association of America, 1963.
VG A. M. Vershik, E. I. Gordon, Groups locally embedded into the class of finite groups, Algebra i Analiz 9 (1997), no. 1, 71–97; English transl., St. Petersburg Math. J. 9 (1998), no. 1, 49–67.
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Additional Information
L. Yu. Glebsky
Affiliation:
IICO-UASLP, Av. Karakorum 1470, Lomas 4ta Session, SanLuis Potosi SLP 78210, Mexico
Email:
glebsky@cactus.iico.uaslp.mx
E. I. Gordon
Affiliation:
Department of Mathematics and Computer Science, Eastern Illinois University, 600 Lincoln Avenue, Charleston, IL 61920-3099
Email:
cfyig@eiu.edu
Keywords:
Approximation,
group,
quasigroup
Received by editor(s):
June 16, 2003
Published electronically:
March 30, 2004
Additional Notes:
The first author was supported in part by CONACyT-NSF Grant #E120.0546 y PROMEP, PTC-62; the second author was supported in part by NSF Grant DMS-9970009
Communicated by:
Efim Zelmanov
Article copyright:
© Copyright 2004
American Mathematical Society