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), 2128
MSC (2000):
Primary 26E35, 03H05; Secondary 28E05, 42A38
Published electronically:
March 30, 2004
MathSciNet review:
2048428
Fulltext PDF Free Access
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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 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.
 1.
S.
Albeverio, E.
I. Gordon, and A.
Yu. Khrennikov, Finitedimensional 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
(2002f:47030), http://dx.doi.org/10.1023/A:1006457731833
 2.
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
(2000j:20050), http://dx.doi.org/10.1023/A:1012485910692
 3.
Trevor
Evans, Some connections between residual finiteness, finite
embeddability and the word problem, J. London Math. Soc. (2)
1 (1969), 399–403. MR 0249344
(40 #2589)
 4.
Trevor
Evans, Word problems, Bull. Amer. Math. Soc. 84 (1978), no. 5, 789–802. MR 0498063
(58 #16240), http://dx.doi.org/10.1090/S000299041978145169
 5.
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.
 6.
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
(98f:03056)
 7.
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
(2003c:03128)
 8.
M.
Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur.
Math. Soc. (JEMS) 1 (1999), no. 2, 109–197. MR 1694588
(2000f:14003), http://dx.doi.org/10.1007/PL00011162
 9.
A.
J. W. Hilton, Outlines of Latin squares, Combinatorial design
theory, NorthHolland Math. Stud., vol. 149, NorthHolland,
Amsterdam, 1987, pp. 225–241. MR 920647
(89a:05037), http://dx.doi.org/10.1016/S03040208(08)728895
 10.
John
von Neumann, Invariant measures, American Mathematical
Society, Providence, RI, 1999. MR 1744399
(2002b:28012)
 11.
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
(2001e:03006)
 12.
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
(93g:20133)
 13.
Algebraic theory of machines, languages, and semigroups, Edited
by Michael A. Arbib. With a major contribution by Kenneth Krohn and John L.
Rhodes, Academic Press, New YorkLondon, 1968. MR 0232875
(38 #1198)
 14.
Herbert
John Ryser, Combinatorial mathematics, The Carus Mathematical
Monographs, No. 14, Published by The Mathematical Association of America;
distributed by John Wiley and Sons, Inc., New York, 1963. MR 0150048
(27 #51)
 15.
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
(98f:20025)
 1.
 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), 3373, October 2000. MR 2002f:47030
 2.
 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), 224262; English transl., Journal of Mathematical Sciences, 107, No. 5 (2001), 43054332. MR 2000j:20050
 3.
 T. Evans, Some connection between residual finiteness, finite embeddability and the word problem, J. Lond. Math. Soc. (2), 1 (1969), 399403. MR 40:2589
 4.
 T. Evans, Word problems, Bull. American Math. Soc., 84, No. 5 (1978), 789802. MR 58:16240
 5.
 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.
 6.
 E. Gordon, Nonstandard Methods in Commutative Harmonic Analysis, AMS, Providence, Rhode Island, 1997. MR 98f:03056
 7.
 E. I. Gordon, O. A. Rezvova, On hyperfinite approximations of the field , Reuniting the AntipodesConstructive and Nonstandard Views of the Continuum, Proceedings of the Symposium in San Servolo/Venice, Italy, May 1720, 2000. B. Ulrich, H. Ossvald and P. Schuster, editors. Synthése Library, volume 306, Kluwer Academic Publishers, Dordrecht, 2001. MR 2003c:03128
 8.
 M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. 1 (1999), 109197. MR 2000f:14003
 9.
 A. J. W. Hilton, Outlines of latin squares, Ann. Discrete Math. 34 (1987), 225242. MR 89a:05037
 10.
 J. von Neumann, Invariant Measures, AMS, Providence, RI, 1998. MR 2002b:28012
 11.
 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. MR 2001e:03006
 12.
 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. MR 93g:20133
 13.
 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. MR 38:1198
 14.
 H. J. Ryser, Combinatorial Mathematics, The Carus Mathematical Monographs, 15, The Mathematical Association of America, 1963. MR 27:51
 15.
 A. M. Vershik, E. I. Gordon, Groups locally embedded into the class of finite groups, Algebra i Analiz 9 (1997), no. 1, 7197; English transl., St. Petersburg Math. J. 9 (1998), no. 1, 4967. MR 98f:20025
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Additional Information
L. Yu. Glebsky
Affiliation:
IICOUASLP, 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 619203099
Email:
cfyig@eiu.edu
DOI:
http://dx.doi.org/10.1090/S107967620400126X
PII:
S 10796762(04)00126X
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 CONACyTNSF Grant #E120.0546 y PROMEP, PTC62; the second author was supported in part by NSF Grant DMS9970009
Communicated by:
Efim Zelmanov
Article copyright:
© Copyright 2004
American Mathematical Society
