Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-27T03:14:30.467Z Has data issue: false hasContentIssue false
  • English
  • Français

GROWTH OF GENERATING SETS FOR DIRECT POWERS OF CLASSICAL ALGEBRAIC STRUCTURES

Part of: Algebraic structures Lie algebras and Lie superalgebras Special aspects of infinite or finite groups

Published online by Cambridge University Press:  21 September 2010

MARTYN QUICK  and
N. RUŠKUC
MARTYN QUICK
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK (email: martyn@mcs.st-and.ac.uk)
N. RUŠKUC*
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK (email: nik@mcs.st-and.ac.uk)
*
✉For correspondence; e-mail: nik@mcs.st-and.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.

Keywords

generating setsgrowthdirect productsalgebraic structuresuniversal algebra

MSC classification

Secondary: 08A40: Operations, polynomials, primal algebras 17B99: None of the above, but in this section 20F05: Generators, relations, and presentations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 1 , August 2010 , pp. 105 - 126
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Berman, J. and Idziak, P. M., ‘Generative complexity in algebra’, Mem. Amer. Math. Soc. 175 (2005).Google Scholar
[2]Berman, J., Idziak, P., Marković, P., McKenzie, R., Valeriote, M. and Willard, R., ‘Varieties with few subalgebras of powers’, Trans. Amer. Math. Soc. 362 (2010), 1445–1473.CrossRefGoogle Scholar
[3]Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra (Springer, New York, 1981).CrossRefGoogle Scholar
[4]Byleen, K., ‘Embedding any countable semigroup in a 2-generated congruence-free semigroup’, Semigroup Forum 41 (1990), 145–153.CrossRefGoogle Scholar
[5]Chen, H., ‘Quantified constraint satisfaction and the polynomially generated powers property’, in: ICALP 2008, Part II, Lecture Notes in Computer Science, 5126 (eds. Aceto, L.et al.) (Springer, Berlin–Heidelberg, 2008), pp. 197–208.Google Scholar
[6]Gaines, F., ‘Some generators for the algebra of n×n matrices’, Linear and Multilinear Algebra 5 (1977/78), 95–98.CrossRefGoogle Scholar
[7]Gaschütz, W., ‘Zu einem von B. H. und H. Neumann gestellten Problem’, Math. Nachr. 14 (1955), 249–252.CrossRefGoogle Scholar
[8]Goodearl, K. R. and Warfield, R. B. Jr, An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts, 16 (Cambridge University Press, Cambridge, 1989).Google Scholar
[9]Grätzer, G. and Kisielewicz, A., ‘A survey of some open problems on p n-sequences and free spectra of algebras and varieties’, in: Universal Algebra and Quasigroup Theory (Jadwisin, 1989), Research and Exposition in Mathematics, 19 (Heldermann, Berlin, 1992), pp. 57–88.Google Scholar
[10]Herstein, I. N., Noncommutative Rings, Carus Mathematical Monographs, 15 (Mathematical Association of America, New York, 1968).Google Scholar
[11]Higman, G., Finitely Presented Infinite Simple Groups, Notes on Pure Mathematics, 8 (The Australian National University, Canberra, 1974).Google Scholar
[12]Hobby, D. and McKenzie, R., The Structure of Finite Algebras, Contemporary Mathematics, 76 (American Mathematical Society, Providence, RI, 1988).CrossRefGoogle Scholar
[13]Humphreys, J. E., Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, 9 (Springer, New York–Berlin, 1980).Google Scholar
[14]Hyde, J., Loughlin, N., Quick, M., Ruškuc, N. and Wallis, A., ‘On generating direct powers of semigroups’, in preparation.Google Scholar
[15]Istinger, M., Keiser, H. K. and Pixley, A. P., ‘Interpolation in congruence permutable algebras’, Math. Colloq. 42 (1979), 229–239.CrossRefGoogle Scholar
[16]Maurer, W. D. and Rhodes, J. L., ‘A property of finite simple nonabelian groups’, Proc. Amer. Math. Soc. 16 (1965), 552–554.CrossRefGoogle Scholar
[17]McKenzie, R. N., McNulty, G. F. and Taylor, W. F., Algebras, Lattices, Varieties, Vol. I (Wadsworth, Monterey, CA, 1987).Google Scholar
[18]Meier, D. and Wiegold, J., ‘Growth sequences of finite groups V’, J. Aust. Math. Soc. (Ser. A) 31 (1981), 374–375.CrossRefGoogle Scholar
[19]Ol’šanskiĭ, A. Yu., ‘An infinite group with subgroups of prime orders’, Izv. Akad. Nauk. SSSR Ser. Mat. 44 (1980), 309–321 (in Russian); English translation: Math. USSR-Izv. 16 (1981), 279–289.Google Scholar
[20]O’Meara, K. C., Vinsonhaler, C. I. and Wickless, W. J., ‘Identity-preserving embeddings of countable rings into 2-generator rings’, Rocky Mountain J. Math. 19 (1989), 1095–1105.Google Scholar
[21]Ribes, L. and Zalesskii, P., Profinite Groups (Springer, Berlin, 2000).CrossRefGoogle Scholar
[22]Shelah, S., Classification Theory and the Number of Nonisomorphic Models, Studies in Logic and the Foundations of Mathematics, 92 (North Holland, Amsterdam, 1990).Google Scholar
[23]Stewart, A. G. R. and Wiegold, J., ‘Growth sequences of finitely generated groups II’, Bull. Aust. Math. Soc. 40 (1989), 323–329.CrossRefGoogle Scholar
[24]Werner, H., ‘Congruences on products of algebras and functionally complete algebras’, Algebra Universalis 4 (1974), 99–105.CrossRefGoogle Scholar
[25]Wiegold, J., ‘Growth sequences of finite groups: collection of articles dedicated to the memory of Hanna Neumann, VI’, J. Aust. Math. Soc. 17 (1974), 133–141.CrossRefGoogle Scholar
[26]Wiegold, J., ‘Growth sequences of finite groups II’, J. Aust. Math. Soc. 20 (1975), 225–229.CrossRefGoogle Scholar
[27]Wiegold, J., ‘Growth sequences of finite groups III’, J. Aust. Math. Soc. (Ser. A) 25 (1978), 142–144.CrossRefGoogle Scholar
[28]Wiegold, J., ‘Growth sequences of finite groups IV’, J. Aust. Math. Soc. (Ser. A) 29 (1980), 14–16.CrossRefGoogle Scholar
[29]Wiegold, J., ‘Growth sequences of finite semigroups’, J. Aust. Math. Soc. (Ser. A) 43 (1987), 16–20.CrossRefGoogle Scholar
[30]Wiegold, J. and Wilson, J. S., ‘Growth sequences of finitely generated groups’, Arch. Math. (Basel) 30 (1978), 337–343.CrossRefGoogle Scholar