Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T09:24:00.387Z Has data issue: false hasContentIssue false
  • English
  • Français

GROWTH RATES OF ALGEBRAS, I: POINTED CUBE TERMS

Part of: Varieties Algebraic structures

Published online by Cambridge University Press:  22 January 2016

KEITH A. KEARNES ,
EMIL W. KISS  and
ÁGNES SZENDREI
KEITH A. KEARNES*
Affiliation:
Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, USA email [email protected]
EMIL W. KISS
Affiliation:
Department of Algebra and Number Theory, Loránd Eötvös University, Pázmány Péter stny 1/c., H-1117 Budapest, Hungary email [email protected]
ÁGNES SZENDREI
Affiliation:
Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, USA email [email protected]
*
✉[email protected]
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.

We investigate the function $d_{\mathbf{A}}(n)$, which gives the size of a least size generating set for $\mathbf{A}^{n}$.

Keywords

growth ratebasic identitypointed cube term

MSC classification

Primary: 08A40: Operations, polynomials, primal algebras
Secondary: 08A55: Partial algebras 08B05: Equational logic, Mal′cev (Mal′tsev) conditions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 101 , Issue 1 , August 2016 , pp. 56 - 94
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Berman, J., Idziak, P., Marković, P., McKenzie, R., Valeriote, M. and Willard, R., ‘Varieties with few subalgebras of powers’, Trans. Amer. Math. Soc. 362(3) (2010), 1445–1473.Google Scholar
Berman, J. and McKenzie, R., ‘Clones satisfying the term condition’, Discrete Math. 52 (1984), 7–29.Google Scholar
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), 197–208.Google Scholar
Dey, I. M. S., ‘Embeddings in non-Hopf groups’, J. Lond. Math. Soc. (2) 1 (1969), 745–749.Google Scholar
Erfanian, A., ‘A problem on growth sequences of groups’, J. Aust. Math. Soc. Ser. A 59(2) (1995), 283–286.Google Scholar
Erfanian, A., ‘A note on growth sequences of alternating groups’, Arch. Math. (Basel) 78(4) (2002), 257–262.CrossRefGoogle Scholar
Erfanian, A., ‘A note on growth sequences of PSL(m, q)’, Southeast Asian Bull. Math. 29(4) (2005), 697–713.Google Scholar
Erfanian, A., ‘On the growth sequences of free product of PSL(m, q)’, Ital. J. Pure Appl. Math. 22 (2007), 19–26.Google Scholar
Erfanian, A., ‘Growth sequence of free product of alternating groups’, Int. J. Contemp. Math. Sci. 2(13–16) (2007), 685–691.Google Scholar
Erfanian, A. and Rezaei, R., ‘On the growth sequences of PSp(2m, q)’, Int. J. Algebra 1(1–4) (2007), 51–62.Google Scholar
Erfanian, A. and Wiegold, J., ‘A note on growth sequences of finite simple groups’, Bull. Aust. Math. Soc. 51(3) (1995), 495–499.Google Scholar
Foster, A. L., ‘On the finiteness of free (universal) algebras’, Proc. Amer. Math. Soc. 7 (1956), 1011–1013.Google Scholar
Freese, R. and McKenzie, R., Commutator Theory for Congruence Modular Varieties, London Mathematical Society Lecture Note Series, 125 (Cambridge University Press, Cambridge, 1987).Google Scholar
Glass, A. M. W. and Riedel, H. H. J., ‘Growth sequences—a counterexample’, Algebra Universalis 21(2–3) (1985), 143–145.Google Scholar
Hall, P., ‘The Eulerian functions of a group’, Q. J. Math. 7 (1936), 134–151.Google Scholar
Hyde, J. T., Loughlin, N. J., Quick, M., Ruskuc, N. and Wallis, A. R., ‘On the growth of generating sets for direct powers of semigroups’, Semigroup Forum 84 (2012), 116–130.CrossRefGoogle Scholar
Kearnes, K. and Kiss, E., ‘Finite algebras of finite complexity’, Discrete Math. 207 (1999), 89–135.Google Scholar
Kearnes, K. and Kiss, E., ‘The shape of congruence lattices’, Mem. Amer. Math. Soc. 222(1046) (2013).Google Scholar
Kearnes, K., Kiss, E. and Szendrei, Á., ‘Growth rates of algebras, II: Wiegold dichotomy’, Internat. J. Algebra Comput. 25(4) (2015), 555–566.Google Scholar
Kearnes, K., Kiss, E. and Szendrei, Á., ‘Growth rates of algebras, III: finite solvable algebras’, Algebra Universalis , to appear.Google Scholar
Kearnes, K. and Szendrei, Á., ‘Clones of algebras with parallelogram terms’, Internat. J. Algebra Comput. 22 (2012).CrossRefGoogle Scholar
Kelly, D., ‘Basic equations: word problems and Mal’cev conditions, Abstract 701-08-04’, Notices Amer. Math. Soc. 20 (1972), A-54.Google Scholar
Kimmerle, W., ‘Growth sequences relative to subgroups’, in: Groups – St. Andrews 1981 (St. Andrews, 1981), London Mathematical Society Lecture Note Series, 71 (Cambridge University Press, Cambridge–New York, 1982), 252–260.Google Scholar
Lennox, J. C. and Wiegold, J., ‘Generators and killers for direct and free products’, Arch. Math. (Basel) 34(4) (1980), 296–300.Google Scholar
Meier, D. and Wiegold, J., ‘Growth sequences of finite groups. V’, J. Aust. Math. Soc. Ser. A 31(3) (1981), 374–375.Google Scholar
Obraztsov, V. N., ‘Growth sequences of 2-generator simple groups’, Proc. Roy. Soc. Edinburgh Sect. A 123(5) (1993), 839–855.Google Scholar
Pollák, G., ‘Growth sequence of globally idempotent semigroups’, J. Aust. Math. Soc. Ser. A 48(1) (1990), 87–88.Google Scholar
Quick, M. and Ruškuc, N., ‘Growth of generating sets for direct powers of classical algebraic structures’, J. Aust. Math. Soc. 89 (2010), 105–126.CrossRefGoogle Scholar
Riedel, H. H. J., ‘Growth sequences of finite algebras’, Algebra Universalis 20(1) (1985), 90–95.Google Scholar
Stewart, A. G. R. and Wiegold, J., ‘Growth sequences of finitely generated groups. II’, Bull. Aust. Math. Soc. 40(2) (1989), 323–329.Google Scholar
Wiegold, J., ‘Growth sequences of finite groups’, J. Aust. Math. Soc. 17 (1974), 133–141; Collection of articles dedicated to the memory of Hanna Neumann, VI.Google Scholar
Wiegold, J., ‘Growth sequences of finite groups. II’, J. Aust. Math. Soc. 20(2) (1975), 225–229.CrossRefGoogle Scholar
Wiegold, J., ‘Growth sequences of finite groups. III’, J. Aust. Math. Soc. Ser. A 25(2) (1978), 142–144.Google Scholar
Wiegold, J., ‘Growth sequences of finite groups. IV’, J. Aust. Math. Soc. Ser. A 29(1) (1980), 14–16.Google Scholar
Wiegold, J., ‘Growth sequences of finite semigroups’, J. Aust. Math. Soc. Ser. A 43(1) (1987), 16–20.Google Scholar
Wiegold, J. and Wilson, J. S., ‘Growth sequences of finitely generated groups’, Arch. Math. (Basel) 30(4) (1978), 337–343.Google Scholar
Wise, D. T., ‘The rank of a direct power of a small-cancellation group’, Proc. Conf. Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), Geom. Dedicata 94 (2002), 215–223.Google Scholar