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Growth of linear semigroups

Part of: Chain conditions, growth conditions, and other forms of finiteness Semigroups

Published online by Cambridge University Press:  09 April 2009

Jan Okniński
Jan Okniński
Affiliation:
Institute of Mathematics Warsaw UniversityBanacha 2 02-097 WarsawPoland e-mail: [email protected]
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Abstract

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We show that the growth function of a finitely generated linear semigroup S ⊆ Mn(K) is controlled by its behaviour on finitely many cancellative subsemigroups of S. If the growth of S is polynomially bounded, then every cancellative subsemigroup T of S has a group of fractions G ⊆ Mn (K) which is nilpotent-by-finite and of finite rank. We prove that the latter condition, strengthened by the hypothesis that every such G has a finite unipotent radical, is sufficient for S to have a polynomial growth. Moreover, the degree of growth of S is then bounded by a polynomial f(n, r) in n and the maximal degree r of growth of finitely generated cancellative T ⊆ S.

MSC classification

Secondary: 20M20: Semigroups of transformations, etc. 16P90: Growth rate, Gelfand-Kirillov dimension
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 60 , Issue 1 , February 1996 , pp. 18 - 30
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Boffa, M. and Bryant, R. M., ‘Les groupes linéaires vérifiant une identité monoidale’, C. R. Acad. Sci. Paris Ser I Math. 308 (1989), 127–128.Google Scholar
[2]Grigorchuk, R. I., ‘Cancellative semigroups of power growth’, Mat. Zametki 43 (1988), 305–319.Google Scholar
[3]Gromov, M., ‘Groups of polynomial growth and expanding maps’, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73.CrossRefGoogle Scholar
[4]Jacob, G., ‘La finitude de representations linéaires des semigroupes est décidable’, J. Algebra 52 (1978), 437–459.CrossRefGoogle Scholar
[5]Jespers, E. and Okniński, J., ‘Nilpotent semigroups and semigroups algebras’, J. Algebra 169 (1994), 984–1011.CrossRefGoogle Scholar
[6]Krause, G. R. and Lenagan, T. H., Growth of algebras and Gelfand-Kirillov dimension (Pitman, London, 1985).Google Scholar
[7]de Luca, A. and Varricchio, S., ‘A finiteness condition for semigroups generalizing a theorem of Coudrain and Schutzenberger’, preprint.Google Scholar
[8]Malcev, A. I., ‘Nilpotent semigroups’, Uč. Zap. Ivanovsk. Ped. Inst. 4 (1953), 107–111.Google Scholar
[9]Merzljakov, Ju. I., Rational groups (Nauka, Moscow, 1987) (in Russian).Google Scholar
[10]Okniński, J., Semigroup algebras (Marcel Dekker, New York, 1991).Google Scholar
[11]Okniński, J., ‘Linear representations of semigroups’, in: Monoids and semigroups with applications (World Sci. Publ., Singapore, 1991) pp. 257–277.Google Scholar
[12]Okniński, J., ‘Strongly π-regular closure of linear semigroups’, in: Semigroups with applications, Oberwolfach 1991 (World Sci. Publ., Singapore, 1992) pp. 90–100.Google Scholar
[13]Okniński, J., ‘Gelfand-Kirillov dimension of noetherian semigroup algebras’, J. Algebra 162 (1993), 302–316.CrossRefGoogle Scholar
[14]Okniński, J., ‘Linear semigroups with identities’, in: Semigroups-Algebraic theory and applications to formal languages and codes (World Sci. Publ., Singapore, 1993), 201–211.Google Scholar
[15]Okniński, J. and Putcha, M. S., ‘PI semigroup algebras of linear semigroups’, Proc. Amer. Math. Soc. 109 (1990), 39–46.CrossRefGoogle Scholar
[16]Putcha, M. S., Linear algebraic monoids, London Math. Soc. Lecture Note Ser. 133 (Cambridge Univ. Press, London, 1988).CrossRefGoogle Scholar
[17]Rowen, L. H., Polynomial identities in ring theory (Academic Press, New York, 1980).Google Scholar
[18]Wehrfritz, B. A. F., Infinite linear groups (Springer, Berlin, 1973).CrossRefGoogle Scholar
[19]Zariski, O. and Samuel, P., Commutative algebra, vols. I, II (Van Nostrand, Princeton, 1958, 1960).Google Scholar