Dynamics of implicit operations and tameness of pseudovarieties of groups

Almeida, Jorge

doi:10.1090/S0002-9947-01-02857-4

Dynamics of implicit operations and tameness of pseudovarieties of groups
HTML articles powered by AMS MathViewer

by Jorge Almeida

Trans. Amer. Math. Soc. 354 (2002), 387-411

DOI: https://doi.org/10.1090/S0002-9947-01-02857-4

Published electronically: August 20, 2001

PDF | Request permission

Abstract:

This work gives a new approach to the construction of implicit operations. By considering “higher-dimensional” spaces of implicit operations and implicit operators between them, the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, it is shown how they can be used to deduce from results of Ribes and Zalesskiĭ, Margolis, Sapir and Weil, and Steinberg that the pseudovariety of $p$-groups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups $\mathbf {V}$, if it can be decided whether a finitely generated subgroup of the free group with the pro-$\mathbf {V}$ topology is dense, then $\mathbf {V}$ is tame.

References

Jorge Almeida, Residually finite congruences and quasi-regular subsets in uniform algebras, Portugal. Math. 46 (1989), no. 3, 313–328. MR 1021193
Jorge Almeida, Finite semigroups and universal algebra, Series in Algebra, vol. 3, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. Translated from the 1992 Portuguese original and revised by the author. MR 1331143
Jorge Almeida, Hyperdecidable pseudovarieties and the calculation of semidirect products, Internat. J. Algebra Comput. 9 (1999), no. 3-4, 241–261. Dedicated to the memory of Marcel-Paul Schützenberger. MR 1722629, DOI 10.1142/S0218196799000163
J. Almeida and M. Delgado, Sur certains systèmes d’équations avec contraintes dans un groupe libre—addenda, Portugal. Math. To appear.
J. Almeida and M. Delgado, Sur certains systèmes d’équations avec contraintes dans un groupe libre, Portugal. Math. 56 (1999), no. 4, 409–417 (French, with English and French summaries). MR 1732025
Jorge Almeida and Benjamin Steinberg, On the decidability of iterated semidirect products with applications to complexity, Proc. London Math. Soc. (3) 80 (2000), no. 1, 50–74. MR 1719180, DOI 10.1112/S0024611500012144
S. Losinsky, Sur le procédé d’interpolation de Fejér, C. R. (Doklady) Acad. Sci. URSS (N.S.) 24 (1939), 318–321 (French). MR 0002001
Jorge Almeida and Pascal Weil, Reduced factorizations in free profinite groups and join decompositions of pseudovarieties, Internat. J. Algebra Comput. 4 (1994), no. 3, 375–403. MR 1297147, DOI 10.1142/S0218196794000051
J. Almeida and P. Weil, Relatively free profinite monoids: an introduction and examples, Semigroups, formal languages and groups (York, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 466, Kluwer Acad. Publ., Dordrecht, 1995, pp. 73–117. MR 1630619
Anatolij W. Anissimow and Franz D. Seifert, Zur algebraischen Charakteristik der durch kontext-freie Sprachen definierten Gruppen, Elektron. Informationsverarb. Kybernet. 11 (1975), no. 10-12, 695–702 (German, with English and Russian summaries). MR 422436
C. J. Ash, Inevitable graphs: a proof of the type $\textrm {II}$ conjecture and some related decision procedures, Internat. J. Algebra Comput. 1 (1991), no. 1, 127–146. MR 1112302, DOI 10.1142/S0218196791000079
Gilbert Baumslag, Residual nilpotence and relations in free groups, J. Algebra 2 (1965), 271–282. MR 179239, DOI 10.1016/0021-8693(65)90008-6
Jean Berstel, Transductions and context-free languages, Leitfäden der Angewandten Mathematik und Mechanik [Guides to Applied Mathematics and Mechanics], vol. 38, B. G. Teubner, Stuttgart, 1979. MR 549481, DOI 10.1007/978-3-663-09367-1
M. Delgado, On the hyperdecidability of pseudovarieties of groups, Int. J. Algebra Comput. To appear.
Samuel Eilenberg, Automata, languages, and machines. Vol. A, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York, 1974. MR 0530382
Samuel Eilenberg, Automata, languages, and machines. Vol. A, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York, 1974. MR 0530382
Samuel Eilenberg and M. P. Schützenberger, On pseudovarieties, Advances in Math. 19 (1976), no. 3, 413–418. MR 401604, DOI 10.1016/0001-8708(76)90029-3
Paul Flavell, Finite groups in which every two elements generate a soluble subgroup, Invent. Math. 121 (1995), no. 2, 279–285. MR 1346207, DOI 10.1007/BF01884299
Michael D. Fried and Moshe Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 11, Springer-Verlag, Berlin, 1986. MR 868860, DOI 10.1007/978-3-662-07216-5
Dion Gildenhuys and Luis Ribes, A Kurosh subgroup theorem for free pro-${\scr C}$-products of pro-${\scr C}$-groups, Trans. Amer. Math. Soc. 186 (1973), 309–329. MR 340433, DOI 10.1090/S0002-9947-1973-0340433-3
Dion Gildenhuys and Luis Ribes, Profinite groups and Boolean graphs, J. Pure Appl. Algebra 12 (1978), no. 1, 21–47. MR 488811, DOI 10.1016/0022-4049(78)90019-1
Rita Gitik, On the profinite topology on negatively curved groups, J. Algebra 219 (1999), no. 1, 80–86. MR 1707664, DOI 10.1006/jabr.1998.7847
Rita Gitik and Eliyahu Rips, On separability properties of groups, Internat. J. Algebra Comput. 5 (1995), no. 6, 703–717. MR 1365198, DOI 10.1142/S0218196795000288
F. Grunewald, B. Kuniavskii, D. Nikolova, and E. Plotkin, Two-variable identities in groups and Lie algebras, Zapiski Nauch. Seminarov POMI 272 (2000), 161–176. To appear also in J. Math. Sciences.
Bernhard Herwig and Daniel Lascar, Extending partial automorphisms and the profinite topology on free groups, Trans. Amer. Math. Soc. 352 (2000), no. 5, 1985–2021. MR 1621745, DOI 10.1090/S0002-9947-99-02374-0
Algebraische Theorie abstrakter Automaten, formaler Sprachen und Halbgruppen, Akademie-Verlag, Berlin, 1973 (German). Unter Mitarbeit von E. F. Assmus, Jr., Jane M. Day, J. J. Florentin, Seymour Ginsburg, Kenneth Krohn, Robert McNaughton, Seymour Papert, John L. Rhodes, Eliahu Shamir, Bret R. Tilson und H. Paul Zeiger; Deutsche Übersetzung aus dem Englischen von J. Kunze, K. Latt, H.-D. Modrow, H.-J. Pohl, K. A. Zech; Herausgeber der deutschen Ausgabe: Lothar Budach, Helmut Thiele. MR 0345756
Kenneth Krohn and John Rhodes, Complexity of finite semigroups, Ann. of Math. (2) 88 (1968), 128–160. MR 236294, DOI 10.2307/1970558
S. Margolis, M. Sapir, and P. Weil, Closed subgroups in pro-V topologies and the extension problem for inverse automata, Int. J. Algebra Comput. To appear.
Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
J. P. McCammond, Normal forms for free aperiodic semigroups, Int. J. Algebra Comput. To appear.
Jean-Eric Pin and Christophe Reutenauer, A conjecture on the Hall topology for the free group, Bull. London Math. Soc. 23 (1991), no. 4, 356–362. MR 1125861, DOI 10.1112/blms/23.4.356
Jan Reiterman, The Birkhoff theorem for finite algebras, Algebra Universalis 14 (1982), no. 1, 1–10. MR 634411, DOI 10.1007/BF02483902
John Rhodes, Undecidability, automata, and pseudovarieties of finite semigroups, Internat. J. Algebra Comput. 9 (1999), no. 3-4, 455–473. Dedicated to the memory of Marcel-Paul Schützenberger. MR 1723477, DOI 10.1142/S0218196799000278
—, Complexity $c$ is decidable for finite automata and semigroups, Tech. report, Univ. California at Berkeley, 2000.
Luis Ribes and Pavel A. Zalesskii, The pro-$p$ topology of a free group and algorithmic problems in semigroups, Internat. J. Algebra Comput. 4 (1994), no. 3, 359–374. MR 1297146, DOI 10.1142/S021819679400004X
John R. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), no. 3, 551–565. MR 695906, DOI 10.1007/BF02095993
B. Steinberg, Inevitable graphs and profinite topologies: some solutions to algorithmic problems in monoid and automata theory, stemming from group theory, Int. J. Algebra Comput. 11 (2001), 25–71.
John G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (1968), 383–437. MR 230809, DOI 10.1090/S0002-9904-1968-11953-6
I. Yu. Zhil’tsov, On identities of finite aperiodic epigroups, Tech. report, Ural State Univ., 1999.