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Transactions of the American Mathematical Society

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Dynamics of implicit operations and tameness of pseudovarieties of groups

Author: Jorge Almeida
Journal: Trans. Amer. Math. Soc. 354 (2002), 387-411
MSC (1991): Primary 20E18, 20M05, 20M07; Secondary 20F10, 20E07, 20E05
Published electronically: August 20, 2001
MathSciNet review: 1859280
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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 Zalesski{\u{\i}}\kern.15em, 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.

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  • 1. J. Almeida, Residually finite congruences and quasi-regular subsets in uniform algebras, Portugal. Math. 46 (1989), 313-328. MR 90m:08001
  • 2. -, Finite semigroups and universal algebra, World Scientific, Singapore, 1995, English translation. MR 96b:20069
  • 3. -, Hyperdecidable pseudovarieties and the calculation of semidirect products, Int. J. Algebra Comput. 9 (1999), 241-261. MR 2001a:20102
  • 4. J. Almeida and M. Delgado, Sur certains systèmes d'équations avec contraintes dans un groupe libre--addenda, Portugal. Math. To appear.
  • 5. -, Sur certains systèmes d'équations avec contraintes dans un groupe libre, Portugal. Math. 56 (1999), 409-417. MR 2001b:20039
  • 6. J. Almeida and B. Steinberg, On the decidability of iterated semidirect products and applications to complexity, Proc. London Math. Soc. 80 (2000), 50-74. MR 2000j:20109
  • 7. -, Syntactic and global semigroup theory, a synthesis approach, Algorithmic Problems in Groups and Semigroups (J. C. Birget, S. W. Margolis, J. Meakin, and M. V. Sapir, eds.), Birkhäuser, 2000, pp. 1-23. MR 2001e:20120
  • 8. J. Almeida and P. Weil, Reduced factorizations in free profinite groups and join decompositions of pseudovarieties, Int. J. Algebra Comput. 4 (1994), 375-403. MR 95m:20066
  • 9. -, Relatively free profinite monoids: an introduction and examples, Semigroups, Formal Languages and Groups (Dordrecht) (J. B. Fountain, ed.), vol. 466, Kluwer Academic Publ., 1995, pp. 73-117. MR 2000f:20095
  • 10. A. W. Anissimov and F. D. Seifert, Zur algebraischen charakteristik der durch kontext-freie sprachen definierten gruppen, Elektron. Informationsverarb. Kybernet. 11 (1975), 695-702. MR 54:10425
  • 11. C. J. Ash, Inevitable graphs: a proof of the type II conjecture and some related decision procedures, Int. J. Algebra Comput. 1 (1991), 127-146. MR 92k:20114
  • 12. G. Baumslag, Residual nilpotence and relations in free groups, J. Algebra 2 (1965), 271-282. MR 31:3487
  • 13. J. Berstel, Transductions and context-free languages, B. G. Teubner, Stuttgart, 1979. MR 80j:68056
  • 14. M. Delgado, On the hyperdecidability of pseudovarieties of groups, Int. J. Algebra Comput. To appear.
  • 15. S. Eilenberg, Automata, languages and machines, vol. A, Academic Press, New York, 1974. MR 58:26604a
  • 16. -, Automata, languages and machines, vol. B, Academic Press, New York, 1976. MR 58:26604b
  • 17. S. Eilenberg and M. P. Schützenberger, On pseudovarieties, Advances in Math. 19 (1976), 413-418. MR 53:5431
  • 18. P. Flavell, Finite groups in which every two elements generate a soluble group, Invent. Math. 121 (1995), 279-285. MR 96i:20018
  • 19. M. D. Fried and M. Jarden, Field arithmetic, Springer, Berlin, 1986. MR 89b:12010
  • 20. D. Gildenhuys and L. Ribes, A Kurosh subgroup theorem for free pro- $\mathcal{C}$products of pro- $\mathcal{C}$ groups, Trans. Amer. Math. Soc. 186 (1973), 309-329. MR 49:5188
  • 21. -, Profinite groups and Boolean graphs, J. Pure and Appl. Algebra 12 (1978), 21-47. MR 81g:20059
  • 22. R. Gitik, On the profinite topology on negatively curved groups, J. Algebra 219 (1999), 80-86. MR 2000g:20072
  • 23. R. Gitik and E. Rips, On separability properties of groups, Int. J. Algebra Comput. 5 (1995), 703-717. MR 97e:20059
  • 24. 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. CMP 2001:08
  • 25. B. Herwig and D. Lascar, Extending partial automorphisms and the profinite topology on free groups, Trans. Amer. Math. Soc. 352 (2000), 1985-2021. MR 2000j:20036
  • 26. K. Krohn and J. Rhodes, Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines, Trans. Amer. Math. Soc. 116 (1965), 450-464. MR 49:10487
  • 27. -, Complexity of finite semigroups, Ann. of Math. (2) 88 (1968), 128-160. MR 38:4591
  • 28. 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.
  • 29. S. W. Margolis and J. C. Meakin, Free inverse monoids and graph immersions, Int. J. Algebra Comput. 3 (1993), 79-99. MR 94e:20105
  • 30. J. P. McCammond, Normal forms for free aperiodic semigroups, Int. J. Algebra Comput. To appear.
  • 31. J.-E. Pin and C. Reutenauer, A conjecture on the Hall topology for the free group, Bull. London Math. Soc. 23 (1991), 356-362. MR 92g:20035
  • 32. J. Reiterman, The Birkhoff theorem for finite algebras, Algebra Universalis 14 (1982), 1-10. MR 84c:08008
  • 33. J. Rhodes, Undecidability, automata and pseudovarieties of finite semigroups, Int. J. Algebra Comput. 9 (1999), 455-473. MR 2000j:20112
  • 34. -, Complexity $c$ is decidable for finite automata and semigroups, Tech. report, Univ. California at Berkeley, 2000.
  • 35. L. Ribes and P. A. Zalesski{\u{\i}}\kern.15em, The pro-$p$ topology of a free group and algorithmic problems in semigroups, Int. J. Algebra Comput. 4 (1994), 359-374. MR 96e:20046
  • 36. J. R. Stallings, Topology of finite graphs, Inventiones Mathematicae 71 (1983), 551-565. MR 85m:05037a
  • 37. 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.
  • 38. J. G. Thompson, Non-solvable groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (1968), 383-437. MR 37:6367
  • 39. I. Yu. Zhil'tsov, On identities of finite aperiodic epigroups, Tech. report, Ural State Univ., 1999.

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Additional Information

Jorge Almeida
Affiliation: Centro de Matemática da Universidade do Porto, P. Gomes Teixeira, 4099-002 Porto, Portugal

Keywords: Profinite topology, implicit operation, pseudovariety, free group, extension closed, finite semigroup, semidirect product
Received by editor(s): February 10, 2000
Received by editor(s) in revised form: March 28, 2001
Published electronically: August 20, 2001
Additional Notes: The author gratefully acknowledges support by FCT through the Centro de Matemática da Universidade do Porto, by the project Praxis/2/2.1/MAT/63/94 (Portugal), and by NSERC (United Kingdom)
Article copyright: © Copyright 2001 American Mathematical Society

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