Startseite Mathematik On the group of a rational maximal bifix code
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

On the group of a rational maximal bifix code

  • Jorge Almeida , Alfredo Costa EMAIL logo , Revekka Kyriakoglou und Dominique Perrin
Veröffentlicht/Copyright: 21. Dezember 2019

Abstract

We give necessary and sufficient conditions for the group of a rational maximal bifix code Z to be isomorphic with the F-group of ZF, when F is recurrent and ZF is rational. The case where F is uniformly recurrent, which is known to imply the finiteness of ZF, receives special attention. The proofs are done by exploring the connections with the structure of the free profinite monoid over the alphabet of F.


Communicated by Manfred Droste


Award Identifier / Grant number: UID/MAT/ 00144/2013

Award Identifier / Grant number: UID/MAT/00324/2013

Award Identifier / Grant number: SFRH/BSAB/142872/2018

Funding statement: J. Almeida and A. Costa acknowledge partial funding by CMUP (UID/MAT/ 00144/2013) and CMUC (UID/MAT/00324/2013), respectively, which are funded by FCT (Portugal) with national (MCTES) and European structural funds (FEDER) under the partnership agreement PT2020. The work of J. Almeida was carried out in part at Masaryk University, whose hospitality is gratefully acknowledged, with the support of the FCT sabbatical scholarship SFRH/BSAB/142872/2018.

References

[1] J. Almeida, Finite Semigroups and Universal Algebra, World Scientific, Singapore, 1994. 10.1142/2481Suche in Google Scholar

[2] J. Almeida, Profinite groups associated with weakly primitive substitutions (in Russian), Fundam. Prikl. Mat. 11 (2005), no. 3, 13–48; translation in J. Math. Sci. 144 (2007), no. 2, 3881-3903. 10.1007/s10958-007-0242-ySuche in Google Scholar

[3] J. Almeida, Profinite semigroups and applications, Structural Theory of Automata, Semigroups, and Universal Algebra, NATO Sci. Ser. II Math. Phys. Chem. 207, Springer, Dordrecht (2005), 1–45. 10.1007/1-4020-3817-8_1Suche in Google Scholar

[4] J. Almeida, A. Cano, O. Klíma and J.-E. Pin, On fixed points of the lower set operator, Internat. J. Algebra Comput. 25 (2015), no. 1–2, 259–292. 10.1142/S021819671540010XSuche in Google Scholar

[5] J. Almeida and A. Costa, Infinite-vertex free profinite semigroupoids and symbolic dynamics, J. Pure Appl. Algebra 213 (2009), no. 5, 605–631. 10.1016/j.jpaa.2008.08.009Suche in Google Scholar

[6] J. Almeida and A. Costa, Presentations of Schützenberger groups of minimal subshifts, Israel J. Math. 196 (2013), no. 1, 1–31. 10.1007/s11856-012-0139-4Suche in Google Scholar

[7] J. Almeida and A. Costa, A geometric interpretation of the Schützenberger group of a minimal subshift, Ark. Mat. 54 (2016), no. 2, 243–275. 10.1007/s11512-016-0233-7Suche in Google Scholar

[8] J. Almeida and A. Costa, Equidivisible pseudovarieties of semigroups, Publ. Math. Debrecen 90 (2017), no. 3–4, 435–453. 10.5486/PMD.2017.7634Suche in Google Scholar

[9] J. Almeida, A. Costa, J. C. Costa and M. Zeitoun, The linear nature of pseudowords, Publ. Mat. 63 (2019), no. 2, 361–422. 10.5565/PUBLMAT6321901Suche in Google Scholar

[10] J. Almeida and M. V. Volkov, Subword complexity of profinite words and subgroups of free profinite semigroups, Internat. J. Algebra Comput. 16 (2006), no. 2, 221–258. 10.1142/S0218196706002883Suche in Google Scholar

[11] J. Berstel, C. De Felice, D. Perrin, C. Reutenauer and G. Rindone, Bifix codes and Sturmian words, J. Algebra 369 (2012), 146–202. 10.1016/j.jalgebra.2012.07.013Suche in Google Scholar

[12] J. Berstel, D. Perrin and C. Reutenauer, Codes and Automata, Encyclopedia Math. Appl. 129, Cambridge University, Cambridge, 2010. 10.1017/CBO9781139195768Suche in Google Scholar

[13] V. Berthé, C. De Felice, F. Dolce, J. Leroy, D. Perrin, C. Reutenauer and G. Rindone, Acyclic, connected and tree sets, Monatsh. Math. 176 (2015), no. 4, 521–550. 10.1007/s00605-014-0721-4Suche in Google Scholar

[14] V. Berthé, C. De Felice, F. Dolce, J. Leroy, D. Perrin, C. Reutenauer and G. Rindone, Bifix codes and interval exchanges, J. Pure Appl. Algebra 219 (2015), no. 7, 2781–2798. 10.1016/j.jpaa.2014.09.028Suche in Google Scholar

[15] V. Berthé, C. De Felice, F. Dolce, J. Leroy, D. Perrin, C. Reutenauer and G. Rindone, Maximal bifix decoding, Discrete Math. 338 (2015), no. 5, 725–742. 10.1016/j.disc.2014.12.010Suche in Google Scholar

[16] V. Berthé, C. De Felice, F. Dolce, J. Leroy, D. Perrin, C. Reutenauer and G. Rindone, The finite index basis property, J. Pure Appl. Algebra 219 (2015), no. 7, 2521–2537. 10.1016/j.jpaa.2014.09.014Suche in Google Scholar

[17] A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups. Vol. I, American Mathematical Society, Providence, 1961. 10.1090/surv/007.1Suche in Google Scholar

[18] A. Costa, Conjugacy invariants of subshifts: an approach from profinite semigroup theory, Internat. J. Algebra Comput. 16 (2006), no. 4, 629–655. 10.1142/S0218196706003232Suche in Google Scholar

[19] A. Costa and B. Steinberg, Profinite groups associated to sofic shifts are free, Proc. Lond. Math. Soc. (3) 102 (2011), no. 2, 341–369. 10.1112/plms/pdq024Suche in Google Scholar

[20] M. Delgado, S. Linton and J. Morais, Automata: A GAP package on finite automata, http://www.gap-system.org/Packages/automata.html. Suche in Google Scholar

[21] M. Delgado and J. Morais, SgpViz: A GAP package to visualize finite semigroups, 2008, http://www.gap-system.org/Packages/sgpviz.html. Suche in Google Scholar

[22] F. Dolce and D. Perrin, Eventually dendric shifts, Computer Science—Theory and Applications, Lecture Notes in Comput. Sci. 11532, Springer, Cham (2019), 106–118. 10.1007/978-3-030-19955-5_10Suche in Google Scholar

[23] N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Math. 1794, Springer, Berlin, 2002. 10.1007/b13861Suche in Google Scholar

[24] A. Glen and J. Justin, Episturmian words: A survey, Theor. Inform. Appl. 43 (2009), no. 3, 403–442. 10.1051/ita/2009003Suche in Google Scholar

[25] W. Krieger, On the uniqueness of the equilibrium state, Math. Systems Theory 8 (1974/75), no. 2, 97–104. 10.1007/BF01762180Suche in Google Scholar

[26] R. Kyriakoglou and D. Perrin, Profinite semigroups, preprint (2017), https://arxiv.org/abs/1703.10088. Suche in Google Scholar

[27] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University, Cambridge, 1995. 10.1017/CBO9780511626302Suche in Google Scholar

[28] M. Lothaire, Combinatorics on Words, Cambridge University, Cambridge, 1997. 10.1017/CBO9780511566097Suche in Google Scholar

[29] D. Perrin, Groups, languages and dendric shifts, Developments in Language Theory, Lecture Notes in Comput. Sci. 11088, Springer, Cham (2018), 60–73. 10.1007/978-3-319-98654-8_5Suche in Google Scholar

[30] J.-E. Pin, A variety theorem without complementation, Russian Math. (Iz. VUZ) 39 (1995),80–90. Suche in Google Scholar

[31] J.-E. Pin, Polynomial closure of group languages and open sets of the Hall topology, Theoret. Comput. Sci. 169 (1996), no. 2, 185–200. 10.1007/3-540-58201-0_87Suche in Google Scholar

[32] J.-E. Pin, Profinite methods in automata theory, 26th International Symposium on Theoretical Aspects of Computer Science—STACS 2009, LIPIcs. Leibniz Int. Proc. Inform. 3, Schloss Dagstuhl, Wadern (2009), 31–50. Suche in Google Scholar

[33] J. Rhodes and B. Steinberg, Profinite semigroups, varieties, expansions and the structure of relatively free profinite semigroups, Internat. J. Algebra Comput. 11 (2001), no. 6, 627–672. 10.1142/S0218196701000784Suche in Google Scholar

[34] J. Rhodes and B. Steinberg, The q-theory of Finite Semigroups, Springer Monogr. Math., Springer, New York, 2009. 10.1007/b104443Suche in Google Scholar

[35] M.-P. Schützenberger, Une théorie algébrique du codage, Séminaire Dubreil–Pisot. Algèbre et théorie des nombres. 9e année 1955/56, Faculté des Sciences de Paris, Paris (1956), 1–24. Suche in Google Scholar

[36] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.6, 2013, (http://www.gap-system.org). Suche in Google Scholar

Received: 2018-11-09
Revised: 2019-10-04
Published Online: 2019-12-21
Published in Print: 2020-05-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 4.2.2026 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2018-0270/pdf
Button zum nach oben scrollen