A new algebraic invariant for weak equivalence of sofic subshifts
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 42 (2008) no. 3, pp. 481-502.

It is studied how taking the inverse image by a sliding block code affects the syntactic semigroup of a sofic subshift. The main tool are $\zeta$-semigroups, considered as recognition structures for sofic subshifts. A new algebraic invariant is obtained for weak equivalence of sofic subshifts, by determining which classes of sofic subshifts naturally defined by pseudovarieties of finite semigroups are closed under weak equivalence. Among such classes are the classes of almost finite type subshifts and aperiodic subshifts. The algebraic invariant is compared with other robust conjugacy invariants.

DOI: 10.1051/ita:2008015
Classification: 20M07,  37B10,  20M35
Keywords: sofic subshift, conjugacy, weak equivalence, $\zeta$-semigroup, pseudovariety
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Chaubard, Laura; Costa, Alfredo. A new algebraic invariant for weak equivalence of sofic subshifts. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 42 (2008) no. 3, pp. 481-502. doi : 10.1051/ita:2008015. http://www.numdam.org/articles/10.1051/ita:2008015/

[1] J. Almeida, Finite semigroups and universal algebra, World Scientific, Singapore (1995), English translation. | MR | Zbl

[2] M.-P. Béal, F. Fiorenzi, and D. Perrin, A hierarchy of shift equivalent sofic shifts. Theoret. Comput. Sci. 345 (2005) 190-205. | MR | Zbl

[3] M.-P. Béal, F. Fiorenzi, and D. Perrin, The syntactic graph of a sofic shift is invariant under shift equivalence. Int. J. Algebra Comput. 16 (2006), 443-460. | MR | Zbl

[4] M.-P. Béal and D. Perrin, A weak equivalence between shifts of finite type. Adv. Appl. Math. 29 (2002) 2, 162-171. | MR | Zbl

[5] D. Beauquier, Minimal automaton for a factorial transitive rational language. Theoret. Comput. Sci. 67 (1985), 65-73. | MR | Zbl

[6] F. Blanchard and G. Hansel, Systèmes codés. Theoret. Comput. Sci. 44 (1986), 17-49. | MR | Zbl

[7] M. Boyle and W. Krieger, Almost markov and shift equivalent sofic systems, Proceedings of Maryland Special Year in Dynamics 1986-87 (J.C. Alexander, ed.). Lect. Notes Math. 1342 (1988), pp. 33-93. | MR | Zbl

[8] M.-P. Béal, Codage symbolique, Masson (1993).

[9] O. Carton, Wreath product and infinite words. J. Pure Appl. Algebra 153 (2000), 129-150. | MR | Zbl

[10] L. Chaubard, L‘équivalence faible des systèmes sofiques, Master's thesis, LIAFA, Université Paris VII, July 2003, Rapport de stage de DEA.

[11] A. Costa, Pseudovarieties defining classes of sofic subshifts closed under taking shift equivalent subshifts. J. Pure Appl. Algebra 209 (2007), 517-530. | MR | Zbl

[12] S. Eilenberg, Automata, languages and machines, vol. B, Academic Press, New York, 1976. | MR | Zbl

[13] R. Fischer, Sofic systems and graphs. Monatsh. Math. 80 (1975), 179-186. | EuDML | MR | Zbl

[14] G. A. Hedlund, Endomorphims and automorphisms of the shift dynamical system. Math. Syst. Theor. 3 (1969), 320-375. | MR | Zbl

[15] W. Krieger, On sofic systems. i. Israel J. Math. 48 (1984), 305-330. | MR | Zbl

[16] D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge (1996). | MR | Zbl

[17] M. Nasu, Topological conjugacy for sofic systems. Ergod. Theory Dyn. Syst. 6 (1986), 265-280. | MR | Zbl

[18] D. Perrin and J.-E. Pin, Infinite words, Pure and Applied Mathematics, No. 141, Elsevier, London, 2004. | Zbl

[19] J.-E. Pin, Varieties of formal languages, Plenum, London (1986), English translation. | MR | Zbl

[20] J.-E. Pin, Syntactic semigroups, Handbook of Language Theory (G. Rozenberg and A. Salomaa, eds.), Springer (1997). | MR

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