Abstract
We give an overview of the theory of rings satisfying a polynomial identity and use this to give a proof of a characterization due to Berstel and Reutenauer of automatic and regular sequences in terms of two properties, which we call the shuffle property and the power property. These properties show that if one views an automatic sequence f as a map on a free monoid on k-letters to a finite subset of a ring, then the values of f are closely related to values of f on related words obtained by permuting letters of the word. We use this characterization to give answers to three questions from Allouche and Shallit, two of which have not appeared in the literature. The final part of the chapter deals more closely with the shuffle property, and we view this as giving a generalization of regular sequences. We show that sequences with the shuffle property are closed under the process of taking sums and taking products; in addition we show that there is closure under a noncommutative product, which turns the collection of shuffled sequences into a noncommutative algebra. We show that this algebra is very large, in the sense that it contains a copy of a free associative algebra on countably many generators. We conclude by giving some open questions, which we hope will begin a more careful study of shuffled sequences.
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References
Allouche, J.-P., Scheicher, K., Tichy, R.F.: Regular maps in generalized number systems. Math. Slovaca 50, 41–58 (2000)
Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)
Allouche, J.-P., Shallit, J.O.: The ring of k-regular sequences. In: Choffrut, C., Lengauer, T. (eds.) STACS 90, Proceedings of the 7th Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 415, pp. 12–23. Springer, Berlin (1990)
Amitsur, A.S., Levitzki, J.: Minimal identities for algebras. Proc. Am. Math. Soc. 1, 449–463 (1950)
Amitsur, S.A., Small, L.W.: Affine algebras with polynomial identities. Rend. Circ. Mat. Palermo (2) Suppl. 31, 9–43 (1993). Recent developments in the theory of algebras with polynomial identities (Palermo, 1992)
Berstel, J., Reutenauer, C.: Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications, vol. 137. Cambridge University Press, Cambridge (2011)
Braun, A.: The nilpotency of the radical in a finitely generated PI ring. J. Algebra 89(2), 375–396 (1984)
Conway, J.H.: On Numbers and Games. Academic Press, New York (1976)
Dekking, F.M., Mendès France, M., Poorten, A.J.v.d.: Folds! Math. Intelligencer 4, 130–138, 173–181, 190–195 (1982). Erratum, 5 (1983), 5
Hansel, G.: A simple proof of the Skolem-Mahler-Lech theorem. In: Brauer, W. (ed.) Proceedings of the 12th International Conference on Automata, Languages, and Programming (ICALP). Lecture Notes in Computer Science, vol. 194, pp. 244–249. Springer, Berlin (1985)
Mahler, K.: An unsolved problem on the powers of 3/2. J. Aust. Math. Soc. 8, 313–321 (1968)
Moshe, Y.: On some questions regarding k-regular and k-context-free sequences. Theor. Comput. Sci. 400(1-3), 62–69 (2008)
Pirillo, G.: A proof of Shirshov’s theorem. Adv. Math. 124(1), 94–99 (1996)
Regev, A.: Existence of identities in A ⊗ B. Isr. J. Math. 11, 131–152 (1972)
Rosset, S.: A new proof of the Amitsur-Levitski identity. Isr. J. Math. 23(2), 187–188 (1976)
Shallit, J.O.: A generalization of automatic sequences. Theor. Comput. Sci. 61, 1–16 (1988)
Acknowledgements
I thank Jean-Paul Allouche and Jeffrey Shallit for many helpful comments. I also thank Jean-Paul Allouche for raising Question 4.9.4.
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Bell, J. (2018). Some Applications of Algebra to Automatic Sequences. In: Berthé, V., Rigo, M. (eds) Sequences, Groups, and Number Theory. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-69152-7_4
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