Some Applications of Algebra to Automatic Sequences
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.
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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