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Normal forms under Simon’s congruence

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Abstract

Simon’s congruence, denoted by \(\sim _k\), relates the words having the same subwords of length at most k. In this paper a normal form is presented for the equivalence classes of \(\sim _k\). The length of this normal form is the shortest possible. Moreover, a canonical solution of the equation \(pwq\sim _k r\) is also shown (the words p, q, r are parameters), which can be viewed as a generalization of giving a normal form for \(\sim _k\). In this paper, there can be found an algorithm with which the canonical solution can be determined in \(O((L+n)n^{k})\) time, where L denotes the length of the word pqr and n is the size of the alphabet.

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Acknowledgements

We would like to thank the anonymous referee for the careful reading of the manuscript and useful comments.

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Authors and Affiliations

  1. Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Magyar Tudósok Körútja 2, 1117, Budapest, Hungary

    Péter Pál Pach

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  1. Péter Pál Pach
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Correspondence to Péter Pál Pach.

Additional information

Communicated by Dominique Perrin.

This research was supported by the National Research, Development and Innovation Office NKFIH (Grant Numbers PD115978 and K124171) and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

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Pach, P.P. Normal forms under Simon’s congruence. Semigroup Forum 97, 251–267 (2018). https://doi.org/10.1007/s00233-017-9910-5

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  • DOI: https://doi.org/10.1007/s00233-017-9910-5

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