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Classification of finitely based words in a class of words over a \(3\)-letter alphabet

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Abstract

Let \(W\) be a finite set of words over an alphabet \(X\). The discrete syntactic monoid \(S(W)\) of \(W\) is a monoid whose elements are \(0, 1\) and all subwords of words in \(W\) with the multiplication: for \(\mathbf {w}_1, \mathbf {w}_2 \in W\), \(\mathbf {w}_1\cdot \mathbf {w}_2=\mathbf {w}_1\mathbf {w}_2\) if \(\mathbf {w}_1\mathbf {w}_2\) is a subword of a word in \(W\) and \(0\) otherwise. The set of words \(W\) is called finitely based if the monoid \(S(W)\) is finitely based. A single word \(\mathbf {w}\) is called finitely based if \(\{\mathbf {w}\}\) is finitely based. In this paper, we classify all finitely based words in a class of words over the alphabet \(\{x,y,z\}\) without subwords of the form \(xz\), \(yx\) or \(zy\).

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Acknowledgments

The authors are very grateful to the anonymous referees for comments and suggestions that have been helpful to improve the quality of this paper. The authors are very grateful to Dr. Marcel Jackson for suggesting this problem to them. This research is partially supported by the National Natural Science Foundation of China (Nos. 11371177, 11401275).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Jian-Rong Li & Yan-Feng Luo

  2. Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou, Gansu, People’s Republic of China

    Yan-Feng Luo

Authors
  1. Jian-Rong Li
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  2. Yan-Feng Luo
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Correspondence to Jian-Rong Li.

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Communicated by Mikhail Volkov.

Dedicated to Professor Mikhail V. Volkov on the occasion of his 60th birthday.

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Li, JR., Luo, YF. Classification of finitely based words in a class of words over a \(3\)-letter alphabet. Semigroup Forum 91, 200–212 (2015). https://doi.org/10.1007/s00233-015-9712-6

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