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On the finite basis problem for the monoids of partial extensive injective transformations

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Abstract

Let \(PEI_n (POEI_n)\) be the monoid of all partial (order-preserving) extensive and injective transformations over a chain of order n. We give a sufficient condition under which a semigroup is nonfinitely based and apply this condition to show that the monoid \(PEI_3 (POEI_3)\) is nonfinitely based. This together with the results of Edmunds and Goldberg gives a complete answer to the finite basis problem for the monoid \(PEI_n (POEI_n)\): the monoid \(PEI_n (POEI_n)\) is nonfinitely based if and only if \(n\geqslant 3\). Furthermore, it is shown that the monoid \(PEI_n (POEI_n)\) is hereditarily finitely based if and only if \(n\leqslant 2\).

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Acknowledgments

The authors are very grateful to the anonymous referees for comments and suggestions, especially for suggesting the concepts of uniquely scattered subwords and separators, which have simplified the presentation. They also thank Mikhail Volkov for his comments, which have been helpful in improving the final version of this paper. This research has been partially supported by the National Natural Science Foundation of China (Nos. 11371177, 11401275) and the Fundamental Research Funds for the Central Universities of China (No. lzujbky-2015-78).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, China

    Xun Hu, Yuzhu Chen & Yanfeng Luo

  2. School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400033, China

    Xun Hu

  3. Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou, 730000, Gansu, China

    Yanfeng Luo

Authors
  1. Xun Hu
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  2. Yuzhu Chen
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  3. Yanfeng Luo
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Correspondence to Yanfeng Luo.

Additional information

Communicated by Mikhail V. Volkov.

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Hu, X., Chen, Y. & Luo, Y. On the finite basis problem for the monoids of partial extensive injective transformations. Semigroup Forum 91, 524–537 (2015). https://doi.org/10.1007/s00233-015-9744-y

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  • DOI: https://doi.org/10.1007/s00233-015-9744-y

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