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Finite basis problem for involution monoids of unitriangular boolean matrices

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Abstract

Let \((BU_n, \,^*\,)\) be the involution monoid of all Boolean upper triangular \(n\times n\) matrices with 1s on the main diagonal under the ordinary matrix multiplication and the skew transposition. The involution monoid \((BU_2, \,^*\,)\) is easily seen to be finitely based. In this paper, we shown that \((BU_n, \,^*\,)\) is non-finitely based for each \(n \ge 3\), which answers an open question posed by Auinger et al. Therefore involution monoid \((BU_n, \,^*\,)\) is finitely based if and only if \(n = 2\).

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Acknowledgements

The authors would like to express their gratitude to E. W. H. Lee for advice, and also to the referees for their valuable comments and suggestions.

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Authors and Affiliations

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

    Wen Ting Zhang, Yan Feng Luo & Nan Wang

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  1. Wen Ting Zhang
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  2. Yan Feng Luo
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  3. Nan Wang
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Correspondence to Wen Ting Zhang.

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Presented by Edmond W. H. Lee.

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This research was partially supported by the National Natural Science Foundation of China under Grant Nos. 11771191, 11401275, 11371177.

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Zhang, W.T., Luo, Y.F. & Wang, N. Finite basis problem for involution monoids of unitriangular boolean matrices. Algebra Univers. 81, 7 (2020). https://doi.org/10.1007/s00012-019-0637-x

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