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On varieties of flat nil-semirings

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Abstract

We study some varieties of flat nil-semirings and provide a sufficient condition for a finite flat semiring to be nonfinitely based. As applications, we show that a finite flat nil-semiring of the form \(S_c(W)\) is finitely based if and only if W consists of cubes of letters and words of length at most 2. Also, we prove that the finite flat nil-semirings S(W) are nonfinitely based for some sets W of special words in \(X^+\). This affirmatively answers two problems raised by Jackson et al. (J Algebra 611:211–245, 2022).

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Acknowledgements

The authors would like to thank Prof. Marcel Jackson for his valuable suggestions and for helping to prepare the final version of this paper. The authors also thank the anonymous referee for an unusually careful reading of the paper that has led to a substantial improvement of this paper. This paper is supported by National Natural Science Foundation of China (11971383, 11571278) and Natural Science Foundation of Shaanxi Province (2022 JM-009).

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Authors and Affiliations

  1. School of Mathematics, Northwest University, Xi’an, 710127, Shaanxi, People’s Republic of China

    Y. N. Wu, X. Z. Zhao & M. M. Ren

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  1. Y. N. Wu
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  2. X. Z. Zhao
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  3. M. M. Ren
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Correspondence to X. Z. Zhao.

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Communicated by Mikhail Volkov.

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Wu, Y.N., Zhao, X.Z. & Ren, M.M. On varieties of flat nil-semirings. Semigroup Forum 106, 271–284 (2023). https://doi.org/10.1007/s00233-023-10337-2

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