On a hereditarily finitely based ai-semiring variety

  • Miaomiao RenEmail author
  • Lingli Zeng


Let \(\mathbf{W}\) denote the join of the ai-semiring variety axiomatized by \(x^2\approx x\) and the ai-semiring variety axiomatized by \(xy\approx zt\). We show that the lattice of subvarieties of \(\mathbf{W}\), \(\mathcal{L}(\mathbf{W})\), is a distributive lattice of order 312. Also, all members of this variety are finitely based and finitely generated. Thus, \(\mathbf{W}\) is hereditarily finitely based.


Ai-semiring Variety Lattice Identity Hereditarily finitely based 



The authors thank the anonymous referees for an unusually careful reading of the paper that has led to a substantial improvement of this paper. The authors also thank Professor Xianzhong Zhao for discussions contributed to this paper. The authors are supported by National Natural Science Foundation of China (11701449). The first author is supported by National Natural Science Foundation of China (11571278), Natural Science Foundation of Shaanxi Province (2017JQ1033), Scientific Research Program of Shaanxi Provincial Education Department (16JK1754) and Scientific Research Foundation of Northwest University (15NW24).

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Conflicts of interest

All authors declare that they have no conflicts of interest

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This article does not contain any studies with human participants or animal.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsNorthwest UniversityXi’anPeople’s Republic of China

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