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On a hereditarily finitely based ai-semiring variety

  • Miaomiao RenEmail author
  • Lingli Zeng
Foundations
  • 14 Downloads

Abstract

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.

Keywords

Ai-semiring Variety Lattice Identity Hereditarily finitely based 

Notes

Acknowledgements

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).

Compliance with ethical standards

Conflicts of interest

All authors declare that they have no conflicts of interest

Ethical approval

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

References

  1. Burris S, Sankappanavar HP (1981) A course in universal algebra. Springer, New YorkCrossRefzbMATHGoogle Scholar
  2. Ghosh S, Pastijn F, Zhao XZ (2005) Varieties generated by ordered bands I. Order 22:109–128MathSciNetCrossRefzbMATHGoogle Scholar
  3. Głazek K (2001) A guide to the literature on semirings and their applications in mathematics and information science. Kluwer Academic Publishers, DordrechtzbMATHGoogle Scholar
  4. Golan JS (1992) The theory of semirings with applications in mathematics and theoretical computer science. Longman Scientific and Technical, HarlowzbMATHGoogle Scholar
  5. Howie JM (1995) Fundamentals of semigroup theory. Clarendon Press, LondonzbMATHGoogle Scholar
  6. Kuřil M, Polák L (2005) On varieties of semilattice-ordered semigroups. Semigroup Forum 71(1):27–48MathSciNetCrossRefzbMATHGoogle Scholar
  7. McKenzie R, Romanowska A (1979) Varieties of \(\cdot \)-distributive bisemilattices. In: Contributions to general algebra (Proc. Klagenfurt Conf. 1978), vol 1, pp 213–218Google Scholar
  8. Pastijn F (2005) Varieties generated by ordered bands II. Order 22:129–143MathSciNetCrossRefzbMATHGoogle Scholar
  9. Pastijn F, Zhao XZ (2005) Varieties of idempotent semirings with commutative addition. Algebra Univers 54:301–321MathSciNetCrossRefzbMATHGoogle Scholar
  10. Petrich M, Reilly NR (1999) Completely regular semigroups. Wiley, New YorkzbMATHGoogle Scholar
  11. Ren MM, Zhao XZ (2016) The varieties of semilattice-ordered semigroups satisfying \(x^3\approx x\) and \(xy\approx yx\). Period Math Hungar 72:158–170MathSciNetCrossRefzbMATHGoogle Scholar
  12. Ren MM, Zhao XZ, Shao Y (2016a) On a variety of Burnside ai-semirings satisfying \(x^{n}\approx x\). Semigroup Forum 93(3):501–515MathSciNetCrossRefzbMATHGoogle Scholar
  13. Ren MM, Zhao XZ, Volkov MV (2016b) The Burnside ai-semiring variety defined by \(x^n \approx x\), manuscriptGoogle Scholar
  14. Ren MM, Zhao XZ, Wang AF (2017) On the varieties of ai-semirings satisfying \(x^{3}\approx x\). Algebra Univers 77:395–408CrossRefzbMATHGoogle Scholar
  15. Shao Y, Ren MM (2015) On the varieties generated by ai-semirings of order two. Semigroup Forum 91(1):171–184MathSciNetCrossRefzbMATHGoogle Scholar
  16. Zhao XZ, Guo YQ, Shum KP (2002) \(\cal{D}\)-subvarieties of the variety of idempotent semirings. Algebra Colloq 9(1):15–28MathSciNetzbMATHGoogle Scholar
  17. Zhao XZ, Shum KP, Guo YQ (2001) \(\cal{L}\)-subvarieties of the variety of idempotent semirings. Algebra Univers 46:75–96MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© 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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