Skip to main content

On a hereditarily finitely based ai-semiring variety

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Burris S, Sankappanavar HP (1981) A course in universal algebra. Springer, New York

    Book  MATH  Google Scholar 

  • Ghosh S, Pastijn F, Zhao XZ (2005) Varieties generated by ordered bands I. Order 22:109–128

    Article  MathSciNet  MATH  Google Scholar 

  • Głazek K (2001) A guide to the literature on semirings and their applications in mathematics and information science. Kluwer Academic Publishers, Dordrecht

    MATH  Google Scholar 

  • Golan JS (1992) The theory of semirings with applications in mathematics and theoretical computer science. Longman Scientific and Technical, Harlow

    MATH  Google Scholar 

  • Howie JM (1995) Fundamentals of semigroup theory. Clarendon Press, London

    MATH  Google Scholar 

  • Kuřil M, Polák L (2005) On varieties of semilattice-ordered semigroups. Semigroup Forum 71(1):27–48

    Article  MathSciNet  MATH  Google Scholar 

  • McKenzie R, Romanowska A (1979) Varieties of \(\cdot \)-distributive bisemilattices. In: Contributions to general algebra (Proc. Klagenfurt Conf. 1978), vol 1, pp 213–218

  • Pastijn F (2005) Varieties generated by ordered bands II. Order 22:129–143

    Article  MathSciNet  MATH  Google Scholar 

  • Pastijn F, Zhao XZ (2005) Varieties of idempotent semirings with commutative addition. Algebra Univers 54:301–321

    Article  MathSciNet  MATH  Google Scholar 

  • Petrich M, Reilly NR (1999) Completely regular semigroups. Wiley, New York

    MATH  Google Scholar 

  • Ren MM, Zhao XZ (2016) The varieties of semilattice-ordered semigroups satisfying \(x^3\approx x\) and \(xy\approx yx\). Period Math Hungar 72:158–170

    Article  MathSciNet  MATH  Google Scholar 

  • Ren MM, Zhao XZ, Shao Y (2016a) On a variety of Burnside ai-semirings satisfying \(x^{n}\approx x\). Semigroup Forum 93(3):501–515

    Article  MathSciNet  MATH  Google Scholar 

  • Ren MM, Zhao XZ, Volkov MV (2016b) The Burnside ai-semiring variety defined by \(x^n \approx x\), manuscript

  • Ren MM, Zhao XZ, Wang AF (2017) On the varieties of ai-semirings satisfying \(x^{3}\approx x\). Algebra Univers 77:395–408

    Article  MATH  Google Scholar 

  • Shao Y, Ren MM (2015) On the varieties generated by ai-semirings of order two. Semigroup Forum 91(1):171–184

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao XZ, Guo YQ, Shum KP (2002) \(\cal{D}\)-subvarieties of the variety of idempotent semirings. Algebra Colloq 9(1):15–28

    MathSciNet  MATH  Google Scholar 

  • Zhao XZ, Shum KP, Guo YQ (2001) \(\cal{L}\)-subvarieties of the variety of idempotent semirings. Algebra Univers 46:75–96

    Article  MathSciNet  MATH  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Miaomiao Ren.

Ethics declarations

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.

Additional information

Communicated by A. Di Nola.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ren, M., Zeng, L. On a hereditarily finitely based ai-semiring variety. Soft Comput 23, 6819–6825 (2019). https://doi.org/10.1007/s00500-018-03719-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-018-03719-0

Keywords