Skip to main content
SpringerLink
Log in

Non-generators in complete lattices and semilattices

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

As is well-known, in a finitary algebraic structure the set \(\Gamma\) of all the non-generators is the intersection of all the maximal proper substructures. In particular, \(\Gamma\) is a substructure.

We show that the corresponding statements hold for complete semilattices but fail for complete lattices, when as the notion of substructure we take complete subsemilattices and complete sublattices, respectively.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Bergman and G. Slutzki, Computational complexity of generators and nongenerators in algebra, Internat. J. Algebra Comput., 12 (2002), 719–735.

  2. G. Grätzer, On the family of certain subalgebras of a universal algebra, Nederl. Akad. Wetensch. Proc. Ser. A 68 = Indag. Math., 27 (1965), 790–802.

  3. G. Grätzer, Lattice Theory: Foundation, Birkhäuser/Springer Basel AG (Basel, 2011).

  4. G. E. Hansoul, The Frattini subalgebra of an infinitary algebra, Bull. Soc. Roy. Sci. Liège, 49 (1980), 423–424.

  5. G. Janelidze, Frattini subobjects and extensions in semi-Abelian categories, Bull. Iranian Math. Soc., 44 (2018), 291–304.

  6. E. W. Kiss and S. M. Vovsi, Critical algebras and the Frattini congruence, Algebra Universalis, 34 (1995), 336–344.

  7. K. Koh, On the Frattini sub-semilattice of a semilattice, Nanta Math., 5 (1971), 22–33.

  8. P. Lipparini, Non-generators in extensions of infinitary algebras, submitted, available at https://art.torvergata.it/handle/2108/277569.

Download references

Acknowledgement

I thank the anonymous referee for many useful comments which helped to improve the paper.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Viale della Ricerca Genereticolata, I-00133, Rome, Italy

    P. Lipparini

Authors
  1. P. Lipparini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Lipparini.

Additional information

Work performed under the auspices of G.N.S.A.G.A.Work partially supported by PRIN 2012 "Logica, Modelli e Insiemi". The author acknowledges the MIUR Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lipparini, P. Non-generators in complete lattices and semilattices. Acta Math. Hungar. 166, 423–431 (2022). https://doi.org/10.1007/s10474-022-01232-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-022-01232-3

Key words and phrases

Mathematics Subject Classification

Search

Navigation