Algebra universalis

, Volume 70, Issue 1, pp 71–94 | Cite as

Fully invariant and verbal congruence relations

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Abstract

A congruence relation θ on an algebra A is fully invariant if every endomorphism of A preserves θ. A congruence θ is verbal if there exists a variety \({\mathcal{V}}\) such that θ is the least congruence of A such that \({{\bf A}/\theta \in \mathcal{V}}\) . Every verbal congruence relation is known to be fully invariant. This paper investigates fully invariant congruence relations that are verbal, algebras whose fully invariant congruences are verbal, and varieties for which every fully invariant congruence in every algebra in the variety is verbal.

2010 Mathematics Subject Classification

Primary: 08B15 Secondary: 08A30 08A35 

Key words and phrases

verbal congruence fully invariant congruence verbose 
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References

  1. 1.
    Bergman, C.: Universal Algebra. Chapman and Hall/CRC, Boca Raton (2012)Google Scholar
  2. 2.
    Bergman C., McKenzie R.: Minimal varieties and quasivarieties. J. Austral. Math. Soc. Ser. A 48, 133–147 (1990)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Berman J.: Interval lattices and the amalgamation property. Algebra Universalis 12, 360–375 (1981)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Clark, D.M., Davey, B.A.: Natural Dualities for the Working Algebraist. Cambridge Studies in Advanced Mathematics, vol. 57. Cambridge University Press, Cambridge (1998)Google Scholar
  5. 5.
    Davey, B.A., Werner, H.: Dualities and equivalences for varieties of algebras. In: Contributions to Lattice Theory, Colloq. Math. Soc. János Bolyai, vol. 33, pp. 101–275. North-Holland, Amsterdam (1983)Google Scholar
  6. 6.
    Freese, R., Ježek, J., Nation, J.B.: Free Lattices. AMS Surveys and Monographs, vol. 42. Amer. Math. Soc., Providence (1995)Google Scholar
  7. 7.
    Freese, R., McKenzie, R.: Commutator Theory for Congruence Modular Varieties. London Mathematical Society Lecture Notes, no. 125. Cambridge University Press, Cambridge (1987)Google Scholar
  8. 8.
    Jónsson B.: Equational classes of lattices. Math. Scand. 22, 187–196 (1968)MathSciNetMATHGoogle Scholar
  9. 9.
    Kearnes K., Szendrei Á.: A characterization of minimal locally finite varieties. Trans. Amer. Math. Soc. 349, 1749–1768 (1997)Google Scholar
  10. 10.
    Keimel, K., Werner, H.: Stone duality for varieties generated by quasiprimal algebras. Mem. Amer. Math. Soc. 148, 59–85 (1974)MathSciNetMATHGoogle Scholar
  11. 11.
    Kiss, E.: On the Loewy rank of infinite algebras. Algebra Universalis 29, 437–440 (1992)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kovács, L., Newman, M.: Minimal verbal subgroups. Proc. Cambridge Philos. Soc. 62, 347–350 (1966)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Kovács, L., Newman, M.: On critical groups. J. Austral. Math. Soc. 6, 237–250 (1966)Google Scholar
  14. 14.
    McKenzie, R.: Equational bases and nonmodular lattice varieties. Trans. Amer. Math. Soc. 174, 1–43 (1972)Google Scholar
  15. 15.
    McKenzie, R., McNulty, G., Taylor, W.: Algebras, Lattices, Varieties, vol. I. Wadsworth and Brooks/Cole, Belmont (1987)Google Scholar
  16. 16.
    Monk, D.: On pseudo-simple universal algebras. Proc. Amer. Math. Soc. 13, 543–546 (1962)Google Scholar
  17. 17.
    Post, E.: Two-valued Iterative Systems of Mathematical Logic. Princeton University Press, Princeton (1941)Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Department of Mathematics, Statistics and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

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