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Algebra universalis

, Volume 75, Issue 2, pp 221–230 | Cite as

Naturally dualizable algebras omitting types 1 and 5 have a cube term

  • Matthew Moore
Article

Abstract

An early result in the theory of Natural Dualities is that an algebra with a near unanimity (NU) term is dualizable. A converse to this is also true: if \({\mathcal{V}(\mathbb{A})}\) is congruence distributive and \({\mathbb{A}}\) is dualizable, then \({\mathbb{A}}\) has an NU term. An important generalization of the NU term for congruence distributive varieties is the cube term for congruence modular (CM) varieties, and it has been thought that a similar characterization of dualizability for algebras in a CM variety would also hold. We prove that if \({\mathbb{A}}\) omits tame congruence types 1 and 5 (all locally finite CM varieties omit these types) and is dualizable, then \({\mathbb{A}}\) has a cube term.

Keywords and phrases

natural duality congruence modular cube term 

2010 Mathematics Subject Classification

Primary: 08C20 Secondary: 08B10 

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References

  1. 1.
    Berman J., Idziak P., Marković P., McKenzie R., Valeriote M., Willard R.: Varieties with few subalgebras of powers. Trans. Amer. Math. Soc. 362, 1445–1473 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Clark, D.M., Davey, B.A.: Natural Dualities for the Working Algebraist. Cambridge University Press, Cambridge (1998)Google Scholar
  3. 3.
    Clark D.M., Idziak P.M., Sabourin L.R., Szabó C., Willard R.: Natural dualities for quasivarieties generated by a finite commutative ring. Algebra Universalis 46, 285–320 (2001)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Davey B.A., Haviar M., Priestley H.A.: The syntax and semantics of entailment in duality theory. J. Symbolic Logic 60, 1087–1114 (1995)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Davey B.A., Heindorf L., McKenzie R.: Near unanimity: an obstacle to general duality theory. Algebra Universalis 33, 428–439 (1995)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Davey B.A., Jackson M., Pitkethly J.G., Talukder M.R.: Natural dualities for semilattice-based algebras. Algebra Universalis 57, 463–490 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Davey B.A., Idziak P.M., Lampe W.A., McNulty G.F.: Dualizability and graph algebras. Discrete Mathematics 214, 145–172 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Davey, B.A., Werner, H.: Dualities and equivalences for varieties of algebras. In: Huhn, A.P., Schmidt, E.T. (eds.) Contributions to Lattice Theory (Szeged, 1980). Coll. Math. Soc. Janos Bolyai, vol. 33, pp. 101–275. North-Holland, Amsterdam (1983)Google Scholar
  9. 9.
    Day A.: A characterization of modularity for congruence lattices of algebras. Canad. Math. Bull. 12, 167–173 (1969)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hobby, D., McKenzie, R.: The Structure of Finite Algebras. Contemp. Math. 76, Amer. Math. Soc., Providence (1988)Google Scholar
  11. 11.
    Kearnes, K.A., Kiss, E.W.,: The Shape of Congruence Lattices. Mem. Amer. Math. Soc. 222, Providence (2013)Google Scholar
  12. 12.
    Kearnes, K.A., Szendrei, Á.: Clones of algebras with parallelogram terms. Internat. J. Algebra Comput. 22, 1250005-1–1250005-30 (2012)Google Scholar
  13. 13.
    Marković P., Maróti M., McKenzie R.: Finitely related clones and algebras with cube terms. Order 29, 345–359 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Maróti M., McKenzie R.: Existence theorems for weakly symmetric operations. Algebra Universalis 59, 463–489 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Nickodemus, M.H.: Natural Dualities for Finite Groups With Abelian Sylow Subgroups. PhD thesis, University of Colorado at Boulder (2007)Google Scholar
  16. 16.
    Quackenbush R., Szabo Cs.: Nilpotent groups are not dualizable. J. Aust. Math. Soc. 72, 173–179 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Quackenbush R., Szabo Cs.: Strong duality for metacyclic groups. J. Aust. Math. Soc. 73, 377–392 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Zádori L.: Natural duality via a finite set of relations. Bull. Austral. Math. Soc. 51, 469–478 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Vanderbilt UniversityNashvilleU.S.A

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