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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Clark, D.M., Davey, B.A.: Natural Dualities for the Working Algebraist. Cambridge University Press, Cambridge (1998)
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)
Davey B.A., Haviar M., Priestley H.A.: The syntax and semantics of entailment in duality theory. J. Symbolic Logic 60, 1087–1114 (1995)
Davey B.A., Heindorf L., McKenzie R.: Near unanimity: an obstacle to general duality theory. Algebra Universalis 33, 428–439 (1995)
Davey B.A., Jackson M., Pitkethly J.G., Talukder M.R.: Natural dualities for semilattice-based algebras. Algebra Universalis 57, 463–490 (2007)
Davey B.A., Idziak P.M., Lampe W.A., McNulty G.F.: Dualizability and graph algebras. Discrete Mathematics 214, 145–172 (2000)
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)
Day A.: A characterization of modularity for congruence lattices of algebras. Canad. Math. Bull. 12, 167–173 (1969)
Hobby, D., McKenzie, R.: The Structure of Finite Algebras. Contemp. Math. 76, Amer. Math. Soc., Providence (1988)
Kearnes, K.A., Kiss, E.W.,: The Shape of Congruence Lattices. Mem. Amer. Math. Soc. 222, Providence (2013)
Kearnes, K.A., Szendrei, Á.: Clones of algebras with parallelogram terms. Internat. J. Algebra Comput. 22, 1250005-1–1250005-30 (2012)
Marković P., Maróti M., McKenzie R.: Finitely related clones and algebras with cube terms. Order 29, 345–359 (2012)
Maróti M., McKenzie R.: Existence theorems for weakly symmetric operations. Algebra Universalis 59, 463–489 (2008)
Nickodemus, M.H.: Natural Dualities for Finite Groups With Abelian Sylow Subgroups. PhD thesis, University of Colorado at Boulder (2007)
Quackenbush R., Szabo Cs.: Nilpotent groups are not dualizable. J. Aust. Math. Soc. 72, 173–179 (2002)
Quackenbush R., Szabo Cs.: Strong duality for metacyclic groups. J. Aust. Math. Soc. 73, 377–392 (2002)
Zádori L.: Natural duality via a finite set of relations. Bull. Austral. Math. Soc. 51, 469–478 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by A. Szendrei.
Rights and permissions
About this article
Cite this article
Moore, M. Naturally dualizable algebras omitting types 1 and 5 have a cube term. Algebra Univers. 75, 221–230 (2016). https://doi.org/10.1007/s00012-016-0371-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-016-0371-6