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PRINCIPAL AND SYNTACTIC CONGRUENCES IN CONGRUENCE-DISTRIBUTIVE AND CONGRUENCE-PERMUTABLE VARIETIES

Part of: Varieties

Published online by Cambridge University Press:  01 August 2008

BRIAN A. DAVEY ,
MARCEL JACKSON ,
MIKLÓS MARÓTI  and
RALPH N. MCKENZIE
BRIAN A. DAVEY
Affiliation:
Department of Mathematics, La Trobe University, Victoria, Australia (email: b.davey@latrobe.edu.au)
MARCEL JACKSON*
Affiliation:
Department of Mathematics, La Trobe University, Victoria, Australia (email: m.g.jackson@latrobe.edu.au)
MIKLÓS MARÓTI
Affiliation:
János Bolyai Mathematical Institute, University of Szeged, Szeged, Hungary (email: mmaroti@math.u-szeged.hu)
RALPH N. MCKENZIE
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, TN, USA (email: mckenzie@math.vanderbilt.edu)
*
✉For correspondence; e-mail: m.g.jackson@latrobe.edu.au
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Abstract

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We give a new proof that a finitely generated congruence-distributive variety has finitely determined syntactic congruences (or, equivalently, term finite principal congruences), and show that the same does not hold for finitely generated congruence-permutable varieties, even underthe additional assumption that the variety is residually very finite.

Keywords

finitely determined syntactic congruencesterm finite principal congruencesfinite principal lengthcongruence distributivecongruence modularcongruence permutable

MSC classification

Secondary: 08B10: Congruence modularity, congruence distributivity
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 1 , August 2008 , pp. 59 - 74
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

The second author was supported by ARC Discovery Project Grant DP0342459. The third author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA), grant nos. T 37877 and T 48809.

References

[1]Almeida, J. and Weil, P., ‘Relatively free profinite monoids: an introduction and examples’, in: Semigroups, Formal Languages and Groups, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 466 (ed. J. Fountain) (Kluwer, Dordrecht, 1995), pp. 73–117.Google Scholar
[2]Baker, K. A., McNulty, G. F. and Wang, J., ‘An extension of Willard’s finite basis theorem: congruence meet-semidistributive varieties of finite critical depth’, Algebra Universalis 52 (2005), 289–302.Google Scholar
[3]Baker, K. A. and Wang, J., ‘Approximate distributive laws and finite equational bases for finite algebras in congruence-distributive varieties’, Algebra Universalis 54 (2005), 385–396.Google Scholar
[4]Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra, Graduate Texts in Mathematics, 78 (Springer, Berlin, 1980).Google Scholar
[5]Choe, T. H., ‘Zero-dimensional compact associative distributive universal algebras’, Proc. Amer. Math. Soc. 42 (1974), 607–613.Google Scholar
[6]Clark, D. M., Davey, B. A., Freese, R. S. and Jackson, M., ‘Standard topological algebras: syntactic and principal congruences and profiniteness’, Algebra Universalis 52 (2004), 343–376.Google Scholar
[7]Clark, D. M., Davey, B. A., Haviar, M., Pitkethly, J. G. and Talukder, M. R., ‘Standard topological quasi-varieties’, Houston J. Math. 29 (2003), 859–887.Google Scholar
[8]Clark, D. M., Davey, B. A., Jackson, M. and Pitkethly, J. G., ‘The axiomatizability of topological prevarieties’, Adv. Math. 218 (2008), 1604–1653.Google Scholar
[9]Clinkenbeard, D. J., ‘Simple compact topological lattices’, Algebra Universalis 9 (1979), 322–328.Google Scholar
[10]Day, B. J., ‘On profiniteness of compact totally disconnected algebras’, Bull. Austral. Math. Soc. 20 (1970), 71–76.Google Scholar
[11]Freese, R. and McKenzie, R., Commutator Theory for Congruence Modular Varieties, London Mathematical Society Lecture Note Series, 125 (Cambridge University Press, Cambridge, 1987).Google Scholar
[12]Jackson, M., ‘Residual bounds for compact totally disconnected algebras’, Houston J. Math. 34 (2008), 33–67.Google Scholar
[13]Johnstone, P., Stone Spaces (Cambridge University Press, Cambridge, 1982).Google Scholar
[14]Jónsson, B., ‘Algebras whose congruence lattices are distributive’, Math. Scand. 21 (1967), 110–121.Google Scholar
[15]McKenzie, R., ‘Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties’, Algebra Universalis 8 (1978), 336–348.Google Scholar
[16]McKenzie, R., ‘The residual bound of a finite algebra is not computable’, Internat. J. Algebra Comput. 6 (1996), 29–48.CrossRefGoogle Scholar
[17]Wang, J., ‘A proof of the Baker conjecture’, Acta Math. Sin. 33 (1990), 626–633 (in Chinese).Google Scholar