Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-10-28T15:17:13.734Z Has data issue: false hasContentIssue false
  • English
  • Français

Monotone clones, residual smallness and congruence distributivity

Published online by Cambridge University Press:  17 April 2009

Ralph McKenzie
Ralph McKenzie
Affiliation:
Department of MathematicsUniversity of CaliforniaBerkeley, California 94720United States of America
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Corresponding to each ordered set there is a variety, determined up to equivalence, generated by an algebra whose term operations are all the monotone operations on the ordered set. We produce several characterisations of the finite bounded ordered sets for which the corresponding variety is congruence-distributive. In particular, we find that congruence-distributivity, congruence-modularity, and residual smallness are equivalent for these varieties.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 41 , Issue 2 , April 1990 , pp. 283 - 300
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Baker, K. and Pixley, A., ‘Polynomial interpolation and the Chinese Remainder Theorem for algebraic systems’, Math. Z. 43 (1975), 165–174.CrossRefGoogle Scholar
[2]Davey, B., ‘Monotone clones and congruence modularity’, (preprint).Google Scholar
[3]Davey, B.Quackenbush, R.W. and Schweigert, D., ‘Monotone clones and the varieties they determine’, (preprint).Google Scholar
[4]Demetrovics, J., Hannák, L. and Rónyai, L., ‘Near unanimity functions of partial orderings’, in Proceedings of the XIVth International Symposium on Multiple Valued Logic, pp. 52–56 (Manitoba, 1984).Google Scholar
[5]Demetrovics, J., Hannák, L. and Rónyai, L., ‘On algebraic properties of monotone clones’, Order 3 (1986), 219–225.CrossRefGoogle Scholar
[6]Demetrovics, J., Rónyai, L., ‘Algebraic properties of crowns and fences’, (preprint of the Computer and Automation Institute, Hungarian Academy of Sciences, Budapest).Google Scholar
[7]Denecke, K., Preprimal Algebras (Math. Forschung Bd. 11 Akademie Verlag, Berlin, 1982).Google Scholar
[8]Duffus, D. and Rival, I., ‘A structure theory for ordered sets’, Discrete Math. 35 (1981), 53–118.CrossRefGoogle Scholar
[9]Freese, R. and McKenzie, R., Commutator theory for congruence modular varieties (London Math. Soc. Lecture Note No. 125, 1987).Google Scholar
[10]Hagemann, J. and Herrmann, C., ‘A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity’, Arch. Math. (Basel) (1979), 234–245.CrossRefGoogle Scholar
[11]Hobby, D. and McKenzie, R., The Structure of Finite Algebras (AMS Contemporary Mathematics Series, 1988).Google Scholar
[12]Jónsson, B., ‘Algebras whose congruence lattices are distributive’, Math. Scand. 21 (1967), 110–121.Google Scholar
[13]Knoebel, A., The equational classes generated by single functionally precomplete algebras (Amer. Math. Soc. Memoir No. 57, 1985).Google Scholar
[14]Lau, D., ‘Bestimmung der Ordnung maximaler Klassen von Functionen der k-wertigen Logik’, Z. Math. Logik Grundlag. Math. 24 (1978), 79–96.Google Scholar
[15]Martynjuk, V.V., ‘Investigation of classes of functions in many-valued logics’, (Russian), Problemy Kibernet. 3 (1960), 49–60.Google Scholar
[16]Nevermann, P., ‘A note on axiomatizable order varieties’, Algebra Universalis 17 (1983), 129–131.CrossRefGoogle Scholar
[17]Nevermann, P. and Rival, I., ‘Holes in ordered sets’, Graphs Combin. 1 (1985), 339–350.Google Scholar
[18]Nevermann, P. and Wille, R., ‘The strong selection property and ordered sets of finite length’, Algebra Universalis 18 (1984), 18–28.CrossRefGoogle Scholar
[19]Quackenbush, R., Rival, I. and Rosenberg, I.G., ‘Clones, order varieties, near unanimity functions and holes’, (preprint), Order.Google Scholar
[20]Rosenberg, I., ‘Über die funktionale Vollständigkeit in den mehrwertigen Logiken’, Rozpravy Ceskoslovenské Akad. Ved 80 (1970).Google Scholar
[21]Rosenberg, I., ‘Near unanimity orders’, (preprint C.R.M. Université de Montréal, 1985).Google Scholar
[22]Tardos, G., ‘A not finitely generated maximal clone of monotone operations’, Order 3 (1986), 211–218.Google Scholar