Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T06:23:44.672Z Has data issue: false hasContentIssue false
  • English
  • Français

Atomicity And Nilpotence

Published online by Cambridge University Press:  20 November 2018

Keith A. Kearnes
Keith A. Kearnes*
Affiliation:
University of Hawaii at Manoa, Honolulu, Hawaii
Rights & Permissions [Opens in a new window]

Extract

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.

There is a body of results for lattices known as “Decomposition Theory” which is aimed at proving certain existence and uniqueness theorems concerning irredundant representations of elements of a compactly generated lattice. The motivation for these results is certainly the quest for sufficient conditions on congruence lattices to insure irredundant subdirect representations of algebras. These theorems usually include some kind of modularity or distribut i v e hypothesis (for uniqueness) and some atomicity hypothesis (for existence); the precise details can be found in [3]. The atomicity condition is usually the hypothesis that the lattice in question is strongly atomic or at least atomic. Now, it is well-known that every algebra has a weakly atomic congruence lattice.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 2 , 01 April 1990 , pp. 365 - 382
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Burris, S. and Sankappanavar, H.P., A course in universal algebra, Graduate Texts in Mathematics 78 (Springer-Verlag, New York, 1981).Google Scholar
2. Camillo, V. and Fuller, K., A note on Loewy rings and chain conditions on primitive ideals, in Module theory, Lecture Notes in Mathematics 700 (Springer-Verlag, New York, 1979), 75–86.Google Scholar
3. Crawley, P. and Dilworth, R.P., Algebraic theory of lattices (Prentice-Hall, Engelwood Cliffs, N.J., 1973).Google Scholar
4. Freese, R. and McKenzie, R., Commutator theory for congruence modular varieties, LMS Lecture Note Series 125, Cambridge (1987).Google Scholar
5. Hall, M. Jr., Theory of groups, 2nd ed. (Chelsea Publishing Co., New York, 1976).Google Scholar
6. McKenzie, R., Finite equational bases for congruence modular varieties, Algebra Universalis 24, (1987), 224–250.Google Scholar
7. Vaughan-Lee, M.R., Nilpotence in permutable varieties, in Universal algebra and lattice theory, Lecture Notes in Mathematics, 1004 (Springer-Verlag, New York, 1983), 293–308.Google Scholar