Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-20T04:23:14.643Z Has data issue: false hasContentIssue false
  • English
  • Français

The Lattice of Subalgebras of a Boolean Algebra

Published online by Cambridge University Press:  20 November 2018

David Sachs
David Sachs*
Affiliation:
University of Illinois
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.

It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 451 - 460
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Birkhofï, Garrett, Lattice Aeory, rev. ed., New York: Amer. Math. Soc. (1948).Google Scholar
2. Dubreil-Jacotin, M. L., L. Lesieur, and Croisot, R., Théorie des Treillis, Paris: Gauthier- Villars (1953).Google Scholar
3. McLaughlin, J. E., Structure theorems for relatively complemented lattices, Pac. J. Math., 3 (1953), 197–208.Google Scholar
4. Sachs, David, Partition and modulated lattices, Pac. J. Math., 11 (1961), 325–345.Google Scholar
5. Sikorski, Roman, Boolean algebras, Berlin: Springer-Verlag (1960).Google Scholar