Vol. 30, No. 1, 1969

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 294: 1
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Desargues’ law and the representation of primary lattices

G. S. Monk

Vol. 30 (1969), No. 1, 175–186
Abstract

In an earlier work of B. Jónsson and the author it was shown that an Arguesian primary lattice of geometric dimension at least 3 can be represented as the submodule lattice of a finitely generated module over a completely primary uniserial ring. Inasmuch as the class of primary lattices includes the class of subspace lattices of (nondegenerate) projective geometries, two questions then naturally arise: (1) Is a primary lattice of geometric dimension at least 4 Arguesian? (2) Is an Arguesian primary lattice of geometric dimension 2 representable ? The first question is answered in the affirmative in §1, thus showing that the abovementioned paper subsumes the results of E. Inaba on the representation of primary lattices of geometric dimension at least 4. A counter example is given in §2 showing that an Arguesian lattice of geometric dimension 2 cannot, in general, be represented, but for reasons far deeper than the cardinality arguments given for the representability or nonrepresentability of subspace lattices of l-dimensional projective geometries.

Mathematical Subject Classification
Primary: 06.30
Secondary: 50.00
Milestones
Received: 24 June 1968
Published: 1 July 1969
Authors
G. S. Monk