Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-15T08:17:53.244Z Has data issue: false hasContentIssue false
  • English
  • Français

A Semimodular Imbedding of Lattices

Published online by Cambridge University Press:  20 November 2018

D. T. Finkbeiner
D. T. Finkbeiner*
Affiliation:
Kenyon College
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.

The study of structural or arithmetic properties of a general lattice often can be facilitated by imbedding as a sublattice of a lattice of a more restricted type whose properties are known. However, if is too restricted, a general imbedding is impossible; for example, cannot be modular because , as a sublattice of , would then have to be modular. One of the best results of this nature has been given by Dilworth in an unpublished work in which he shows that any finite dimensional lattice is isomorphic to a sublattice of a semi-modular point lattice (1, pp. 105 and 110). In the present paper Dilworth's imbedding process is modified to obtain a sharper result: Any finite dimensional lattice is isometrically isomorphic to a sublattice of a semi-modular lattice which has the same number of points as and which preserves basic properties of the join-irreducible arithmetic of .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 582 - 591
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Birkhoff, G., Lattice Theory (rev. éd.), Amer. Math. Soc. Coll. Pub., 25 (1948).Google Scholar
2. Dilworth, R.P., Arithmetic theory of Birkhoff lattices, Duke Math. J., 8 (1941), 286–299.Google Scholar
3. Finkbeiner, D.T., A general dependence relation for lattices, Proc. Amer. Math. Soc, 2 (1951), 756–759.Google Scholar