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Homomorphisms of a modular lattice

Citation

Edmondson, D E (1954) Homomorphisms of a modular lattice. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-11212003-113012

Abstract

This thesis is an algebraic study of the irreducible ideals of a modular lattice and their application to the characterization of the homomorphisms or congruence relations of the lattice.

First, an arithmetic characterization of modularity is given in terms of the irreducible ideals of the lattice. This is a new structure result for modular lattices which, since it characterizes general modular lattices, is more fundamental in the structure of modular lattices than the Kurosh-Ore Theorem.

Second, through the arithmetic characterization developed, subsets of the irreducible ideals are used to define congruence relations on the lattice and its lattice of ideals. It is then shown that every congruence relation on a modular lattice can be so characterized.

In conclusion, a generalization of the theorem that the congruence relations of a finite dimensional modular lattice form a Boolean algebra is given by proving that the congruence relations on the lattice of ideals of a modular lattice form a Boolean algebra if and only if the lattice is finite dimensional.

Item Type:Thesis (Dissertation (Ph.D.))
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Dilworth, Robert P.
Thesis Committee:
  • Unknown, Unknown
Defense Date:1 January 1954
Record Number:CaltechETD:etd-11212003-113012
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-11212003-113012
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4616
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:21 Nov 2003
Last Modified:26 Dec 2012 03:10

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