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A Decomposition Theory for Lattices without Chain Conditions


Crawley, Peter Linton (1961) A Decomposition Theory for Lattices without Chain Conditions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/3RKX-HF58.


In this thesis we are concerned with representing an element of a lattice as an irredundant meet of elements which are irreducible in the sense that they are not proper meets, and with certain arithmetical properties of these decompositions. A theory is developed for the class of compactly generated atomic lattices which extends the classical theory for finite dimensional lattices. The principal results are the following. Every element of an arbitrary compactly generated atomic lattice has an irredundant meet decomposition into irreducible elements. These decompositions are unique in distributive lattices. In a modular lattice the decompositions of an element have the Kurosh-Ore replacement property, that is, for any two decompositions of an element, each irreducible in the first decomposition can be replaced by a suitable irreducible in the second decomposition. Moreover, characterizations are obtained of those lattices having unique decompositions and those lattices having the replacement property.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:(Mathematics)
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 1961
Record Number:CaltechETD:etd-03172006-084230
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:977
Deposited By: Imported from ETD-db
Deposited On:17 Mar 2006
Last Modified:11 Nov 2023 00:19

Thesis Files

PDF (Crawley_pl_1961.pdf) - Final Version
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