Citation
Greene, Curtis (1969) Combinatorial Properties of Finite Geometric Lattices. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/0VKW5375. https://resolver.caltech.edu/CaltechTHESIS:02222016103219152
Abstract
Let L be a finite geometric lattice of dimension n, and let w(k) denote the number of elements in L of rank k. Two theorems about the numbers w(k) are proved: first, w(k) ≥ w(1) for k = 2, 3, ..., n1. Second, w(k) = w(1) if and only if k = n1 and L is modular. Several corollaries concerning the "matching" of points and dual points are derived from these theorems.
Both theorems can be regarded as a generalization of a theorem of de Bruijn and Erdös concerning ʎ= 1 designs. The second can also be considered as the converse to a special case of Dilworth's theorem on finite modular lattices.
These results are related to two conjectures due to G. C. Rota. The "unimodality" conjecture states that the w(k)'s form a unimodal sequence. The "Sperner" conjecture states that a set of noncomparable elements in L has cardinality at most max/k {w(k)}. In this thesis, a counterexample to the Sperner conjecture is exhibited.
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): 
 
Thesis Committee: 
 
Defense Date:  7 April 1969  
Funders: 
 
Record Number:  CaltechTHESIS:02222016103219152  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:02222016103219152  
DOI:  10.7907/0VKW5375  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9572  
Collection:  CaltechTHESIS  
Deposited By:  INVALID USER  
Deposited On:  22 Feb 2016 21:55  
Last Modified:  29 Apr 2024 21:11 
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