Citation
Freese, Ralph Stanley (1972) Varieties generated by modular lattices of width four. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:04082016123408947
Abstract
A variety (equational class) of lattices is said to be finitely based if there exists a finite set of identities defining the variety. Let M^{∞}_{n} denote the lattice variety generated by all modular lattices of width not exceeding n. M^{∞}_{1} and M^{∞}_{2} are both the class of all distributive lattices and consequently finitely based. B. Jónsson has shown that M^{∞}_{3} is also finitely based. On the other hand, K. Baker has shown that M^{∞}_{n} is not finitely based for 5 ≤ n ˂ ω. This thesis settles the finite basis problem for M^{∞}_{4}. M^{∞}_{4} is shown to be finitely based by proving the stronger result that there exist ten varieties which properly contain M^{∞}_{4} and such that any variety which properly contains M^{∞}_{4} contains one of these ten varieties.
The methods developed also yield a characterization of subdirectly irreducible width four modular lattices. From this characterization further results are derived. It is shown that the free M^{∞}_{4} lattice with n generators is finite. A variety with exactly k covers is exhibited for all k ≥ 15. It is further shown that there are 2^{Ӄo} sub varieties of M^{∞}_{4}.
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:  13 December 1971  
Funders: 
 
Record Number:  CaltechTHESIS:04082016123408947  
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:04082016123408947  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9661  
Collection:  CaltechTHESIS  
Deposited By:  Leslie Granillo  
Deposited On:  08 Apr 2016 20:23  
Last Modified:  08 Apr 2016 20:23 
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