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Varieties generated by modular lattices of width four


Freese, Ralph Stanley (1972) Varieties generated by modular lattices of width four. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/W67C-JR90.


A variety (equational class) of lattices is said to be finitely based if there exists a finite set of identities defining the variety. Let Mn denote the lattice variety generated by all modular lattices of width not exceeding n. M1 and M2 are both the class of all distributive lattices and consequently finitely based. B. Jónsson has shown that M3 is also finitely based. On the other hand, K. Baker has shown that Mn is not finitely based for 5 ≤ n ˂ ω. This thesis settles the finite basis problem for M4. M4 is shown to be finitely based by proving the stronger result that there exist ten varieties which properly contain M4 and such that any variety which properly contains M4 contains one of these ten varieties.

The methods developed also yield a characterization of sub-directly irreducible width four modular lattices. From this characterization further results are derived. It is shown that the free M4 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 2Ӄo sub- varieties of M4.

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:13 December 1971
Funding AgencyGrant Number
Ford FoundationUNSPECIFIED
Record Number:CaltechTHESIS:04082016-123408947
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9661
Deposited On:08 Apr 2016 20:23
Last Modified:09 Nov 2022 19:20

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