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Rings and Modules Seminar
~ Abstracts ~

R. Padmanabhan
Ranganathan(dot)Padmanabhan(at)umanitoba(dot)ca

University of Manitoba

Tuesday, January 12, 2021

Non-modular Huntington Varieties
Abstract:

Classical algebras like groups and rings offer a rich source of lattice identities and implications. The most famous example of this is the concept of modularity discovered by Richard Dedekind from the lattice of all ideals of a ring (or the lattice of all normal subgroups of a group). Here we consider a new lattice implication inspired by the theory of modules: \[ CM(\wedge):\ a\wedge b= 0\mbox{ and } (a\vee b)\wedge c = 0\rightarrow a\wedge(b\vee c)= 0 \] Calugareanu has applied this weaker form of modularity to his investigations on the lattice of submodules (see Lemma 6.3, page 60 in [2]). Following [6], a variety of lattices \(K\) is called Huntington if every uniquely complemented lattice in \(K\) is distributive. A classical theorem of Birkhoff and von Neumann says that the variety of all modular lattices is Huntington. In this note we prove that any uniquely complemented lattice satisfying the implication \(CM(\wedge)\) is distributive, thus generalizing the Birkhoff-von Neumann theorem. We further prove that there are continuumly many non-modular Huntington varieties of lattices.

References:

  1. M. E. Adams and J. Sichler, Lattices with unique complementation., Pac. J. Math. 92:1–13, 1981.
  2. Calugareanu, Grigore Lattice concepts of module theory, Kluwer Texts in the Mathematical Sciences, Dordrecht, 2000.
  3. R. P. Dilworth, Lattices with unique complements., Trans. AMS, 57:123–154, 1945.
  4. G. Gratzer, Two problems that shaped a century of lattice theory, AMS Notices, Vol 54, 2007.
  5. E. V. Huntington, Sets of independent postulates for the algebra of logic, Trans. AMS, 5:288–309 1904. 400, 1981.
  6. R. Padmanabhan, W. McCune, and R. Veroff, Lattice laws forcing distributivity under unique complementation, Houston J of Math. 933 (2007) 391–401.
  7. V. N. Salii, Lattices with Unique Complements, American Mathematical Society Providence, RI, 1988.


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