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Rings and Modules Seminar
~ Abstracts ~

R. Padmanabhan
padman(at)cc(dot)umanitoba(dot)ca

Department of Mathematics
University of Manitoba

Wednesday, May 20, 2009

Ternary Modular Algebras
Abstract:

A well-known theorem of Garrett Birkhoff and Jon von Neumann says that every uniquely complemented modular lattice is distributive (and hence a Boolean algebra). J.R. Isbell has characterized the variety of all ternary modular algebras (A; m) satisfying all the identities true in modular lattices where

m(x,y,z) = (x ∧ (y ∨ z)) ∨ (y ∧ z).
There are non-lattice examples in this variety. We define the concept of "uniquely complemented ternary algebras" in the first order language of m such that a lattice satisfies this sentence iff it is uniquely complemented in the usual sense. Also, Isbell defines distributivity by the symmetry condition m(x,y,z) = m(y,x,z). Again, this becomes the usual distributive law if the algebra (A; m) happens to be a lattice. In other words, the essence of "unique complementation" as well as that of distributivity can be captured purely in the language of the ternary operation. Now it is but natural to ask whether the resulting version of Birkhoff-von Neumann theorem is still valid in this new context. We prove that this is indeed the case.

Theorem: If a ternary modular algebra (A; m) is uniquely complemented (in the ternary sense), then it is distributive (in the ternary sense).

This work is part of a joint collaboration with Bill McCune and Bob Veroff of the University of New Mexico.

References:

  1. G. Birkhoff, Lattice Theory, AMS Colloquium Publications, 1967, page 122.
  2. J.R. Isbell, Median algebras, Trans. Amer. Math. Soc. 260 (1980), no. 2, 319--362.

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