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.
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