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A variety \( K \) of lattices is called Huntington if every uniquely complemented lattice in \( K \) is distributive. In 1904, E.V. Huntington conjectured that every uniquely complemented lattice was distributive. In fact, the conjecture had been verified for several special classes of lattices. However, in 1945, Dilworth shattered this conjecture by proving that any lattice can be embedded into a uniquely complemented lattice. In 1981, Adams and Sichler strengthened the original embedding theorem of Dilworth by showing the existence of continuumly many varieties in which each lattice can be embedded in a uniquely complemented lattice of the same variety!

In spite of these deep theorems, it is still hard to find "nice" and "natural" examples of uniquely complemented lattices that are not Boolean. The reason is that uniquely complemented lattices having a little extra structure most often turn out to be distributive. This seems to be the essence of Huntington's original conjecture. Accordingly, we plan to attack the problem backwards: that is, by finding additional (albeit, mild) conditions that, if added, would solve the problem in the affirmative. Many such conditions were already discovered during 1930's and 40's. The most notable among such conditions - due to Birkhoff and von Neumann - is modularity which is the only known variety defining a Huntington variety. Here we present several lattice implications forcing a uniquely complemented lattice to be distributive. Since some of these sentences are consequences of modularity, we obtain generalizations of the classical result of Birkhoff and von Neumann that every uniquely complemented modular lattice is a Boolean algebra. In particular, we prove that every uniquely complemented lattice in the least nonmodular variety containing the variety of all modular lattices is distributive. Finally we show that there are continuumly many non-modular Huntington varieties of lattices.

References:

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