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In 1904 Huntington [4] conjectured that every uniquely complemented lattice must be distributive (and hence a Boolean algebra). In 1945, R. P. Dilworth shattered this conjecture by proving [2] that every lattice can be embedded in a uniquely complemented lattice. For a much powerful version of the same result, see Adams and Sichler [1]. In spite of these deep results, it is still hard to find ÒniceÓ examples of uniquely complemented lattices that are not distributive. 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 HuntingtonÕs conjecture. For example, we have the theorem of Garrett Birkhoff and von Neumann that every uniquely complemented modular lattice is Boolean. Following [5], we call a lattice property P a Huntington property if every uniquely complemented P-lattice is distributive. Similarly, a lattice variety K is said to be a Huntington variety if every uniquely complemented lattice in K is distributive. In this terminology, the class of all modular lattices was the largest previously known Huntington variety. A monograph by Salii [6] gives a comprehensive survey of known Huntington properties. Among these, modularity is the only known condition that is a lattice identity. In this paper, we show that there are uncountably many nonmodular Huntington varieties. Since some of these varieties include the modular lattices, these results may be construed as generalizations of the classical von Neumann-Birkhoff theorem.

1. M. E. Adams and J. Sichler. Lattice with unique complementation. Pac. J. Math., 92:1Ð13, 1981.
2. R. P. Dilworth. Lattices with unique complements. Trans. AMS, 57:123Ð154, 1945.
3. G. Gratzer. Two problems that shaped a century of lattice theory, Notices AMS, Vol 54, July 2007.
4. E. V. Huntington. Sets of independent postulates for the algebra of logic. Trans. AMS, 5:288Ð309, 1904. 400, 1981.
5. R. Padmanabhan, W. McCune, and R. Veroff. Lattice laws forcing distributivity under unique complementation, Houston J of Math. 33 (2007) 391-401.
6. V. N. Salii. Lattices with Unique Complements. American Mathematical Society, Providence, RI, 1988.