Huntington's Conjecture, Revisited. In 1904, E. V. Huntington conjectured that any uniquely complemented lattice was distributive. In fact, the conjecture had been verified for several special classes of lattices. However, in 1945, Dilworth disproved this conjecture by showing 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 a contiuum of varieties in which each lattice can be embedded in a uniquely complemented lattice of the same variety! These constructions are obtained by a limit process. It is still hard to find "nice" and "natural" 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 heart of Huntington's 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 Garrett Birkhoff and John von Neumann - is modularity. A lattice property P is called a Huntington Property if every uniquely complemented P-lattice is distributive. Similarly, a lattice identity f=g is said to be Huntington if every uniquely complemented lattice satisfying f=g is distributive. In his 1988 Monograph, V.N. Salii has compiled a number of known Huntington properties. Among these, modularity is the only one that is equational. In this project, we give a number of new non-modular Huntington identities. First we discover several new Huntington implications and then program a first-order theorem prover to find proper non-trivial lattice identities which formally imply these Huntington implications. In particular, using Jonsson's Lemma, we show that every primitive lattice generates a non-modular Huntington variety. Open Problems and Conjectures. 1. If A and B are Huntington, is A v B Huntington? (False for infinite join, but I believe this to be true for finite join). 2. If a variety K does not contain the 5-element lattice M3 then it is Huntington. References Adams, M. E.; Sichler, J. Lattices with unique complementation. Pacific J. Math. 92 (1981), no. 1, 1--13. Dilworth, R. P. Lattices with unique complements. Trans. Amer. Math. Soc. 57, (1945). 123--154. Huntington, E. V. Sets of independent postulates for the algebra of logic, Trans. Amer. Math. Soc. 5 (1904) 288-309. Jónsson, Bjarni. Algebras whose congruence lattices are distributive. Math. Scand. 21 1967 110--121 (1968). McKenzie, Ralph. Equational bases and nonmodular lattice varieties. Trans. Amer. Math. Soc. 174 (1972), 1--43. Padmanabhan, R.; McCune, William.; Veroff, Robert. Lattice laws forcing distributivity under unique complementation, (accepted for publication in Houston Journal of Mathematics). Salii, V. N. Lattices with unique complements, A.M.S. Translations vol. 69, Amer. Math. Soc., Providence, R. I., 1988.