Department of Mathematics
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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, R. P. Dilworth dropped a bombshell by proving that any lattice can be embedded into a uniquely complemented lattice. In other words, no lattice property preserved under the formation of sublattices can be derived from unique com0lementation. In 1969, Gratzer and Chen showed that this particular result can be obtained without making use of some of the more difficult machinery developed in the above paper (e.g. an extra unary operator). 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!

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 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. Let us call a lattice property P a Huntington property if every uniquely complemented P-lattice is Boolean. Similarly, a lattice variety K is said to be a Huntington variety if every uniquely complemented member of K is a Boolean algebra. In this talk, we prove that there are continuum many such varieties.