George Gratzer - On the endomorphism monoids of (uniquely) complemented lattices

In 1970, the authors proved the following result:

Theorem 1   Every monoid can be represented as the -endomorphism monoid of a suitable bounded lattice L.

Now we can prove the following two results:

Theorem 2   Every monoid can be represented as the -endomorphism monoid of a suitable complemented lattice L. Moreover, if is finite, then L can be chosen as a finite complemented lattice.

Theorem 3   Every monoid can be represented as the -endomorphism monoid of a suitable uniquely complemented lattice L.

Theorem  solves Problem VI.24 of G. Grätzer's General Lattice Theory, (1978).

Recall that uniquely complemented lattices are very difficult to construct. R. P. Dilworth in 1945 solved a long standing conjecture of lattice theory by proving that not every uniquely complemented lattice is distributive (Boolean). He proved this by examining free lattices with a ``free'' complement operation. Free algebras have very special -endomorphism monoids since every map of the generators can be extended to a -endomorphism. So we were quite surprised that Theorem  could be sharpened to Theorem .

The proof of these results relies on several results in the literature, due to C. C. Chen and G. Grätzer 1969, H. Lakser 1972, M. E. Adams and J. Sichler 1977, V. Koubek and J. Sichler 1984.