A family of finite De Morgan algebras

The algebra of truth values for fuzzy sets of type-2 consists of all mappings from the unit interval into itself, with operations certain convolutions of these mappings with respect to pointwise max and min. This algebra has been studied rather extensively in the last few years, both from an applications point of view and a theoretical one. Most of the theory goes through when is replaced by any two finite chains, in which case interesting finite algebras arise-De Morgan algebras and Kleene algebras in particular-and a basic question is just where these algebras fit into the world of all such finite algebras. We investigate one particularly interesting family of such De Morgan algebras.