On the structure of De Morgan monoids with corollaries on relevant logic and theories (Q1119633)

A De Morgan monoid is constant iff it is generated by its identity alone. It is shown that the only nontrivial proper homomorphic image of a prime De Morgan monoid in a constant one is the 4-element algebra C4. Moreover, the only element mapped by such a homomorphism to the lattice 0 of C4 is the lattice 0 of the original. These facts are used to obtain results on De Morgan monoids with idempotent generators, including a complete description of the free De Morgan monoid with one idempotent generator. The paper concludes with some applications to the relevant logic R and particularly to the arithmetic R{\#}.