Positive derivations on archimedean lattice-ordered rings

It is known that the only positive derivation on a reduced archimedean f-ring is the zero derivation. We investigate derivations on general archimedean lattice-ordered rings. First, we consider semigroup rings over cyclic semigroups and show that, in the finite case, the only derivation that is zero on the underlying ring is the zero derivation and that, in the infinite case, such derivations are always based on the derivative. Turning our attention to lattice-ordered rings, we show that, on many algebraic extensions of totally ordered rings, the only positive derivation is the zero derivation and that, for transcendental extensions, derivations that are lattice homomorphisms are always translations of the usual derivative and derivations that are orthomorphisms are always dilations of the usual derivative. We also show that the only positive derivation on a lattice-ordered matrix ring over a subfield of the real numbers is the zero derivation, and we prove a similar result for certain lattice-ordered rings with positive squares. The second author thanks Hamilton College for its support of his visits to the first author in Houston. He also thanks John Miller for his friendship and hospitality over the last thirty years.