  We extend the usual de&#64257;nition of coherence, for modules over rings, to
partially ordered right modules over a large class of partially ordered rings,
called po-rings. In this situation, coherence is equivalent to saying that
solution sets of &#64257;nite systems of inequalities are &#64257;nitely
generated semimodules. Coherence for ordered rings and modules, which we call
po-coherence, has the following features: (i) Every subring of Q, and every
totally ordered division ring, is po-coherent. (ii) For a partially ordered
right module A over a po-coherent poring R, A is po-coherent if and only if A
is a &#64257;nitely presented R-module and A^+ is a &#64257;nitely generated
R^+-semimodule. (iii) Every &#64257;nitely po-presented partially ordered right
module over a right po-coherent po-ring is po-coherent. (iv) Every
&#64257;nitely presented abelian lattice-ordered group is po-coherent.
