Finitely presented and coherent ordered modules and rings

We extend the usual de？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 ？nite systems of inequalities are ？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 ？nitely presented R-module and A^+ is a ？nitely generated R^+-semimodule. (iii) Every ？nitely po-presented partially ordered right module over a right po-coherent po-ring is po-coherent. (iv) Every ？nitely presented abelian lattice-ordered group is po-coherent.