Finitely presented and coherent ordered modules and rings

We extend the usual definition 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 finite systems of inequalities are finitely 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 po-ring R, A is po-coherent ii and only ii A is a finitely presented R-module and A(+) is a finitely generated R+-semimodule. (iii) Every finitely po-presented partially ordered right module over a right po-coherent po-ring is po-coherent. (iv) Every finitely presented abelian lattice-ordered group is po-coherent.