Order-coherent Archimedean \(f\)-algebras. (Q1771964)

A partially ordered directed commutative ring \(A\) with positive core \(A^{+}\) is po-coherent if for any \(a_{1},\dots ,a_{n} \in A\) the solution of the system \(a_{1}x_{1} + \cdots + a_{n}x_{n} \geq 0\), \(x_{1}, \dots ,x_{n} \geq 0\) is a finitely generated \(A^{+}\)-subsemimodule of the module of all \(n \times 1\)-matrices over \(A\). It is known (C.\ B.\ Huijsmans and B. de Pagter) that every Archimedean uniformly complete \(f\)-algebra with unit which is po-coherent is also Dedekind \(\sigma \)-complete. The aim of the paper is to prove that also the converse is valid.