We introduce a notion of separativeness for positively ordered monoids (P.O.M.'s), similar in definition to the notion of separativeness for commutative semigroups but which has a simple categorical equivalent, weaker that injectivity, the transfer property. We show that existence in a separative extension of the ground P.O.M. of a solution of a given linear system is equivalent to the satisfaction by the ground P.O.M. of a certain set of equations and inequations, the resolvent. We deduce in particular a characterization of the P.O.M.'s which are infective relatively to the class of embeddings of countable P.O.M.'s; those include in particular divisible weak-cardinal algebras. We also deduce that finitely additive positive non-standard measures invariant relatively to a given exponentially bounded group separate equidecomposability types modulo this group.