### Abstract

Over an ordered field, the class of solution sets of finite systems of homogeneous weak linear inequalities is closed under projection, and this fact yields a simple proof of Farkas' theorem characterizing the nonnegative solvability of systems of linear equations. If one generalizes the notion of inequality to that of congruence inequality--which combines an inequality with a congruence in a special way--then one may exploit techniques from model theory to prove that over any ordered Abelian group, the class of solution sets of finite systems of congruence inequalities is closed under projection. Thus ordered rings other than fields obey versions of Farkas' theorem, and this talk will describe such results for dense subrings of the reals and for the integers.