A discrete variant of Farkas' lemma and related results

We generalize Farkas' lemma to the setting of a module over a linearly ordered commutative ring. We prove the result in full generality in a purely linear-algebraic way, without making any additional hypothesis about the ring, which may even contain zero divisors. We also present some related results in the new discrete setting (Tucker's key theorem, Motzkin's transposition theorem and Tucker's transposition theorem). Finally, we briefly discuss some of the possible applications of the results and we raise several questions for further research.