ABSTRACT

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.

中文翻译：

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Farkas 引理的离散变体和相关结果

摘要

我们将 Farkas 引理推广到线性有序交换环上的模块设置。我们以纯线性代数的方式证明了结果的完全通用性，而不对环做出任何额外的假设，环甚至可能包含零除数。我们还在新的离散设置中展示了一些相关结果（Tucker 的关键定理、Motzkin 的转置定理和 Tucker 的转置定理）。最后，我们简要讨论了结果的一些可能应用，并提出了一些问题以供进一步研究。