arXiv 1102.25782011-02math.CO

半环上的平面流与二次关系

Planar flows and quadratic relations over semirings

Vladimir I. Danilov, Alexander V. Karzanov, Gleb A. Koshevoy

备注：: 35 pages. This is the revised version accepted for publication in J. Algebraic Comb. (The final publication is available at springerlink.com.) Also in the Appendix we add the assertion that the function of minors of any matrix over a field is generated (using Lindström method) by flows in a weighted planar acyclic directed graph

摘要：

适应Lindström的著名的建设，我们考虑了一个广泛的功能，这是由流生成的平面无环有向图的顶点（或边）采取的权重在任意交换半环。我们给出了一个组合描述的“通用”的二次关系有效的这样的功能。它们对特定半环的专门化涉及大量已知的矩阵子式的二次关系（例如，Plücker关系）和这种关系的热带对应物。并讨论了一些应用和相关问题。

英文摘要：

Adapting Lindström's well-known construction, we consider a wide class of functions which are generated by flows in a planar acyclic directed graph whose vertices (or edges) take weights in an arbitrary commutative semiring. We give a combinatorial description for the set of "universal" quadratic relations valid for such functions. Their specializations to particular semirings involve plenty of known quadratic relations for minors of matrices (e.g., Plücker relations) and the tropical counterparts of such relations. Also some applications and related topics are discussed.

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