Multiorder, Kleene stars and cyclic projectors in the geometry of max cones
Multiorder, Kleene stars and cyclic projectors in the geometry of max cones
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DOI：
10.1090/conm/495/09704
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发表时间：
2008-07
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影响因子：
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通讯作者：
Sergeĭ Sergeev
中科院分区：
文献类型：
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作者：
Sergeĭ Sergeev
本文综述了极大-加凸几何中的一些问题，主要是关于多重序、Kleene星和循环投影的作用的结果，并将它们与极大代数中的一些问题联系起来。多阶原理导致了有限维凸几何中某些命题的极大加类似，并与极大代数中的集合覆盖条件有关。Kleene星是极大代数的基础，因为它们累积最优路径的权重并描述矩阵的特征空间。另一方面，热带凸性方法将一个半模分解为若干凸区域，这些凸区域是唯一定义的Kleene星的列跨度。另一个最近的几何结果，即几个零交集的半模可以通过最大加半空间相互分离，导致了对称为循环投影仪的特定非线性算子的研究。这些非线性算子可用于求解齐次多边极大线性方程组。结果在最大锥的设置中呈现，即，极大次半环上的半模
This paper summarizes results on some topics in the max-plus convex geometry, mainly concerning the role of multiorder, Kleene stars and cyclic projectors, and relates them to some topics in max algebra. The multiorder principle leads to max-plus analogues of some statements in the finite-dimensional convex geometry and is related to the set covering conditions in max algebra. Kleene stars are fundamental for max algebra, as they accumulate the weights of optimal paths and describe the eigenspace of a matrix. On the other hand, the approach of tropical convexity decomposes a finitely generated semimodule into a number of convex regions, and these regions are column spans of uniquely defined Kleene stars. Another recent geometric result, that several semimodules with zero intersection can be separated from each other by max-plus halfspaces, leads to investigation of specific nonlinear operators called cyclic projectors. These nonlinear operators can be used to find a solution to homogeneous multi-sided systems of max-linear equations. The results are presented in the setting of max cones, i.e., semimodules over the max-times semiring.