单调函数和映射
Monotone functions and maps
摘要：
在[S. Basu，A. Gabrielov，N. Vorobjov，Semi-monotone sets. arXiv：1004.5047v2（2011）]我们定义半单调集，作为开有界集，可定义在实数上的o-极小结构中，并且与R^n中的所有平移坐标锥有连通的交点。本文通过定义单调函数和映射，并研究它们的基本几何性质，进一步发展了这一理论。证明了有界连续可定义函数或映射单调的几个等价条件。证明了一类单调映射图在与仿射坐标子空间相交和投影到坐标子空间时是闭的。证明了单调映射的图是拓扑正则胞腔。这些结果推广和发展了Basu等人关于半单调集的相应结果。
英文摘要：
In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all translated coordinate cones in R^n. In this paper we develop this theory further by defining monotone functions and maps, and studying their fundamental geometric properties. We prove several equivalent conditions for a bounded continuous definable function or map to be monotone. We show that the class of graphs of monotone maps is closed under intersections with affine coordinate subspaces and projections to coordinate subspaces. We prove that the graph of a monotone map is a topologically regular cell. These results generalize and expand the corresponding results obtained in Basu et al. for semi-monotone sets.
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