Monotone functions and maps

TL;DR

This paper extends the theory of semi-monotone sets by defining and analyzing monotone functions and maps within o-minimal structures, establishing their geometric properties and topological regularity.

Contribution

It introduces the concept of monotone maps, provides equivalent conditions for monotonicity, and proves their graphs are regular cells, expanding prior semi-monotone set results.

Findings

Graphs of monotone maps are topologically regular cells.

Monotone maps are closed under intersections and projections.

Equivalent conditions characterize monotonicity in bounded continuous definable functions.

Abstract

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.