Lattice Theory of the Poset of Regions

N. Reading
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N. Reading: North Carolina State University, Department of Mathematics

Chapter Chapter 9 in Lattice Theory: Special Topics and Applications, 2016, pp 399-487 from Springer

Abstract: Abstract Hyperplane arrangements (collections of codimension-1 subspaces) have long been an object of study in combinatorics, topology, and geometry. This chapter explores the lattice theory of the poset of regions of a (real) hyperplane arrangement. We discuss the open problem, first posed by Björner, Edelman, and Ziegler [70], of characterizing by local geometric conditions which posets of regions are lattices. We give a local geometric characterization (“tightness”) of which posets of regions are semidistributive lattices. Along the way, we discuss a local condition for checking that a partially ordered set is a lattice, along with analogous local conditions for determining lattice-theoretic properties. In the case of simplicial arrangements (which are in particular tight), we characterize the regions of the arrangement in terms of two notions of combinatorial convexity.

Keywords: Lattice Theory; Maximal Chain; Lattice Congruence; Relative Interior; Hyperplane Arrangement (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-44236-5_9

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DOI: 10.1007/978-3-319-44236-5_9

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