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Lattice and order properties of the poset of regions in a hyperplane arrangement. (English) Zbl 1092.06006

An arrangement \(\mathcal A\) is a finite nonempty collection of hyperplanes in \({\mathbb R}^d\). The complement of the union of the hyperplanes is disconnected and the closures of its connected components are called regions. In the paper arrangements are central, meaning that every hyperplane contains the origin. A central hyperplane arrangement is called simplicial if every region is a simplicial cone. The lattice of intersections \(L(\mathcal A)\) of a central arrangement \(\mathcal A\) consists of all arbitrary intersections of hyperplanes, partially ordered by reverse inclusion. A unique minimal element is adjoined to make \(L(\mathcal A)\) a lattice.
A lattice is called supersolvable if it possesses a maximal chain \(C\) such that for any maximal chain \(C'\), the sublattice generated by \(C\cup C'\) is distributive. A central arrangement is called supersolvable if \(L(\mathcal A)\) is a supersolvable lattice. A lattice \(L\) is called congruence normal if \(L\) can be obtained by a finite sequence of doublings of convex sets.
The fundamental object of study in the paper is the poset \(P(\mathcal A,B)\) of regions of a central arrangement \(\mathcal A\) with respect to a fixed region \(B\). It is shown that the poset of regions of a supersolvable hyperplane arrangement is a congruence normal lattice. Order dimensions of these posets are also discussed. It is proved that the poset of regions of a simplicial arrangement is a semi-distributive lattice. The congruence normality of a lattice is given in terms of edge-labellings. This gives a simple criterion to determine whether a given simplicial arrangement has a congruence uniform lattice of regions.

MSC:

06A07 Combinatorics of partially ordered sets
20F55 Reflection and Coxeter groups (group-theoretic aspects)
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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