In two dimensions (i.e., in the real affine plane) each region is a convex polygon (if it is bounded) or a convex polygonal region which goes off to infinity. Adding an extra top element to the face semilattice gives the face lattice. The face semilattice of an arrangement is the set of all faces, ordered by inclusion. The faces of codimension 1 may be called the facets of A. The regions are faces because the whole space is a flat. Each flat of A is also divided into pieces by the hyperplanes that do not contain the flat these pieces are called the faces of A. In real affine space, the complement is disconnected: it is made up of separate pieces called cells or regions or chambers, each of which is either a bounded region that is a convex polytope, or an unbounded region that is a convex polyhedral region which goes off to infinity. The Orlik-Solomon algebra is then the quotient of E by the ideal generated by elements of the form (where have an empty intersection) and by boundaries of elements of the same form for which has codimension less than p. A chain complex structure is defined on E with the usual boundary operator. To define it, fix a commutative subring K of the base field, and form the exterior algebra E of the vector space The intersection semilattice determines another combinatorial invariant of the arrangement, the Orlik–Solomon algebra. (The empty set is excluded from the semilattice in the affine case specifically so that these relationships will be valid.) The Orlik–Solomon algebra The Whitney-number polynomial of A is similarly related to that of L( A). Thus it is good to know that p A( y) = y i p L( A)( y), where i is the smallest dimension of any flat, except that in the projective case it equals y i + 1 p L( A)( y). Summed over B ⊆ C ⊆ A such that f( B) is nonempty.īeing a geometric lattice or semilattice, L( A) has a characteristic polynomial, p L( A)( y), which has an extensive theory (see matroid). Another polynomial associated with A is the Whitney-number polynomial w A( x, y), defined by (The dimension of the empty set is defined to be −1.) This polynomial helps to solve some basic questions see below. Summed over all subsets B of A except, in the affine case, subsets whose intersection is empty. The characteristic polynomial of A, written p A( y), can be defined by Polynomialsįor a subset B of A, let us define f( B) := the intersection of the hyperplanes in B this is S if B is empty. In general, when L( A) is a semilattice, there is an analogous matroid-like structure that might be called a semimatroid, which is a generalization of a matroid (and has the same relationship to the intersection semilattice as does the matroid to the lattice in the lattice case), but is not a matroid if L( A) is not a lattice. When L( A) is a lattice, the matroid of A, written M( A), has A for its ground set and has rank function r( S) := codim( I), where S is any subset of A and I is the intersection of the hyperplanes in S. (This is why the semilattice must be ordered by reverse inclusion-rather than by inclusion, which might seem more natural but would not yield a geometric (semi)lattice.) If the arrangement is linear or projective, or if the intersection of all hyperplanes is nonempty, the intersection lattice is a geometric lattice. The intersection semilattice L( A) is a meet semilattice and more specifically is a geometric semilattice. General theory The intersection semilattice and the matroid 1.1 The intersection semilattice and the matroid.If S is 3-dimensional one has an arrangement of planes. Historically, real arrangements of lines were the first arrangements investigated.
If the whole space S is 2-dimensional, the hyperplanes are lines such an arrangement is often called an arrangement of lines. L( A) is partially ordered by reverse inclusion. These subspaces are called the flats of A. (excluding, in the affine case, the empty set). The intersection semilattice of A, written L( A), is the set of all subspaces that are obtained by intersecting some of the hyperplanes among these subspaces are S itself, all the individual hyperplanes, all intersections of pairs of hyperplanes, etc. One may ask how these properties are related to the arrangement and its intersection semilattice. Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M( A), which is the set that remains when the hyperplanes are removed from the whole space. In geometry and combinatorics, an arrangement of hyperplanes is a finite set A of hyperplanes in a linear, affine, or projective space S.
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