The Face Semigroup of Hyperplane Arrangements

This research combines concepts from Abstract Algebra and Geometry. We start with a finite collection of hyperplanes in the Euclidean Space Rⁿ. Arrangements of hyperplanes partition the plane into several sections.We call these sections faces. A binary operation is defined on the set of faces of the arrangement. This algebraic structure of faces with the binary operation forms a semigroup but not a group. We study other properties of this semigroup. As a special case we study hyperplane arrangements in R² that come from invariant theory. Several examples will be given along with the presentation.