On left regular bands and real conic–line arrangements

Abstract

An arrangement of curves in the real plane divides it into a collection of faces. In the case of line arrangements, there exists an associative product which gives this collection a structure of a left regular band. A natural question is whether the same is possible for other arrangements. In this paper, we try to answer this question for the simplest generalization of line arrangements, that is, conic-line arrangements.Investigating the different algebraic structures induced by the face poset of a conic-line arrangement, we present two different generalizations for the product and its associated structures: an alternative left regular band and an associative aperiodic semigroup. We also study the structure of sub left regular bands induced by these arrangements. We finish with some chamber counting results for conic-line arrangements.

On left legal semigroups

“…Many important combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a left regular band (see, for example, [1,8,10,21]). A band is a semigroup in which every element is an idempotent element, and a left regular band is a band which satisfies the identity aba = ab.…”