On Left regular bands and real Conic-Line arrangements

TL;DR

This paper explores algebraic structures on the faces of conic-line arrangements in the real plane, generalizing known structures from line arrangements, and introduces new products and semigroup concepts.

Contribution

It introduces two new algebraic structures, an alternative left regular band and an associative aperiodic semigroup, for conic-line arrangements.

Findings

Defined new algebraic structures for conic-line arrangements.

Established chamber counting formulas for these arrangements.

Analyzed substructure properties of the introduced bands.

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.