  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.
