Implement the face semigroup of a hyperplane arrangement and provide a custom class for the algebra

[tscrim]

The face semigroup algebra of a hyperplane arrangement is implemented as an instance of the FiniteDimensionalAlgebra. We should first provide a direct implementation of the face semigroup as a stand-alone parent. Then using that, we should implement a custom class for the algebra (possibly a subclass of FiniteDimensionalAlgebra, but the most likely candidate would be as a subclass of CombinatorialFreeModule whose basis is the semigroup). This will give it more features and a direct link to the hyperplane arrangement.

[saliola]

Hi Travis. I have code along these lines. It implements the face semigroup as a semigroup, and then the algebra is just its semigroup algebra (with special methods handled by a category for left regular bands).

[tscrim]

@saliola It would be great if we could use that code as a base, or if you are willing to make it into a full PR, even better! The algebra probably still needs a custom class for backwards compatibility, but not 100% sure of this. I can take care of this part in either case.