Computational Treatment of the Hierarchy of General and Evolution Algebras

The concept of the hierarchy of algebras is not too covered in the literature. Indeed, there are very few works on the hierarchy of algebras in general and almost none for any particular type of algebras. In a previous work by one of the authors, the hierarchyof evolution and Lie algebras was studied and that study was used to extend that concept to the hierarchy of associated graphs, although the computational treatment given to that study was not too extensive. This paper delves into this last aspect, theoretically estimating the worst space and time complexity for general algebras and verifying it empirically. Examples near the worst case have been proposed and executed several times per dimension to get a neat scatter-plot. The polynomial fits which optimize their goodness values follow the predicted theoretical laws. Time complexity has been tested for evolution algebras too, yielding no relevant changes w.r.t. the general case. Finally, a deep study on dependences between complexity and the structure of both the input and the output has been carried out. It has been noticed that intricacy of output structure and execution times are highly correlated. Effect of input structure is much weaker.