A lineage table is a hashtable whose values are a Gr\"obner basis for the ideal I or the ideal generated by L, and keys are the lineages of the corresponding elements.
A lineage of a polynomial is a natural number, or an ordered pair of lineages, tracing its history in the given Gr\"obner basis computation. Lineages that are natural numbers are assigned to the original input polynomials.
The object LineageTable is a type, with ancestor classes HashTable < Thing.