Algebra B
Geffroy, Sylvain · Zenodo (CERN European Organization for Nuclear Research) · 2025
The foundation of this work is the development of Algebra B, a novel combinatorial theory of rooted trees whose algebraic structure is motivated by the conceptual separation of 'container' from 'content' in number theory. A basket tree is a finite rooted tree whose nodes carry immutable capacities and mutable content. Capacities are positive integers that strictly decrease along edges and can only be changed by a uniform homogeneous enlargement operation. This capacity defines the allowed structural relationships, while the content (subtrees) may be freely added or removed. We introduce a new notion of content division based on the maximal number of pairwise disjoint, capacity-preserving pattern embeddings of a pattern tree S into a host tree P. This process yields the content quotient and the content remainder. We establish strong structural theorems, including the invariance of the tree's shape (depth and width profile) under homogeneous enlargement. Finally, we construct and prove the uniqueness of canonical configurations for admissible capacity profiles: a greedy algorithm finds the unique minimal-depth basket tree realization. Algebra B occupies a distinct position from existing frameworks by rigorously enforcing the fundamental asymmetry between capacity and content