In the current technical note we provide a topological generalization of hierarchy of partitions(HOP) structure, and the implications over the axioms of h-increasingness and scale increasingness [13]. Further in this study we will explicit the Lagrange optimization in the optimal cuts problem and the conditions necessary on the energy to obtain a global optimum using the a dynamic program. Further a general multi-constraint optimization problem is considered with multiple Lagrangian multipliers, leading to a general version of scale increasingness that orders cuts, by ordered tuples of multipliers. The report also differentiates Inf-Modularity and Submodularity and their space of application. The final demonstration on wavering hierarchies show how one can relax conditions on the hierarchical structure.
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