Hierarchical Set Constructions via Multi-Iterated Powersets and the Signed Iterated Power Multiset

This paper develops a unified, hierarchy-aware framework for set constructions that scale from items to templates, libraries, catalogs, and beyond. We introduce the Signed Power Multiset and the Signed Iterated Power Multiset, defined by coordinatewise factorization. We prove that the signed constructions reduce to classical powersets and power multisets when signs are nonnegative or repetitions disappear, and that multiplicities factor across disjoint supports. For finite bases we obtain size recurrences forming exponential towers and identify cancellation laws triggered by negative multiplicities. We also formalize multi–iterated powersets indexed by block vectors and establish a flattening law showing that only the total height matters. Worked examples from inventory reconciliation and planning illustrate how the framework captures layered selections, recalls, and multi-stage decisions. We also extend our investigation to the concept of Named Sets, considering their generalizations such as the Named Power Set and the Named Iterated PowerSet.