Let (H,·) be a monoid.
The power monoid of H, first studied in full generality by Y. Fan and S. Tringali, is the collection P_fin(H) of finite, nonempty subsets of H, with the operation of setwise multiplication given by $X·Y := \{x·y: x∈X, y∈Y\}.
This is a highly non-cancellative monoid in which many standard factorization questions (e.g., for which H is P_fin(H) BF, or which sets occur as sets of factorization lengths) have complicated and interesting answers.
We pivot to the submonoid P_{fin,1}(H) consisting of finite subsets containing 1, which is equimorphic to P_fin(H) when H is a group, but is also deserving of study in its own right.
We determine exact conditions on H for which P_{fin,1}(H) is atomic (resp. BF).
Due to its non-cancellative nature, P_{fin,1}(H) eludes characterization by some of the usual tools of factorization theory.
To respond in a systematic way to non-cancellative phenomena, we formulate the notion of "minimal'' factorizations and the "minimal'' versions of the usual properties BmF, FmF,HmF, and UmF (corresponding, respectively to BF, FF, HF, and UF or factoriality).
With this in hand, we can give exact conditions on those H which make P_{fin,1}(H) BmF (resp. FmF, HmF, UmF).
As a further application, we show that all intervals of the form [2,k] are realized as sets of factorization lengths in P_{fin,0}(ℤ/nℤ) for k\in[2,n-1].
Even P_{fin,0}(ℕ), the reduced power monoid of the naturals, is a rich object of study.
Of particular interest are the quantifiable differences between the intervals [0,n] and the other elements of P_{fin,0}(ℕ).
It is already known, due to Fan and Tringali, that L([0,n]) = [2,n].
We refine this result by introducing the \textit{partition type} of a factorization and showing that [0,n] has factorizations of almost every partition type, and that non-intervals sharply fail to do so.
Intervals are further distinguished by giving an exponential lower bound on |Z([0,n])|, the number of factorizations of [0,n].
Following the idea of partition type beyond the realm of power monoids, we take a detour to show that the density of atoms of a given degree in any numerical semigroup algebra over a finite field is asymptotically zero (as we let the degree approach infinity).
Returning to power monoids, we end by focusing in particular on sets of factorization lengths in P_{fin,0}(ℕ).
The study of which sets occur as sets of lengths in P_{fin,0}(ℕ) is fairly difficult, and requires some new tools.
To this end, we show that all factorization phenomena that occur in P_{fin,0}(ℕ^d), for d>1, also occur in P_{fin,0}(ℕ) (and vice-versa).
Consequently, we may leverage the intuition and geometry of the integer lattice.
After developing the necessary methods, we recover some known results on sets of lengths; namely, that \{n\} and \{2,n+1\} occur as sets of lengths for any n\ge 2.
Finally, we demonstrate that [2,m+2]∪\{m+n+1\} can be realized as a set of factorization lengths for all m≥1 and n≥2, representing progress toward the conjecture, made by Fan and Tringali, that P_{fin,0}(ℕ) realizes all feasible sets of lengths.