Let
$H$
be a multiplicatively written monoid with identity
${1}_{H}$ (in particular, a
group), and denote by
${\mathcal{\mathcal{P}}}_{fin,\times}\left(H\right)$
the monoid obtained by endowing the collection of all finite subsets of
$H$
containing a unit with the operation of setwise multiplication
$\left(X,Y\right)\mapsto \left\{xy:x\in X,y\in Y\right\}$. We study
fundamental features of the arithmetic of this and related structures, with a focus on the
submonoid,
${\mathcal{\mathcal{P}}}_{fin,1}\left(H\right)$, of
${\mathcal{\mathcal{P}}}_{fin,\times}\left(H\right)$ consisting of all
finite subsets of
$H$
containing the identity.
Among others, we establish that
${\mathcal{\mathcal{P}}}_{fin,1}\left(H\right)$
is atomic (i.e., each nonunit is a product of atoms) if and only if
${1}_{H}\ne {x}^{2}\ne x$ for every
$x\in H\setminus \left\{{1}_{H}\right\}$. Then we
prove that
${\mathcal{\mathcal{P}}}_{fin,1}\left(H\right)$
is BF (i.e., it is atomic and every element has factorizations of bounded length) if and
only if
$H$
is torsionfree; and we show how to transfer these conclusions from
${\mathcal{\mathcal{P}}}_{fin,1}\left(H\right)$ to
${\mathcal{\mathcal{P}}}_{fin,\times}\left(H\right)$
through the machinery of equimorphisms.
Next, we introduce a suitable notion of “minimal factorization” (and investigate its
behavior with respect to equimorphisms) to account for the fact that monoids may
have nontrivial idempotents, in which case standard definitions from factorization
theory degenerate. Accordingly, we obtain necessary and sufficient conditions for
${\mathcal{\mathcal{P}}}_{fin,\times}\left(H\right)$
to be BmF (meaning that each nonunit has at least one minimal
factorization and all such factorizations are bounded in length); and for
${\mathcal{\mathcal{P}}}_{fin,1}\left(H\right)$ to be
BmF, HmF (i.e., a BmFmonoid where all the minimal factorizations of a given
element have the same length), or minimally factorial (i.e., a BmFmonoid
where each nonunit element has an essentially unique minimal factorization).
Finally, we prove how to realize certain intervals as sets of minimal lengths in
${\mathcal{\mathcal{P}}}_{fin,1}\left(H\right)$.
Many proofs come down to considering sumset decompositions in cyclic groups, so
giving rise to an intriguing interplay with arithmetic combinatorics.
