Title: Power Monoids: Factorization of Subsets
Speaker: Austin Antoniou (Ohio State University)
Abstract: Factorization Theory is concerned with studying how objects (usually elements of a monoid) decompose into irreducible objects. We are most familiar with how integers decompose into primes, but the study of factorization is most interesting in settings where factorizations are highly non-unique.
Let $H$ be a monoid, and let $\mathcal{P}(H)$ be the collection of finite, nonempty subsets of $H$ with the operation given by $XY = \{ xy : x\in X, y\in Y \}$. This gives $\mathcal{P}(H)$ the structure of a monoid, and we call this the Power Monoid of $H$. This turns out to give a class of accessible examples of monoids with some unruly behaviors. The aim of this talk is to discuss some of the complications of $\mathcal{P}(H)$, characterize the $H$ for which the unruliness of $\mathcal{P}(H)$ is tamed, and see some deeper results for particular choices of $H$.
