Balint Rago: On isomorphisms between power monoids

On isomorphisms between power monoids

Balint Rago, Graz University

Abstract:

Let $S$ be a multiplicatively written semigroup. The family $\mathcal{P}(S)$ of non-empty subsets of $S$ endowed with the binary operation of setwise multiplication [ (X,Y)\mapsto {xy:x\in X, y\in Y} ] induced by $S$, is called the \textit{large power semigroup} of $S$. Although power semigroups were already introduced in the 1950s, the study of arithmetical and algebraic properties of these objects received a lot of attention in the last decade. In this talk, we give a survey of recent developments in this area and present some new results involving the \textit{reduced finitary power monoid} $\mathcal{P}{\text{fin},1}(H)$ of a monoid $H$, which consists of all finite subsets of $H$ containing the identity element. To give an example, a central question, called the \textit{isomorphism problem}, is whether an isomorphism between $\mathcal{P} monoids. In this talk, we give a full characterization of the isomorphism problem for the class of commutative and cancellative monoids.},1}(H_1)$ and $\mathcal{P}_{\text{fin},1}(H_2)$ for two monoids $H_1$ and $H_2$ implies that $H_1$ and $H_2$ are isomorphic. This question was answered in the negative for arbitrary monoids, yet remained open for \textit{cancellative