On the isomorphism problem for power semigroups

Abstract

Let $\mathcal{P}(S)$ be the semigroup obtained by equipping the family of all non-empty subsets of a (multiplicatively written) semigroup $S$ with the operation of setwise multiplication induced by $S$ itself. We call a subsemigroup $P$ of $\mathcal{P}(S)$ downward complete if any element of $S$ lies in at least one set $X\in P$ and any non-empty subset of a set in $P$ is still in $P$. We obtain, for a commutative semigroup $S$, a characterization of the cancellative elements of a downward complete subsemigroup of $\mathcal{P}(S)$ in terms of the cancellative elements of $S$. Consequently, we show that, if $H$ and $K$ are cancellative semigroups and either of them is commutative, then every isomorphism from a downward complete subsemigroup of $\mathcal{P}(H)$ to a downward complete subsemigroup of $\mathcal{P}(K)$ restricts to an isomorphism from $H$ to $K$. This solves a special case of a problem of Tamura and Shafer from the late 1960s and generalizes a recent result by Bienvenu and Geroldinger, where it is assumed, among other conditions, that $H$ and $K$ are numerical monoids.

arXiv comment9 pages, no figures. Final version to appear in M. Brešar, A. Geroldinger, B. Olberding, and D. Smertnig (eds.), Recent Progress in Ring and Factorization Theory, Springer Proc. Math. Stat., Springer, 202?