What is it all about?

Power semigroups arise from a very simple idea: instead of studying the elements of a (multiplicatively written) semigroup \(S\) one by one, we consider sets of elements and multiply them in a natural way. More precisely, we can define a multiplication on the power set of \(S\) by setting

\[ XY := \{ xy : x \in X,\; y \in Y \}, \qquad \text{for all } X, Y \subseteq S. \]

Endowed with this operation, suitable families of subsets of \(S\) themselves form semigroups, collectively referred to as the power semigroups of \(S\). Most notably, the family of all non-empty subsets of \(S\) forms the large power semigroup of \(S\), whereas the family of all non-empty finite subsets of \(S\) forms the finitary power semigroup of \(S\); the former is commonly denoted by \(\mathcal P(S)\), and the latter by \(\mathcal P_{\mathrm{fin}}(S)\).

These constructions are extremely natural and appear in several areas of mathematics and theoretical computer science. In particular, power semigroups play a role in the theory of formal languages and automata, and they have been studied from different perspectives, including semigroup theory, universal algebra, and more recently factorization theory and arithmetic combinatorics.

Historically, the first appearances of power semigroups as objects of interest in their right can be traced back at least to a 1950 paper by Ballieu [Bal-1950]. A systematic investigation began in the late 1960s and was pursued intensively throughout the 1980s and 1990s. A catalyst for these early developments was the role of power semigroups in the study of formal languages and automata [Alm-2002].