The Cauchy-Davenport theorem for semigroups
Résumé
We generalize the Davenport transform and use it to prove that, for $\mathfrak A = (A, +)$ a cancellative unital semigroup and $X,Y$ two subsets of $\mathfrak A$ such that $\langle Y \rangle_\mathfrak{A}$ is commutative, one has that $|X + Y| \ge \Omega_\mathfrak{A}(X,Y)$, where $\Omega_\mathfrak{A}(X,Y)$ is given by \begin{displaymath} \textstyle \min \! \left( |X| + |Y| - 1, \sup_{y_0 \in Y^{\times}} \min_{y \in Y \setminus \{y_0\}} |\langle y - y_0 \rangle_\mathfrak{A}| \right) \end{displaymath} if $2 \le |X|,|Y| < \infty$. While extending the Cauchy-Davenport theorem to the more general and abstract setting of (possibly non-commutative) semigroups, this also strengthens a previous generalization by G. Karolyi relating to the sum-set of non-empty finite sets in commutative groups, where $\Omega_\mathfrak{A}(X,Y)$ in the above estimate is replaced with $\min(p(\mathfrak A),|X| + |Y| - 1)$, with $p(\mathfrak A)$ being the order of the smallest non-trivial subgroup of $\mathfrak A$, which is never greater than $\Omega_\mathfrak{A}(X,Y)$, and indeed (much) smaller in a number of cases.
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