CAUCHY–DAVENPORT TYPE THEOREMS FOR SEMIGROUPS

Let $\mathbb{A}=(A,+)$ be a (possibly non-commutative) semigroup. For $Z\subseteq A$, we define $Z^{\times }:=Z\cap \mathbb{A}^{\times }$, where $\mathbb{A}^{\times }$ is the set of the units of $\mathbb{A}$ and $$\begin{eqnarray}{\it\gamma}(Z):=\sup _{z_{0}\in Z^{\times }}\inf _{z_{0}\neq z\in Z}\text{\text{ord}}(z-z_{0}).\end{eqnarray}$$ The paper investigates some properties of ${\it\gamma}(\cdot )$ and shows the following extension of the Cauchy–Davenport theorem: if $\mathbb{A}$ is cancellative and $X,Y\subseteq A$, then $$\begin{eqnarray}|X+Y|\geqslant \min ({\it\gamma}(X+Y),|X|+|Y|-1).\end{eqnarray}$$ This implies a generalization of Kemperman’s inequality for torsion-free groups and strengthens another extension of the Cauchy–Davenport theorem, where $\mathbb{A}$ is a group and ${\it\gamma}(X+Y)$ in the above is replaced by the infimum of $|S|$ as $S$ ranges over the non-trivial subgroups of $\mathbb{A}$ (Hamidoune–Károlyi theorem).