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).