Addition theorems in acyclic semigroups

Abstract

We give a necessary and sufficient condition on a given family A of finite subsets of integers for the Cauchy-Davenport inequality |A + B|<= |A| + |B| − 1, to hold for any family B of finite subsets of integers. We also describe the extremal families for this inequality. We prove this result in the general context of acyclic semigroups, which contain also the semigroup of sequences of elements in an ordered group.