A Cauchy-Davenport theorem for semigroups

الأصل · EN

We generalize the Davenport transform and use it to prove that, for a (possibly non-commutative) cancellative semigroup A = (A, +) and non-empty subsets X,Y of A such that the subsemigroup generated by Y is commutative, we have |X + Y| ≥ (ω(Y), |X| + |Y| - 1), where ω(Y):= ₀ ∈ Y ∩ ₐ× ∈ Y {y₀} |<y - y₀>|. This carries over the Cauchy-Davenport theorem to the broader setting of semigroups, and it implies, in particular, an extension of I. Chowla's and S.S. Pillai's theorems for cyclic groups and a notable strengthening of another generalization of the same Cauchy-Davenport theorem to commutative groups, where ω(Y) in the above is replaced by the minimal order of the non-trivial subgroups of A.