Cauchy-Davenport type theorems for semigroups

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Let A = (A, +) be a (possibly non-commutative) semigroup. For Z A we define Z×:= Z ∩ A×, where A× is the set of the units of A, and γ(Z):= ₀ ∈ Z× ₀ ≠ z ∈ Z ord(z - z₀). The paper investigates some properties of γ(·) and shows the following extension of the Cauchy-Davenport theorem: If A is cancellative and X, Y A, then |X+Y| ≥ (γ(X+Y),|X| + |Y| - 1). This implies a generalization of Kemperman's inequality for torsion-free groups and strengthens another extension of the Cauchy-Davenport theorem, where A is a group and γ(X+Y) in the above is replaced by the infimum of |S| as S ranges over the non-trivial subgroups of A (Hamidoune-Károlyi theorem).