A generalization of the Dubovitskii-Milyutin separation theorem for commutative semigroups (Q1106425)

Let S be an Abelian semigroup. If A is a subsemigroup of S, then define the algebraic interior of A by \[ cor A:=\{a\in A| \forall s\in S\exists n\in {\mathbb{N}}:na+s\in A\}. \] A typical result of the paper states that: If \(A_ 0,A_ 1,...,A_ n\) are disjoint subsemigroups of S such that \[ A_ 0\cap (cor A_ 1+S)\cap...\cap (cor A_ n+S)\neq \emptyset \] then there exist additive functions \(f_ 1,...,f_ n:S\to [-\infty,\infty [\), not all zero, such that \[ f_ 1|_{A_ 1}\leq 0,....,f_ n|_{A_ n}\leq 0\quad and\quad 0\leq (f_ 1+...+f_ n)_{A_ 0}. \] The case \(n=1\) yields a Hahn-Banach-type separation theorem, the general case can be considered as a generalization of a well-known result of Dubovitskii and Milyutin.