Extensions of semigroup valued, finitely additive measures (Q1188331)

A commutative semigroup \(E\) with 0 is said to have the charge extension property (CEP) if every charge \(\mu: {\mathfrak C}\to E\) defined on an algebra of subsets of a set \(X\) can be extended to a charge \(\nu: {\mathcal P}(X)\to E\), the power set of \(X\). The authors prove that \(E\) has CEP if \(E\) is a compact semigroup or if \(E\) is a positively ordered semigroup satisfying certain completeness and distributive conditions. Moreover, if \((E,\lor,\land)\) is a lattice with 0, then \((E,\lor)\) has CEP iff the lattice \((E,\lor)\) is Dedekind complete and join-continuous.