Common extensions of semigroup-valued charges (Q1340550)

Let \(E\) be a positively preordered commutative semigroup (\(= pp\)- semigroup). Two \(E\)-valued charges \(\mu\) and \(\nu\) defined, respectively, on algebras \(\mathfrak A\) and \(\mathfrak B\) of subsets of a set \(X\) are called consistent if for all \(A\in {\mathfrak A}\) and \(B\in {\mathfrak B}\), \(A\subseteq B\) (resp. \(A\supseteq B\), \(A= B\)) implies \(\mu(A)\leq \nu(B)\) (resp. \(\mu(A)\geq \nu(B)\), \(\mu(A)= \nu(B)\)). \(E\) is said to have the 2-CHEP (two-charge-extension-property) when any two consistent \(E\)-valued charges have a common extension; when one restricts oneself to charges defined on finite algebras, then \(E\) is said to have the grid property. \(E\) has the 1-CHEP when for every subalgebra \(\mathfrak A\) of an algebra \(\mathfrak B\), every \(E\)-valued charge on \(\mathfrak A\) extends to an \(E\)-valued charge on \(\mathfrak B\). The main results: (1) 2-CHEP implies 1-CHEP. (2) The \(pp\)-semigroups with the grid property are characterized. (3) If \(G\) is a directed p.o. Abelian group and \(G_ +\) has the 2-CHEP, then \(G\) is Dedekind complete. If \(G\) is an \(\ell\)-group, then \(G_ +\) has the 1-CHEP iff \(G\) is Dedekind complete.