Common extensions of semigroup-valued charges
Résumé
Let $A$ and $B$ be fields of subsets of a nonempty set $X$ and let $\mu:\,A\to E$ and $\nu:\,B\to E$ be finitely additive measures (``charges'') taking values in a commutative semigroup $E$. We assume that $\mu$ and $\nu$ are consistent, that is, they agree on $A\cap B$, $a\leq b$ implies that $\mu(a)\leq\nu(b)$ (for $a\in A$, $b\in B$), and symmetrically (where $x\leq y$ means that there exists $z$ such that $x+z=y$, for all $x$, $y\in E$). We investigate conditions on $E$ such that any two consistent $E$-valued measures on certain types of fields of sets have a common extension on a larger Boolean algebra. In particular, if the only condition is that $A$ and $B$ are finite, then we obtain the so-called "grid property" on $E$, and we prove that this grid property is finitely axiomatizable.