Common extensions of semigroup- valued charges

Let A and B be fields of subsets of a nonempty set X and let μ : A → E and ν : B → E be finitely additive measures (“charges”) taking values in a commutative semigroup E. We assume that μ and ν are consistent (e.g. μ = ν on A ∩ B) and ask whether they have a common extension to a charge ρ : A ∨ B → E. Now, we shall see (proposition 2.3) that the most natural consistency condition which we can formulate involves a partial preordering (which may not be an ordering) on E. Furthermore, the formulation of our results will be much clearer when expressed with the preordering than without (see for example theorem 3.2); note that this situation is reminiscent of [12]. More generally, the consideration of the preordering seems fundamental in the study of homomorphism extension properties of commutative semigroups when these are rather viewed as positive cones (of, say, ordered groups), see [14] to [17]. For all these reasons, we shall consider charges with values in what we will call here a pp-semigroup (definition 1.1) rather than just a semigroup. Note that in [14] to [17], where completeness with respect to the ordering plays an important role, these structures are called P.O.M.’s. We say that a pp-semigroup E has the 2-charge extension property (from now on 2CHEP) when any two consistent E-valued charges μ, ν have a common extension ρ; when one restricts oneself to finite Boolean algebras A and B, then we will say that E has the grid property. Finally, by considering only one algebra, one gets the following definition of the 1-CHEP: a pp-semigroup E has the 1-CHEP when for every Boolean subalgebra A of a Boolean algebra B, every E-valued charge on A extends to a E-valued charge on