A functional equation proof of the distributive-triples theorem

with domains M,V , and S. The substances are assumed to be homogeneous. In this case, mass has two decompositions. The first is the conjoint one 〈V × S,%M 〉 underlying the usual multiplicative relation, where %M is a qualitative weak ordering determined by placing masses on the two scales of an equal-arm balance. A standard monotonicity assumption induces orderings %V on volumes and %S on homogenous substances. The second is a concatenation structure for the masses, 〈M,%M , ◦M 〉, which formalizes the idea of placing two masses on the same pan of the balance. And there is a volume concatenation structure 〈V ,%V , ◦V 〉. When certain axioms are satisfied, each structure gives rise to a measure of mass: From ◦M , asm1 a homomorphic mapping onto the nonnegative real numbers