Difference measurement and simple scalability with restricted solvability

Let A, B be two sets, with B ? A × A, and ≤ a binary relation on B. The problem analyzed here is that of the existence of a mapping u: A → R, satisfying:

$\text{(a,b) ? (}\text{a}\text{?}\text{,}\text{b}\text{?}\text{)}\text{iff}\text{∨∧ μ(b) ? μ(a) ? μ(}\text{b}\text{?}\text{) ? μ(}\text{a}\text{?}\text{)}$

whenever (a, b), (a′, b′) ∈ B. In earlier discussions of this problem, it is usually assumed that B is connected on A. Here, we only assume that B satisfies a certain convexity property. The resulting system provides an appropriate axiomatization of Fechner's scaling procedures. The independence of axioms is discussed. A more general representation is also analyzed:

$\text{(a,b) ? (}\text{a}\text{?}\text{,}\text{b}\text{?}\text{)}\text{iff}\text{∨∧ F[μ(b), μ(a)] ? F[μ}\text{b}\text{?}\text{]}$

, where F is strictly increasing in the first argument, and strictly decreasing in the second. Sufficient conditions are presented, and a proof of the representation theorem is given.