Some representation problems for semiorders

Suppose we have a number representation of a semiorder 〈A, P〉 such that aPb iff f(a)+δ(a) < f(b), for all a, b ∈ A, where δ is a nonnegative function describing the variable jnd. Such an f (here called a closed representation) may not preserve the simple order relation $\text{R}{}^1$ generated by 〈A, P〉, i.e., $\text{aR}{}^1\text{b}$ but f(a) > f(b) for some f, δ and a, b ∈ A. We show that this “paradox” can be eliminated for closed and closed interval representations. For interval representations it appears to be impossible. That is why we introduce a new type of representation (an R-representation) which is of the most general form for number representations that preserve the linear structure of the represented semiorders. The necessary and sufficient condition for an R-representation is given. We also give some independent results on the semiorder structure. Theorems are proved for semiorders of arbitrary cardinality. The Axiom of Choice is used in the proofs.