Theorem 3.1, p. 15 — a semimetric on acyclic orders satisfies Axioms 1, 3, 4 iff it is a µ-top-difference semimetric

Let $X$ be a nonempty finite set and let $d$ be a semimetric on the set $\mathcal A(X)$ of acyclic orders on $X$, with values in $[0,\infty)$. Then $d$ satisfies Axioms 1, 3 and 4 if and only if there is a positive measure $\mu$ on $2^X$ such that

The axioms are: (1) additivity along orders in-between two given orders, where "in-between" means reachable by one-step perturbations obeying the consistency condition (2) of p. 10; (3) the distance of a single addition or deletion of $(a,b)$ is $2^{N(b,\succsim)-1}$, resp. $2^{N(b,\succsim)}$, times $d(\succsim_{ab},\succsim^+_{ab})$; (4) orders with the same strict part are at distance $0$. The theorem identifies the $\mu$-top-difference semimetrics, which compare two preferences by the maximal elements they select from every menu, as exactly the semimetrics with these three properties.

Formalization Note Relations are X → X → Prop and $d$ is a real function on all of them; IsSemimetric d (nonnegativity, $d(R,R)=0$, symmetry, triangle inequality on acyclic orders) is the pseudometric reading of "semimetric", and the conclusion asserts equality only on acyclic orders. One measure $\mu$, given by point weights $w\ge0$ with $\mu(T)=\sum_{x\in T}w(x)$, serves all pairs (the zero measure is allowed, as in footnote 14). Axiom 3's $\oplus$ clause applies only when $\succsim\oplus(a,b)$ is acyclic, as on p. 8. Axiom 1 uses the paper's "in-between", not the literature's weaker betweenness.