Weakly weighted generalised quasi-metric spaces and semilattices

The paper proves that for invariant generalised quasi-metric semilattices, (DPC) holds iff the space is componentwise weakly weighted, with the canonical potentials wx(y)=d(x,y)−d(y,x) yielding the weights on components and uniqueness up to componentwise constants (Theorem 6.2). It also shows, in the quasi-metric case, that an invariant quasi-metric semilattice is weakly weighted iff it satisfies (DPC) (Cor. 5.5), with one direction already true without invariance (Prop. 5.2) and the reverse built via wx (Thm. 5.4). The candidate solution reproduces exactly these statements and proves them by the same core idea: define the canonical potentials wx, use invariance and (DPC) along meet-chains to show d(y,z)+wx(y)=d(z,y)+wx(z), and conversely derive (DPC) from the weak-weight identity along ordered chains, handling ∞-cases by triangle inequality. The remaining characterization “w ≈ wX” is also established exactly as in the paper. Hence both are correct, and the proofs are substantially the same in structure and ideas (canonical potentials, meet reduction, and chain decompositions) as in Theorem 5.4 and Theorem 6.2 of the paper .