Hockey-Stick Domination and Distributional Comparison on Finite Posets

We develop a framework for comparing probability measures on finite posets via hockey-stick domination, an order relation defined through interval-counting test functions. The theory introduces poset integrals, derivatives, power functions and the associated moment functionals, all of which are invariant under poset isomorphisms. We prove that hockey-stick domination admits an exact quantitative characterization: whenever $\mu$ is dominated by $\nu$ in the hockey-stick order, the corresponding Zolotarev-type distance is equal to one half of the second-order poset moment of $\nu-\mu$. We further develop a constructive theory for generating such domination relations. In particular, we show that hockey-stick domination is preserved under direct products, disjoint unions, ordinal sums, and suitable ideal restrictions, yielding natural families of examples on chains, Boolean posets, rectangular lattices, rooted trees, and Young diagrams.