Arrow's Theorem for incomplete relations

Let U be a set with three or more elements, let W be the set of weak orderings of U, let T be the set of total orderings of U, and let f be an n-ary function mapping Wn to W. Arrow’s Impossibility Theorem asserts that if f satisfies Arrow’s Condition P (“Pareto”) and Condition 3 (“independence of irrelevant alternatives”) then f is a projection function on total orderings, i. e., there is some k ∈ {1, . . . , n} such that f (R1, . . . ,Rn) = Rk for all total orderings R1, . . . ,Rn ∈ T . Th. If a transitive-valued multivariate relational operator f satisfies “unidirectional” (and stronger) versions of Arrow’s Conditions P and 3, and maps all profiles from a diverse set R of binary relations on U to transitive relations on U, then f must be the unanimous consent function for some set of input variables, and if R is very diverse then f is a projection function. Cor. Characterizations of intersection and projection functions; Arrow’s Theorem. R. D. Maddux Arrow’s Theorem for Incomplete Relations