Continuity and completeness of strongly independent preorders

We show that a strongly independent preorder on a possibly infinite dimensional convex set that satisfies two of the following conditions must satisfy the third: (i) the Archimedean continuity condition; (ii) mixture continuity; and (iii) comparability under the preorder is an equivalence relation. In addition, if the preorder is nontrivial (has nonempty asymmetric part) and satisfies two of the following conditions, it must satisfy the third: (i′) a modest strengthening of the Archimedean condition; (ii) mixture continuity; and (iii′) completeness. Applications to decision making under conditions of risk and uncertainty are provided, illustrating the relevance of infinite dimensionality.