A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expected utility axiom of strong independence. We consider when a mixture preorder has a multi-representation that consists of real-valued, mixture-preserving functions. If it does, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity is sufficient for a mixture-preserving multi-representation when the dimension of the mixture space is countable, but not when it is uncountable. Our strongest positive result is that mixture continuity is sufficient in conjunction with a novel axiom we call countable domination, which constrains the order complexity of the mixture preorder in terms of its Archimedean structure. We also consider what happens when the mixture space is given its natural weak topology. Continuity (having closed upper and lower sets) and closedness (having a closed graph) are stronger than mixture continuity. We show that continuity is necessary but not sufficient for a mixture preorder to have a mixture-preserving multi-representation. Closedness is also necessary; we leave it as an open question whether it is sufficient. We end with results concerning the existence of mixture-preserving multi-representations that consist entirely of strictly increasing functions, and a uniqueness result.

混合预序是混合空间（如凸集）上与混合操作兼容的预序。在决策理论方面，它满足强独立性的中心期望效用公理。我们考虑混合预购何时具有由实值、混合保留函数组成的多重表示。如果是，它必须满足 Herstein 和 Milnor (1953) 的混合连续性公理。当混合空间的维数是可数时，混合连续性对于保持混合的多重表示来说是足够的，但当它是不可数时则不然。我们最强的积极结果是，混合连续性与我们称为可数支配的新公理相结合就足够了，该公理根据其阿基米德结构限制了混合前序的顺序复杂性。我们还考虑了当混合空间被赋予其自然的弱拓扑时会发生什么。连续性（具有封闭的上下集）和封闭性（具有封闭图）强于混合连续性。我们表明，连续性是必要的，但对于混合预序具有混合保留的多重表示来说还不够。封闭性也是必要的；我们把它是否足够作为一个悬而未决的问题。我们以关于完全由严格递增函数和唯一性结果组成的混合保留多重表示的存在作为结束。我们表明，连续性是必要的，但对于混合预序具有混合保留的多重表示来说还不够。封闭性也是必要的；我们把它是否足够作为一个悬而未决的问题。我们以关于完全由严格递增函数和唯一性结果组成的混合保留多重表示的存在作为结束。我们表明，连续性是必要的，但对于混合预序具有混合保留的多重表示来说还不够。封闭性也是必要的；我们把它是否足够作为一个悬而未决的问题。我们以关于完全由严格递增函数和唯一性结果组成的混合保留多重表示的存在作为结束。