The main contribution of the paper is the proof that any element in the convex hull of a decomposably antichain-convex set is Pareto dominated by at least one element of that set. Building on this result, the paper demonstrates the disjointness of the convex hulls of two disjoint decomposably antichain-convex sets, under the assumption that one of the two sets is upward. These findings are used to obtain a number of consequences on: the structure of the set of Pareto optima of a decomposably antichain-convex set; the separation of two decomposably antichain-convex sets; the convexity of the set of maximals of an antichain-convex relation; the convexity of the set of maximizers of an antichain-quasiconcave function. Emphasis is placed on the invariance of the solution set of a problem under its “convexification.” Some entailments in the field of mathematical economics of the results of the paper are briefly discussed.

中文翻译：

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关于可分解反链凸集的帕累托优势

该论文的主要贡献是证明可分解反链凸集的凸包中的任何元素都是帕累托由该集的至少一个元素支配的。在此结果的基础上，本文证明了两个不相交的可分解反链凸集的凸包的不相交性，假设两个集合中的一个是向上的。这些发现用于获得以下方面的许多结果： 可分解反链凸集的帕累托最优集的结构；两个可分解反链凸集的分离；反链凸关系的极大值集合的凸性；反链拟凹函数的极大值集合的凸性。强调问题的解集在其“凸化”下的不变性。