A basic problem in the theory of partially ordered vector spaces is to characterise those cones on which every order-isomorphism is linear. We show that this is the case for every Archimedean cone that equals the inf-sup hull of the sum of its engaged extreme rays. This condition is milder than existing ones and is satisfied by, for example, the cone of positive operators in the space of bounded self-adjoint operators on a Hilbert space. We also give a general form of order-isomorphisms on the inf-sup hull of the sum of all extreme rays of the cone, which extends results of Artstein-Avidan and Slomka to infinite dimensional partially ordered vector spaces, and prove the linearity of homogeneous order-isomorphisms in a variety of new settings.

中文翻译：

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关于有序-同构的线性

偏序向量空间理论中的一个基本问题是刻画那些每个阶同构都是线性的锥体。我们表明，对于每个阿基米德锥体都是这种情况，它等于其参与的极端射线总和的 inf-sup 船体。这个条件比现有条件更温和，并且可以通过例如希尔伯特空间上有界自伴随算子空间中的正算子锥来满足。我们还给出了圆锥的所有极端射线之和的 inf-sup 外壳上的阶同构的一般形式，将 Artstein-Avidan 和 Slomka 的结果扩展到无限维偏序向量空间，并证明齐次的线性各种新设置中的顺序同构。