The first aim of this paper is to give a description of the (not necessarily linear) order isomorphisms \(C(X)\rightarrow C(Y)\) where X, Y are compact Hausdorff spaces. For a simple case, suppose X is metrizable and T is such an order isomorphism. By a theorem of Kaplansky, T induces a homeomorphism \(\tau :X\rightarrow Y\). We prove the existence of a homeomorphism \(X\times \mathbb {R}\rightarrow Y\times \mathbb {R}\) that maps the graph of any \(f\in C(X)\) onto the graph of Tf. For nonmetrizable spaces the result is similar, although slightly more complicated. Secondly, we let X and Y be compact and extremally disconnected. The theory of the first part extends directly to order isomorphisms \(C^{\infty }(X)\rightarrow C^{\infty }(Y)\). (Here \(C^{\infty }(X)\) is the space of all continuous functions \(X\rightarrow [-\infty ,\infty ]\) that are finite on a dense set.) The third part of the paper considers order isomorphisms T between arbitrary Archimedean Riesz spaces E and F. We prove that such a T extends uniquely to an order isomorphism between their universal completions. (In the absence of linearity this is not obvious.) It follows, that there exist an extremally disconnected compact Hausdorff space X, Riesz isomorphisms \(\hat{}\) of E and F onto order dense Riesz subspaces of \(C^{\infty }(X)\) and an order isomorphism \(S:C^{\infty }(X)\rightarrow C^{\infty }(X)\) such that \(\hat{Tf}=S\hat{f}\) (\(f\in E\)).

中文翻译：

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Riesz空间之间的有序同构。

本文的首要目的是给出（不一定是线性）阶同构\（C（X）\ rightarrow C（Y）\）的描述，其中X，  Y是紧凑的Hausdorff空间。对于一个简单的情况，假设X是可量化的，而T是这样的一个同构。根据Kaplansky定理，T诱导同胚\（\ tau：X \ rightarrow Y \）。我们证明了同胚\（X \ times \ mathbb {R} \ rightarrow Y \ times \ mathbb {R} \）的存在，该映射将任何\（f \ in C（X）\）的图映射到f。对于不可度量的空间，结果相似，尽管稍微复杂一些。其次，我们让X和Y变得紧凑且极端分离。第一部分的理论直接扩展到有序同构\（C ^ {\ infty}（X）\ rightarrow C ^ {\ infty}（Y）\）。（这里\（C ^ {\ infty}（X）\）是在密集集合上有限的所有连续函数\（X \ rightarrow [-\ infty，\ infty] \）的空间。）本文认为，为了同构牛逼任意阿基米德里斯空间之间Ë和˚F。我们证明了这样的T唯一地扩展到它们的通用完成之间的有序同构。（在不存在线性的，这不是显而易见的。）因此，即存在一个extremally断开紧Hausdorff空间X，中Riesz同构\（\帽子{} \）的Ë和˚F到顺序致密的中Riesz子空间\（C ^ {\ infty}（X）\）和阶同构\（S：C ^ {\ infty}（X）\ rightarrow C ^ {\ infty}（X）\）使得\（\ hat {Tf} = S \ hat {f} \）（\（f \ in E \））。