van Engelen B L, van Rooij A C M
Zijpendaalseweg 25, 6814 CC Arnhem, The Netherlands.
2Department of Mathematics, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands.
Positivity (Dordr). 2018;22(4):1081-1096. doi: 10.1007/s11117-018-0560-y. Epub 2018 Jan 27.
The first aim of this paper is to give a description of the (not necessarily linear) order isomorphisms where , are compact Hausdorff spaces. For a simple case, suppose is metrizable and is such an order isomorphism. By a theorem of Kaplansky, induces a homeomorphism . We prove the existence of a homeomorphism that maps the graph of any onto the graph of . For nonmetrizable spaces the result is similar, although slightly more complicated. Secondly, we let and be compact and extremally disconnected. The theory of the first part extends directly to order isomorphisms . (Here is the space of all continuous functions that are finite on a dense set.) The third part of the paper considers order isomorphisms between arbitrary Archimedean Riesz spaces and . We prove that such a 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 , Riesz isomorphisms of and onto order dense Riesz subspaces of and an order isomorphism such that ( ).
本文的首要目的是描述（不一定是线性的）序同构，其中(X)、(Y)是紧致豪斯多夫空间。对于一个简单的情形，假设(X)是可度量化的且(\varphi)是这样一个序同构。根据卡普兰斯基定理，(\varphi)诱导出一个同胚(\hat{\varphi})。我们证明存在一个同胚(\psi)，它将任何(\varphi)的图像映射到(\hat{\varphi})的图像上。对于不可度量化空间，结果类似，尽管稍微复杂一些。其次，我们令(X)和(Y)是紧致且极不连通的。第一部分的理论直接推广到序同构(\varphi: C(X) \to C(Y))。（这里(C(X))是所有在一个稠密子集上有限的连续函数(f: X \to \mathbb{R})的空间。）本文的第三部分考虑任意阿基米德里斯空间(E)和(F)之间的序同构(\varphi: E \to F)。我们证明这样的(\varphi)唯一地扩展为它们的通用完备化之间的序同构。（在没有线性的情况下，这并不明显。）由此可知，存在一个极不连通的紧致豪斯多夫空间(Z)，里斯同构(\alpha: E \to E_1)和(\beta: F \to F_1)，将(E)和(F)映到(C(Z))的序稠密里斯子空间(E_1)和(F_1)上，以及一个序同构(\hat{\varphi}: E_1 \to F_1)，使得(\hat{\varphi} \circ \alpha = \beta \circ \varphi)（(\varphi)的扩展）。
