Abstract This paper approaches the construction of the universal completion of the Riesz space C ( L ) of continuous real functions on a completely regular frame L in two different ways. Firstly as the space of continuous real functions on the Booleanization of L. Secondly as the space of nearly finite Hausdorff continuous functions on L. The former has no counterpart in the classical theory, as the Booleanization of a spatial frame is not spatial in general, and it offers a lucid way of representing the universal completion as a space of continuous real functions. As a corollary we obtain that C ( L ) and C ( M ) have isomorphic universal completions if and only if the Booleanization of L and M are isomorphic and we characterize frames L such that C ( L ) is universally complete as almost Boolean frames. The application of this last result to the classical case C ( X ) of the space of continuous real functions on a topological space X characterizes those spaces X for which C ( X ) is universally complete. Finally, we present a pointfree version of the Maeda-Ogasawara-Vulikh representation theorem and use it to represent the universal completion of an Archimedean Riesz space with weak unit as a space of continuous real functions on a Boolean frame.

中文翻译：

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论pointfree函数空间的泛完备性

摘要 本文以两种不同的方式在完全规则的坐标系 L 上构造连续实函数的 Riesz 空间 C ( L ) 的普遍完备性。首先是L的布尔化上的连续实函数空间。其次是L上的近有限Hausdorff连续函数的空间。前者在经典理论中没有对应物，因为空间框架的布尔化一般不是空间的，并且它提供了一种清晰的方式来表示作为连续实函数空间的普遍完备性。作为推论，我们得到 C ( L ) 和 C ( M ) 具有同构全能完成当且仅当 L 和 M 的布尔化是同构的，并且我们表征帧 L 使得 C ( L ) 作为几乎布尔帧全能完成。将最后一个结果应用于拓扑空间 X 上连续实函数空间的经典情况 C ( X ) 表征了那些 C ( X ) 普遍完备的空间 X。最后，我们提出了 Maeda-Ogasawara-Vulikh 表示定理的无点版本，并用它来表示具有弱单元的阿基米德 Riesz 空间的普遍完备性，作为布尔框架上的连续实函数空间。