Epicomplete archimedean l-groups via a localic Yosida theorem. (English) Zbl 0718.06005
An object X of a category is called epicomplete if every bimorphism with domain X is invertible. The key result in this paper is that in archimedean \(\ell\)-groups with weak order unit the epicomplete ones are monoreflective. The purpose of the paper is more broadly to describe the epicomplete objects and to use this as an illustration of the value of the authors’ reformulation of the Yosida representation for weak-unital archimedean \(\ell\)-groups. The point is that the somewhat awkward extended real-valued functions finite almost everywhere, of the classical Yosida theorem [see, e.g., A. Hager and L. Robertson, Symp. Math. 21, 411-431 (1977; Zbl 0382.06018)], become ordinary maps to the real line, whose domain is in general a ‘pointless space’ or locale. Among the further results, the authors observe that the epicompletion is not always an essential extension and obtain some conditions for its essentiality.
Reviewer: J.R.Isbell (Buffalo)
MSC:
| 06F15 | Ordered groups |
Keywords:
epicompleteness; archimedean \(\ell \)-groups; Yosida representation; locale; epicompletionCitations:
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