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Characterizations of frame quasi-uniformities. (English) Zbl 0792.54026
The authors continue to develop a theory of entourage-like quasi- uniformities for frames. In this theory a quasi-uniformity U is defined on a frame $$A$$ and has associated with it a conjugate frame quasi-uniformity $$\widehat{\mathbf{U}}$$ and a uniformity $$\mathbf{U}^*$$, which is the coarsest frame quasi-uniformity on $$A$$ that contains $$\mathbf{U} \cup \widehat{\text\textbf{U}}$$. They show that if $$(X,{\mathcal U})$$ is a quasi-uniform space and U is the frame quasi-uniformity on $${\mathcal T}({\mathcal U}^*)$$ corresponding to $$\mathcal U$$, then $$\widehat{\mathbf{U}}$$ and $$\mathbf{U}^*$$ are the frame quasi-uniformities corresponding to $${\mathcal U}^{-1}$$ and $${\mathcal U}^*$$. They use this result to establish that the classical theory of quasi-uniform spaces is comprised by their theory of frame quasi-uniformities.
Each frame quasi-uniformity $$\mathbf U$$ on a frame $$A$$ determines a subframe, the frame of $$\mathbf U$$, which they denote by $$Fr(\mathbf{U})$$. Considering the biframe $$(Fr(\mathbf{U}^*),Fr (\text\textbf{U}),Fr(\widehat{\text\textbf{U}}))$$ they prove that, in the setting of biframes, their theory is equivalent to J. L. Frith’s [Structured frames, Ph. D. Thesis, University of Cape Town, 1987] theory of covering quasi-uniformities.
Reviewer: H.P.Künzi (Bern)

MSC:
 54E15 Uniform structures and generalizations 06D20 Heyting algebras (lattice-theoretic aspects) 54E55 Bitopologies
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References:
 [1] DOI: 10.1080/16073606.1983.9632289 · Zbl 0513.06005 [2] DOI: 10.1007/BF01351768 · Zbl 0736.54023 [3] Fletcher P., Frame quasi-uniformities · Zbl 0796.54037 [4] Fletcher P., Quasi-Uniform Spaces (1982) · Zbl 0501.54018 [5] Frith J. L., Structured Frames (1987) [6] Frith J. L., Comment. Math. Univ. Carol. [7] DOI: 10.1112/jlms/s2-5.1.48 · Zbl 0241.54023 [8] Johnstone P. T., Stone Spaces (1982)
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