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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)

54E15 Uniform structures and generalizations
06D20 Heyting algebras (lattice-theoretic aspects)
54E55 Bitopologies
Full Text: DOI
[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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