Geodesic
Completeness properties of function rings in pointfree topology
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 2, pp. 245-259.

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This note establishes that the familiar internal characterizations of the Tychonoff spaces whose rings of continuous real-valued functions are complete, or $\sigma$-comp\-lete, as lattice ordered rings already hold in the larger setting of pointfree topology. In addition, we prove the corresponding results for rings of integer-valued functions.
Classification : 06D22, 54C30, 54G05
Keywords: frame of reals; lattice ordered rings of real valued continuous functions and integer valued continuous functions; extremally disconnected frame; basically disconnected frame; cozero map 
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Banaschewski, Bernhard; Hong, Sung Sa. Completeness properties of function rings in pointfree topology. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 2, pp. 245-259. http://geodesic.mathdoc.fr/item/CMUC_2003__44_2_a5/