Pointfree pseudocompactness revisited (Q886296)

The concept of pseudocompactness in the setting of pointfree topology, due to \textit{D. Baboolal} and \textit{B. Banaschewski} [``Compactification and local connectedness of frames'', J. Pure Appl. Algebra 70, No.~1--2, 3--16 (1991; Zbl 0722.54031)], is studied in detail in this paper. Several internal and also external characterizations of pseudocompactness for general frames are given. Some of these are pointfree extensions of analogous characterizations in topological spaces due to \textit{R. M. Stephenson jun.} [``Pseudocompact spaces'', Trans. Am. Math. Soc. 134, 437--448 (1968; Zbl 0169.53903)]. Nicely, in the case of internal characterizations, whereas the proofs of Stephenson use external tools like maps into the reals, all the proofs in the present paper use only things residing within the frames in question, with no reference to the reals.