Another Note on Precompact Uniform Frames

In [17] pointfree compactness and precompactness are characterized by the convergence and clustering of classical filters, ultrafilters and Cauchy filters. Banaschewski in [2] introduced the notion of general filters as bounded meet semilattice homomorphisms. Together with Hong in [4], [5], [6], [7] they described the notion of convergence of such filters as those that are cover preserving. In this paper, we revisit the notion of general filters in a frame and introduce the concepts of clustering general filters, maximal general filters and general ultrafilters. These concepts have hitherto not been considered for general filters. We use these concepts to characterize, among other things, almost compact frames, Boolean frames and precompact uniform frames.