Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58537Special Issue on the Occasion of Banaschewski's 90th Birthday (II)20170701One-point compactifications and continuity for partial frames578843180ENJohnFrithDepartment of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701,
South Africa.AnnelieseSchauerteDepartment of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.Journal Article20161020Locally compact Hausdorff spaces and their one-point compactifications are much used in topology and analysis; in lattice and domain theory, the notion of continuity captures the idea of local compactness. Our work is located in the setting of pointfree topology, where lattice-theoretic methods can be used to obtain topological results.<br />Specifically, we examine here the concept of continuity for partial frames, and compactifications of regular continuous such.<br /><br />Partial frames are meet-semilattices in which not all subsets need have joins.<br />A distinguishing feature of their study is that a small collection of axioms of an elementary nature allows one to do much that is traditional for frames or locales. The axioms are sufficiently general to include as examples $sigma$-frames, $kappa$-frames and frames.<br /><br />In this paper, we present the notion of a continuous partial frame by means of a suitable ``way-below'' relation; in the regular case this relation can be characterized using separating elements, thus avoiding any use of pseudocomplements (which need not exist in a partial frame). Our first main result is an explicit construction of a one-point compactification for a regular continuous partial frame using generators and relations. We use strong inclusions to link continuity and one-point compactifications to least compactifications. As an application, we show that a one-point compactification of a zero-dimensional continuous partial frame is again zero-dimensional. <br /><br />We next consider arbitrary compactifications of regular continuous partial frames. In full frames, the natural tools to use are right and left adjoints of frame maps; in partial frames these are, in general, not available. This necessitates significantly different techniques to obtain largest and smallest elements of fibres (which we call balloons); these elements are then used to investigate the structure of the compactifications. We note that strongly regular ideals play an important r^{o}le here. The paper concludes with a proof of the uniqueness of the one-point compactification.http://cgasa.sbu.ac.ir/article_43180_02e474fcbfa63e236d1fbd237390dba8.pdf