One-point compactifications and continuity for partial frames | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 6، دوره 7، Special Issue on the Occasion of Banaschewski's 90th Birthday (II) - شماره پیاپی 1، مهر 2017، صفحه 57-88 اصل مقاله (1.08 M) | ||
| نوع مقاله: Research Paper | ||
| نویسندگان | ||
| John Frith؛ Anneliese Schauerte | ||
| Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa. | ||
| چکیده | ||
| Locally 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. Specifically, we examine here the concept of continuity for partial frames, and compactifications of regular continuous such. Partial frames are meet-semilattices in which not all subsets need have joins. 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. 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. 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. | ||
تازه های تحقیق | ||
Dedicated to Bernhard Banaschewski on the occasion of his $90^{th}$ birthday | ||
| کلیدواژهها | ||
| frame؛ partial frame؛ $sels$-frame؛ $kappa$-frame؛ $sigma$-frame؛ $mathcal{Z}$-frame؛ compactification؛ one-point compactification؛ strong inclusion؛ strongly regular ideal؛ continuous lattice؛ locally compact | ||
| مراجع | ||
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