Frith, J., Schauerte, A. (2017). One-point compactifications and continuity for partial frames. Categories and General Algebraic Structures with Applications, 7(Special Issue on the Occasion of Banaschewski's 90th Birthday (II)), 57-88.

John Frith; Anneliese Schauerte. "One-point compactifications and continuity for partial frames". Categories and General Algebraic Structures with Applications, 7, Special Issue on the Occasion of Banaschewski's 90th Birthday (II), 2017, 57-88.

Frith, J., Schauerte, A. (2017). 'One-point compactifications and continuity for partial frames', Categories and General Algebraic Structures with Applications, 7(Special Issue on the Occasion of Banaschewski's 90th Birthday (II)), pp. 57-88.

Frith, J., Schauerte, A. One-point compactifications and continuity for partial frames. Categories and General Algebraic Structures with Applications, 2017; 7(Special Issue on the Occasion of Banaschewski's 90th Birthday (II)): 57-88.

One-point compactifications and continuity for partial frames

^{}Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.

Abstract

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.

Highlights

Dedicated to Bernhard Banaschewski on the occasion of his $90^{th}$ birthday

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