a r t i c l e i n f o a b s t r a c tArticle history: Dedicated to Bob Lowen on the occasion of his 60th birthday MSC: 06D22 54E05 54E15 Keywords: Quasi-uniformity Proximity Strong inclusion Frame Biframe Samuel compactification Proximal biframe Quasi-uniform biframeThe paircover approach is used to explore the links between quasi-uniform and proximal biframes. The Samuel compactification for quasi-uniform biframes is constructed and its universal property discussed.
This paper concerns the notions of closed and open maps in the setting of partial frames, which, in contrast to full frames, do not necessarily have all joins. Examples of these include bounded distributive lattices, $$\sigma $$
σ
- and $$\kappa $$
κ
-frames and full frames. We define closed and open maps using geometrically intuitively appealing conditions involving preservation of closed, respectively open, congruences under certain maps. We then characterize them in terms of algebraic identities involving adjoints. We note that partial frame maps need have neither right nor left adjoints whereas frame maps of course always have right adjoints. The embedding of a partial frame in either its free frame or its congruence frame has proved illuminating and useful. We consider the conditions under which these embeddings are closed, open or skeletal. We then look at preservation and reflection of closed or open maps under the functors providing the free frame or the congruence frame. Points arise naturally in the construction of the spectrum functor for partial frames to partial spaces. They may be viewed as maps from the given partial frame to the 2-chain or as certain kinds of filters; using the former description we consider closed and open points. Any point of a partial frame extends naturally to a point on its free frame and a point on its congruence frame; we consider the closedness or openness of these.
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