Abstract
We address classical questions concerning the existence and properties of completions in a new context, namely, that of uniform partial frames.
A partial frame is a meet-semilattice in which certain joins exist and finite meets distribute over these joins. We specify these joins by means of a so-called selection function, which must satisfy certain axioms to produce a useful theory. The axioms we use here were introduced in [9] and are sufficiently general to encompass in the resulting theory frames, \({\kappa}\)-frames and \({\sigma}\)-frames.
Using covers to describe uniform structures on partial frames, we develop the notion of completeness for a uniform partial frame, using the frame-theoretic version of the well-known fact that a complete uniform space is isomorphic to any uniform space in which it is densely embedded.
In constructing a completion, we make substantial use of the functor which takes \({\mathcal{S}}\)-ideals and the functor which takes \({\mathcal{S}}\)-cozero elements, as well as the category equivalence that these functors induce. Our strategy involves the transfer of important properties concerning the completion from the category of uniform frames to that of uniform partial frames.
As a final application, we provide two constructions of the Samuel compactification. One involves the completion of the totally bounded coreflection; the other uses the functors mentioned above to transfer the corresponding compactification from uniform frames.
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Frith, J., Schauerte, A. Completions of uniform partial frames. Acta Math. Hungar. 147, 116–134 (2015). https://doi.org/10.1007/s10474-015-0514-9
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DOI: https://doi.org/10.1007/s10474-015-0514-9
Key words and phrases
- frame
- uniform \({\mathcal{S}}\)-frame
- uniform partial frame
- \({\sigma}\)-frame
- \({\kappa}\)-frame
- meet-semilattice
- complete
- completion
- compact
- Samuel compactification
- cozero
- \({\mathcal{S}}\)-cozero
- ideal
- \({\mathcal{S}}\)-ideal
- \({\mathcal{S}}\)-Lindelöf
- \({\mathcal{S}}\)-separable
- selection function