Acta Mathematica Hungarica

, Volume 147, Issue 1, pp 116–134 | Cite as

Completions of uniform partial frames

  • J. Frith
  • A. Schauerte


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.

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 

Mathematics Subject Classification

06A12 06B10 06D22 18A40 54B30 54D20 54D35 54E15 


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© Akadémiai Kiadó, Budapest, Hungary 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of Cape TownRondeboschSouth Africa

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