On the description of frame asymmetric uniformities via paircovers
A quasi-uniformity on a frame may be equivalently described either in terms of paircovers or in terms of entourages. The former is defined as a structure U on a biframe (L0,L1,L2) and the latter directly as a structure E on a frame L which induces two subframes L1(E) and L2(E) of L such that the triple (L,L1(E),L2(E)) is a biframe (this is the pointfree analogue of the bitopological space (X,T1(E),T2(E)) induced by any quasi-uniformity E on the set X). While the approach via paircovers is most convenient for calculations, it does not faithfully reflects the spatial original notion since it is not formulated directly on frames.
Here, it will be shown that it is possible to describe frame quasi-uniformities by defining the paircovering structure U directly on a frame L without requiring the prior knowledge of the underlying biframe. It turns out that the biframe structure (L,L1(E),L2(E)) appears only a posteriori, induced by the structure in a very natural way. In addition, we indicate how to do the same for strong relations on biframes and the corresponding quasi-proximal frames.
Pré-publicações do Departamento de Matemática da Universidade de Coimbra