Epimorphisms of Uniform Frames

It is shown that some familiar properties of epimorphisms in the category of frames cary over to the category of uniform frames. This is achieved by suitably enriching certain frame homomorphisms to uniform frame homomorphisms. This note deals with the natural question, apparently never considered so far, whether certain familiar facts concerning epimorphisms, and specifically epi-extensions, of frames also hold for uniform frames. We shall show this is indeed the case by establishing the following results. There are uniform frames with arbitrarily large epi-extensions. Whenever the underlying frame of a uniform frame has arbitrarily large epi-extensions, the same holds for the uniform frame itself. A uniform frame has no proper epi-extensions iff it is a Boolean frame with its largest uniformity. Of course, the first of these assertions may readily be obtained as a consequence of the second but since its proof is considerably more direct than that of the latter it seemed worthwhile to include it. For general background of frames we refer to Johnstone [4] or Vickers [8], and for uniform frames to the original paper by Isbel [3] or the more recent Banaschewski [1]. We begin by recalling the relevant basic facts concerning epimorphisms of frames. The crucial construction here is the embedding L → CL of any frame L into the frame CL of its congruences, these being the equivalence relations on L which are subframes of L × L, otherwise characterized as the kernel relations of the homomorphisms Thanks go to the National Sciences and Engineering Research Council of Canada for continuing support of the first author by means of a research grant, and to the Ministry of Education of the Czech Republic for the support of the third author by the project LN 00A056.